AC electrical theory
An introduction to phasors, impedance and admittance, with emphasis on radio frequencies. By David W. Knight !ersion ".11. #th $une %"1%. & D. W. Knight, %""# ' %"1%.
(ttery )t *ary, Devon, +ngland. ,lease chec- the author.s we/site to ensure that you have the most recent versions of this article and its associated documents0 http://www.g3ynh.info/
Table of Contents
,reface.............................................................% 1. 1ield electricity............................................2 %. Circuit analysis overview..........................1" 2. Basic electrical formulae...........................13 3. 4esonance..................................................15 #. 6mpedance, 4esistance, 4eactance............17 8. !ectors 9 )calars.......................................%" 5. Balanced !ector +quations........................%2 :. ,hasors.......................................................%# 7. !oltage *agnification 9 ;.......................%: 1". ,ower 1actor 9 )calar ,roduct ..............2% 11. ,hasor dot product...................................23 1%. Comple< =um/ers...................................2# 12. Comple< arithmetic.................................2: 13. 6mpedances in ,arallel.............................27 1#. ,arallel resonance....................................31 18. Dynamic 4esistance................................32 15. Dou/le'slash notation..............................33 1:. ,arallel'to')eries transformation.............38 17. )eries'to',arallel transformation.............35 %". ,arallel resonator in parallel form...........3: %1. 6maginary resonance................................#" %%. ,hase analysis..........................................#% %2. 4esistance tuned >C resonator?..............## %3. ,hasor theorems.......................................#: %#. @eneralisation of (hm.s >aw..................8% %8. @eneral statement of $oule.s >aw............82 %5. Bandwidth................................................8# %:. deciBels 9 logarithms.............................88 %7. Bandwidth of a series resonator..............87 2". >ogarithmic frequency............................52 21. A proper definition for resonant ;...........53 2%. Bandwidth in terms of ;.........................5# 22. >orentAian line'shape function................58 23. *a<imum power transfer........................5: 2#. Bhe potential divider...............................:% 28. (utput impedance of potential divider....:2 25. BhCvenin.s Bheorem................................:3 2:. *easuring source resistance....................:3 27. +rror analysis...........................................:# 3". Antenna system ;....................................7" 31. Basic impedance transformer..................71 3%. Auto transformers....................................7# 32. ,rototype D'matching networ-................75 33. Admittance, conductance, susceptance....7: 3#. ,arallel resonator B,1...........................1"1 38. Enloaded ; of parallel resonator..........1"3 35. Current magnification............................1"8 3:. Controlling loaded ;.............................1":
Bhis document provides an introduction to the su/Fect of AC circuit analysis, with particular emphasis on radio'frequency applications. 6t was developed as part of a collection of writings on the su/Fect of radio'frequency impedance matching and measurementG which were first made availa/le via the 6nternet in %""#, and were grouped under the wor-ing title .1rom Bransmitter to Antenna.. 6ts purpose was Hand still isI to widen the audience for the other articles /y providing essential /ac-ground material, /ut it can Fust as well /e read /y those who have a more general interest. Bhe approach adopted is that of starting with the /asic laws of DC electricity and e<panding them to deal with AC. Bhe modified laws are then used to derive and e<plore results which are normally accepted without proof, there/y e<plaining the origins of various standard formulae and demonstrating the general method /y which linear circuit design equations are o/tained. Bhe level of treatment is one which does not demand a high level of mathematical s-ill at the outsetG /ecause the required techniques are introduced as the narrative progresses. Jence the discussion should /e accessi/le to anyone who has some -nowledge of /asic alge/ra and is reasona/ly familiar with circuit diagrams and electrical terminology. Apart from providing a conventional introduction to AC theory however, there is also a su/te<t. Bhis relates to the author.s concerns as a scientist0 one /eing that there appears to /e an almost universal pu/lic misconception regarding the nature of electricityG and the other /eing a lac- of mathematical rigour in the way in which phasor techniques are commonly used. Both of these issues can Hand shouldI /e addressed at this stage in the development of wor-ing -nowledge, and so the accompanying discussion attempts to do that. As all e<perienced engineers and physicists -now, our understanding of electricity comes from *a<well.s equations. Bhe pro/lem for those who wish to teach electrical su/Fects however, is that the electromagnetic field approach requires advanced mathematics and does not lead directly to the practicalities of circuit design. Jence it is sensi/le to hold /ac- on the more a/stract ideas until they /ecome unavoida/leG /ut that leaves the pro/lem of how to dispel the notion that electricity is synonymous with electrons flowing through wires. Bhe author.s solution is to provide an e<tended pream/leG which gives a purely qualitative e<planation of electricity in terms of fieldsG and is intended to leave the reader with the same mental picture as will /e held /y those who are familiar with *a<well.s theory. (n the matter of mathematical rigourG it is not the intention to wrap the su/Fect in formalism, /ut merely to eliminate certain /ad practices. Bo this end, we pay particular attention to the definitions and properties of the mathematical o/Fects involved, and develop a way of wor-ing which identifies and preserves the alge/raic signs of the circuit parameters. Bhe outcome is an internally'consistent theory of circuits, which produces results which are correct in /oth magnitude and phase. 6n this way, we eliminate the need for the so'called .physical considerations. traditionally used to resolve am/iguitiesG and we discover which of the two commonly used definitions of admittance is actually the correct one. )ome, of course, will as-0 .why should we /other to learn phasor analysis when we can model circuits using ),6C+?. 6n fact, the use of ),6C+ is highly recommendedG /ut simulation is essentially a way of chec-ing e<isting design wor-. 6t does not offer a systematic approach to the /usiness of optimising circuits or inventing new ones. Bhe techniques discussed here, on the other hand, allow the equations descri/ing the /ehaviour of a circuit to /e written down e<plicitly. We might then, for e<ample, separate out the terms descri/ing unwanted /ehaviour with a view to ma-ing alterations which will eliminate them. )uch is the /asis for the development of precision measuring instruments and all manner of other high'performance circuitry. David Knight. $une %"1%.
1. Field electricity
When AC theory is introduced, and especially when there is a /ias towards radio frequencies, the very first new idea required H/y many people at leastI is a correct understanding of the word .electricity.. Bhe teaching of /asic science often involves what are -nown as .lies to children., and the one a/out electricity /eing Kelectrons flowing through wiresK is an intellectual dead'end. +lectricity is actually an invisi/le form of light. )pecifically, it is electromagnetic energy of very long wavelength Hin comparison to visi/le lightIG which is why we can /uild devices called .radio transmitters. which cause electricity to propagate off into space. Jence we will never understand high'frequency electricity /y counting electronsG and we must first refine our ideas of voltage and current /y thin-ing a/out certain mysterious entities -nown as .fields.. Bhe term .field. has a vernacular meaning0 Kregion of influenceK, and this is the route /y which it came into the language of physics. 6n scientific parlance however, a field is more rigorously defined as .a quantity which can ta-e on different values, and possi/ly also different directions of ma<imum action, at different points in space and time.. Bhe geometric field idea can /e used HsayI to descri/e the 2D temperature gradient around a hot o/Fect, or the average velocities of molecules in a flowing liquidG /ut the fields which are the most perple<ing, and which ultimately reveal the deepest secrets of the Eniverse, are those which appear to produce action at a distance. (f these so' called .force fields.G the gravitational, electric and magnetic are the most familiarG and of course, it is the latter two which concern us here. Bhe /eginning of what is loosely called .modern physics. can /e traced to a single deduction made /y $ames Cler- *a<well in the latter part of the 17th Century. *a<well collected the details of every -nown scientific result concerning electricity and magnetism, and lent his phenomenal mathematical s-ill to the pro/lem of finding a single theory. Bhis led him to discover inconsistencies in the laws of induction Hi.e., those laws which govern the effects of time'varying electric and magnetic fieldsI which would nowadays /e interpreted as violations of the principle of conservation of charge. +lectrons had not /een discovered at the time, and electric charge was thought to /e some -ind of fluidG /ut whatever it was, the physical ideas of the day did not permit it to disappear from one place and reappear in another. Je fi<ed the pro/lem /y ma-ing a /old and unprecedented stepG which was to postulate the e<istence of a new -ind of electric current, not associated with flowing charges, which he called .displacement current.. )peculation is one thingG /ut *a<well had a test for his theory. With the inclusion of displacement current, the modified laws still allow the e<istence of electricity when all of the terms relating to physical matter are deleted. Bhis .free electricity. has to /e in the form an oscillating electric field com/ined with an oscillating magnetic field, with the directions of action of the fields disposed at right'angles. 6t had turned out that the fields do not represent mysterious action at a distance after all Halthough it too- some years /efore that point was fully acceptedI. Bhey are instead stores of and agents for the transfer of pure energy. Bhe li/eration of electricity is however su/Fect to a strict conditionG which is that the energy e<ists /y virtue of continuous transfer /etween the two fields according to the laws of inductionG i.e., a decaying electric field gives rise to a magnetic field, and a decaying magnetic field gives rise to an electric fieldG and so the energy swaps /ac-wards and forwards. Bhe mechanism only wor-s if the energy is propagating through space in a direction at right'angles to the crossed fields with a velocity given /y the e<pression0 v L 1MNHOPI where O [email protected]
lower case KmuKI is the magnetic permeability and P HKepsilonKI is the electric permittivity of the surrounding medium. ,ermea/ility O is a constant of proportionality o/tained from the force of magnetic attraction or repulsion which occurs /etween wires carrying an electric current. ,ermittivity P is a constant o/tained from the relationship /etween the physical dimensions of a capacitor and its capacitance. *a<well found that the /est availa/le measurements for the permea/ility and permittivity of vacuum, O" and P" HKmu noughtK and Kepsilon noughtKI gave a propagation velocity for free energy,
3 c L 1MNHO" P"I , which turned out to /e the same as the speed of light. Bhus he was a/le to confirm a suggestion put forward /y *ichael 1araday some years /efore, which is that light is composed of electromagnetic waves. *a<well had also shown, of course, that electrical energy is a form of lightG and that older ideas derived from DC e<periments were no longer tena/le. *a<well died in 1:57 at the age of 38, only si< years after the pu/lication of his great treatise on electricity and magnetism. Bhus it was left to others to e<plore the ramifications of his wor-. 6n the latter part of the 17th Century, there were two great interpreters of *a<well.s electromagnetic theory0 (liver Jeaviside and Jeinrich JertAG /oth of whom were /rilliant mathematicians in their own right. Bhese two scientists independently cleared'up *a<well.s notation and reduced a nest of alge/raic clutter to a set of four equations which descri/e the fields. Bhe four .*a<well.s equations. which we -now today are actually a variant of the form preferred /y Jeaviside He<tra terms, which are Aero for the Eniverse in its present state, are nowadays usually deletedI. Bhe clima< of JertA.s wor- was the creation and detection of *a<wellian waves under la/oratory conditions1G which means that JertA is the father of radio telecommunications, and also the inventor of the first radio antennas. Jis clarification of *a<well.s theory was also the /asis of the wor- of one Al/ert +instein, a Durich patent e<aminer with a ha/it of daydreaming a/out o/Fects in relative motion. +instein realised that *a<well.s separation of light and matter implies that the speed of light is constant regardless of any motion on the part of the o/server. Bhis led to the )pecial and @eneral Bheories of 4elativity, which overturned all 17th Century notions of space and time. Je also gave us the e<plicit unification of electricity and magnetism, /y showing that electromagnetic induction is a relativistic phenomenon. *ost readers will /e aware that an electro'mechanical generator wor-s /y moving a coil of wire relative to a strong magnetic field. Bhe changing magnetic field Has seen from the coil.s viewpointI gives rise to an electric field, which manifests itself as a voltage across the ends of the coil. +instein tells us that the magnetic field does not so much create an electric fieldG it is an electric field when seen from a moving frame of reference. >i-ewise, an electric field is a magnetic field when viewed /y a moving o/server. Bhis means that generators Hand /y a converse principle, electric motorsI ma-e use of relativistic effects when they convert energy /etween its electrical and mechanical forms. Jeaviside.s e<tended version of *a<well.s equations was /ac-ground to the wor- of ,aul Dirac, who later went on to predict the e<istence of anti'matter. Jeaviside.s most important wor- however was carried out /efore the advent of radio as a technology, and was primarily related to the pro/lems of long'distance electrical communication Htelegraphs and telephonesI. 6t is Jeaviside who gave us the correct picture of electricity, /y way of another corollary of *a<well.s theory called .the principle of continuity of energy. Hnot to /e confused with the principle of conservation of energyI. Bhe principle of continuity dictates that energy cannot simply disappear from one location and reappear in another, it must, in some sense, ma-e a Fourney. Bhis, incidentally, is not the same as imagining that energy follows a specific routeG /ecause we can only e<plain phenomena such as optical diffraction Hand remain consistent with quantum theoryI if we allow that even the very smallest quantity of energy can follow a multiplicity of paths during flight. =evertheless, it retains a form of integrity Hit is conserved%I, which means that electric and magnetic fields from different energy sources cannot com/ine to ma-e electromagnetic radiation. 6t is intriguing to note that, were such com/ination possi/le, every stray field would interact and the Eniverse would e<plode, perhaps to e<pand to a state in which such interaction can no longer occur. Bhe continuity principle allows us to /rea- reality down into separate energy transfer processesG and so without it, we would not /e a/le to understand the Eniverse. (n a more immediate level however, it tells us e<actly how
1 Hertz, the isco!erer of "lectric #a!es. $ulian Blanchard, Bell )ystem Bechnical $ournal, $uly 172:, !ol. 15, =o. 2, p2%8 ' 225. QAvaila/le from http0MM/stF./ell'la/s.comM R % 6s the $ni!erse lea%ing energy& Bamara * Davis. )cientific American, $uly %"1", p%1'%5.
# the energy flows in an electrical circuit, and indeed, how it flows into space from a radio antenna. Bhe e<planation for the principle of continuity comes, once again, from the wor- of +instein, this time from his investigation of the photoelectric effect. 6t also follows logically from *a<well.s equationsG the issue /eing that there must /e a reason for the rate HfrequencyI at which energy swaps /etween the fields in a propagating electromagnetic wave. (n the assumption that energy is the simplest thing in the Eniverse, the only possi/le governing factor is the amount of energy /eing transported. Jence it was /ound to /e discovered that light is made up of .particles. Hi.e., discrete units, not solid o/FectsI, each oscillating at a frequency dictated /y the amount of energy it contains. Bhe particles, of course, are nowadays called .photons., and the relationship /etween energy and frequency is -nown to /e a direct proportionality0 +Lhf where h is .,lan-.s constant., and has a value of 8.8%8"8:78S1"'23 $oule seconds. Bhis introduces another strange concept, -nown as .wave'particle duality.G which is sometimes claimed to /e a parado< /ut is actually nothing of the sort. 1or a gli/ e<planation, we can say that it would only /e parado<ical if a /atch of propagating energy was not made up of discrete units, /ecause then the energy would have no way of -nowing its own frequency and so would not /e a/le to form a wave. 1or a more formal way of thin-ing a/out this issue howeverG note that an electromagnetic wave is defined in relation to a route through a field. Bhe wave'li-e nature of the energy flow is detected /y inserting pro/es Hmeasuring devicesI into the field and /uilding up a picture /y intercepting photons. 1rom this we infer that photons travel as waves, even though we can only discern that /y adding together the small pac-ets of energy delivered /y them. Bhis, incidentally, raises a general point in relation to scientific o/servationG which is that there are no parado<es in nature. ,arado<es e<ist only in the mind of the o/server, and result from attempts to interpret information using faulty starting assumptions. When we detect waves, we do so on the assumption that our pro/es measure field strength. Bhis is very convenient, /ecause it allows us solve pro/lems using field theoryG /ut when the meaning of an o/servation is in dou/t, or when discrepancies /egin to accrue, it is important to remem/er that all measurements are ultimately purely dependent on what can /e inferred from the a/sorption and emission of energy. We rarely need to thin- a/out the granularity of light when wor-ing at electrical frequencies, /ecause the amount of energy in each photon is e<ceedingly small. Bhe wave property is instead the most dominant feature, and is often reinforced /y a /ehaviour called .coherence.G which is the a/ility of identical photons traversing the same set of paths to synchronise their fields. 6n this way, we see smooth waves in the collective /ehaviour of many particles. Jeaviside -new nothing of this, /ut the photon theory e<plains his continuity principle /y identifying the energy carrier. Bhe fields e<tend throughout space, /ecause they represent the propensity to e<change energy, and the photons can turn up anywhere that the fields have finite intensityG /ut the photons themselves undergo no additional e<change processes during transit. (ur ideas on the flow of electromagnetic energy are nowadays associated with $ohn Jenry ,oynting, who formalised the continuity principle in a theorem which /ears his name. Jeaviside however, had already /een using the principle for some time, and had a far more elegant derivation lodged with his pu/lisher at the time of ,oynting.s first pu/lic presentation. ,oynting also, did not interpret his findings correctly, whereas Jeaviside had no such trou/leG and so it was the latter who first descri/ed the underlying mechanism2. What follows may come as a shoc- to those who have /een taught the .lies'to'children. version. 6t turns out that the inside of a good conductor is the one part of a circuit where the transmission of electrical energy does not ta-e place. We can understand Jeaviside.s e<planation /y using 1araday.s Klines of forceK, which provide a way of visualising the electric and magnetic fields. )ome field patterns relevant to to the wor-ings of circuits are shown /elow0
2 'li!er Hea!iside, ,aul $ =ahin. %nd edition Hpaper/ac-I. $ohn Jop-ins Eniversity ,ress %""%. 6)B= "':"1:'87"7'7. Ch. 5, Bech note 2, p1%7'121.
Bhe left'hand diagram represents an electric field as it might e<ist /etween two charged spheres, or /etween two electrical conductors seen in cross'section. 4ecall that li-e charges repel, and opposite charges attractG and so a positively charged particle will /e repelled /y the HTI electrode and attracted to the H'I electrode. Jence, depending on the starting point, the arrows show the direction in which a positive charge will /e accelerated, and the lines show the path which will /e followed. Bhere are, of course, an infinite num/er of possi/le starting positions, and so the field has an infinite num/er of linesG /ut a sparse representation is sufficient to give the general idea. Bhe curvature of the lines arises /ecause the mutual force /etween pairs of charged /odies is governed /y an inverse'square law, i.e., the attraction or repulsion is strong when the /odies are close, /ut falls off rapidly with distance. Jence a particle close to one electrode will have a traFectory almost perpendicular to the surface, /ut the field line /ecomes curved further away /ecause the particle is then influenced /y /oth electrodes. Bhe middle diagram shows the lines of the magnetic field surrounding a wire when a current of positive charges is flowing away from the o/server. Bhe wire is shown in cross section, and the cross within its /oundary represents the flow direction as the tail fins of a receding dart. Bhe right' hand diagram shows the field when the current is flowing towards the o/serverG the dot /eing interpreted as the point of an approaching dart. =otice that we use a convention esta/lished /efore the discovery of the electron H/y $ $ Bhomson in 1:75IG which is that current flows from HTI to H'I. +lectrons flow the other wayG /ut the continued use of conventional current ma-es no difference to the theory and serves to preserve the intelligi/ility of past scientific literature. 6n the case of a magnetic field, the arrows drawn on the lines of force show the direction in which a compass will point when placed in pro<imity to the wire Hpresuming that the current is large enough to overcome the +arth.s magnetic fieldI. Bhe cloc-wise .rotation. of the field lines when the HconventionalI current is flowing away is -nown as .*a<well.s cor-screw rule.. Bhis rule derives from the convention that the field lines around a /ar magnet Hor compass needleI emerge from the =orth'see-ing pole and return to the )outh'see-ing pole. *agnetic /odies repel when their force lines are in opposition H=orth pole to =orth poleI, and attract when their force'lines point in the same direction H=orth pole to )outh poleI. Jence a compass needle is repelled /y the field lines coming towards it and attracted to the field lines going away from it. Bhe *a<wellian fields have the same geometric properties as 1araday linesG and so we can now forget a/out forces on charged /odies and magnets and thin- a/out electromagnetic energy. 4ecall that *a<well discovered that light travels with its electric field oscillating at right angles to its magnetic field, the direction of propagation /eing at right angles to /oth fields. Jeaviside and ,oynting now tell us that the transport of energy in electrical circuits occurs in e<actly the same way. An ar/itrarily chosen point in an electric or magnetic field has /oth intensity Hi.e., magnitude, or strengthI and a direction of action. )uch points are -nown as .vectors., and a region of space filled with vectors is called a .vector field.. Bhere is also a mathematical operation called .vector multiplication., which can /e applied at points where two fields cross to produce a new vector at right'angles to the original two. =otional directions were assigned to the electric and magnetic vectors in the discussion a/oveG and the adopted convention is, of course, one which gives the
5 correct direction of energy transport when the fields are com/ined using the vector product Hor Kcross productK, as it is also -nownI. Bhe electric field is usually given the sym/ol E, and the magnetic field the sym/ol H. Bhe cross product is then written0 PLESH where P is -nown as the .,oynting vector., and gives the intensity and average direction of the energy flow at some point in the com/ined electric and magnetic Hi.e., electromagneticI field. =ow consider the electromagnetic field around a wire in a circuit Has shown on the rightI. +lectric field lines emerge perpendicular to the surface, magnetic field lines encircle the conduction currentG and there are an infinite num/er of points at which they cross at right angles. Bhus, assuming that the E and H fields are related in accordance with the continuity principle, we can wor- out the direction in which energy is travelling Hand also the rate of flowI at any location. Bhe -ey to the right of the diagram gives the direction of the ,oynting vector in relation to the E and H fields. 6t follows that if the electric field is strictly perpendicular to the wire, then the ,oynting vector lies parallel to the wireG and the fields as they are depicted have it running away from the o/server. =otice also, that all of the propagating energy is on the outside of the wire Hal/eit in greatest concentration close to the surface where the magnetic field is strongestIG and it transpires that if the wire is a perfect conductor, there is none on the inside at all. 6t requires /oth an electric field and a magnetic field for the transportation of energy. A perfect conductor, however, is a material which, /y definition, cannot sustain an electric field. Bhis can /e understood /y noting that the electrical resistance /etween any two points within the /ody of a perfectly conducting o/Fect is Aero, in which case there can /e no voltage difference and so no electric field. Jence electrical energy cannot flow inside a good conductor. Bhis understanding, incidentally, gives rise to a semantic difficulty regarding whether there is a difference /etween .electricity. and .electrical energy.. 6t is hard to Fustify the preservation of different meanings for the two terms, and yet people will persist in saying that electricity Kflows throughK conductors. We can sidestep the issue /y saying that electricity flows along the wires, /ut that does little to rectify the /asic misconception. Bhe general consensus now seems to /e that unqualified use of the word .electricity. should /e avoided altogether in any rigorous scientific conte<t. Bhe electricity for which the utility company demands payment however, is definitely of the Jeaviside, rather than the electrons in wires, variety. =ow that we have esta/lished the location of the electrical energyG it must /e added that a small amount does flow into H/ut not throughI practical conductors. Bhis is /ecause metal Hpresuming that the temperature is too high for it to /e superconductingI always has some resistance. Bhe inflowing energy is, of course, lost from the fields and converted into heat. Bhe mechanism of energy delivery can /e understood, once again, using the ,oynting vectorG and it e<plains not only unwanted losses, /ut also what happens in relation to devices which are deli/erately made resistive so that they can a/sor/ large amounts of energy. We start /y imagining a small particle of resistance, such as an infinitesimal resistive region in an otherwise perfectly conducting wire. We -now of course, that resistance is distri/uted throughout conducting materialsG /ut the continuity principle allows us to /rea- energy transfer processes down into separate components, which can later /e com/ined to give the overall picture. When a current flows along a conductor, a resistive region gives rise to a voltage drop or .potential difference.. Jence an electric field e<ists /etween a point Fust upstream and and a point Fust downstream of the o/stacle. Bhe diagram on the right shows the interaction /etween the magnetic field encircling the wire and a
: single electric field line Hother lines are left to the imaginationI. Esing the sense of the ,oynting vector given earlier, we see that energy flows into the resistance from /oth the upstream and downstream sides. =ow, mentally, rotate the diagram a/out the wire a<is and it /ecomes a spherical wave'front converging onto the point resistance. A further important o/servation arises when we consider what happens when the direction of the current is reversed. 6n that case, the directions of the electric and the magnetic vectors are /oth reversed, and so energy continues to flow into the resistance. Bhe fields e<plain the strange fact that the direction of energy flow cannot /e reversed /y swapping the polarity of the power supply. 6t can /e reversed however, /y replacing the resistance with a device which is a source of energy Ha /attery or generatorI or /y devices which can store energy Hinductors and capacitorsIG this /eing a matter which we will e<plore in detail later. When we thin- of the ,oynting vector in relation to a complete electrical system, we are really thin-ing of the average of a large num/er of microscopic energy transfer processes. 6n the case of a simple circuit consisting only of a generator and a resistive loadG the ,oynting vector is directed along the wires, from generator to loadG and the direction is the same on /oth sides of the generator. )hould we e<amine the average energy flow close to a wire however, we will see that the direction is tilted very slightly towards the wire, on account of the distri/uted resistive losses. )o now we have the /asic field picture of electricity, /ut there remain a few issues which need to /e e<plained. ,articularly, we need to loo- again at electric current, and the matter of why it is defined in terms of moving charges. As it says in every school physics te<t/oo-, an Ampere is a current of one Coulom/ per secondG and since the charge of an electron is '1.8"%1:7%S1"'17 Coulom/s, an Amp flowing from HTI to H'I corresponds to 1M1.8"%1:7%S1"'17 L 8.%3138"1%%S1"1: electrons per second flowing from H'I to HTI. Bhis is correct Hassuming that the current is due to electronsI, /ut only in the special limiting case when the frequency of the electromagnetic energy /eing transferred tends to Aero. 6n other wordsG it is only strictly true for DC electricity. As the frequency increases, the correspondence /etween conduction current and effective current /ecomes progressively less accurateG which is why *a<well invented displacement current. Bhe role of the electrons in conduction is actually an optical one. Bhey interact with the electromagnetic field in such a way as to increase the amount of energy which can /e stored in the region of space immediately surrounding the conductor. Bhis creates a duct through which the energy prefers to flowG in a manner analogous to the way in which mirages are sometimes seen in hot deserts, and over'the'horiAon !J1 radio communication /ecomes possi/le on hot days. Ducting occurs when the refractive inde< of the medium increases with distance from the surface, i.e., a light ray which tries to move away is /ent towards the parallel direction. Bhe e<istence of electrons was hypothesised, and indeed the name was coined, some time /efore $ $ Bhomson identified them as the current'carriers in cathode'ray tu/es H1:75I. 6t must then have seemed to many that the electrical fluid theory was confirmedG /ut the discovery was actually its nemesis. Epon estimating the num/er of free electrons in a given volume HsayI of copper, it turns out that the average velocity of propagation of an electron current through a solid medium is of the order of a few millimetres per second. Bhe mass of the electron is also so small that the amount of energy transferred /y collisions with the atoms of the conductor comes nowhere near to the amount transferred /y the electromagnetic field. 6n fact, the collision energy is merely the resistive loss which occurs in imperfect conductors. Bhe electrons do not carry the electrical energy, /ut they do e<tract a small ta< for the service they provide in guiding it along the outside surfaces. )o how are we now to thin- of electric current? Bhe current which conveys energy would seem to /e /est descri/ed as a displacement current, at least in the sense that it is not carried /y electrons. 6t merely /ecomes correlated with the num/er of electrons passing a given point in a conductor per second when the generator frequency is low. 1or historical reasons however Hi.e., /ecause *a<well
7 modified the definition of current instead of replacing itI we thin- of current as the sum of conduction and displacement currents. Bhis turns out to /e a reasona/le approach, /ecause there are many situations in which conduction current is important. +lectronics, for e<ample, is the art of controlling electricity /y controlling the conduction current. Charged particles are also involved in the wor-ings of chemical energy sources H/atteries and fuel cellsI and electrochemical processes in general He.g., electro'platingI. 1or the vast range of AC electrical pro/lems however Hincluding the design of electronic circuitsI conduction current is a source of misconceptions, and needs to /e distinguished from current in the general electromagnetic sense. We shall proceed /y thin-ing in terms of a type of current called .current.G which may or may not /e correlated with the movements of charged particles, /ut for most of the time we don.t care whether it is or not. Bhis might seem to imply that we have turned current into an a/stract idea, /ut actually we have simply unloaded some unnecessary /aggage. *any readers will /e aware that voltage is sometimes referred to as .electromotive force. H+*1I. 6ts unit of measurement is not a pure force in the =ewtonian sense, /ut it is proportional to the force e<erted on a charged particleG and so the ha/it of referring to it as a force is loosely Hand widelyI accepted. >i-ewise, current is proportional to the force e<erted on a magnet in the field, and its unit, the Ampere, is the measure of magnetomotive force. Bhe electric field in a circuit is everywhere proportional to the driving voltage. >i-ewise, the magnetic field is everywhere proportional to the effective currentG and the two fields /etween them set the intensity of the ,oynting vector. Also, it has to /e said that AC ammeters are cali/rated according to the amount of energy delivered, and so read the true, or magnetomotive, quantity. Before we move on, it is perhaps worth ma-ing a few o/servations on the use of the term .displacement current.. Bhere are occasional Hnon peer'reviewedI pu/lications which agonise a/out the supposedly deep philosophical implications of this strange quantity. Bhat is unfortunate, /ecause it doesn.t actually e<ist. *a<well coined the term /ecause he initially imagined it as distortions of the Uther, the latter /eing an elastic medium supposed to permeate all space and there/y .e<plain. the parado<ical phenomenon of action at a distance. Bhe 17th Century luminiferous Uther3 has now gone the way of the +arth'centred Eniverse Hand good riddanceIG and )cience has come to e<plain all electromagnetic phenomena in terms of fields and particles. 6t will do no harm to thin- of displacement current as a convenient fudgeG which allows us to e<tend the laws of DC electricity to higher frequencies and there/y avoid having to su/Fect every pro/lem to the full electromagnetic treatment. Jence, .displacement current. is that which has to /e evo-ed /ecause electromagnetic energy doesn.t always follow the wires. 6t is not a physical current. 6t is Fust a quantity which corrects for the difference /etween the magnetomotive force and the conduction current. *agnetomotive force Hor .**1.I is incidentally, not completely synonymous with current. Were it so, we would gladly drop the misleading concept .current. altogetherG /ut unfortunately, we are stuc- with it. Bhe reason is that **1 and current are only identical in circuits formed of a single conducting loop. When the circuit is composed of overlapping loops, disposed in such a way that adFacent conductors carry current in the same direction, the **1 is increased due to a phenomenon called .magnetic flu< lin-age.. )uch overlapping structures are, of course, -nown as coils or inductors, and have the property that they allow the amount of magnetic energy which can /e stored in a given volume of space to /e magnified. )till, for the greater Hnon'overlappingI part of an electrical circuit, current and **1 are practically the sameG and to a good appro<imation, we can dispense with the details and treat coils as separate o/Fects having a single magnetic concentrating property called .inductance.. Certainly, it is very useful to -now how to calculate inductance from the num/er of turns and the physical dimensions of a coilG /ut it is a matter which can separated from the general /usiness of designing electrical systems.
3 Bhe term .+ther. has however come /ac- into favour in the discussion of the properties of the quantum vacuum. see0 The (ightness of )eing, 1ran- WilcAe-. %"":, H,enguin edn. 6)B= 75:"131"3213%I, especially ch. :.
*. Circ+it analysis o!er!iew
Bhe /asic theory of electrical circuits is -nown as .lumped component analysis.. Bhe ver/ .to analyse., incidentally, means .to /rea- down into simpler or more'fundamental parts.G and in this case, analysis is the art of descri/ing and predicting the /ehaviour of circuits /y treating them as networ-s of interconnected resistances, capacitances, inductances and generators. Bhis turns out to /e an e<traordinarily accurate technique when correctly appliedG /ut it has limitations and idiosyncrasies of which the practitioner needs to /e aware. A peculiarity, which is often introduced without comment, is that AC generators Hof the analytical varietyI are considered to produce sinusoidal outputs. *any practical generators He.g., mechanical alternators, radio transmittersI do indeed produce something appro<imating a voltage or current sine'waveG /ut the reason goes somewhat deeper than that. 6f we ta-e, for e<ample, a moving'coil microphone Hwhich is a type of generator which produces electricity from air'pressure variationsI, we will find that its output in response to HsayI the sound of the human voice, is e<tremely complicated. A technique -nown as .1ourier analysis. however, shows that all waveforms can /e /uilt'up /y adding'together sinusoidal waves of different frequenciesG and physical investigation shows that these separate frequency components actually e<ist. A set of one or more frequency components is -nown as a signal. 6t transpires that no new frequencies will /e added to a signal when it is processed He.g., passed from the input to the output of an electrical networ-I, provided that the materials encountered in the transmission path /ehave in a linear manner. 6n general, a material is said to /e .linear. when its change in response to some force is proportional to the intensity of that force Hi.e., the graph of change versus force is a straight lineI, and the change is reversi/le. 4esistance, of course, o/eys a straight'line law called .(hm.s law.. Bhe AC voltage versus current laws laws governing inductance and capacitance are also linear. Bhus, the /asic circuit elements com/ine to ma-e linear networ-sG which lac- the a/ility to produce new signal components. Bhis means, overall, that a linear networ- treats each frequency component as if it e<ists in isolation. We can therefore quantify its /ehaviour one frequency at a time, which is why the /asic generator of circuit analysis produces only a single frequency. Bhe response of a circuit to more'complicated waveforms can always /e /uilt'up Hwhen requiredI /y adding the results of analyses carried out at the component frequencies. 1or many purposes however, the focus of interest when several frequencies are involved is the frequency response, which is Fust a stepwise application of the one'frequency'at'a'time approach. Despite the simplicity offered /y single'frequency analysis, there are, of course, numerous electrical and electronic components which /ehave in a non'linear fashion. )emiconductor devices Hdiodes, transistors, etc.I are an o/vious e<ampleG /ut there are also materials which change their characteristics according to field strength. Bhis might seem to place a limitation on linear networtheoryG /ut actually, there is a straightforward solution. A non'linear device is one which accepts energy at one or more e<citation frequencies and converts it into energy at one or more new frequencies. 1rom a circuit analysis point'of'view, anything which a/sor/s energy is a resistance, and anything which which creates a new frequency component is a generator. Jence we can put the /ehaviour of an alien device into a metaphorical ./lac- /o<.G with one or more two'terminal connections called .ports., which loo- to the outside world li-e networ-s of /asic circuit elements. Bhe rule /y which energy disappears into a resistance inside the /o<, and reappears from one or more generators inside the /o< is called the .transfer function.. Bhe fact that anything with electrical connections and a -nown transfer function can /e incorporated into circuit theory confers enormous power upon the method. As mentioned previously howeverG circuit analysis does have its limitations. 6t is after all, not a general theory, /ut a proFection or .degenerate form. of electromagnetic theory. =aturally, a price is paid for the simplification, and it is instructive to consider what that is. )o, loo- at a circuit diagram and try to find where the lengths of the wires and the physical dimensions of the components are written. Bhat information is conspicuous /y its a/sence, /ecause a circuit diagram
11 is a purely topological representation Hli-e the famous >ondon Enderground mapI. 6t was a curious and usually unremar-ed discovery of the early circuit e<perimenters, that it doesn.t matter how the equipment is laid out, or whether a component is large or small, or how long the wires are Hprovided that they are a lot more conductive than any of the designated resistancesI. Bhis, of course, ceases to /e true as the frequency is increasedG and this /rea-down of DC theory is partly due to the finite speed of light. Consider a sine'wave generator connected /y means of relatively long wires to a resistive load. 6f we measure the voltage difference /etween any two points in the circuit, we o/tain a quantity which is proportional to the total electric field e<isting /etween those points. 6n this case, the field is associated with electromagnetic energy propagating from the generator to the load. )ince it ta-es a finite time for the energy to ma-e the Fourney, this means that the voltage measured across the generator will not /e identical to the voltage measured across the load. 6f we presume that the resistive loss in the wires is negligi/le, the main difference will /e in the relative phases of the two sine wavesG i.e., if we ta-e some reference point on the waveform, such as the Aero'crossing'point on going from negative to positive, we will find that the load waveform is delayed relative to the generator waveform. Bhis will not /e noticea/le if the measurements are made using an ordinary AC voltmeter, /ut the time difference can certainly /e demonstrated using a dual'channel oscilloscopeG and the same effect will give rise to performance deviations in more complicated Hi.e., phase criticalI circuits. Bhere are ways of dealing with such pro/lems Hand in the e<ample case, it is to represent the wires as inductances with some capacitance /etween themI, /ut it is important to understand that the .truth. of circuit diagrams is contingent upon unspecified factors. Knowing the difference /etween representation and reality is the art Has opposed to the scienceI of circuit analysis. =o one would want to apply the full electromagnetic theory to routine circuit pro/lemsG and indeed, success in the solving of *a<well.s equations for some particular class of pro/lems is often regarded as a scientific event. Jence electricity is primarily associated with circuits, rather than fields and waves, and /eing .good at it. requires a level of understanding which is difficult to formalise. +<perience comes with time, /ut we can at least invite entry to the @uild /y offering a straightforward rule of thum/. A light wave in vacuum completes one cycle of variation of its electric and magnetic fields upon travelling a distance given /y the e<pression0 VLcMf where f is the frequencyG c L %7757%3#: metresMsecond is the speed of lightG and V [email protected]
/daKI is the wavelength. Due to the essentially refractive nature of circuits, the apparent velocity of signal propagation through an electrical networ- is never e<actly c, /ut it rarely deviates from c /y more than a few W. Jence we can easily o/tain an idea of the phase errors which will accumulate as a result of constructing a circuit on a particular physical scale. A given amount of phase error does not necessarily translate into the same error in some other quantity, /ut if we confine ourselves to thin-ing of the order of the error Hi.e., its magnitude therea/outsI, it is a fairly good guide. (n that /asis, we can answer the question0 K6f 6 /uild the circuit as drawn, how well will its performance agree with my analysis?K Bhe scale on which attention to physical detail Hlayout, wire lengths, component siAe, etc.I will /e required, for a given level of agreement, is shown for various frequencies in the ta/le /elow Husing the mental'arithmetic appro<imation c L 2S1": mMsI.
Fre,+ency f 2""KJA 2*JA 2"*JA 2""*JA [email protected]
#a!elength VLcMf 1Km 1""m 1"m 1m 1"cm
Constr+ction scale for a gi!en acc+racy 1-. 1. -.1. 1""m 1"m 1m 1"m 1m 1"cm 1m 1"cm 1cm 1"cm 1cm 1mm 1cm 1mm ".1mm
Assuming that most signal'processing circuits are constructed on a scale of a/out 1"cm, we can see that there will /e no serious discrepancies /etween analysis and practical results for frequencies up to a/out 2*JA Hignoring displacement currents for the time /eingI. Beyond that, we are definitely into the realm of radio'frequency engineering, where layout is importantG /ut note that this does not mean that analysis will fail. 4ather, as was alluded'to earlier, it requires a modified approach, where some physical varia/les have to /e turned into theoretical circuit components. As we approach ultra'high frequencies HEJ1I however, the struggle to adapt the lumped component representation will /ecome increasingly difficultG and the need to resort to *a<well.s equations Hor at least, to standard solutions o/tained from the scientific literatureI will /ecome more and more apparent. While on the su/Fect of scale incidentallyG note that the wavelength range of visi/le light runs from ".5 to ".3 microns Hwhere 1 micron L 1Om L 1"'8 mI. 6f we were to represent the interaction of visi/le light and matter using circuit diagrams, the circuits would have to /e /uilt on the molecular scale Haround 1nm L 1"'7 mI. Jence visi/le light has no measura/le tendency to /e guided /y ordinary electrical circuitryG /ut it does have the convenient ha/it of reflecting from the components, so that we can see them. We can now draw together two threads from the preceding discussion. 4epresenting a circuit diagrammatically /egins with a /asic assumptionG that either the circuit is infinitesimally small, or that the speed of light is infinite. +ither way, it means that every part of the circuit is assumed HinitiallyI to /e in instantaneous communication with every other part. 6t is also assumed that the electrical energy always follows the wires, whereas it is actually distri/uted in the fields surrounding the circuit. 6n /oth cases, in the a/sence of corrective measures, this can result in disagreement /etween the /ehaviour calculated from circuit theory and the measured performance of the actual circuit. 6n resolving the potential for discrepancy, we must first recognise that circuit diagrams fall into two categories0 those which are used for production engineering and end up in service manualsG and theoretical diagrams used /y designers. As we will see in this and su/sequent articlesG a theoretical diagram is actually a type of mathematical statementG which can /e e<tended to descri/e the /ehaviour of a physical circuit to an almost ar/itrary degree of precision. A production diagram, on the other hand, is Fust a record of the interconnections in a set of manufactured su/'assem/lies Hresistors, transistors, coils, etc.I. As has already /een implied0 for equipment operating at audio frequencies and /elow, there may /e a great deal of similarity /etween the diagram used /y the design engineer and the diagram in the service manualG /ut for well'designed radio equipment, this will not necessarily /e the case. (n the su/Fect of circuit diagramsG it will /e noted that the =orth American or $apanese preferred HAig'Aag lineI sym/ol for resistance is used here. Bhis convention is adopted Hor, in the author.s case, was never un'adoptedI /ecause the rectangular /o< sym/ol was already in use /y theoreticians long /efore +uropean standards Happarently intended solely for the convenience of draughtsmenI were put forward. 6n this, and all of the other documents produced /y this author, the /o< sym/ol is
12 used strictly to represent a generalised electrical networ-. 6t is also, in particular Hand in -eeping with long'standing practiceI, used to represent a generalised two'terminal linear networ- called an .impedance. Ha mathematical construct with which we are a/out to /ecome very familiarI. 6n the following sections of this article, we will derive the /asic AC theory, which deals with notionally discrete resistances, inductances and capacitances, these /eing -nown as ideal components. As will /e shown, the /ehaviour of networ-s of these components can /e determined /y starting with a handful of empirical electrical formulae Hi.e., formulae determined /y e<perimentI and then using the properties of the ,oynting vector to determine the rules of com/ination. We will not, incidentally, need to -now how to carry out 2D vector multiplication e<plicitlyG we simply need to -now that, for multiplication operations of any type, when the two quantities /eing multiplied have the same alge/raic sign, the answer is positive, and when they have opposite signs, the answer is negative. Bhis will result in an e<tendi/le theory of linear networ-sG e<tension /eing a matter of incorporating circuit modules H/lac- /o<esI called .component models. or .equivalent circuits., which have ports defined as networ-s of ideal components. Bhese modules were mentioned previously as a way of dealing with non'linear devicesG /ut the approach can /e applied to any electrical device which requires the use of a more sophisticated theory Hsuch as electromagneticsI, or which is not well descri/ed /y treating it as a discrete ideal component. 6t will /e noticed that we have tal-ed of .resistances, capacitances and inductances., rather than .resistors, capacitors, and inductors.. Bhere may /e no great difference /etween the two conceptions from the point'of'view the audio engineerG /ut for the radio engineer, ideal components and practical components are not the same. Whether a practical component can /e regarded as an ideal component is a matter of internal dimensions and wavelength. )ome capacitors, for e<ample, are made /y rolling'up long lengths of metallised plastic film, giving an assem/ly which /ehaves quite li-e a pure capacitance at low frequencies, /ut turns into an inductance at radio frequencies Hthus radio engineers have a certain fondness for capacitors which have small electrodesI. )imilarly, an inductor is made /y coiling'up a length of wire, and its property of pure inductance is modified /y the resistance of the wire and /y the time it ta-es for an electromagnetic wave to propagate along it. +ven the hum/le resistor is not perfectG and in general0 every practical two'terminal device is replaced /y an equivalent circuit, which may HsometimesI correspond to a discrete ideal linear component at low frequencies, /ut at high frequencies will always mutate into a networ- of resistances, capacitances and inductances. Bhe su/Fect of component models is developed in detail in later articles in this seriesG /ut for now, the important point is that circuits designed on the assumption of ideal /ehaviour may require e<tra wor- if they are to /e realised in practice. =eglecting possi/le H/ut usually minorI non'linearitiesG the difference /etween ideal and practical components can always /e attri/uted to a com/ination of three factors0 time delaysG displacement currentsG and the finite resistance of conductors at ordinary temperatures. Bhe same can also /e said of wiring and layout. Jence, an accurate theoretical model for a practical electrical system may also require components to represent spurious effects in the circuit at large. 6n the case of e<treme time delays Hi.e., long connecting wiresI, we can introduce a two'port module called a .transmission line., which is a solution of *a<well.s equations in a /o<. *inor effects can /e accounted for /y including .stray. or .parasitic. resistances, capacitances and inductances here and there. 6n the a/sence constructional details and related corrections howeverG it is implicit in any theoretical circuit0 that the components will /e grouped on a scale which is small in comparison to wavelengthG that interconnection resistance is negligi/le in comparison to specified resistanceG and that there may /e a need for shielding to prevent energy from turning'up in places where it is not wanted. Javing cautioned against the pitfalls howeverG we should also caution against over'modelling. +ven at radio frequencies, there need not always /e a great deal of difference /etween the idealised and the practical representations. Bhis is /ecause, firstly0 circuits will often wor- adequately when /uilt according to the assumption that practical components are nearly idealG and secondly, nominal
13 component values are su/Fect to manufacturing tolerances, which means that accurate performance can only /e achieved /y ma-ing some components adFusta/le. AdFustment not only serves to correct the nominal component value to the required value, it can also a/sor/ deficiencies in the original analysis. Jence, it is important to understand that a simple approach will often do the tric-G and e<treme attention to su/tleties is usually only needed when attempting to achieve the most e<acting standards of radio frequency H41I performance.
3. )asic electrical for/+lae
Bhe theory which will /e developed in the sections to follow is of a type classed as .steady'state analysis.. Bhis does not mean that nothing changes, /ut that everything which does change is assumed to do so sinusoidallyG i.e., there are no sudden or .transient. events. We can always represent more complicated phenomena Hif necessaryI /y adding'together sine waves of different frequencies, or we can loo- to other /odies of theory /etter suited to systems which undergo non' periodic Hi.e., non cyclicalI changeG /ut for a large range of pro/lems, the steady'state approach is all that is needed. Bhe translation from DC to the AC steady'state is a matter of replacing the /attery with a sine'wave generator, and then adopting definitions for voltage and current such that the esta/lished DC laws continue to /e true Hinsofar as that is possi/leI. Bhe necessary choice is to use 4*) values of voltage and currentG where the 4*) Hwhich stands for0 the square'4oot of the *ean of the )quaresI is an average chosen so that AC and DC electricity /oth have the same heating Hor long'term energy'deliveryI effect#. By using 4*) values, we conscript various formulae and diagrammatic conventions into AC serviceG e<cept that, due to the alternation of the power supply, it no longer ma-es any sense to write the sym/ols HTI and H'I against the generator terminals, and it no longer ma-es any sense to equate current to the flow of electrons. Bhe solution is to ensure that the polarities of voltages and currents are defined in a way which gives the correct sense to the ,oynting vector Hwhich is what the DC conventions achieve /y rote instead of reason in any caseI. An accepted usage, which is adopted here, is to draw an arrow across the generator terminals to indicate the direction in which electric potential is assumed to increase. Current then .flows away. from the high'potential terminal. 6f /oth current and voltage are then ta-en to /e positive Hor /oth negative if you li-e, it amounts to the same thingI, the chosen convention is that energy Hor powerI flowing away from a generator is positive. As was mentioned earlier, AC analysis is the art of representing circuits as networ-s of interconnected resistances, inductances and capacitances. Jence we have the tas- of developing the general rules of com/ination for those elements. What we have to com/ine is summarised in the ta/le /elowG which gives the /asic electrical formulae much as they appear in numerous te<t/oo-s.
# KThe 012 3!erageK D. W. Knight. QAvaila/le from g2ynh.infoR.
(/serve that only the entries in the left'hand column contain fundamental scientific information. Bhe uppermost entry is a statement of (hm.s lawG which is that the electrical current 6 is proportional to the voltage applied across the ends of a conductor, the constant of proportionality /eing -nown as the resistance. Bhe entry /elow it is the power lawG which represents the o/servation that a conductor HresistorI heats up as a consequence of an electric current Hi.e., it dissipates energyI and the power consumed Hi.e., the energy delivered per unit'of'timeI is the product of the applied voltage and the current, i.e., ,L!6 QWattsR. Also, /y using the su/stitutions K6L!M4K and K!L64K, we o/tain two alternative power laws0 K,L!XM4K and K,L6X4K. Bhe e<pression K,L6X4K is -nown as Joule's law, and is a statement of the fundamental relationship /etween electricity and thermodynamics. (ne important point to note a/out the standard power and resistance formulae however, is that they are all derived from e<periments with DC electricity. Bhey represent incomplete statements of (hm.s law and $oule.s law, /ecause they can only /e applied to AC circuits when the load on the generator is a pure resistance. >ater'on we will show how to state these laws in a completely general way, /ut some groundwor- will /e required /efore that can /e done. =otice incidentally, the correspondence /etween the power law ,L!6 and the earlier'given definition of the ,oynting vector P L E S H. Bhe former is a dimensionally'reduced version of the latter, as can /e seen /y noting that the unit of electrical field strength is .!olts per metre., and the unit of magnetic field strength is .Amperes per metre.. Also, even though we do not need to -now how to perform 2'dimensional vector multiplicationG it is not difficult to understand that any type of multiplication also multiplies the units of measurement. Jence the unit of the ,oynting vector is .Watts per metre'squared., which is a measure of illuminationG i.e., the delivery of electrical power is a matter of illuminating the receiving o/Fect with electromagnetic energy. 6n the case of inductors and capacitors, the entries in the left'hand column tell us that they also o/ey (hm.s law when connected to a generator of alternating voltage /ut, insofar as we can construct them without inadvertently including resistance, they consume no power. Bhe reason why the ideal versions of these components cannot dissipate energy is that they have no resistance /y definition, i.e., they cannot convert energy into heat or wor-. 6nstead, over the course of a generator cycle, the amount of energy which flows into the component is e<actly equal to the amount which
18 flows out. Bhis property forces a Y7"Z phase difference /etween the voltage and current waveformsG a Y['cycle or .quadrature. offset /eing that which causes the ,oynting vector to reverse its direction four times per cycle. Jence, although the average or steady'state power consumption is Aero, the instantaneous power'flow is alternating at twice the generator frequency. 6n the case of an inductor, the AC resistance or reactance \> Hmeasured in (hmsI is directly proportional to the inductance Hin JenrysI and to the frequency f Hin JertA, i.e., cycles per secondII of the applied voltage. Bhe quantity %]f is -nown as the angular frequency Hi.e., the frequency in radians per second, where %] radians corresponds to 28"ZI and is often given the sym/ol ^ [email protected]
'case KomegaKI. 6n the case of a capacitor, the reactance \C is inversely proportional to the angular frequency, and also inversely proportional to the capacitance Hin 1aradsI. =ote also that capacitive reactance is shown as /eing negativeG /ecause it transpires that when capacitors and inductors are connected to form resonant circuits, the reactance of the inductor, in some sense, cancels the reactance of the capacitor. Bhis means that one of the types of reactance has to /e considered to /e negative and, as will /e e<plained later, we choose it to /e the capacitive variety in order to /e consistent with the conventions of trigonometry. Bhe other entries in the ta/le are derived from the formulae in the left'hand column, using only (hm.s law and a /asic electrical rule -nown as Kirchhoff's first law Hpronounced0 K-ir'-hovKI. Kirchhoff.s law tells us that the sum of all the currents flowing into a given point in a circuit is equal to the sum of all the currents flowing out. Bhis law was originally regarded as proof of the principle of conservation of charge in DC circuits Hwhat goes in must come outIG /ut it also turns out to /e true of current in the general HmagnetomotiveI sense, provided that we use the correct rules of addition Hto /e determined shortlyI in circuits involving /oth resistance and reactance. Bhe entries in the middle and right'hand columns are, of course, the well'-nown series and parallel com/ination formulae for passive electrical components. Bhese e<pressions may all /e regarded as e<amples of simple mathematical models Hin this case, in the sense that a single component can serve to represent a com/ination of several componentsI. (f these, the formula for resistances in series is the simplest of all, and tells us that whenever we encounter two resistors in series, we can treat them as a single resistor with a value equal to the sum of the two resistances. Bhat this statement is derived from e<isting physical laws can /e seen /y applying some /asic techniques of circuit analysis to the circuit shown /elow0 0esistors in 2eries Bo analyse this circuit, we first o/serve that, as a requirement of Kirchhoff.s first law, the current in the two resistors must /e the same. (hm.s law then tells us that !1 L 6 41 and !% L 6 4%. =ow, since voltage is analogous to pressure, common sense Hotherwise -nown as Kirchhoff.s second lawI tells us that the total pressure'drop is equal to the sum of the pressure'drops across the two resistors, i.e., ! L !1 T !% ,utting these ideas together we have0 ! L 6 41 T 6 4 % L 6 H41 T 4%I =ow, if we postulate a hypothetical resistance 4 which represents the series com/ination of 41 and 4%, it must /e possi/le to replace 41 and 4% with this resistance and o/tain the same current for a given voltage, i.e., ! L 6 4 L 6 H41 T 4%I Jence0 4 L 41 T 4 %
15 0esistors in Parallel 6n the case of two resistors in parallel, the voltage across the two resistors is the same. Jence (hm.s law tells us that0 ! L 6 1 41 L 6 % 4% . . . . . . . H3.1I and Kirchhoff.s first law tells us that0 6 L 61 T 6% =ow, if we postulate a hypothetical resistance 4 which represents the parallel com/ination of 41 and 4%, we have0 ! L 6 4 L H61 T 6%I 4 We can eliminate 61 and 6% /y using equation H3.1I a/ove, i.e., 61L!M41 and 6%L!M4%, hence0 !L Q H!M41I T H!M4%I R 4 Bhe voltage can then /e factored out and cancelled to give0 1 L Q H1M41I T H1M4%I R 4 and dividing each side of the equation /y 4 gives0 1M4 L H1M41I T H1M4%I Bhis is one form of the standard e<pression for resistors in parallel, and a little rearrangement will give us the other. 6nverting the e<pression a/ove gives0 4 L 1 M Q H1M41I T H1M4%I R We then arrange the terms inside the square /rac-ets to have a common denominator Hmultiply the 1M41 term /y 4%M4% and multiply the 1M4% term /y 41M41I, i.e., 4 L 1 M Q _H4%MH414%I` T _41MH414%I` R hence0 4 L 1 M Q H41 T 4% I M 41 4% R which, upon inversion, gives0 4 L 41 4% M H41 T 4% I Bhe formulae for inductors and capacitors in series and parallel may also /e derived /y using e<actly the same approach as was used a/oveG the only difference /eing that inductive reactance \>L%]f>, or capacitive reactance \CL '1MH%]fCI, is su/stituted in place of resistance. Bhe %]f factors and any minus signs disappear /y cancellation, leaving formulae involving only inductance or capacitance. =ote incidentally, that the inductors in the illustrations in the previous ta/le are shown orientated at right'angles to each other, this /eing done as a reminder that the formulae are only true when there is no magnetic coupling /etween the coils. =ote also, that the capacitor formulae ta-e on the opposite forms of their resistance counterpartsG this /eing due to the reciprocal HinverseI relationship /etween capacitance and capacitive reactance.
Bhe com/ination rules discussed a/ove allow us to deal with resistors, or capacitors, or inductors in series and parallel, /ut for reasons which will /ecome clear in the following sections, they do not provide a method for dealing with com/inations of resistance and reactance Hif we try to add resistance to reactance directly, our calculations will not agree with our measurementsI. We can however deal with com/inations of inductive and capacitive reactance, provided that we o/serve the convention that capacitive reactance is negative. We may therefore add to our repertoire of standard formulae /y writing general e<pressions for pure reactances in series and parallel, i.e.0 4eactances in series \ L \1 T \ % 4eactances in parallel \ L \1 \% M H\1 T \%I
1: =ow, since inductive reactance is positive and increases with frequency, and capacitive reactance is negative and decreases with frequencyG if an inductance is placed in series or parallel with a capacitance, there will occur a frequency at which the two reactances cancel. Bhat frequency, of course, is the resonant frequency of the com/ined reactances. A resonant frequency is usually denoted /y the sym/ol . f". HKf noughtKI 6n the case of an inductor and a capacitor in series, the reactance goes to zero, i.e., the com/ination /ehaves li-e a short'circuit Hneglecting resistanceI, when \>T\CL". 6n the case of an inductor and a capacitor in parallel, since the term \>T\C is on the /ottom of a fraction, it would appear that the reactance goes to infinity, i.e., the com/ination /ehaves li-e an open'circuit when \>T\CL". A complete open'circuit does not appear in practice however, /ecause in the parallel case, it transpires that we are not at li/erty to neglect the resistances of the coil and the capacitor. We therefore cannot calculate the exact resonant frequency of a practical parallel tuned circuit, nor the resistance which remains when the reactance has /een cancelled, until we have developed a more comprehensive theoryG and so that is another matter which we must leave until later. We can say however, that the e<act resonant frequency of a series tuned circuit, and the appro<imate resonant frequency of a typical parallel tuned circuit occurs when0 \> L '\C i.e., %] f" > L 1MH%] f" C I =ow, if we rearrange this equation to put /oth instances of f" on one side we have0 f"X L 1MH 3]X > C I and ta-ing the square'root gives0 f" L 1MQ %] NH> CI R 4.1 Hpronounced0 Kf nought equals one over two pi root >CK, or in rhyme0 K(ne over two pi root >C gives the resonant frequencyKI. +quation H4.1I is, of course, is the standard resonance formulaG /ut /efore accepting it we should note that, /ecause it contains a square root, every com/ination of > and C has two resonant frequencies associated with it. +very equation involving a square root has two solutions /ecause the square root of a num/er is, /y definition0 .a quantity which, when multiplied /y itself, gives the num/er in question.. When two negative num/ers are multiplied, the result is a positive num/er. Jence, if < is a positive num/er, we must note not only that0 <X L < S < /ut also that0 <X L H'<I S H'<I hence0 NH<XI L Y< )o, the resonance formula stated e<plicitly /ecomes0 f" L Y1MQ %]NH> CI R and there are two solutions, numerically identical /ut of opposite sign. By convention, we usually assume the positive resultG /ut since there were no restrictions on the validity of the arguments used in deriving the formula, the negative frequency solution must e<ist and must mean something. AC electrical theory, as we delve more deeply into it, will present us with various little conundrums Hoften involving square rootsIG and although negative frequency is one of the most trivial, we will /e ill prepared for the others if we simply let it pass. Bhe negative frequency solution arises /ecause a sinusoidal waveform is derived from circular motion, and there are two possi/ilities for this0 cloc-wise and anti'cloc-wise. Bhis does not mean that the positive and negative frequencies are identical however, as the following argument will illustrate0 Consider an alternator Hmechanical AC generatorI stopped at a position where it would give Aero output voltage Hor currentI if it were spinning, and where it will give an initially positive output if its shaft is turned
17 anti'cloc-wise Hsee illustration /elowI. =ow, if the shaft is turned clockwise, the output will initially go negative. 6t follows that the difference /etween the positive and negative frequency outputs is that while the voltage Hor currentI associated with one is positive the voltage associated with the other is always negative, and vice versa. Jence changing the sign of a frequency has the effect of shifting the phase of the associated waveform /y 1:"Z. 6ncidentally, for anyone who might insist on ta-ing the direction of rotation analogy too seriouslyG it is of course o/vious that if the generator is an electronic oscillator, the concept of rotation is meaningless. 6n that case, the negative frequency solution can /e o/tained /y swapping the connections Has it can with any generator or resonatorI.
5. 6/pedance, 0esistance, 0eactance
Bhe /asic electrical laws discussed earlier tell us that resistors consume power when connected across a generator /ut that perfect inductors and capacitors do not. Bhe com/ination formulae then tell us how to deal with resistances or reactances in series and parallelG /ut they do not tell us how to deal with com/inations of resistance and reactance. Bhis is a serious limitation, which can only /e overcome /y introducing the generalised concept of impedance, i.e., the theory of two'terminal devices which o/ey (hm.s law /ut do not necessarily consume all of the electrical power delivered to them. A concept which needs to /e formalised at this point is that of a linear, passive, two terminal networ-. An electrical device is linear if its graph of voltage versus current is a straight line, i.e., if it o/eys (hm.s lawG and it is passive if it contains no sources of energy. Bhe general term .networ-. is used /ecause, although the theory we are a/out to develop covers .simple. devices li-e capacitors and resistors, it also covers any com/ination Hactual or hypotheticalI of resistances and reactances in series and parallel connected to a single pair of terminals. A networ- can /e hypothetical in the sense that it /ehaves in the same way as Hi.e., serves as a model forI an actual two terminal device. 1or e<ample, when an antenna system is connected as a load to a radio transmitter, we can treat it as a hypothetical networ- of resistances and reactances. An antenna, incidentally, is not completely passive, /ecause it also receives radio signals, /ut we can model the receiving case /y considering it to /e e<actly the same networ- as in the transmitting case, /ut with one or more generators connected in series with it. Any linear passive two'terminal networ- can /e regarded as an impedance. Bhis means that its electrical /ehaviour at a particular frequency can /e e<plained /y invo-ing two Hand only twoI mutually independent propertiesG namely resistance and reactance. 4esistance 4 is that property of the networ- which ena/les it to dissipate Hi.e., consume or dispose ofI energy, and reactance \ is that property which ena/les it to store energy. 6t is also a special property of our Eniverse thatG while energy dissipation is, on average, a one'way processG the electromagnetic energy storage mechanisms come in the form of a complementary pair. Bhis means that true resistance is always
%" positive Ha statement which we will qualify laterI, /ut there are two opposing types of reactance, which of course we -now as inductive reactance, \>L%]f>, and capacitive reactance, \CL'1MH%]fCI. 6nductive reactance arises through the storage of energy in a magnetic field, and capacitive reactance through the storage of energy in an electric field. When inductance, capacitance, and resistance are com/ined within the same two'terminal ./lac- /o<., the opposing reactances will always tend to cancel'out to some degree, and so the two types of reactance ma-e only a single contri/ution to the impedance at any particular frequency. Bhere is however, no way in which resistance and reactance can /e com/ined to form a single numerical quantity, /ecause the physical processes they represent turn out to /e mutually e<clusive. A natural distinction arises /etween resistance and reactance /ecause perfect energy dissipation implies that the ,oynting vector never reversesG whereas perfect storage and return implies alternation, with the ,oynting vector spending equal amounts of time in the two possi/le flow directions. Jence, for a resistanceG when the instantaneous voltage is positive, the current is positiveG and when the voltage is negative, the current is negativeG i.e., the voltage and current waveforms are perfectly in phase. 1or the ,oynting vector to alternate and give Aero average power delivery however, there must /e a Y['cycle difference /etween the voltage and current waveforms. 6t is the "Z difference in the resistive case, and the Y7"Z difference in the reactive case, which gives rise to a condition of mathematical independence, or othogonality, which we can e<ploit to o/tain a generalised form of (hm.s law. (nce we have that generalisation, the rules of com/ination follow and give rise to a complete and internally'consistent AC theory. 1or DC circuits, we can write (hm.s law as K!L64K. 1or AC circuits therefore, we must suspect that we can write something along the lines of K!L6DK, as long as we recognise that impedance, D, the generalised attri/ute of o/Fects which o/ey (hm.s law, must /e represented /y some composite quantity containing two distinct elements 4 and \. 6n circumstances such as this, it is traditional to see if anyone has developed a /ranch of mathematics which suits the pro/lem, and the clue regarding where to loo- lies in the independence of 4 and \. 6f two quantities are completely independent, they must in some sense e<ist in different dimensions Hi.e., they always move at right' angles to each otherI. Bhis means that impedance cannot /e represented /y an ordinary num/er, i.e., a one'dimensional quantity lying on a line /etween 'a and Ta, it must /e represented /y a point on a two'dimensional plane, which is another way of saying that D can /e plotted as a point on a graph of 4 against \. With regard then to solving pro/lems involving impedance, it so happens that we are spoilt for choice, /ecause there are no less than two appropriate /ranches of mathematics, namely vectors and complex numbers. Bhe vector approach traditionally preferred /y engineers is that of ma-ing s-etches or graphs, and using trigonometry to wor- out the actual num/ersG whereas the comple< num/er approach is alge/raic, in that it allows equations involving two'dimensional o/Fects to /e written'down and re'arranged. Both approaches are equivalent however, and sometimes one can clarify the other, and so we will adopt a notation and a way of thin-ing which ena/les us to switch freely /etween them.
7. 8ectors 9 2calars
A vector is, /y definition, a mathematical o/Fect which must /e descri/ed /y two or more independently varia/le num/ers. 6mpedances, as we have noted, fall into this categoryG and so vectors can /e used to descri/e them. (ne very useful property of vectors is that they can /e mi<ed with ordinary num/ers and manipulated using the normal rules of arithmetic, provided that the rules are generalised to accommodate them. Because vectors are different from ordinary num/ers however, it is helpful to note each one as a letter in a bold typeface Hor in handwriting /y putting a little arrow a/ove the sym/olI, and an optional comma'separated list in /rac-ets may /e included to denote its e<tent in its various dimensions. Bhus we can represent an impedance as :H4 , \I , /y
%1 which we mean that : is characterised /y an amount 4 in the resistance dimension and an amount \ in the reactance dimension. 6n the conte<t of vectors, ordinary num/ers are -nown as scalars, /ecause the effect of multiplying a vector /y a scalar is to scale it Hi.e., magnify or shrin- itI without otherwise changing it. Bhus, if s is a scalar, we can write0 s:H4 , \I L :;Hs4 , s\I =ote also, a widely used mathematical notation, which is to use an apostrophe or KprimeK H . I to indicate that an o/Fect has /een modified. We can immediately deduce a rule for adding vectors /y o/serving that two quantities will only add together if they e<ist in the same dimension Hyou can.t increase the length of an o/Fect /y adding to its widthI. Bhus, if we want to add two impedance vectors :1H41 , \1I and :%H4% , \%I , i.e., find out what happens when the impedances are placed in series, all we have to do is add the 4 parts and the \ parts separately to find the new impedance :H41T4% , \1T\%I . Bhis operation is indicated /y the .T. sym/ol, Fust as in ordinary arithmetic, i.e. if :H4 , \I L :1H41 , \1I T :%H4% , \%I then 4L41T4% and \L\1T\% (ne point in treating impedances as vectors, is that it ena/les us to draw diagrams in order to visualise what is going on. We can do this /y representing an impedance as a line in a plane, with a particular length and orientation. 6n this sense, a vector diagram is li-e a navigation chart, with the distances, in this case, measured in (hms. *athematicians calls such maps .spaces., /y analogy with ordinary spaceG and a space in which distance is measured in (hms is called impedance space. =ow o/serve that although the 4 and \ parts of an impedance e<ist in different dimensions, they /oth e<ist in the same space /ecause they are connected /y the fact that they are measured using the same units Hi.e., (hmsI. We may therefore deduce that the difference /etween a space and a graph is that all of the a<es in a space must /e la/elled in the same unitsG whereas the a<es of a graph can have different units He.g., temperature vs. timeI. bou may, of course, have heard of four' dimensional space'time, which appears to diso/ey the rule Fust stated, /ut in fact the unit of the fourth physical dimension is not time /ut the speed of light multiplied by time, i.e., ct. Bhe units of ct are metres per second S seconds, i.e., metres, and so +insteinian space has four dimensions with units of length. Wor-ing in impedance spaceG if we adopt the standard convention that resistance increases to the right and reactance increases upwards, we can o/tain the line representing an impedance /y plotting a point, then moving right /y a distance 4 and upwards /y a distance \ Hor downwards if \ is negativeI, and plotting another point. Bhe length of the line which Foins the two points is called the magnitude or .modulus. of :, and is written c:c Hand pronounced Kmod DKI. Bhe magnitude is always positive /y definition, and is o/tained /y using ,ythagoras. theorem Hthe square on the hypotenuse of a right' angled triangle is equal to the sum of the squares of the other two sidesI. Jence0 c:c L TNH4X T \XI 7.1 =otice also that the definition of magnitude has a meaning for Band L \ M 4 ordinary num/ers /ecause they can /e regarded as a one'dimensional Cosd L 4 M c:c vectors. Jence, if s is a scalar0 )ind L \ M c:c csc L TNHsXI i.e., the effect of ta-ing the magnitude of an ordinary num/er is simply to remove the sign HT or 'I. Bhe direction of : is given /y the angle d Hlower'case KphiKI it ma-es with the horiAontal HresistanceI a<is, which is the angle whose tangent is \M4, i.e.,
%% \M4LBand Jence0 dLArctanH\M4I 7.* H .Arctan. is sometimes written . Ban'1 . I. =ote that d can /e positive or negativeG and in particular, if we adopt the standard trigonometric convention that a positive angle is o/tained /y going anti'cloc-wise from Aero Hsee diagram rightI, d will /e positive for an impedance with an inductive reactance and negative for an impedance with a capacitive reactance. =otice also that c:c and d, ta-en together, provide a complete characterisation of a two'dimensional vector and so give us an alternative way of recording its properties. Bhe form introduced earlier0 :H4,\I is -nown as the rectangular form /ecause it contains a list of values in dimensions chosen to /e at right'angles to each other. Bhe alternative0 :Hc:c,dI is -nown as the polar form, /ecause it uses polar co'ordinates Hdistance and /earingI. Bhe polar form uses different units in its two dimensions H(hms, degrees or radiansIG whereas the rectangular form has the same units in /oth dimensions H(hms, (hmsI. Bhere is no am/iguity /etween the rectangular and polar forms /ecause the list in /rac-ets is optional, and a vector has the same properties regardless of how it is defined. Also, if a specific vector quantity is to /e noted /y putting actual num/ers into the /rac-ets, a degrees HZI sym/ol ne<t to the angle will indicate that the polar form is intended. We can now regard equations H7.1I and H7.*I as the transformations which ta-e a two' dimensional vector from the rectangular to the polar form. Bhe reverse transformations are o/tained from the standard trigonometric relations0 CosdL4Mc:c and )indL\Mc:c, i.e., 4 L c:c Cosd and \ L c:c )ind Bhe full set of transformations is summarised in the following ta/le0 4ectangular form :H 4 , \ I :H c:cCosd , c:c)ind I ,olar form
:H NQ4X T \XR , ArctanQ\M4R I : H c: c , d I
6f we want to add two impedances graphically, we simply place the /eginning of the second against the end of the first, and draw a new line from the /eginning of the first to the end of the second. Bhus we get a new impedance, with a new magnitude and a new direction. Bhis might all seem rather unnecessary in view of the simple addition rule given earlier, /ut the meaning of vector addition is HhopefullyI o/vious when it is visualised in this way. Whatever the method used in performing the arithmetic however, the point in doing it, as we shall see, is that it allows us to -eep trac- of the relationship /etween the voltage applied across an impedance and the corresponding current.
>. )alanced 8ector ",+ations
6t is here that we must o/serve the principal property of the .equals. sym/ol, which is that if a given type of mathematical o/Fect lies on one side of it, then e<actly the same type of o/Fect must lie on the other. Bhus, now that we -now that impedances are vectors, we must re'write (hm.s law in such a way that equality is never violated. Bhere are numerous ways in which that can /e done, /ut for the moment we will e<amine three possi/ilities0 8L6: 8L6: 6L!M: 6t is /y no means o/vious that all of these e<pressions must /e trueG /ut, as we shall see, they all are when interpreted correctly. )ince impedance is a vector, then either voltage is a vector, or current is a vectorG or Hpresuming that the product of two vectors is also a vectorI /oth current and voltage are vectors. 6n fact /oth voltages and currents are vectors /ecause they each have associated with them a magnitude, a frequency, and a phaseG the phase /eing defined as0 the time at which a chosen event in the wave cycle occurs He.g., the time of Aero'crossing from negative to positive in the illustration /elowI. Bhe generator frequency is not an independent varia/le in the definition of impedance, /ecause it already appears in the reactance H \>L%]f> , \CL '1MQ%]fCR I, and so we may deduce that the direction of the impedance vector HdI constitutes phase information, i.e., it gives us the time difference Hin degrees or radiansI /etween corresponding events in the voltage and current waveforms. Jence d is -nown as the phase angle, and can /e converted into a time difference in seconds /y dividing a complete cycle of the waveform into 28"Z or %] radians and noting that the time'per'cycle or period of the waveform is 1Mf.
=ow note that since 6 and 8 are vectors, we can write them in polar or rectangular forms using the transformations H7.3I given earlier, i.e., 6H c6c , d I L 6H c6cCosd , c6c)ind I 8H c8c , d I L 8H c8cCosd , c8c)ind I 6n general, it is natural to thin- of currents and voltages in their polar formsG /ut the rectangular form is important for understanding what happens when the phase angle is either "Z or 1:"Z. Ba-ing a current vector as an e<ample0 6H c6c , "Z I L 6H c6cCos"Z, c6c)in"Z I L 6H Tc6c , " I and 6H c6c , 1:"Z I L 6H c6cCos1:"Z, c6c)in1:"Z I L 6H 'c6c , " I When a two dimensional vector lies along the "Z direction, either pointing with it or in opposition, its e<tent in one of its spatial Hi.e., rectangular formI dimensions is AeroG and, as in our interpretation of negative frequency given in section 4, the minus H'I sym/ol is associated with a 1:"Z phase shift Hor phase reversalI.
%3 )o, now that we -now that /oth current and voltage are vectors, we must conclude that 8L6: is the general statement of (hm.s law. 6t transpires however, that we may admit the validity of the other possi/ilities 6L!M: and 8L6: under certain circumstances. Bhe point is that, in AC theory, we are usually interested not in the a/solute phases of the voltages and currents Hi.e., the phases relative to some e<ternal referenceI, /ut in their phases relative to each other. Bhis means that we are often at li/erty to choose the direction of one of the vectors in order to learn the directions of the others relative to it. Bhe direction chosen for this special reference vector is in principle ar/itraryG /ut a simplification occurs if we choose it to /e either "Z or 1:"Z /ecause )ind goes to Aero in either case, and a vector which is Aero in one of its spatial dimensions /ehaves, in this context, as though it has one less dimension. A two'dimensional vector which drops a dimension in this way, of course, /ecomes a one'dimensional vector, i.e., a scalar. Jence, whenever a voltage or current appearing in a mathematical e<pression is written as a scalar, the sym/ol can /e Hand, as we shall see later, must beI interpreted to mean that the corresponding vector is lying along the "Z a<is. A vector which transforms as a scalar in some specific conte<t is called a pseudoscalar. A pseudoscalar has the property that when its space co'ordinates are reflected with respect to the origin H","I it changes sign, whereas a true scalar remains unchanged8. Jence voltages and currents /ecome pseudoscalars when we choose their directions to /e "Z or 1:"Z. Another electrical pseudoscalar is resistanceG a special -ind of impedance which can /e treated as a scalar, /ut which /ecomes negative if the co'ordinates of impedance space are reversed. 6f we choose the current in (hm.s law to /e our reference vector, and set its phase angle to "Z, it /ecomes a pseudoscalar of value equal to its e<tent in the "Z directionG i.e., it is identifia/le as the quantity c6cCos"Z or Tc6c. Bhus, in the relationship 8L6:, we can recognise 6 as the reference vector against which the phase of 8 will /e determined0 6 L 6H c6c , "Z I L Tc6c and 8 L 6 : L HTc6cI : Bhe pseudoscalar current 6 is therefore equal to the current magnitude c6c, the latter /eing the quantity registered /y an ordinary AC ammeter Han device ignorant of phaseI. 6t is however not identical to the magnitude /ecause it can /e negative in principle, even if not usually in practiceG i.e., if, for some reason, the reference phase is chosen to /e 1:"Z, then0 6 L 6H c6c , 1:"Z I L 'c6c An AC ammeter must /e considered to register magnitude c6c rather than pseudoscalar current 6 /ecause swapping the connections ma-es no difference to the reading, i.e., the instrument can never give a negative indication Hand putting a minus sign in front of each of the num/ers on the scale won.t help, /ecause then it will never /e a/le to give a positive indicationI. We can however equate the meter reading c6c with the reference vector 6 if we want to -now the phase of the voltage relative to "Z. A similar logic applies in the case of the relationship 6L!M:, where, on the HcorrectI assumption that the reciprocal of a vector Hi.e., 1M:I is also a vector, we can identify the pseudoscalar voltage ! as the reference vector against which the phase of 6 will /e determined0 ! L 8H c8c , "Z I L Tc8c c8c is, of course, the quantity registered /y an ordinary AC voltmeter, and we can equate it to ! if we want to -now the phase of the current relative to "Z. )o, what we have seen here is that if one of a set of voltage or current vectors is replaced /y its magnitude, it /ecomes a reference vector pointing at "Z. We may also deduce the converse, which is that if a vector should happen to /e pointing at "Z /y virtue of a choice made elsewhere, then it too can /e replaced /y its magnitude. 6t is however important to understand that there is a difference /etween a vector which has dropped a dimension and a magnitude, /ecause there will /e
8 "le/entary Particles, +nrico 1ermi, )illiman memorial lecture series, bale Eniversity ,ress, 17#1. Definition of pseudoscalar0 p7.
%# many circumstances in which we will want to use the magnitudes of vectors which are free to point in any direction. 6n particular, we will need this distinction later in order to generalise $oule.s law. 6t will however /ecome apparent that adoption of the convention that vectors written as scalars Hi.e., un'/oldI are pointing at "Z Hor 1:"ZI preserves the meaning of most of the DC and pure'resistance' only formulae which appear in standard te<t/oo-s. Bhe correspondence arises /ecause, whenever a vector is written as a scalar, a statement is made to the effect that the phase of that vector He<cept for the signI can /e ignored. A DC formula wor-s for AC when the circuit contains only pure resistance /ecause, in that case, rotating one vector to point at "Z rotates all of the others to point at "Z, and so they can all drop a dimension. Jence 8L6: /ecomes !L64 Hfor e<ampleI. (ne consequence of all of this is that, in formulae, we should avoid writing voltages and currents as scalars unless we really mean them to /e pointing at "Z or 1:"Z. We must however permit a common convention, without which the notation will appear very cum/ersomeG which is that whenever we refer to a current or a voltage without mentioning phase we mean magnitude, i.e., the o/serva/le quantity which can /e measured with a two'terminal meter. 6n other words, a measurement ta-en from a voltmeter may /e written in isolation HsayI !o+tL%5!, /ut as soon as it is inserted into a formula with other vectors it acquires a phase, even if we don.t need to -now what that is, and must then /e identified as c8o+tc. )o, mindful of the warning that reference vectors and magnitudes are not quite the same thing, the e<pression 8L6:, now tells us that if we multiply an impedance /y the magnitude of the current passing through it, we will o/tain a vector representing the magnitude of the applied voltage and its phase relative to the phase of the current. Bhis is an e<tremely useful result, and stems from the fact that the vector representation has captured the physics of the situation e<actly. 6n effect, having o/served that resistance and reactance act independently on the current, and that inductive and capacitive reactances act in oppositionG we have elected to represent pure inductive reactance as a vector pointing at T7"Z, pure resistance as a vector pointing at "Z, and pure capacitive reactance as a vector pointing at '7"Z. Bhus we have satisfied the requirement that the ,oynting vector must alternate for reactance, /ut not for resistance, and we have incorporated it into the definition of impedance itself. =ow however, it follows, that when some mixture of resistance and reactance is connected across a generator, the angle for the voltage'current phase difference will lie at some intermediate value, and the use of vectors allows this angle, the phase angle d, to /e determined from simple geometry. *ore to the point, a phase angle of "Z implies that an impedance will a/sor/ all of the power delivered to it, and a phase angle of Y7"Z implies that an impedance will not accept any power. Bhus we can o/serve that the phase angle represents not only the relationship /etween voltage and current for an impedance, /ut also the effectiveness with which power can /e delivered to it.
As we have Fust shownG one of the interpretations of (hm.s law is that, if an impedance vector is scaled /y a current magnitude, it is transformed into a voltage vector. )ince the act of scaling a vector does not change its direction, it transpires that /oth the impedance vector and the voltage vector contain the same phase information, and that this information is conserved after multiplication /y a scalar. ,ut in plain language, this means that, although the current through an impedance will change according to (hm.s law as the applied voltage is changed, the 8'6 phase relationship will not change provided that the frequency is held constant. 6t is for this reason that vectors used in impedance related applications are -nown as .Phasors. Hi.e., phase'vectors, or .carriers of phase.I, and diagrams involving them as .Phasor Diagrams.. Bhe special properties of phasors Has distinct from vectors in generalI are as follows0
%8 ● Bhe phase co'ordinate is defined in relation to other phasors rather than to an a/solute time reference. ● Bime is measured in degrees or radians relative to one cycle of the frequency at which the analysis is /eing carried out. ● A phasor deemed to /e pointing at "Z may /e replaced /y its magnitude, and a phasor deemed to /e pointing at 1:"Z may /e replaced /y the negative of its magnitude. ● ,hasors are strictly two'dimensionalG i.e., the vector cross product Hwhich produces a new vector at right angles to the original twoI has no meaning for phasors. )hown /elow is a phasor diagram illustrating what happens when an impedance consisting of a resistance, an inductance, and a capacitance in series is connected across a generator. We can easily deduce the total impedance /y inspection in this caseG /ut notionally, it is o/tained /y regarding the individual series elements as phasors0 :4H4,"I, :>H",\>I, and :CH",\CIG and adding them together. Bhus0 :4H4, "I T :>H", \>I T :CH", \CI L :H4, \>T\CI We can draw the resultant phasor : /y moving along /y a distance 4 and moving up /y a distance \>T\C Hor down if \>T\C is negativeI, /ut notice that in the diagram, the resistances and reactances have all /een scaled /y a reference phasor 6, which is equal to the magnitude of the current. By so doing, all of the quantities have /een turned into voltages, and so the diagram has /ecome a voltage phasor diagram.
With regard to the physical phenomena represented hereG o/serve that, since 4, C, and > are in series, they must all carry the same current. We can deduce the magnitudes of the voltages across across the three components using (hm.s law, i.e., c8>cL6\>, c8CcL6\C, and !4L64 Hthe latter /eing written as a scalar /ecause it is in phase with 6 and therefore pointing at "ZI. We also -now the relative phases of these voltages /ecause they are all lin-ed to the phase of the common currentG i.e., the voltage 64 across the resistance is in phase with the current, the voltage 6\> across the inductance is at T7"Z relative to the current, and the voltage 6\C across the capacitance is at '7"Z relative to the current. We can therefore add these three voltages as vectors to o/tain the magnitude of the generator voltage and its phase relative to the phase of the currentG although in the diagram the voltages across the two reactances have /een added first to produce the more diagrammatically convenient quantity 6\>T6\C Hthis /eing the voltage across the total reactance in the systemI. =ote that the voltages across the two reactances always tend to cancel /ecause there is a fi<ed 1:"Z phase difference /etween them, and so the magnitude of the voltage across the total reactance is always smaller than the magnitude of either 6\> or 6\C . Bhe relationship /etween the phase'angle o/tained from a phasor diagram and the waveforms which can /e o/served using a two'channel oscilloscope is shown /elow Hwhere e means Kgreater thanK and f means Kless thanKI 0
Jere we have o/tained a waveform which is e<actly in phase with the current /y measuring the voltage across the resistive component H/ottom traceI. When this is compared against the waveform of the total voltage 8 Husing the upward Aero'crossing as an ar/itrary reference pointI, we
find that 8 is advanced in time Hi.e., leadingI relative to 6 when the impedance is inductive H\>T\Ce"I, and 8 is retarded HlaggingI relative to 6 when the impedance is capacitive H\>T\Cf"I. 6f we call the time difference o/served on the oscilloscope gt Hwhere .g. is upper'case KDeltaK, a sym/ol normally used to mean Kthe difference inKI, then the ratio of gt to the time of a complete cycle is the same as the ratio of the phase angle d to a complete circle. Bhe time'per'cycle Halso -nown as the period of the waveformI is of course the reciprocal of the frequency H1MfI, hence, if d is measured in radians0 gtMH1MfI L dMH%]I i.e., gt L d M H%]fI =ote incidentally, that it is impossi/le Hneglecting the use of superconductorsI to ma-e a series >C4 networ- from which all of the resistance can /e isolatedG /ecause practical inductors and capacitors always have some internal resistance. A measurement made across any part of the total series resistance will however always produce a voltage which is in phase with 6. A device which measures current /y sampling the voltage across a resistance is, of course, an ammeter. 6t was stated earlier that capacitive reactance is defined as a negative quantity in order to ma-e AC theory consistent with trigonometry. Bhe convention we follow is, of course, that which says that the phase angle of a vector increases as it rotates in the anti'cloc-wise direction. Jence the choice of \C as the negative reactance stems from the fact that voltage lags Hi.e., pea-s 7"Z later
%: thanI current for a capacitor, whereas voltage leads Hpea-s 7"Z ahead ofI current for an inductor. Bhis can /e remem/ered /y considering what happens when a capacitor in series with a resistor is connected to a /attery0 a large inrush of current precedes the /uild'up of the voltage across the capacitor terminals. 6f the capacitor is replaced /y a coil, the opposite happensG the /uild'up of current is delayed /y a /ac-'voltage produced /y the growing magnetic field. Bhere is no need for convoluted reasoning in AC theory howeverG Fust remem/ering the sign of \C ta-es care of everything. =ote however, that many technical articles follow the hallowed tradition of treating \C as negative in some statements and positive in others. Bhis is done as an aid to comprehension, /ecause it encourages the reader to re'derive all of the mathematics in order to find out what the writer was trying to say.
@. 8oltage 1agnification 9 A
(ne of the curious properties of the series >C4 networ- discussed a/ove is that the voltages across the reactances can /e much larger than the applied voltage. Ba-e for e<ample, an impedance consisting of a 1h resistance in series with an inductance having \>L1""h and a capacitance having \CL '1""h, all connected across a generator giving 1! output. 6n this case, the system is resonant /ecause \>T\CL"G and so the voltage across the total reactance, c8\cL" and the phase angle, dL"Z. Because the two reactances have cancelled each other out, the impedance loo-s li-e a pure 1h resistance, /ut there is a current of 1A flowing through each reactance, and so each has a voltage of 1""! across it. Bhis voltage magnification H1""01I is also, /y definition, the ; or KqualityK of the series tuned circuit formed /y >, C, and 4, i.e., ;L\>M4 and also ;L '\CM4 Hor c\CcM4I. Bhe smaller the series resistance, the /etter the quality. 6n the case of an impedance such as an antenna, of course, we cannot get inside it and measure the voltages across the individual components Hand a simple series >C4 com/ination is nowhere near complicated enough to account for the way in which antenna impedance varies with frequencyI. When ma<imising the power delivered to an antenna however, we frequently need to place the antenna impedance in series with another impedance in such a way as to create a pure resistance into which the transmitter can deliver all of its power. We would of course, li-e to cancel the reactance of the antenna /y placing a pure opposite reactance in series with it, /ut pure reactance is unattaina/le, and so our compensating Hor conjugateI reactance always /rings some e<tra HlossI resistance with it. 6n this case, although the voltage appearing across the terminals of the com/ined impedance may /e very low, the voltage across the antenna terminals can /e enormous, and we must choose the voltage ratings of our matching networ- components accordingly. 1or an illustration of the voltage magnification effect, consider the short vertical antenna system depicted in the diagram /elow0
%7 6n order to avoid misconceptions it is important to /e aware that the vertical rod itself is not the antenna. Bo use the rod as a radiator, we must apply a voltage to the pair of terminals formed /y it and the ground'plane, and so the antenna is the com/ination of the rod and the ground'plane. Bhe input impedance of an electrically short Hless than a quarter'wavelength long, i.e., fVM3I vertical antenna loo-s predominantly li-e a very small capacitor, which is essentially the capacitance which e<ists /etween the vertical section and the ground. A small capacitor has a large negative reactance Hrecall \CL '1M%]fCI, and so we need to place a large inductive reactance H\>L%]f>I in series with the antenna to ma-e the whole thing loo- li-e a resistor. 6f we now fill'in some of the details a/out the antenna and the loading coil, we are in a position to calculate the voltages across the antenna terminals and the loading coil for a given generator power, and also the overall efficiency of the complete antenna system Hi.e., the proportion of the applied power which is actually radiatedI. Any mechanism which dissipates energy, i.e., consumes power, must loo- electrically li-e a resistance. Bhe resistive part of the antenna input impedance is shown to contain two components 4a and 4r . 4a is the electrical losses of the antenna, due mainly to the 41 resistance of the metal conductors used to ma-e the rod and the ground plane and the dielectric losses H41 heatingI of any insulating materials used. 4r is the radiation resistance of the antenna, i.e., a resistive component associated with energy radiated into space. Both 4a and 4r are in some sense distri/uted over the whole antenna, /ut they appear as a single resistive component H4aT4rI in the antenna input impedance. Ba-e for e<ample an antenna with a radiation resistance of %h and an input reactance of '2"""h. Bhese are the appro<imate values to /e e<pected for vertical section and ground'plane radial lengths of a/out 5W of the wavelength at the frequency of operation, i.e., "."5V Hgraphs for estimation of radiation resistance for short antennas are given in the references listed /elow5 :I. 6f we are very careful a/out the materials used in the antenna system, we might -eep the loss component 4a down to a/out ".#h, so that the input impedance of the antenna will loo- li-e %.#h of resistance and '2"""h of reactance. Bo cancel the antenna reactance H\aL'2"""hI, we need to place a coil having \>LT2"""h in series with it. )uch a .loading coil. is normally placed out with the antenna, mainly /ecause coils inside metal /o<es have more losses than coils mounted in wide' open spaces, /ut even so, the coil will not /e perfect and will have a distri/uted 41 resistance which loo-s li-e another resistive component in the antenna input impedance. Bhe amount of coil resistance is given /y the ; of the coil, which is the ratio of reactance to loss resistance, i.e., ;>L\>M4>. A well'made loading coil might have a ; of a/out 3"", and so 4>L\>M;>L5.#h. With the reactance of the antenna now cancelled /y the coil, the input impedance of the whole antenna system now loo-s li-e a pure resistance of %.#T5.#L1"h. )uppose we now decide to deliver 1" Watts H1"WI from a generator HtransmitterI into this 1"h resistance. Knowledge of the power level ena/les us to calculate the antenna current and hence the voltages which appear /etween the various terminals, /ut a word of caution is in order /efore using the standard power formulae for this purpose. Bhe e<pressions0 K,L6!K, K,L6X4K, and K,L!XM4K are all deeply suspect /ecause, if we try to convert them into vector e<pressions simply /y changing the voltages and currents into vectors, then the equations which result will /e nonsense /ecause power Henergy per unit'of'timeI is scalar Hstrictly, pseudoscalar, as we will see laterI. We need to develop some additional ideas on the su/Fect of vectors /efore this matter can /e fully resolved, /ut the standard e<pressions will /alance if all of the vectors involved can drop a dimension. Jence we can concede that the e<pressions are true for voltage and current magnitudes, provided that the generator is driving a purely resistive load Ha somewhat restrictive condition, /ut we happen to satisfy it hereI. Bhus, using the standard formulae, we can wor- out that the current in the antenna will /e
5 K"fficiency of 2hort 3ntennasK, )tan @i/ilisco [email protected]
!, Jam 4adio, )ept 17:%, p1:'%1. @raphs of radiation resistance vs. electrical length for short verticals and dipoles. +fficiency calculations. : KHow long is a piece of wire&K $ $ Wiseman, +lectronics and Wireless World, April 17:#, p%3'%#. Discussion of the efficiency Hor lac- thereofI of electrically short verticals. Bhe effect of top loading.
2" c6c L 6 L NH,M4I L NH1"M1"I L 1A and the voltage at the generator will /e c8c L ! L NH,4I L NH1"S1"I L 1"!. =ote however that the current will result in a voltage of 2"""! H6\I across /oth reactances, and although these voltages are cancelled at the generator, we can certainly e<perience them as real /y touching the Function /etween the rod and the loading coil Hnot recommendedI. 6n fact, the electric field'strength at the top of the loading coil is so great that a neon lamp or a small fluorescent tu/e held there will light without any wires connected to it Hsee photograph /elowI. Bhe actual voltage appearing across the coil, c8>c, can /e o/tained /y using ,ythagoras. Bheorem, i.e., c8>c L 6NH\>XT4>XI L 6NH2"""XT5.#XI L 2"""."1! and the voltage across the antenna terminals Hi.e., the voltage /etween the /ottom of the rod and the ground planeI is, c8ac L 6NH\aXTQ4aT4rRXI L 6NH2"""XT%.#XI L 2""".""1!. Bhese voltages are /arely different from the voltages across the HtheoreticalI pure reactances, and reflect the fact that reactance dominates the impedances of /oth the coil and the antennaG /ut despite the reactive input impedance of the antenna we have nevertheless turned it into an effective radiator. (ne way to loo- at this is to say that /y resonating the antenna with a loading reactance, we e<ploit the voltage magnification of the resulting tuned circuit in order to force power into a reactive load. We could, of course, do the same /y sheer /rute forceG /ut that would involve using a generator with an output of Fust over 2"""! to get a measly 1A into the antenna. (ne further point to note here is that the voltages calculated /y the a/ove methods are all 4*) voltages. 4*), as mentioned earlier, stands for .square'0oot of the 1ean of the 2quares., this /eing a mathematical tric- to find an equivalent constant HDCI voltage or current which gives the same heating effect as an alternating voltage or current. Bhe voltages and currents referred to in the theory of impedance must /e 4*) values /y definition, /ecause that is the only way in which (hm.s law can /e generalised to include /oth DC and AC HDC /ecomes a special case of AC with f L "I. Bhe need for an 4*) average arises /ecause the ordinary average of a sinusoidal alternating voltage is Aero Hthe voltage spends as much time /eing positive as it does negativeI. 6f however, we square the instantaneous voltage, we o/tain a function which is proportional to the power it will deliver to a resistance. 6f we average that power function Hi.e., ta-e the mean of the squaresI and then ta-e the square root, we will o/tain the equivalent direct voltage, i.e., the constant voltage which will deliver the same amount of power to a given resistance. Bhe 4*) average of a sine wave is the instantaneous pea- value divided /y the square'root of %, i.e., !4*)L!,- MN%, and !,-L!4*)SN% . Bhus if we calculate a voltage of 2"""!4*) across the antenna terminals, the ma<imum instantaneous Hpea-I voltage will /e 2"""S1.313%L3%35!, and it is this higher figure which must /e used in calculating the voltage ratings of the components used. Bhe final piece of information we can e<tract from the vertical antenna e<ample under discussion is the efficiency of the system. 6n this case, all we have to do is note that the total input resistance was 1"h, whereas the radiation resistance was %h. Bhis gives an efficiency of %M1" or %"W, i.e., %W radiated for 1"W in, %"W radiated for 1""W in. Bhis incidentally, is not a disaster, and represents a good figure for a short loaded'verticalG around 1"W of ))B radiated from a reasona/ly high location /eing sufficient for worldwide short'wave communication in suita/le atmospheric conditions.
8oltage /agnification in action 1or the antenna in the illustration on the right, the frequency of operation is 1.:3*JA and the physical length of the antenna assem/ly is 1.3#m from the /ottom of the ru//er mounting /ase to the top of the neon lamp soldered to the tip. Bhe section a/ove the loading coil is ".58m long H".""35VI. Bhe clamp holding the fluorescent tu/e is made from acrylic resin H,erspe<I, and there is no electrical connection /etween the tu/e and the whip. Bhe glow from /oth lamps is visi/le at an input level of 1W, /ut since the photograph was ta-en on a /right summer.s day Hal/eit in the shadeI, the power input to the antenna matching networ- HA*EI was turned up to 1""W to overcome the daylight. Bhe antenna is one of the author.s old 18"m mo/ile whips from the early 175"s. 6t is not an optimal design, /ut it gave useful service Ha range of several miles using 1W of A*I despite having an efficiency of considera/ly less than 1W. Bhe long thin shape of the coil does not give ma<imum ;, /ut it does cause the coil to radiate to some e<tent Hsome of its .loss. resistance is actually radiation resistanceI. Bhe 8W fluorescent tu/e was added for this demonstration, /ut the neon /ul/ at the tip was always used as a tuning aid. Bhe generator in the photograph is a Kenwood B)32"s J1 transceiver with its mains power supplyG and the A*E is an *1$7:7C B'networ-. Bhe input to the antenna is resistive when the length is adFusted correctly Ha/out %#h, mainly due to the coilI, and the A*E was used to transform this resistance to #"h, as required /y the generator. Bhose wishing to reproduce this demonstration should note that, apart from the mains lead, there is no proper ground'plane for the set'up, and the author had to tune'up wearing ru//er gloves in order to avoid getting /urnt fingers. *ounting the antenna on a car is safer.
1-. Power Factor 9 2calar Prod+ct
6n the preceding discussion, we o/served that reactance acts as an impediment to the delivery of power into an impedance, and that the applied voltage must /e increased in order to overcome it. Bhis means that the DC power formula K,L6!K, if we interpret it to mean the product of the current and voltage magnitudes, is not generally true for impedances /ecause, e<cept in the special case that \L" Hi.e., when the impedance is a pure resistanceI it will give a result which is larger than the actual power delivered. As mentioned earlier, we can deduce what is wrong with the equation in a purely a/stract way /y noting that 6 and 8 are phasors, whereas , is scalar. =ow we will fi< the pro/lem /y finding a method of vector multiplication which produces a scalar. Bhe first step in doing so is to refer to the product of the magnitudes c6cc8c as the apparent power0 ,apparent L c6c c8c Bhe true power, on the other hand, is the power dissipated in the resistive part of the impedance, which can /e determined from the magnitude of the currentG i.e., using a properly /alanced version of the DC formula0 , L c6cX 4 and if we choose 6 as a "Z reference vector0 , L 6X 4 +arlier in this chapter, we showed how an impedance phasor diagram can /e scaled /y a reference phasor 6 to o/tain a voltage phasor diagram Hi.e., every resistance or reactance in the diagram is multiplied /y 6 I. Bhe phasor diagram /elow has /een scaled /y 6X to o/tain a .power phasor. diagram. Jere we should /e aware that the phrase .power phasor. is an o<ymoron Hi.e., a contradiction in terms li-e Kencrypted /roadcastKI /ecause average power is scalarG /ut apparent power is not power, and we can thin- of it as a vector. 6n particular, having set the phase of the current to /e "Z, the .phase. of the apparent power is given /y the e<pression0 Papparent L 6 8 and since 8 L 6 :H4,\I, then Papparent L 6X :H4,\I which gives the definition of apparent power as0 PapparentH6X4, 6X\I Bhus the phase of 8 relative to 6 is the .phase. of the apparent power relative to the true power H,L6X4IG and the magnitude of the apparent power is the diagonal of the .phasor. diagram shown /elow0
1rom this, we can determine a correction factor for the c6cc8c power formula, particularly /y o/serving that the cosine HadFacent M hypotenuseI of the phase angle is ,MHc86cI, i.e.G , L c8 6c Cosd or, after factoring out the pseudoscalar 60 , L c8c 6 Cosd or, since 6Lc6c0 , L c8c c6c Cosd =otice that Cosd is Aero for dLY7"Z Hno power is delivered to a pure reactanceI, and CosdL1 for dL"Z Hreal and apparent power are the same for a pure resistanceI. Be aware also that the formula a/ove appears in standard te<t/oo-s as0
22 K , L ! 6 Cosd K /ut unfortunately, there is nothing we can do to salvage this traditional version. We will prove later that the un'/old sym/ols ! and 6, when used in an equation, must /e interpreted as phasors pointing at "Z or 1:"Z /ecause that is the only way in which we can incorporate DC and AC into the same theory. 8 and 6 however can only point in the same direction when dL"Z. Bhe standard formula is therefore internally inconsistent Ha mathematical o<ymoronI. Bhe /est that can /e said for it is that there is little choice /ut to assume ! and 6 to /e magnitudes in this instance, since the e<pression is nonsense otherwise. Bhe quantity . c8cc6cCosd . is -nown as the scalar product or dot product of the two vectors, and is defined in the same way for all vectors Hregardless of the num/er of dimensionsI0 3B) L c3c c)c Cosd 6t is the component Hshadow lengthI of ) when proFected onto the direction of 3 multiplied /y the length of 3 Hand vice versa, i.e., 3 and ) are interchangea/leI. Jad we attac-ed the DC power formula K,L6!K with a fore-nowledge of vector theory, we would have failed it on the grounds of dimensional inconsistency H, has too many dimensionsI and deduced that the scalar product is required, i.e., , L 8B 6 L c8c c6c Cosd 1-.1 6nstead, we attac-ed the pro/lem /ac-wards and discovered the definition of the scalar product instead. =ote however, that there is a su/tle difference /etween the general vector dot product and the phasor dot product, which will /e discussed shortly. 6n the conte<t of impedance, Cosd is -nown as the power-factor H,1I, and is of particular interest to electricity generating companies, which prefer their customers to place pure resistances across the supply so that they do not have to run their generators into reactive loads. Bhus if a load such as an electric motor is inductive as well as resistive, a suita/le capacitor placed across it or in series with it can /e used to cancel the reactance and /ring the power'factor to unity, i.e., dL"Z and CosdL1. Bhis /rings the apparent power into coincidence with the actual power consumed, and has the effect of minimising the consumer.s electricity /ill as well as minimising the stress on the generators and power transmission equipment. Bhus power-factor correction, in relation to electricity distri/ution, is equivalent to the /usiness of /ringing an antenna system into resonance. Bhe reactance'cancelling step in antenna matching, and the insertion of a loading coil into a vertical antenna, can /oth perfectly well /e regarded as a forms of power'factor correction. =ow, since power can only /e delivered to the resistive part of an impedance, only that part of voltage'multiplied'/y'current which corresponds to true power Hi.e., the c6cX4 componentI can /e measured in Watts. Bhe reason is that power Hthe amount of energy delivered or wor- done in unit timeI esta/lishes the relationship /etween electricity and thermodynamics, and the connection is through energy dissipation. 6t is therefore the convention in electrical engineering, to e<press apparent power in volt'amps H!AI and only true power in Watts. *any readers will already /e aware that mains transformers and porta/le electric generators Hfor e<ampleI are rated in !AG the implication /eing that to get the full power output without over'stressing the device, it is necessary to ma-e the apparent power in !A equal to the true power in Watts, i.e., to provide the transformer or generator with a resistive load. )ince ma<imum power output will /e associated with a particular value of load resistanceG it transpires that all generators, not Fust radio transmitters, require impedance matching if the ma<imum allowa/le output is to /e o/tained. (ne further point which should /e noted on the su/Fect of power, is that power flowing from a generator to a resistance is, /y convention, positive. 6n DC circuits, this means that, if the voltage applied to a resistance is ta-en to /e positive, then the current in the resistance must also /e ta-en to /e positiveG and if the voltage is ta-en to /e negative, then the current is negative. Also note that, in AC circuits, power is calculated from 4*) voltages and currents. Bhis means that it is already an average or steady'state quantity, and there is therefore never a need to compute the 4*) value Hit can /e done, out of mathematical curiosity, /ut it is numerically not the same as
23 c84*)cc64*)cCosd , see ref 7I. Bhus the term K4*) WattsK, commonly seen in the Ji'1i literature, is nonsense and should /e avoided.
11. Phasor dot prod+ct
When the angle /etween two vectors is ta-en for the purpose of computing the dot product, there are actually two possi/le choices0 the acute angle Hf7"ZIG and the o/tuse angle He7"ZI. Bhese are shown in the diagram on the right as d and d.. =ow, using the trigonometric identity0 CosH1:"'dI L 'Cosd we can see that there are two possi/le solutions to 3B) L c3c c)c Cosd which are numerically identical /ut of opposite sign. 4ecall from the earlier discussion that there are also two possi/le solutions to the ta-ing of a magnitude H/ecause it involves a square'rootI, /ut that /y convention we ta-e the positive answer. )o it is with the dot product, in general vector theoryG /ut in phasor theory it transpires that there is also a meaning to the negative solution. =otice that when computing power from the dot product, we don.t actually specify the acute angle. )trictly, the angle which must /e used is the phase angle, and the power will /e negative if the phase angle should happen to /e o/tuse. =oting that0 , L c6cX 4 it should /e apparent that the magnitude of the phase angle will /e greater than 7"Z, and the power will /e negative, for an impedance which has a negative resistive component. =egative power dissipation does not occur in nature, /ecause it violates the principle of conservation of energy. 6t can occur in circuit analysis howeverG when a networ- which has /een defined as passive turns out to /e active. Bhe point is that power can flow out of an .impedance. if the networ- inside it should happen to include a generator. (ne situation in which negative resistance can /e encountered is when modelling antenna systems with multiple feed points. Due to the coupling /etween the different parts of the antenna, it sometimes occurs that more power flows out of one of the ports than flows inG and the computed input impedance then has a negative resistive component. Bhe same can happen in any networwith multiple ports. =ormally the situation is avoided /y defining the port as active in advanceG /ut when modelling a complicated antenna, there is generally no analytical solution for the input impedance of a port Hi.e., it is not possi/le to find a solu/le alge/raic e<pressionI and the computation involves so'called .numerical methods.. Jence the e<istence of an active element in a port cannot always /e foreseen. 6t follows thatG when writing computer programs for networ- analysis, it is important to code for the possi/ility of negative resistance Hrather than terminating with an error message when it occurs, or worse still, ignoring the signI. Bhere is nothing intrinsically wrong with itG it is Fust an unconventional way of defining an active networ-, and it should not /e ta-en to constitute a fault or inconsistency in the mathematics.
7 C012 watt, or not&C >awrence Woolf, +lectronics World Dec 177:, p1"32'1"3#. Why !4*) × 64*) is not 4*) power.
1*. Co/pleD E+/bers
Although the graphical .phasor diagram. approach outlined in the previous sections is suita/le for pro/lems involving phasor addition and scaling Hi.e., series networ-sIG it is somewhat less tracta/le for solving pro/lems involving phasor multiplication and division, one of particular importance /eing that of how to analyse networ-s involving impedances in parallel. 6n section 3 we derived an e<pression for resistances in parallel, and also /y inference an e<pression for reactances in parallel, i.e., 4 L 41 4% M H41 T 4% I \ L \1 \% M H\1 T \%I 6t should come as no surprise that, if we repeat the e<ercise with impedances instead of reactances or resistances, applying ordinary arithmetical operations to the phasors without -nowing what they mean, we end up with the e<pression0 : L :1 :% M H :1 T :% I Bhe pro/lem now is that of how to interpret this equation, a somewhat inconvenient matter if we continue to define phasors as comma'separated listsG /ut it transpires that there is a short'cut, due to the fact that we are only dealing with two-dimensional vectors, which is that such vectors can /e treated as complex numbers. Comple< num/ers were first discovered as a .necessary evil. in solving quadratic equations, i.e., equations which can /e written in the form0 a<XT/<TcL". Bhey were once descri/ed as the wor- of the Devil, /ut in fact, they merely indicate that ordinary num/ers are not the whole story. Bhose who studied quadratic equations at school, /ut never got as far as comple< num/ers, may /e surprised to learn that all of the e<amples they were given were deli/erately chosen so as not to involve comple< num/ersG and that education systems in general e<pend more effort trying to protect students from the -nowledge of comple< num/ers than they e<pend trying to teach the su/Fect. A derivation of the general solution for all quadratic equations is shown in the /o< /elow0 Feneral 2ol+tion for A+adratic ",+ations: (/taining the general solution to all quadratic equations is a matter of re'arranging the general form a<XT/<TcL" so that < is all alone on one side of the equation. We can start /y su/tracting c from /oth sides, so that0 a<X T /< L 'c and then divide /oth sides /y a, so that0 <X T H/<MaI L 'HcMaI . . . . H1*.1I. We now need to find a su/stitution for the term <XTH/<MaI such that < is on its own. We can do that /y o/serving that <XTH/<MaI loo-s similar to part of the e<pansion of a quantity in the form H<TpIX, Hwhere .p. is Fust an ar/itrarily chosen sym/olI i.e., H<TpIX L <XT%p<TpX . . . . H1*.*I Bo use this su/stitution, we equate the term %p< in equation H1*.*I with the term /<Ma in equation H1*.1I, i.e., we put pL/M%a and rewrite equation H1*.*I thus0 Q < T H/M%aI RX L <X T H/<MaI T H/XM3aXI which can /e rearranged /y su/tracting H/XM3aXI from /oth sides to give0 <X T H/<MaI L Q < T H/M%aI RX ' H/XM3aXI )u/stituting this into e<pression H1*.1I gives0 Q < T H/M%aI RX ' H/XM3aXI L 'HcMaI and adding H/XM3aXI to /oth sides gives0 Q < T H/M%aI RX L H/XM3aXI 'HcMaI We then put the terms of the right'hand side onto a common denominator, thus0 Q < T H/M%aI RX L H/X ' 3acIM3aX =ow we can ta-e the square root of /oth sides to get < on its own, /ut note that when a square'root
28 is ta-en, there are two possi/ilities /ecause qSq is the same as H'qISH'qI, i.e., NHqXILYq. Jence0 < T H/M%aI L YNQ H/X ' 3acIM3aX R L Q YNH/X ' 3acI RM%a finally, we su/tract /M%a from /oth sides to o/tain0 < L Q '/ YNH/X ' 3acI R M %a 1*.3 which is, of course, the standard school formula for solving quadratic equations. Bhe formula H1*.3I loo-s innocuous enough, /ut what happens when 3ac is larger than /X ? 6n that case, the solution for < has a term containing the square'root of a negative num/er Hi.e., a num/er which is negative when multiplied /y itselfI even though the /asic rules of arithmetic demand that when a num/er is squared, the answer must always /e positive. Ba-e, for e<ample, the seemingly innocent quadratic equation <X'<T1L". 6n this case0 aL1, /L '1, and cL1, and the solution is0 < L H1M%I YHNH'2I IM% Bhe /est simplification we can manage is to factor out the square root of '1, i.e., < L ".# Y".:88NH'1I Bhus there are two solutions, < L ".#T".:88NH'1I and < L ".#'".:88NH'1I, /oth of which contain a part which is a real num/er, and a part which is not a real num/er. Bhat which is not real is imaginary, and so the odd/all quantity . NH'1I . was given the sym/ol . i ., H/y >eonhard +uler, 15"5'15:2I and this sym/ol is still used /y mathematicians. When it /ecame apparent to scientists researching into electricity that this /ranch of mathematics might /e useful however, the sym/ol . i . had already /een allocated to represent current, and so the ne<t letter in the alpha/et, . G ., was allocated for use in conFunction with electrical pro/lems Hhere we will write the sym/ol in bold, to ma-e it easier to spotI. Bhus we can write the unsimplifia/le solution to the previous e<ample as0 < L ".# Y".:88G Bhat which is not simplifia/le is complex, and so in this case, < is a complex number. . G . is called the imaginary operator, /ecause it operates on a num/er in such a way as to ma-e it impossi/le to add it to a real num/er. (nce . G . Hor . i . I was discovered, mathematicians went on to find general solutions for cu/ic equations, and quartic equations Hi.e., equations involving <i and <3 I, and it was proved that no other type of imaginary operator was required. Bhis means that all num/ers can /e reduced to the sum of a real part and an imaginary part, and e<pressed in the general form0 < L a T G/ with the proviso that sometimes /L" and the num/er is purely real, and sometimes aL" and the num/er is purely imaginary. Bhus it is not so much that comple< num/ers are peculiar, /ut that real num/ers are a special class of comple< num/ers which Fust happen to have the imaginary part equal to Aero. (nce it was understood that num/ers are in general comple<, the ne<t step was to wor- out what that meant. Bhe clue comes from our earlier discussion of vectors. 1irstly, we may o/serve that all real num/ers must lie on a line stretching /etween 'a and Ta. )econdly we may o/serve that G causes imaginary num/ers to e<ist in a dimension separate from real num/ers. Bherefore the effect of G is to rotate the num/er'line through 7"Z. Bhirdly, we may o/serve that the num/ers " and "TG" are the same, so that the real and imaginary num/er'lines must cross at ". Bhe upshot is that comple< num/ers Hi.e., all num/ersI can /e represented as points in a plane, which is the same as saying that the num/er aTG/ can /e plotted as a point on a graph of a vs. /. Bhat graph is, of course, number space, and maps in this space are -nown as rgand diagrams.
25 We must o/serve, at this point, that comple< num/ers are so li-e impedances that had they /een discovered /y electrical engineers, they might well have /een named after impedances. =aturally, since comple< num/ers are the general class of num/ers to which all num/ers /elong, they are essential for solving all -inds of mathematical pro/lems, /ut nowhere is the association so direct and so profound that all we have to do to convert an impedance into a comple< num/er is to write0 : L 4 T G\ Bhis says that impedance is a quantity with a real part 4 and an imaginary part \. Bhe original terms .real. and .imaginary. are also perfectly appropriate, /ecause the apparent power H ,L6!4 I dissipated in a resistance is indeed real, while the apparent power H ,L6c8\c I dissipated in a pure reactance is entirely imaginary. Bhus it is hard to ma-e a logical distinction /etween the two statements0 Kimpedances can /e represented /y comple< num/ersK and Kimpedances are comple< num/ersK. 6t follows also, from the relationships implicit in (hm.s law, that if impedances can /e treated as comple< num/ers, then so too can voltages and currents. Bhis does not mean that these o/Fects have somehow ceased to /e vectors however, far from it. Bhe comple< num/er form is Fust another two'dimensional vector representation, which complements the rectangular and polar forms we have already met. 6n fact, it is merely a version of the rectangular form in which the 7"Z difference /etween the dimensions is imposed /y the G operatorG and a vector always /ehaves in the same way regardless of how it is defined. Bhis minor change ma-es a huge difference however, /ecause it allows a phasor to /e written as an ordinary alge/raic sum. An e<pression with G in it might not seem ordinary of courseG /ut it is so in the sense that the e<istence of G is required /y the rules of common arithmetic, and so G is /y definition su/Fect to those rules. Bhe comple< form of a phasor ma-es the rectangular form effectively redundant. Bhe transformations from the comple< to the polar form are given /elow, and are very similar to the transformations given earlier in ta/le 7.3. Comple< form : L 4 T G\ : L c:cHCosd T G)ind I ,olar form
:H NQ4X T \XR , ArctanQ\M4R I : H c: c , d I
=otice also that G can /e regarded as a phasor operator, /ecause its effect on an alge/raic e<pression is to turn that e<pression into a phasor Hanother good reason for writing G in /oldI. Jence, in the matter of writing properly /alanced vector equations, we may note that if a live phasor Hi.e., one which has not /een turned into a scalar /y ta-ing a magnitude or a scalar productI e<ists one one side of the .L. sym/ol, then there must /e a live phasor or an e<pression with G in it Hi.e., a live phasorI on the other side.
2: Euler's Formula: 1or those familiar with e<ponents, note that0 Cos d T G)in d L eG H Bhis equation is -nown as !uler's formula, and defines the relationship /etween alge/ra and trigonometryG where .e. is sometimes referred to as !uler's number and is, to more decimal places then you.ll pro/a/ly ever need0 %.51: %:1 :%: 3#7 "3# %2# 28" %:5 351 2#% 88% 375 5#5 %35 "72 877 7#7 #53 788 785 8%5 5%3 "58 82" 2#2 #35 #73 #51 2:% 15: #%# 188 3%5 3%5 388 271 72% ""2 "#7 7%1 :15 312 #78 8%7 "32 #5% 7"" 223 %7# %8" #7# 82" 52: 12% 2%: 8%5 732 37" 582 . . . . etc., etc.
13. Co/pleD arith/etic
Comple< num/ers can /e added in the same way as vectors, i.e., H41 TG\1I T H4% TG\%I L H41 T 4%I TGH\1 T \%I and they can /e scaled in the same way as vectors, i.e., sH4 T G\I L s4 T Gs\ Hit is traditional to move G to the /eginning of the term it operates on, to ma-e its presence more o/viousI. Bhe real power of the representation however, comes from the fact that we -now immediately how to perform multiplication involving comple< num/ers /ecause, although e<pressions having non'Aero real and imaginary parts cannot /e reduced to a single num/er, we can deal with the multiplication cross'terms /y o/serving that GX L '1. Jence0 H41 T G\1IH4% T G\%I L 414% T G\14% T G\%41 TGX\1\% L H414% ' \1\%I TGH41\% T \14%I Bhus we can multiply two comple< num/ers and always o/tain a result which can /e re'arranged into the form . aTG/ .. Bhis outcome demonstrates also that the ordinary alge/raic product of two phasors, 3), is another phasorG and is not the same as the dot HscalarI product 3B). Bhe ordinary product is -nown as the complex product, or the phasor product, Hand is also not the same as the cross product used in general vector theoryI. Bhe statement GX L '1 incidentally, is the same as saying that rotation of a num/er through 7"Z followed /y another rotation through 7"Z has the effect of reversing its original direction, i.e., multiplying it /y '1. We now have part of the solution of how to interpret the e<pression0 :L:1:%MH:1T:%I. (ne further tric- is required in order to cope with the division part of the pro/lem however, and this comes from noticing what happens when the comple< num/er aTG/ is multiplied /y the comple< num/er a'G/0 Ha TG/IHa 'G/I L aX TGa/ 'Ga/ 'GX/X L aX T /X a'G/ is called the complex conjugate of aTG/, and vice versa. An asteris- is normally used to denote the comple< conFugate of a num/er, e.g., if :L4TG\, then : L4'G\ H: is pronounced KD'starKI. When a num/er is multiplied /y its comple< conFugate, the result is always real. Bhus if G appears in the denominator Hthe /ottom partI of a fraction, we can multiply /oth the numerator Hthe top partI and the denominator /y the comple< conFugate of the denominator. *ultiplying /oth the top and /ottom of a fraction /y the same num/er ma-es no difference to the value, /ut the operation ma-es the denominator real, so that the fraction can then /e rearranged into a form which loo-s, once again, li-e aTG/. We now have a complete set of definitions for mathematical operations involving phasors, and thus armed, we are in a position to attac- the parallel impedance pro/lem.
14. 6/pedances in Parallel
6f :1L41TG\1 and :%L4%TG\%, what is the impedance :L4TG\ which results from placing :1 in parallel with :% ? :1 :% :L H :1 T :% I L H 41 T 4% I TGH \1 T \% I H 41 T G\1I H 4% T G\% I
H 414% ' \1\% I TGH 41\% T \14% I :L H 41 T 4% I TGH \1 T \% I =ow multiply numerator and denominator /y the comple< conFugate of the denominator0 Q H 414% ' \1\% I TGH 41\% T \14% I R Q H 41 T 4% I 'GH \1 T \% I R :L Q H 41 T 4% I TGH \1 T \% I R Q H 41 T 4% I 'GH \1 T \% I R and multiply out the terms in the denominator to show that it is now real0 Q H 414% ' \1\% I TGH 41\% T \14% I R Q H 41 T 4% I 'GH \1 T \% I R :L H 41 T 4% IX T H \1 T \% IX Bhe terms in the numerator are now multiplied out and rearranged so as to separate the real and imaginary parts, i.e., the numerator is put into the form aTG/ as follows0 H414%'\1\%IH41T4%ITH41\%T\14%IH\1T\%I TGQH41\%T\14%IH41T4%I'H414%'\1\%IH\1T\%IR :L H 41 T 4% IX T H \1 T \% IX )implification of this e<pression involves multiplying out the /rac-ets and crossing out any pairs of terms which are equal and opposite0 41X4% T414%X '\1\%41 '\1\%4% T\%\141 T41\%X T\1X4% T\1\%4% TGQ41X\% T414%\% T\1414% T\14%X '\1414% '414%\% T\1X\% T\1\%XR :L H 41 T 4% IX T H \1 T \% IX Which leaves us with0 Q 41X4% T4%X41 T41\%X T4%\1X R TGQ 41X\% T4%X\1 T\1X\% T\%X\1R :L H 41 T 4% IX T H \1 T \% IX Bhis solution can /e written in various ways, depending on preferenceG e.g.0
3" Q 414% H41T4%I T41\%X T4%\1X R TGQ \1\% H\1T\%I T\14%X T\%41X R :L H 41 T 4% IX T H \1 T \% IX or0 Q 41H4%XT\%XI T 4%H41XT\1XI R TGQ \1H4%XT\%XI T \%H41XT\1XI R :L H 41 T 4% IX T H \1 T \% IX Bhe real part of e<pression H14.1I is 4, and the imaginary part is \, and so we can write0 414% H41T4%I T41\%X T4%\1X 4L H41 T 4%IX T H\1 T \%IX (r, alternatively, using e<pression H14.1aI0 41H4%XT\%XI T 4%H41XT\1XI 4L H41 T 4%IX T H\1 T \%IX and \L H41 T 4%IX T H\1 T \%IX \1H4%XT\%XI T \%H41XT\1XI and \L H41 T 4%IX T H\1 T \%IX \1\% H\1T\%I T\14%X T\%41X H14.1aI H14.1I
Bhe formula Hand variantsI given a/ove for impedances in parallel, while not e<actly memora/le, has the advantage of /eing completely general. 1irst note that if we put \1L" and \%L", then all of the reactive terms vanish and we are left with the formula for resistors in parallel, i.e., 4L414%MH41T4%I. )imilarly, if we put 41L4%L", we end up with the parallel reactance formula \L\1\%MH\1T\%I. *ore usefully however, we can put only \%L" and find out what happens when a resistance is placed in parallel with an impedance, and we can put 4%L" and find out what happens when a pure reactance is placed in parallel with an impedance. Bhe latter operation is of particular importance in the matter of devising and analysing antenna matching networ-s. i/ensional consistency Bhe solution to the parallel impedance pro/lem is our first e<ample of what might /e called a .messy. mathematical derivation. As such, it is fairly typical of circuit analysis pro/lems, which involve no difficult logical steps, /ut tend to e<pand into large num/ers of terms, many pairs of which su/sequently turn out to /e equal and opposite and so cancel. Bhus the pro/lem e<pands alarmingly, and then contracts again into one or more relatively simple e<pressions. 6t can /e difficult to -eep trac- of the various parts of the equation when carrying out such manipulations, which means that mista-es are li-ely to occur. Bhere is however a simple reality' chec-, which identifies invalid terms and gives an immediate indication of the li-ely correctness of the result. Bhis is the test of dimensional consistency which, with a certain amount of practice, can /e carried out at a glance. Bhe rules are as follows0 ● 6f two quantities are to /e added together Hor su/tractedI it must /e possi/le to e<press them in the same units. 6t would ma-e no sense to add a distance in metres to a temperature in ZC. 6t would also ma-e no sense to add a distance in metres to a distance in centimetres, /ut in that case the distance in centimetres can /e divided /y 1"" to convert it into metres, and then the addition can /e performed.
31 6t follows, that if a .T. or a .'. sym/ol appears anywhere in an equation, the dimensions of the quantities on either side of that sym/ol must /e the same. ● An equation, which supposedly represents a certain quantity, must have dimensions appropriate to that quantity. Ba-e, for e<ample, the e<pression for the real part of two impedances in parallel, as derived a/ove0 414% H41T4%I T41\%X T4%\1X 4L H41 T 4%IX T H\1 T \%IX Bhe truth of this statement is not immediately o/vious, /ut a chec- of dimensional consistency can very quic-ly tell us if it is capa/le of /eing true. 6n this case, the denominator Hthe /ottom partI of the fraction has two /rac-ets each containing quantities having the units of resistance H(hmsI. Jence the terms H41T4%IX and H\1T\%IX have dimensions of QhXR, and the overall dimensions of the denominator are QhXR. 6n the case of the numeratorG there are three terms to /e added, each having the dimensions of QhiR, and the overall dimensions of the numerator are QhiR. Dividing the dimensions of the numerator /y the dimensions of the denominator we o/tain0 QhiRMQhXRLQhRG and so the equation is dimensionally consistent and represents a quantity which can /e e<pressed in (hms. 6t is also possi/le to test the dimensional consistency of equations involving mi<ed units. Bhe point here is that units have aliases, which are composites of other unitsG and so we can chec- any equation, provided that we -now the relationships /etween the units used. 6n the conte<t of circuit analysis, these relationships are easily o/tained, /ecause they are em/edded in the /asic formulae from which the mathematical argument is constructed. (hm.s law, 8M6L:, for e<ample, tells us that (hms are equivalent to !olts divided /y Amperes, and so a quantity having the latter dimensions, i.e., a voltage divided /y a current, may legitimately replace a quantity measured in (hms. )imilarly, the reactance laws \>L%]f> and \CL '1M%]fC, tell us that (hms can also /e replaced with QJenrys S radians M secondR, or /y Q1MH1arads S radians M secondIR. Bhus we should not /e confused /y structures such as0 : L 4 TGI %]f> '1MH%]fCI ` Bhe /rac-et after the G is internally consistent, and represents a quantity measured in (hms.
15. Parallel resonance
6n an earlier section, we said that it is not possi/le to calculate the e<act resonant frequency of a parallel tuned circuit, nor the impedance which it presents at resonance, without ta-ing the resistances of the coil and capacitor into account. =ow, of course, having derived a general equation for impedances in parallel, we are in a position to rectify that omission. Bhe networ- we need to analyse is shown on the rightG where 4C is the so'called equivalent series resistance H+)4I of the capacitor, and 4> is the loss resistance of the coil, which we previously defined as 4>L\>M;> Hhere ; is given the su/script .>. to indicate that it is the ; of the coil, not the overall ; of the tuned circuitI. 1or the purposes of this discussion, we will assume that /oth 4C and 4> are predominantly due to the 41 resistance of the wires and other conducting materials used to ma-e the components, and for reasons which are e<plained in su/sequent articles1", are considera/ly larger than the DC resistances. 1or the types of components used in J1 antenna matching applications, 4C will /e of the order of ".1h, and 4> typically a few ohms.
1" Components and *aterials. www.g2ynh.info
3% 6n the general electronic literature, several different definitions are used for the resonant frequency of a parallel tuned circuitG the alternatives /eing the frequency at which the impedance of the circuit has its largest magnitude, and the frequency at which \>L '\C . Jere however, we will adopt the most straightforward definition, which is the frequency at which the impedance is purely resistive Halso -nown as the .unity power'factor frequency.I. We can find this frequency /y setting the imaginary part equal to Aero in equation H14.1I a/ove, i.e.0 \ L Q\C\>H\C T\>I T\C4>X T\>4CX R M Q H4C T 4>IX TH\C T \>IX R L " HWhere the su/scripts 1 and % have /een changed to C and > as /efits the current pro/lemI. =ow notice, that to ma-e the reactance equal to Aero, we only need to ma-e the numerator of this e<pression equal to Aero, i.e., we can ignore the denominator. Jence0 \C\>H\C T\>I T\C4>X T\>4CX L " . . . H15.1I We now need to ma-e the frequency dependence of this e<pression e<plicit /y using the su/stitutions0 \CL '1M%]f"C , and \>L%]f">, i.e.0 'H%]f"> M %]f"C IH %]f"> ' 1M %]f"C I ' H 4>X M %]f"C I T H %]f"> 4CX I L " Bhe resonant frequency can now /e found /y re'arranging this e<pression to get f" on its own. Also, since we -now that the series'resonance formula is an appro<imation for the e<pression we are a/out to derive, we e<pect the result to loo- li-e the series'resonance formula with an additional correction term or factor. We can /egin /y multiplying'out the first two /rac-ets. Jence0 'H %]f">X M C I T H > M %]f"CX I ' H 4>X M %]f"C I T H %]f"> 4CX I L " =ow we will put all of the terms containing %]f" on one side, and the terms containing 1MH%]f"I on the other. %]f" H >4CX ' >XMC I L H1 M %]f" I QH 4>XMC I ' H >MCX IR Bhen multiply /oth sides /y %]f", and divide /oth sides /y H>4CX'>XMCI0 H%]f"IX L QH 4>XMC I ' H >MCX IR M H >4CX ' >XMC I and factor'out 1M>C from the right'hand side0 H%]f"IX L H 1 M >C I H 4>X ' >MC I M H 4CX ' >MC I Jere we will also multiply top and /ottom /y '1 to put the >MC terms first, >MC generally /eing much larger than the resistance'squared terms, hence0 H%]f"IX L H 1 M >C I H Q>MCR ' 4>X I M H Q>MCR ' 4CX I which rearranges to0 f"= 1 % >C
> / C −4 % > > / C −4 % C
Bhus we find that the resonant frequency of a parallel tuned circuit is the same as that for a series tuned circuit e<cept for a correction factor NQH>MC ' 4>XIMH>MC ' 4CXIR, which is usually close to unity. =otice that this factor is equal to 1 if 4> and 4C are AeroG and also that the factor is 1 when 4>L4C. Example0 A 2OJ coil is connected in parallel with a 3%p1 capacitor. Bhe appro<imate resonant frequency is0 1MH%]NQ>CRIL1MH %]NQ2S1"'8S3%S1"'1%R I L13.15:837*JA. 6n the region of 13*JA, the coil has a loss resistance of %j and the capacitor has an equivalent series resistance H+)4I of ".1j. Bhus >MCL513%:.#5, 4>XL3, and 4CXL"."1. Jence the correction factor is0 NQH513%:.#5'3IMH513%:.#5'"."1IRL".77773313. Bhe precise resonant frequency Hto the nearest 1JAI is therefore ".777733S13.15:837L13.15:%#2*JA. Bhe quantity >MC is called the .K> C 4atioK of the tuned circuit Hand it has units of .(hms squared.I. =ote that0
32 >MC L ' \> \C L c\> \Cc 6t will turn out that the >MC ratio is an important parameter of resonant circuits. Also, there is some precedent for referring to the square root of the >MC ratio as the characteristic resistance of the tuned circuit, /y analogy with the characteristic impedance of a lossless transmission line, which is 4" L NH"M#I where " is inductance per unit of length and # is capacitance per unit of lengthG /ut the lengths cancel and so the characteristic resistance of an ideal transmission line is the square root of its >MC ratio. 6n the e<ample given a/ove, the resonant frequency differed from the ideal case /y only ".""%:W or 278JA, the reason /eing that the >MC ratio was very large in comparison to the squares of the loss resistances. 6n J1 radio applications, the >MC ratios of tuned circuits are generally in the order of several tens of thousands of hX, whereas the value tolerances of radio components are seldom /etter than 1W and often considera/ly worse. 6n order to o/tain an e<act resonant frequency, it is necessary to ma-e either the coil or the capacitor adFusta/leG and the required adFustment range will easily swallow any deviation caused /y using the ideal'case formula f"L1MQ%]NH>CIR. We may therefore conclude that, in normal circumstances, the assumption of Aero losses may /e perfectly accepta/le when calculating the resonant frequency of a parallel'tuned circuitG /ut, as we shall see in the ne<t section, it is not accepta/le when calculating the impedance at resonance.
17. yna/ic 0esistance
1or an ideal parallel tuned circuit Hi.e., 4>L" and 4CL"I, the impedance /ecomes infinite at resonance. Bhis, of course, does not happen in practiceG /ut provided that the loss resistances of the components are small, it does rise to a high value. )ince we have defined resonance as the frequency at which the reactance is cancelled, this impedance is also purely resistive, and it is -nown as the dynamic resistance of the parallel tuned circuit. Jere we will give it the sym/ol 4p" Heffective parallel resistance when f L f"I. 6t is, of course, given /y the real part of equation H14.1I Hthe parallel impedance formula given earlierIG i.e.0 4>4CH4>T4CI T4>\CX T4C\>X 4p" L H17.1I H4>T4CIX T H\>T\CIX 6n the e<ample from the previous section we had0 4>L%h, 4CL".1h, >L2OJ, CL3%p1, f"L13.15:%#2*JA, \CL '%85.%8:511%h and \>L%85.%#255%:h. 6f we apply the a/ove formula to these data, we o/tain0 4p" L Q ".3% T%H5132%.#8277I T".1H513%3.#57"5I R M Q H%.1IX TH'"."1372:3IX R 4p" L 1#""":.""#7 M 3.31"%%21#8 4p" L 23."125 Kh Bhe only pro/lem with equation H17.1I is that it is very cum/ersome Hand resistant to simplificationI. We might therefore /e inclined to loo- for some simplifying assumptionsG and the most o/vious of these is to note that since \> is very nearly equal to '\C, we might as well assume the term \>T\C to /e Aero. Bhis also implies that \CXL\>XL '\C\>L>MC, hence equation H17.1I /ecomes0 4p" L Q 4>4CH4>T4CI TH>MCIH4>T4CI R M H4>T4CIX i.e. 4p" L Q 4>4C T H>MCI R M H4>T4CI 17.*
33 =otice that this formula has lost all of its reactance terms, which is very convenient. 6f we apply it to our e<ample data, where >MCL513%:.#5hX, we o/tain0 4p" L H%S".1M%.1I T 513%:.#5M%.1 L "."7# T 23"12.81 h 4p" L 23."125 Kh Bhe appro<imation is almost e<act for components of moderate ;. Also we may o/serve that the term 4>4CMH4>T4CI is much smaller than the term H>MCIMH4>T4CI, and given that we are unli-ely to -now the component resistances very accurately, we might as well drop the first term. Jence the appropriate formula for calculating the dynamic resistance is0 4p" L H>MCI M H4>T4CI 17.3 Bhis equation is an e<cellent appro<imation for the dynamic resistance, /ut strangely, it is not the one offered in most te<t/oo-s. Bhe usual appro<imation is that, in addition to \>T\C /eing Aero, the +)4 of the capacitor is assumed to /e Aero. Bhis causes all of the terms containing 4C in equation H17.1I to disappear, and gives rise to a considera/le simplification, viA.0 4p" L 4> \CX M 4>X i.e., 4p" L \CX M 4> 6f we apply this formula to our e<ample data we o/tain0 4p" L 5132%.#8277 M % L 2#.5182Kh 6n this case the deviation from the true value is 15"%.8h, or #W, which may /e a reasona/le appro<imation for many purposes, /ut needs to /e treated with caution. Also, the failure to eliminate reactance from the formula ma-es computation more difficult.
1>. o+bleJslash notation
6n geometry, the e<pression0 . ABMMCD . means0 Kthe line drawn from point A to point B lies in parallel with the line drawn from point C to point DK. Jence, /y e<isting convention, the sym/ol . MM . means Kin parallel withK. 6n electrical engineering, of course, we are frequently interested in circuits in which components are connected in parallel, and so we can usefully adapt the dou/le slash notation to have a non'geometric meaning. We can, for e<ample, re'state our /asic parallel component formulae as follows0 41 MM 4% L 41 4% M H 41 T 4% I \1 MM \% L \1 \% M H \1 T \% I :1 MM :% L :1 :% M H :1 T :% I >1 MM >% L >1 >% M H >1 T >% I and possi/ly, /ut /est avoided0 C1 MM C% L C1 T C% Bhis convention is often convenient, /ecause it saves the /other of having to define a temporary varia/le to represent the parallel com/inationG i.e., instead of writing0 K>et 4 represent the parallel com/ination of 41 and 4%K, and then having to remem/er what 4 isG we simply wor- with the quantity H41MM4%I , which can /e e<panded or calculated when necessary, /ut more to the point is Fust a resistance with an o/vious definition. While straightforward however, the use of the MM notation involves a su/tlety which lies in the distinction /etween physical and mathematical o/Fects. 6n descri/ing a test procedure, for e<ample, we might put an entry in a ta/le0 KBest load0 8:h MM 1""p1K. Bhe item K8:h MM 1""p1K is a physical o/Fect, a capacitor in parallel with a resistor, /ut it is not a complete mathematical statement of impedance and cannot /e treated as an impedance in any calculation. 6n order to turn the parallel com/ination into a mathematical o/FectG we must ensure that the quantities on either side of the MM sym/ol are of the same type and that they are e<pressed in the same units. 6n this case we can fi<
3# the pro/lem /y noting that, if the report is to have any useful meaning, a test frequency must /e stated somewhere. 6f that frequency is, say, 13*JA, then the reactance of the capacitor /ecomes '1MH%]fCI L '112.5h, and its impedance Hassuming that losses are negligi/leI is "'G112.5h. Jence we can re'state the test load as H8: MM 'G113Ih . Bhis is the same as saying H8:TG" MM "'G113Ih G and is, of course, a complete statement of the load impedance in the form :1MM:% which can /e converted into the 4TG\ form if so desired. . A particular logic emerges from these o/servations and it is important to /e aware of it0 1>.1I 3 resistance is an i/pedance. 4esistances and impedances are the same type of o/Fect. A resistance in parallel with an impedance is an impedance. A resistance is simply an impedance which happens to have its imaginary part equal to Aero. Bhis means, incidentally, that the preferred pseudoscalar sym/ol for : is usually 4, rather than D. 1>.*I 3 reactance is not an i/pedance. Bhe statement0 : L 8: MM 113 has a completely different meaning to the statement0 : L 8: MM 'G113 Hthe first is a resistance in parallel with a resistance, the second is a resistance in parallel with a reactanceI. *athematically, a reactance cannot /e com/ined directly with an impedance, /ut a reactance can /e converted into an impedance /y multiplying it /y G. >oo-ing at this another way0 impedance and reactance have reference directions which are 7"Z apart. Bo ma-e them compati/le, it is necessary to rotate one of them through 7"Z. 1>.3I 2calability is preser!ed. When the dou/le slash notation is used to create a mathematical o/Fect, i.e., the same type of phasor e<ists on /oth sides of the MM sym/ol, it has the useful property that a common factor can /e multiplied'in or divided'out of the parallel o/Fect, i.e.0 s:1 MM s:% L s H:1 MM :%I $roof% s:1 MM s:% L s:1 s:% M Hs:1 T s:% I L s :1 :% M H:1 T :% I L s H:1 MM :%I 1>.4I The associati!e r+le. Bhe dou/le slash notation can /e e<tended to represent any num/er of impedances in parallel0 :1 MM :% MM :2 MM....MM :n L 1 M Q H1M:1I T H1M:%I T H1M:2I T . . . . . T H1M:nI R and the associative rule of arithmetic Hand linear electrical devices in parallelI is o/eyed, i.e.0 H:1 MM :%I MM :2 L :1 MM :% MM :2 1>.5I o+bleJslash prod+ct definition. Bhe MM notation implies a specialised -ind of phasor multiplication, which we might call the double-slash product or the parallel product of a pair of phasors. )ince its use in conFunction with parallel capacitors is pointless, we will adopt the following strict mathematical definition0 a MM b L abMHaTbI
1?. ParallelJtoJ2eries transfor/ation
6n the discussion so far, we have adopted the ha/it of representing every impedance as a resistance in series with a reactance. 6t ma-es good sense to do so in most circumstances, /ecause it allows the impedance to /e written directly in the form 4TG\. Bhere are many situations however, in which circuit analysis can /e simplified /y representing an impedance as a resistance in parallel with a reactance. Bhe two possi/le representations are equally validG /ut it should /e o/vious from the parallel impedance equation H14.1I derived earlier, that the parallel representation for a particular impedance requires a different com/ination of resistance and reactance to that of the series representation. 6n the ne<t two sections we will e<plore the relationships /etween the two representations, /eginning with the transformation of an impedance from its parallel to its series form0 Bo derive this transformation, we simply regard the parallel elements as two separate impedances 4pTF" and "TG\p , and apply the formula for impedances in parallel Hi.e., Q:1MM:%RL:1:%MQ:1T:%R I. Jence0 4 TG\ L H 4p MM G\p I i.e.0 4 TG\ L G\p 4p M H 4p T G\p I and 4 and \ are simply the real and imaginary parts of the right hand side of this e<pression once it has /een put into the form aTG/. We proceed as usual /y multiplying the top HnumeratorI and /ottom HdenominatorI /y the comple< conFugate of the denominator, thus0 G 4p \p H 4p ' G\p I 4 TG\ L H 4p T G\p I H 4p ' G\p I Which rearranges to0 4p \pX T G \p 4pX 4 TG\ L H 4pX T \pX I
Jence, for the series representation0 4p \pX 4L and H 4pX T \pX I
\p 4pX \L H 4pX T \pX I
1urther pieces of information which we can e<tract from the parallel'to'series transformation, and which will /e useful later, are the phase'angle, magnitude and ; of an impedance in its parallel form0 Phase angle and A of an i/pedance in parallel for/: Bhe phase angle for an impedance in its series form was given earlier as e<pression H8.*I0 d L ArctanH\ M 4I By using e<pression H1?.1I a/ove we can su/stitute for \ and 4 to o/tain0 d L ArctanH\p 4pX M 4p \pXI i.e., d L ArctanH4p M \pI which also tells us that \M4L4pM\p , i.e. the ratio of resistance to reactance of an impedance in its series form is the inverse of the ratio for the impedance in its parallel form. Also, since we -now
35 that c\cM4>oss is an e<pression for the ; of an electrical component, we may further note that component ; can /e e<pressed as0 ;co/p L 4p>oss M c\pc Hthe higher the parallel loss resistance, the higher the ;I. 1agnit+de of an i/pedance in parallel for/0 Bhe magnitude of an impedance in its series form is given /y H7.1I0 c:c L NH4X T \XI. )u/stituting for 4 and \ using e<pression H1?.1I we o/tain0 c:c L NQ_ H4p \pXIX T H\p 4pXIX `M H4pX T \pXIX R L NQ_ 4pX \pX H \pX T 4pXI `M_ H4pX T \pXIX `R We can ta-e the square root of the 4pX\pX term and so factor it out of the square'root part of the e<pression, provided that we only use the positive result Hmagnitudes are always positiveI. Jence0 c:c L c 4p \p M N H 4pX T \pX I c 1?.* A convenient rearrangement of this e<pression can /e o/tained /y forci/ly factoring \p from the denominator0 c:c L c 4p \p M _ \p N Q H4pXM\pXI T 1 R ` c =ow, since 4p and 4pXM\pX are always positive, we can drop the magnitude /rac-ets to o/tain0 c:c L 4p M TN Q H4pM\pIX T 1 R 1?.3 Bhis form is particularly useful for frequency response calculations, /ecause it allows the reactance contri/ution to /e treated as a correction factor0 1 M N Q H4pM\pIX T 1 R which goes to unity Hk1I when the reactance is large in comparison to the resistance.
1rom e<pression H1?.1I in the previous section, we have0 4 L 4p \pX M H 4pX T \pX I . . . . [email protected]
and \ L \p 4pX M H 4pX T \pX I . . . . [email protected]
*I (/taining the series'to'parallel transformation is a matter of using these two equations to o/tain equations for 4p and \p. Bhis would prove to /e a somewhat tric-y pro/lem, had we not noticed from the preceding derivation of the magnitude Hequation 1?.*I that0 c:cX L 4X T \X L 4pX \pX M H 4pX T \pX I Bhe right hand side of this equation can also /e o/tained /y multiplying e<pression [email protected]
/y 4p, or /y multiplying e<pression [email protected]
*I /y \p. Jence0 4 4p L 4X T \X and \ \p L 4X T \X i.e.0 4p L H4X T \XI M 4 [email protected]
and \p L H4X T \XI M \ [email protected]
*-. Parallel resonator in parallel for/
Javing derived the series to parallel transformation, we are now in a position to analyse the parallel resonator in a different way. Bhe outcome should /e mathematically unsurprising, /ecause we are /ound to o/tain the same results as /efore, /ut the technique will give us a new way of thin-ing a/out the circuit.
HBhe sym/ol K l K means0 Kis /y definition equal toK. Bhe sym/ol K MM K means Kin parallel withKI. As the diagram a/ove illustratesG the parallel impedance representation allows us to visualise the circuit as an ideal parallel resonator with a resistance connected across it. Bhis separates the reactive and the resistive parts of the pro/lem and tells us immediately that unity power'factor resonance occurs when \>pL '\Cp, and that the dynamic resistance is given /y the value of 4>pMM4Cp at f". We can, of course, relate the parallel impedance form of the resonator to the series impedance form /y using the transformations given in the previous section Hequations [email protected]
, i.e., 4Cp L H4CX T \CXI M 4C \Cp L H4CX T \CXI M \C 4>p L H4>X T \>XI M 4> \>p L H4>X T \>XI M \> H*-.1I H*-.*I H*-.3I H*-.4I
Esing the appropriate transformations H*-.* and *-.4I, the resonance condition \>pL '\Cp /ecomes0 H4>X T \>XI M \> L 'H4CX T \CXI M \C which can /e rearranged to0 \C H4>X T \>XI T \> H4CX T \CXI L " and then to0 \C\>H\C T\>I T\C4>X T\>4CX L " We have seen this e<pression /efore as equation H15.1I, and so the derivation may continue as in section 15 to give the parallel resonance formula H15.*I. f" L _1MQ %]NH>CI R `_ NQH>MC ' 4>XIMH>MC ' 4CXIR ` Bhe dynamic resistance 4p" is given /y0 4p" L 4>pMM4Cp Jence using the transformations H*-.1I and H*-.3I we have0 Q H4>X T \>XI M 4> R Q H4CX T \CXI M 4C R 4p" L Q H4>X T \>XI M 4> R T Q H4CX T \CXI M 4C R which simplifies to0 H4>X T \>XI H4CX T \CXI 4p" L 4CH4>X T \>XI T 4>H4CX T \CXI
37 Bhus we o/tain another formula for the dynamic resistance of a parallel resonator, and it is interesting to compare it with equation H17.1I, which was our original derivation Hhere we show it rearranged slightlyI0 4CH4>X T \>XI T 4>H4CX T \CXI 4p" L H*-.7I H4> T 4CIX T H\> T \CIX Bhe two formulae are radically different in appearanceG /ut it is easy to verify, /y plugging in the num/ers from the e<ample in section 17, that they /oth give e<actly the same answer. Bhis leaves the issue of which one of them is the /est simplificationG and the answer in this case is that it is equation H*-.7I. We can tell /y loo-ing at the power or degree of the numerator and denominator of each equation. (/serve first that all of the quantities involved in the e<pressions are measured on (hms. Jence the numerator of H*-.5I has dimensions of h3 and the denominator has dimensions of hi. 6n equation H*-.7I however, the numerator has dimensions of hi and the denominator hX. Jence the numerator of H*-.7I is of lower degree than that of H*-.5I, and the denominators li-ewise. Bhis means that H*-.5I can /e simplified further and ultimately transformed into H*-.7IG although for anyone who cares to try it, the manipulations required are la/orious, and require the use of equation H15.1I as a su/stitution. )omething more tracta/le happens however when we multiply equations H*-.5I and H*-.7I and ta-e the square root to o/tain a new e<pression for 4p" , i.e., we ta-e the geometric mean of the two formulae. 6n this case the denominator of H*-.5I cancels the numerator of H*-.7I and we o/tain0 4 p" =
Jere we can ma-e the following simplifying assumptions0 1I )ince \>X is normally much greater than 4>X in radio circuits, 4>X can /e deleted from the numerator without ma-ing much difference. %I )ince \CX is also usually much greater than 4CX, 4CX can /e deleted from the numerator without ma-ing much difference. 2I 6f the ;s of the resonator components are reasona/ly high, H\>T\CI is very nearly Aero at resonance and can therefore /e deleted from the denominator without ma-ing much difference. Bhe result is0 4 p" =
% % % 4% > \ > 4 C \ C
4 > 4 C \ > \ C
Bhis e<pression can /e simplified /y o/serving that everything inside the square root /rac-et is squared, /ut in doing so we must /e mindful of a common fallacy. &he square root of the square of a number is not the number itself. A square root always has two solutions, one positive, one negativeG and if only one of the solutions can /e true, additional information is required for selection of the correct one. 6n this case, we -now that 4p" must /e positive if the networ- is passive, and so we accept the positive square rootsG /ut note that in section 7 we defined the positive square root of a square as a magnitude, i.e.G TNH\XI L c\c Bhis rule must /e strictly applied, /ecause simply deleting the superscripts and the square root sym/ol would have given us a negative value for 4p" /ecause \C is negative. Jence0 4p" L c\>cc\Cc M H4> T 4CI We have noted /efore that0 c\>cc\Cc L >MC hence0
% \% > \C
4 > 4 C
#" 4p" L H>MCI M H4>T4CI which we have seen /efore as equation H17.3I. While it is instructive to attac- a derivation from several directions and verify that all approaches lead to the same conclusion, the point of the parallel impedance representation is that it often ma-es pro/lems easier to solve. Bhe parallel resonator is a good e<ample /ecause the parallel representation gives a direct separation of the resistive and reactive parts of the pro/lem. A further and very important point however, is that we do not use the parallel representation with a view to converting it into the series form at the earliest opportunity. 6t is simply another way of e<pressing impedanceG and it is no less authoritative than the series form. Jence if we have data for an inductor or capacitor in series form, we can transform it into the parallel form and use it li-e that. Bhe parallel form may seem less authoritative than the series form /ecause the e<pression for 4p Hequation [email protected]
has reactance in it, and so e<plicitly varies with frequency. 6n reality however, the resistive component in the series form also varies with frequency, due to a variety of frequency dependent losses such as, s-in effect and dielectric a/sorption Qsee KComponents and *aterialsKK, capacitive and inductive coupling to resistive materials in the vicinity of the component, and of course our old friend radiation. Bhus, when solving pro/lems using simple circuit models, we need to /e aware that resistances inserted to represent losses are e<pected to vary with frequency, regardless of representation. Bhus the practical pro/lem of finding the dynamic resistance of a parallel resonator /ecomes that of measuring the impedances of the components at a frequency reasona/ly close to the desired resonance, transforming the losses into parallel resistances, and ta-ing the parallel com/ination of those.
*1. 6/aginary resonance
6t was o/served in section 3 that the series'resonance formula0 f" L 1MQ %]NH> CI R always gives two solutions for the resonant frequency, one positive and one negative. Bhe parallel resonance formula H15.*I does the same, /ut presents us with a further conceptual challenge, in that it also allows imaginary solutions. 6f we inspect the formula0 % >C we can o/serve that if either 4>X or 4CX should /ecome larger than >MC H/ut not /oth at the same timeI, then the quantity inside the right'most square'root /rac-et will /ecome negative. (nce again, there were no restrictions on the validity of the arguments which went into deriving the formula, and so imaginary resonance is possi/le and must have a physical meaning. Bhe answer to this conundrum can /e o/tained /y considering the parallel resonator as two separate impedances connected across a generator Hsee diagram rightI. 4eal resonance implies a condition where the current from the generator is in phase with the voltage it produces, i.e., it occurs at a frequency where the resonator constitutes a resistive load. Bhe output current 6 is the vector sum of the currents in the two /ranches of the resonator, i.e.G 6 L 6> T 6C and so real resonance occurs when 6>T6C is real. 4eal resonance can only occur however, if the current in one /ranch can /ecome large enough for its imaginary component to cancel the imaginary component of the current in the other /ranch. =otice that if we allow 4> to f"= 1
> / C −4 % > > / C −4 % C
#1 /ecome e<tremely large, then practically no current will flow in the inductive /ranch and the circuit will not resonate. Bhe same argument applies, of course, to the capacitive /ranch. Bhe parallel resonance formula can therefore /e seen to tell us that true Hi.e., realI resonance cannot occur if the resistance in either /ranch rises a/ove a certain critical value, that value /eing the square'root of the >MC ratio Hthe characteristic resistanceI, i.e., 4" L NH>MCI 6f the resistance in one /ranch rises a/ove NH>MCI, then the current in that /ranch will always /e too fee/le to /ring the system into resonance. What happens instead is that the phase of the total current 6 can approach and move away from the phase of the generator voltage as the frequency is varied, /ut it is never a/le to reach it. Bhe .resonant frequency. is simply the imaginary frequency of closest approach Hand it does not e<ist on the real frequency lineI. 6t is imaginary /ecause the com/ined impedance of the two /ranches can never /ecome real Hi.e., resistiveI /y cancellation. =ote however, /y inspecting the circuit, that the com/ined impedance does /ecome resistive at Aero and infinite frequencies, /ut that this is not due to cancellation0 At "JA, \>L" and \Cka H.k. means KapproachesK or Ktends towardsKI, so the impedance is simply 4> G and at infinite frequency, \>ka and \CL", so the impedance is 4C. 6f a real resonant frequency does not e<ist therefore, what will /e o/tained is a networ- which has a voltage'current phase relationship always on one side or the other of Aero degrees, only approaching "Z at Aero or infinite frequency. 6t was mentioned earlier that resonant circuits used in J1 radio applications tend to have large >MC ratios, often greater than 1""""hX. 6n the case of parallel resonance, one reason for this policy should now /e apparentG i.e., we need to o/tain a high characteristic resistance H 4" L TNQ>MCR I in order to ensure that the circuit will function properly with practically realisa/le inductors. A parallel resonator with an >MC ratio of 1""hX, for e<ample, will not wor- if the 41 resistance of the inductive /ranch is greater than 1"h at the e<pected resonant frequency, and it is /y no means impossi/le for a practical inductor to e<ceed such a limit. We may conclude, from this discussion, that a parallel tuned circuit will only resonate usefully if NH>MCI is made larger than the resistance in either of the /ranches. Bhe qualification .usefully. must /e applied however, /ecause if the resistance in both /ranches is allowed to /ecome larger than the critical value, then /oth the numerator and the denominator of the term inside the square'root /rac-et will /ecome negative, and so the term itself will /e positive. Bhus there will /e a real resonance, /ut the current in /oth of the /ranches will /e fee/le, and so the resonance will also /e fee/le and of little practical use. (ne final significance of the characteristic resistance which is worth remem/ering is that it is equal to the magnitudes of the reactances in the circuit at the .ideal case. resonant frequency, i.e., the resonant frequency when the resistance in /oth /ranches is equal. Bhis frequency, as was mentioned earlier, is given /y the series resonance formula, i.e., f"s L 1MQ %]NH> CI R or in radians M sec0 %]f"s L 1MQNH> CI R =ow, if we call the inductive reactance at this frequency \>"s , then0 \>"s L %]f"s > L > M QNH> CI R and, since any num/er is the square of its own square root0 \>"s L TNH>MCI )imilarly, for the capacitive reactance0 \C"s L '1MQ %]f"s C R L 'Q NH> CI R M C \C"s L 'NH>MCI
**. Phase analysis
We can visualise the phase relationship /etween voltage and current in a parallel resonant circuit /y deriving an e<pression for the 6'8 phase angle and plotting it as a graph against frequency for various values of included resistance. Bhis is only one of the many situations in which graphs of phase vs. frequency are instructive, and so this section will serve as a general introduction to the technique of phase analysis as well as a specific investigation of the parallel resonator. Bhe circuit to /e analysed is shown on the right, and we can use (hm.s law straight away to write an e<pression for the current0 6L!M: where : is the parallel com/ination of the impedances in the two /ranches of the resonator, and we choose the phase of 8 to /e "Z and treat it as a scalar. 6f we also define0 :>L4>TG\> , and :CL4CTG\C , then0 : L :> :C M H :> T :C I hence0 6 L ! H:> T :CI M H:> :CI =ow, we noted earlier that the phase angle of a comple< e<pression aTG/ is given /y0 d L ArctanH/MaI so in order to o/tain the 6'8 phase difference we first write0 6 M ! L H:> T :CI M H:> :CI then split the right hand side of the equation into its real and imaginary parts, divide the imaginary /y the real, and ta-e the inverse tangent. +<panding the e<pression a/ove we get0 6 4> T G\> T 4C T G\C L ! H 4> T G\> IH 4C T G\C I and multiplying out the terms in the denominator gives0 6 L ! 4>4C ' \>\C T GH4>\C T \>4CI =ow we multiply numerator and denominator /y the comple< conFugate of the denominator0 6 L ! H4>4C ' \>\CIX T H4>\C T \>4CIX then multiply out the numerator, crossing out equal and opposite terms, to get0 6 L ! H4>4C ' \>\CIX T H4>\C T \>4CIX Bhis is in the form aTG/, so the phase angle is given /y0 \>H4CX T \CXI T \CH4>X T \>XI Band L ' 4>H4CX T \CXI T 4CH4>X T \>XI =ow, there is no need to rearrange this formula any further in order to use it, /ut since we are analysing the phenomenon of parallel resonance, it is interesting to recall that \>\CL '>MC. 6f we 4>H4CX T \CXI T 4CH4>X T \>XI ' GQ \CH4>X T \>XI T \>H4CX T \CXI R Q 4> T 4C T GH\> T \CI RQ 4>4C ' \>\C ' GH4>\C T \>4CI R 4> T 4C T GH\> T \CI
#2 multiply out the numerator, we will o/tain two terms which contain \>\C, and this leads to an alternative e<pressionG i.e.0 \>4CX T \>\CX T \C4>X T \C\>X Band L ' 4>H4CX T \CXI T 4CH4>X T \>XI /ecomes0 \>4CX ' H>MCI\C T \C4>X ' H>MCI\> Band L ' 4>H4CX T \CXI T 4CH4>X T \>XI which rearranges to0 \>H4CX ' >MCI T \CH4>X ' >MCI Band L ' H**.1I 4>H4CX T \CXI T 4CH4>X T \>XI Which, since the >MC ratio is a fi<ed parameter for the resonant circuit, somewhat simplifies calculation. We will now use the e<pression a/ove to evaluate the effect of resistance in a fairly representative parallel resonator. 1or this e<ample we will use an inductance of 1OJ and a capacitance of 1""p1. Bhis com/ination gives an >MC ratio of 1""""hX and hence a critical resistance 4"LNH>MCIL1""h. Bhe .ideal. resonant frequency, i.e., the resonant frequency when 4>L4C is0 f"s L 1MQ %]NH> CI R L 1#.71#37321*JA Hi.e., %]f"s L 1""* radiansMsecI, and at this frequency, \>L '\CLNH>MCIL1""h. )hown /elow is a set of graphs of the 6'8 phase relationship for our e<ample resonator with various values of 4C and 4> /etween 1h and NH>MCI. Bhese graphs were produced using the 'pen 'ffice #alc spreadsheet program Havaila/le free from 'pen'ffice.orgI, the procedure /eing to create columns for frequency, \>, and \C, and use the calculated reactance values in the Arctangent Hinverse tangentI of equation H**.1I given a/ove. =ote that spreadsheets often give the results of inverse trigonometric functions in radians, and so it is necessary to multiply the e<pression /y 1:"M]L#5.%7#557#1 to get the result in degrees Hthere are %] radians in 28"ZI, i.e.0 d L '#5.%7#557#1Arctan_ Q\>H4CX'>MCIT\CH4>X'>MCIRMQ4>H4CXT\CXIT4CH4>XT\>XIR ` Bhe plotted curves /elow were created using the spreadsheet KchartK tool Qsee accompanying spreadsheet file0 parLresLph.odsR.
(f the curves shown, only the e<ample with 4>L1 and 4CL1 constitutes a good healthy resonance. Bhe choice of 1h in each of the /ranches incidentally was made simply so that the resonant frequency would coincide with f"s . Any curve with at total resistance 4>T4CL%h will have an almost identical appearance. Bhe ; of the resonance Has will /e e<plained laterI is #" in this case Hi.e., ;"L\>MQ4>T4CR I, which is fairly highG and so the phase of the current lags the voltage /y nearly 7"Z at frequencies a few percent /elow the resonant frequency, and leads it /y nearly 7"Z at frequencies a few percent a/ove. Jence the circuit provides the generator with a nearly pure inductive load /elow resonance, and a nearly pure capacitive load a/ove. 6n the case where 4>L#" and 4CL#", the ; of the resonance is 1. A large resistive component is present in the impedance at all frequencies, and so the 6'! phase difference never approaches 7"Z in either direction. Bhe curves for 4>L#" and 4CL1, and 4>L1 and 4CL#", are included to show that the resonant frequency Hthe point where the curve crosses the Aero phase'difference a<isI moves to low frequency when 4> e<ceeds 4C, and vice versa. Bhe curves for 4>L1"" and 4CL1, and 4>L1 and 4CL1"" show that the .resonant frequency. goes to Aero when 4>LNH>MCI, and goes to infinity when 4CLNH>MCI. Bhese results seem to indicate that the parallel resonator is infinitely tuna/le /y means of a varia/le resistor, a proposition which warrants careful e<amination.
*3. 0esistance t+ned (C resonator&
Bhe parallel resonator shown on the right was offered as a Kcircuit ideaK in +lectronics World11G it /eing pointed out in the article that the tuning range is " to a if 4LNH>MCI, and the ; of the circuit is sta/le /ecause the total resistance is constant. Both of these claims are true, within the scope of the modelG /ut there are a couple of fatal flaws in the concept and we will address them lest people should start to /elieve that the circuit will wor-. Bhe author of the article was perhaps a little unsure of the " to a claim, and so concluded that a varia/le resistor can give a Kmuch widerK frequency range than a varia/le capacitor or inductor. We however, can straight'away dispense with the infinite upper limit /y drawing the circuit model on the right. We might descri/e the original circuit as Kwhat you try to /uildK, whereas this ma-es some attempt to simulate Kwhat you actually getK. A capacitor is simply two pieces of electrically conducting material in pro<imity. Bhe conductors do not have to /e plates. Capacitance appears whenever two conductors have the a/ility to /e at different relative voltages Hi.e., capacitance is made /y not shorting things togetherI, and so there will always /e some .stray capacitance. across the coil. Bhis precludes resonance at infinite frequency, /ut in fact, a coil /ehaves as though it has considera/ly more parallel capacitance than simple consideration of strays would predict. Bhe reason is that it ta-es a finite amount of time for an electromagnetic wave to ma-e its helical Fourney along the wire in the coil, and the resulting phase shift has to /e represented /y placing a hypothetical capacitance, the coil.s self-capacitance C> in parallel with the the idealised pure inductance. )elf'capacitance is dependent on the length of the winding wire and the effective velocity for a wave travelling along it. Bhis propagation velocity Hthe so'called phase velocityI is frequency dependent, /ut most radio coils are operated in a regime where the velocity is changing in such a way that the self'capacitance appears to ta-e on a definite value. 6n this regime, the apparent self'capacitance turns out to depend only on the e<ternal dimensions of the coil Hthe turn' to'turn spacing and the num/er of turns are practically irrelevantI. Bhe inclusion of self' capacitance in the model allows for the fact that the coil has a self-resonant frequency H)41I even when there is nothing whatsoever connected to it, and it is part of the J1 resonator design procedure to ensure that the )41 is outside the frequency'range of interest. A physically small resonator coil suita/le for radio receiver applications might have a self' capacitance of a/out 1p1. >et us suppose therefore that this applies to the 1OJ coil from the previous e<ample. Bhis amount of unavoida/le capacitance places an upper limit on the ma<imum attaina/le resonant frequency somewhere very roughly around 1MQ%]NH>C>IRL18"*JA. )tray capacitance /etween the connecting wires will reduce this frequency, so if we construct the circuit carefully we should e<pect the inductive /ranch to self'resonate somewhere in a range from a/out 3" to 18"*JA. All electrical conductors have inductance Ha coil is simply a structure designed to enhance inductance /y causing the magnetic fields developed /y adFacent turns to add togetherI. Jence the wires and plates involved in ma-ing up the capacitive /ranch of the resonator will constitute an additional series inductance, which we can model to a good appro<imation /y imagining a small inductor >C in series with the capacitor. 1or the 1""p1 capacitor of our previous e<ample, it will /e very difficult to get this .self'inductance. to /e less than a/out 1"nJ, so we might place the upper limit for the series self'resonance of the capacitive /ranch somewhere very roughly around 1MQ%]NH>CCIRL18"*JA. 6nductance of the connecting wires will reduce this frequency, so we should e<pect the capacitive /ranch also to self'resonate somewhere in a range from a/out 3" to 18"*JA. 6n the days /efore synthesisers, simple Hsingle'conversionI short'wave radio receivers used
11 C3n +n+s+al t+ned circ+itC, ) Che-cheyev, +lectronics World, $an %""3, p31.
#8 wide'range !1(s and varia/le capacitance tuning. 1or full short'wave coverage however, it was necessary to provide the receiver with a /and'switch, one reason /eing that it was very difficult to o/tain a tuning range of much greater than a/out an octave without changing coils. Bhe larger the coil, the larger the self'capacitance, and so the /and'switch selects progressively smaller coils with progressively shorter connecting wires as the frequency is increased. Winding a set of candidate coils and chec-ing them for self'resonance will quic-ly indicate to the designer that a set of frequency ranges li-e 1'%, %'3, 3':, :'18, and 18'2%*JA is easily achieva/le, /ut trying to reduce the num/er of /ands to four He.g., 1'%.2#, %.2#'#.#%, #.#%'12, 12'2".8I requires careful construction, and reducing the num/er to three H1'2.18, 2.18'1", 1"'21.8I is very difficult using a conventional rotary switch. Bhis does not mean that a three'/and solution cannot /e o/tained, /ut it falls close to the /orderline at which it /ecomes prefera/le to use an ela/orate low'capacitance technique such a .turret /andchanger., i.e., a rotating turret which carries the coils and ena/les them to /e connected to the active circuit via very short leads Hsee /elowI.
1otorised bandchanger t+rret from 17#: vintage *arconi AD2"5 aviation transmitter.
6n the matter of ma-ing a resistance tuned parallel resonator thereforeG our rough calculations, and o/servations of what others have /een a/le to achieve in practice, seem to indicate an appro<imately %01 rule'of'thum/ for the upper limit of the frequency range. What this means in this instance however, is that we should not try to push the resonator to much more than a/out twice its ideal'case resonant frequencyG so if we consider our e<ample 1OJ in parallel with 1""p1 resonator, which resonates at a/out 18*JA when the resistance in /oth /ranches is equal, we might reasona/ly e<pect to /e a/le to tune it from "'2%*JA. Bhis, although not infinite, is nevertheless a phenomenal tuning rangeG /ut unfortunately, there is a catch. Bhe pro/lem is that if we use a varia/le resistor of value equal to NH>MCI, the ; of the circuit will /e appro<imately 1. We will investigate the relationship /etween ; and /andwidth shortly, /ut we can pre'empt those findings /y stating that such a circuit will /e completely useless as a /and'/ass filter such as might /e used to provide a radio receiver with selectivity. We might therefore consider raising the circuit ; to 1", /y reducing the value of the varia/le resistor in our e<ample NH>MCIL1""h resonator so that the total resistance in /oth /ranches adds up to 1"h. 6n this way, as we shall see, we sacrifice .some. of the tuning range in order to o/tain a poor /ut possi/ly useful ;. Bo find the tuning range which results, we can use the full parallel resonance formula0 f" L _1MQ %]NH> CI R `_ NQH>MC ' 4>XIMH>MC ' 4CXIR `
#5 i.e., f" L f"s NQH>MC ' 4>XIMH>MC ' 4CXIR =ow if, for the sa-e of simplicity, we assume that all of the resistance is in the inductive /ranch at the low frequency limit, and in the capacitive /ranch at the high frequency limitG the correction factor NQH>MC ' 4>XIMH>MC ' 4CXIR /ecomes ".77377 when 4>L1", and 1.""#"3 when 4CL1". )o, for our 1OJ in parallel with 1""p1 resonator, with its ideal'case resonance of 1#.71#*JA, we o/tain a tuning range of 1#.:28 to 1#.778*JA, a spectacular Y".#W with a ; of 1". (f course, if we dispense with the varia/le resistor and use a varia/le capacitor or inductor instead, we can easily o/tain a tuning range of more than %01, while sustaining a ; of around #". )o much for the resistance'tuned resonator as a varia/le /and'pass filter, /ut perhaps we can use it as the frequency'determining device in an oscillator? Bhat was certainly the suggestion in the Kcircuit ideaK article from whence it came. An oscillator is effectively an amplifier with some of its output fed /ac- into its input via a frequency'selective networ-, and we can easily design an amplifier with sufficient gain to overcome the losses of the candidate circuit. Bhe first pro/lem however, is that the spurious self'resonances of the networ- are of higher ; than the desired resonance, and if nothing is done a/out them the circuit will pro/a/ly oscillate at somewhere around the self'resonance frequency of the inductive /ranch. We might however decide to use an amplifier which is too slow to oscillate at !J1, or implement some -ind of low'pass filter in order to ensure that the system has no gain in the trou/lesome self'resonance region. Bhus /y some artifice we might actually get the oscillator to su/mit to our will, at which point it will deliver its final insult /y producing a signal which appears to /e modulated /y a hissing noise. Bhis ill' mannered /ehaviour will occur /ecause an oscillator is effectively a generator of filtered white noise. =o oscillator produces a pure sine'wave. A practical 41 oscillator always produces a /and of noise centred on a selected resonance of the frequency'determining networ-, with a width determined /y the networ- ; and the amplifier gain. Bhe usual o/Fective in oscillator design for radio applications is therefore to o/tain as high a ; as possi/le, so that the desired output is a spi-e in the amplitude vs frequency domain sufficiently narrow to /e regarded as a sine wave. Bhus the resistively tuned parallel >C resonator is of little practical appeal in situations demanding spectral purity. 6ts significance to this discussion lies instead in the fact that the Kcircuit ideaK is a plausi/le fallacyG and that its appearance in an electronics pu/lication did not generate a flurry of letters pointing out its flaws. We might comment, at this point, that it is necessary to /uild a circuit and try it /efore recommending it to othersG /ut that is no help in finding out what went wrong if the circuit should fail to wor- as e<pected. Bhere is a great deal of difference /etween a resonance and a useful resonanceG and a practical circuit component operated at radio frequencies does not /ear description as a pure inductance, capacitance, or resistance. Wide'range resistance'tuned >C oscillators Hoperating at low frequenciesI have nevertheless /een /uilt1%, /ut the theory of operation depends on more than the simple o/servation that resistance appears in the parallel resonance formula. )uch circuits were used many years ago for resistance' to'frequency converters in scientific instrumentation applications, /ut are nowadays rendered o/solete /y simple Hand spectrally noisyI 4C oscillators such as can /e implemented using C*() logic inverters or the ### timer 6C.
1% KBheory and Application of 4esistance BuningK, C. Brunetti and +. Weiss. ,roc. 64+, $une 1731, p222'233.
*4. Phasor theore/s
+arly in this chapter we o/served that the standard electrical formulae represent incomplete statements of (hm.s >aw and $oule.s >aw. We then went on to generalise (hm.s law, /ut have yet to state all of its implicationsG and we repaired the !6 power law /y introducing the scalar product, /ut have yet to analyse $oule.s law. We also introduced the the idea that if a phasor is pointing at "Z or 1:"Z it can /e treated as a scalar, a tric- which o/viously wor-s, /ut for which we offered no convincing mathematical proof. All of these discomforts arise /ecause of a narrative e<pediency, which is that of delaying the introduction of comple< num/ers until after that of vectors. We will now resolve all of the residual issues, with the aid of a handful of simple theorems which require complicated geometrical arguments if they are to /e proved using vectors, /ut are easy to prove using comple< num/ers. Bhese theorems incidentally are also true for real num/ers, which are effectively one'dimensional vectors. *4.1 1agnit+de ratio theore/: Bhe magnitude of the ratio of two Hcomple<I num/ers is equal to the ratio of their magnitudes. cE1ME%c L cE1c M cE%c *4.1 $roof% >et E1La1TG/1 and E%La%TG/% Bhen0 E1 M E% L H a1 TG/1 I M H a% TG/% I =ow multiply numerator and denominator /y the comple< conFugate of the denominator0 E1 M E% L H a1 T G/1 IH a% ' G/% I M H a%X T /%X I L Q a1a% T /1/% TGH /1a% ' a1/% IR M H a%X T /%X I =ow caTG/c L NHaX T /XI, therefore0 cE1ME%c L N Q _ H a1a% T /1/% IX T H /1a% ' a1/% IX ` M H a%X T /%X IX R L NQ_Ha1a%IX TH/1/%IX T%a1a%/1/% TH/1a%IX TH/%a1IX '%a1a%/1/%`MHa%XT/%XIXR L NQ_ Ha1a%IX TH/1/%IX TH/1a%IX TH/%a1IX `MHa%XT/%XIXR L NQ_ a%XH a1X T /1XI T /%XH a1X T /1XI `M H a%X T /%XIX R L NQ H a1X T /1XI H a%X T /%XI M H a%X T /%XIX R L NQ H a1X T /1XI M H a%X T /%XI R L QNH a1X T /1XIR M QNH a%X T /%XIR L cE1c M cE%c *4.* 1agnit+de reciprocal theore/: Bhe magnitude of the reciprocal of a Hcomple<I num/er is equal to the reciprocal of its magnitude. c1MEc L 1McEc *4.* $roof% >et E1 L 1TG" L 1 =ow cE1ME%c L cE1c M cE%c Bherefore0 c1ME%c L c1c M cE%c c1ME%c L 1 M cE%c
#7 *4.3 1agnit+de prod+ct theore/: Bhe magnitude of the product of two Hcomple<I num/ers is equal to the product of their magnitudes. cE1 E%c L cE1c cE%c *4.3 $roof% >et E1La1TG/1 and E%La%TG/% Bhen0 E1 E% L H a1 TG/1 IH a% TG/% I L a1a% ' /1/% TGHa1/% T a%/1I cE1 E%c L NQHa1a% ' /1/%IX T Ha1/% T a%/1IXR L NQHa1a%IX T H/1/%IX '%a1a%/1/% T Ha1/%IX T Ha%/1IX T%a1a%/1/% R L NQa1XHa%X T /%XI T /1XHa%X T /%XIR L NQHa1X T /1XIHa%X T /%XIR L QNHa1X T /1XIRQNHa%X T /%XIR L cE1c cE%c *4.4 2caling theore/: Bhe magnitude of the product of a scalar and a comple< num/er is equal to the product of the scalar and the magnitude. csEc L scEc *4.4 $roof% >et s /e a scalar, and ELaTG/. sE L sa TGs/ csEc L NQHsaIX T Hs/IXR L NQsXHaX T /XIR L sNHaX T /XI L scEc i.e., a scalar can /e factored out of or multiplied into a magnitude /rac-et in the same way that it can /e done with any other type of /rac-et. *4.5 ropJdi/ension theore/: A phasor with a phase angle of "Z or 1:"Z transforms as a scalar0 EH cEc , "ZI L TcEc *4.5 EH cEc , 1:"ZI L 'cEc $roof% A phasor pointing at "Z can /e represented as a comple< num/er with a positive real part and a Aero imaginary part. A phasor pointing at 1:"Z can /e represented as a comple< num/er with a negative real part and a Aero imaginary part. Jence if0 E L aTG" then0 ELa and cEc L TNHaX T "XI L cac Jence0 E L = L YcEc where = is a pseudoscalar equal in value and sign to the real part of E. Bhis may appear trivial, /ut it shows that our assumption that a phasor which has dropped a
8" dimension can /e treated as a scalar is universal, rather than a special interpretation of a particular phasor e<pression. A further implication however is that = is not identical to the magnitude of E, /ecause magnitudes are always positive whereas = can /e positive or negative. We can force = to /ecome equal to cEc /y stipulating that d L "Z. We can also drop a dimension, i.e., set the imaginary part to Aero, /y choosing d L 1:"Z, /ut in that case we get = L'cEc. Bhus the alleged scalar which results from dropping a dimension is not a magnitude, /ut it is a quantity which is equal in magnitude to a magnitude, and if d L "Z it is positive. Bhis may seem a pedantic distinction, /ut the point in ma-ing it is that if we restrict the scope of our phasor alge/ra through erroneous interpretation, we lose the a/ility to include DC electricity in our theory, and we lose the a/ility to e<plore e<otic ideas such as negative resistance. Bhe pseudoscalar we o/tain /y dropping a dimension can be negative, even if usually it isn.t. *4.7 2,+are /agnit+de theore/: Bhe product of a comple< num/er and its comple< conFugate is the square of the comple< num/er.s magnitude. E E L cEcX *4.7 $roof% >et ELaTG/ and E La'G/ E E L aX T /X /ut cEc L NHaX T /XI therefore E E L cEcX Jence the product of a comple< num/er and its comple< conFugate is a true scalar. 6t is also literally a scalar product. 4ecall that the definition of a scalar product is0 aBb L cac cbc Cosd /ut if a and b are identical, then dL"Z and CosdL1. Jence0 EBE L cEcX L E E *4.> ConG+gate prod+ct theore/: Bhe comple< conFugate of the product of two comple< num/ers is the product of the comple< conFugates. HE1 E%I L E1 E% *4.> $roof% >et E1La1TG/1 and E%La%TG/% Bhen0 E1 E% L Ha1 TG/1IHa% TG/%I L a1a% ' /1/% TGHa1/% T a%/1I Bherefore0 HE1 E%I L a1a% ' /1/% 'GHa1/% T a%/1I L a1Ha% 'G/%I 'G/1Ha% 'G/%I L Ha1 'G/1IHa% 'G/%I L E1 E%
81 *4.? 6nJphase ,+otient theore/: 6f two phasors are in phase, their ratio can /e treated as a scalar. E1HcE1c, dI M E%HcE%c, dI L cE1c M cE%c *4.? proof% Esing the polar to comple< transformation H1*.4I0 E1HcE1c, dI L cE1cHCosd T G)indI E%HcE%c, dI L cE%cHCosd T G)indI where the phase angle d is the same in /oth cases. Bherefore0 E1 M E% L cE1cHCosd T G)indI M Q cE%cHCosd T G)indI R L cE1c M cE%c *[email protected]
1agnit+de Ca!eat: Bhe mathematical operation of .ta-ing a magnitude. destroys information. )pecifically, it is important to /e aware that if cac L cbc, then it is not necessarily true that aLb. Bhe magnitude operation discards the directional information of a vector, and the sign information of a scalar. =ote for e<ample that although0 cac L ca c L c'ac L c'a c any one of the quantities inside magnitude /rac-ets is definitely not identical to any of the others. Bhe magnitude retains only the length of the o/Fect. 6n so doing however, it does retain the unit of measurementG i.e., a magnitude is a length in impedance space, or voltage space, or current space, etc., and so has the units of the space in which it e<ists. *4.1- 1agnit+de ",+i!alence: As noted a/ove, for any comple< num/er : 0 c:c L c: c L c':c L c': c When designing electrical circuits, it is not unusual to meet situations in which the magnitude of a voltage or current needs to /e determined, /ut the phase is unimportant. As the theorems a/ove show, when only the magnitude is needed, all of the impedances involved in the calculation can /e replaced /y their magnitudes Hprovided that the impedances are factors, i.e., multipliers or divisors, not terms in a summationI. What is less o/vious however, is that an impedance : enclosed /etween magnitude /rac-ets can then /e replaced /y one of the alternatives having the same magnitude, namely : , ': and ': . Bhis principle of (agnitude !quivalence allows us to deduce alternative networ-s which will produce the same outcome. 6n particular, it allows us to identify situations in which inductance can /e replaced /y capacitance and vice versa. *4.11 3bo+t these Theore/s0 Bhe theorems given a/ove do not appear in standard engineering te<t/oo-s. Bherefore it is legitimate to as-0 .Why have they /een stated here when everyone else manages without them?. Bhe answer to the question is this0 By stic-ing to the mathematical rules0 particularly /y ensuring that we always use properly /alanced vector equations, and /y using any simplifications which can /e proved in a general wayG we eliminate the need for phasor diagrams. +ssentially, we can let the alge/ra do all of the reasoning. We can still use phasor diagrams for the purpose of e<plaining what is going on, /ut they /ecome merely illustrative and ma-e no difference whatsoever to the outcome of a pro/lem solving e<ercise. Bhe traditional role of the phasor diagram has /een to help in resolving the am/iguities caused /y unrigorous mathematical definitions. But the mathematics is self'consistent. 6f the pro/lem is defined correctly, the hand'waving /ecomes unnecessary.
*5. Feneralisation of 'h/;s (aw
We have already arrived at a general statement of (hm.s law in section > /y o/serving that it can /e written as a phasor equation0 8L6:. We have also o/served that we are at li/erty to treat either 6 or 8 as a scalar equal in value to its own magnitude in order to learn the phase of the other relative to it. Bhe drop'dimension theorem H*4.5I gives our Fustification for doing so, /ut also allows us to generalise our phasor (hm.s law to include DC. We do this /y noting that if a circuit has any series capacitive reactance, then as the frequency goes to Aero, \C goes to infinityG hence the magnitude of the impedance goes to infinity and the impedance /ecomes an open circuit. Bhe only type of reactance which gives DC continuity is, of course, inductive reactance, and at fL", \>L". Bhus :H4,\I drops a dimension and /ecomes :H4,"IL4. Jence we can write 8L64 for DC Hor for pure resistance and ACI, /ut since /oth 8 and 6 are then in phase, they can drop dimensions also. Bhus we o/tain !L64 if we drop dimensions at dL"ZG /ut more to the point, we are also at li/erty to drop dimensions at dL1:"Z and o/tain the perfectly valid alternative0 H'!I L H'6I 4 i.e., we have a theory which covers all aspects of AC electricity and also allows us to have the negative voltages and currents required for the analysis of DC circuits. Bhis is why we must insist that the un'/old sym/ols ! and 6 are not magnitudes, they are pseudoscalars Hor, if you prefer, comple< num/ers in the form aTG"I which can point in either a positive or a negative direction. 6t is only resistance which can never /e negative in a passive networ-, and that is for physical rather than for mathematical reasons. An additional interpretation of (hm.s law is also given to us /y the magnitude ratio and product theorems H*4.1J*4.3I. Bhese allow that if 8L6:, then0 c8c L c6 :c L c6c c:c Hand all possi/le rearrangementsI. Bhis says that the need for comple< arithmetic is removed if all you want to -now is a magnitude, i.e., if the left hand side of an equation is a magnitude, then all of the phasors on the right can /e replaced /y their magnitudes. Bhis o/servation simplifies some pro/lems enormously, since failure to apply the magnitude ratio and product theorems when the situation allows results in unwitting repetition of the wor-ing used in the proofs in sections *4.1 to *4.3. )hown /elow are some of the possi/le interpretations of (hm.s law which stem from the discussion in this chapter. Bhere is no need to memorise these formulae /ecause they are all derived from the statement K8L6:K. What they show is that all manner of complicated arguments involving phasor diagrams are in fact trivial and can /e deduced /y inspection of the master equation. 8L6: 8 L 6 H4TG\I !L6: 8L6: c8c L c6c c:c c8c L 6 c:c ! L c6c c:c !L64 : L 8 M 6 L 8 6 M c6cX 4 T G\ L 8 M 6 : L ! M 6 L !6 M c6cX :L8M6 M:M L M8M M M6M M:M L M8M M 6 M:M L ! M M6M 4L!M6 6 L 8 M : L 8 : M c:cX 6 L 8 M H4TG\I L 8 H4'G\I M H4X T \XI 6 L ! M : L ! : M c:cX 6 L 8 M : L 8 : M c:cX M6M L M8M M M:M 6 L M8M M M:M M6M L ! M M:M 6L!M4
H'!I L H'6I 4 4 L H'!I M H'6I H'6I L H'!I M 4 Where0 8 6 and : are phasors, 8 6 and : are comple< conFugates, c8c c6c and c:c are magnitudes, ! and 6 are phasors pointing at "Z, H'!I and H'6I are negative values of ! and 6 Hand
82 thus are phasors pointing at 1:"ZI, and un'/old D is not normally used, /ecause an impedance pointing at "Z already has the sym/ol 4.
*7. Feneral state/ent of No+le;s (aw
6n section 3, we gave $oule.s law in its standard form0 , L 6X 4 Bhis, now that we -now that 6 should /e interpreted as a phasor pointing at "Z or 1:"Z, proves to /e a correctly /alanced vector equationG /ut it is only so /y an accident of notation and, as we shall see shortly, the restriction on the phase of the current is unnecessary and limits the scope of the formula. $oule.s law, even in its standard form, is a more fundamental statement than ,L6!G /ecause the squaring of the current prevents the direction of the current from having any effect on the direction of the power. 6t therefore tells us that power is positive when resistance is positive, i.e., the dissipation of energy is a uni'directional process. Bhe direction in question is that of entropy, the general spreading out and cooling down of the Eniverse, which is associated with the irreversi/ility of time. Bhus, assuming that an impedance is /y definition a strictly passive networ-G our relationship with impedance space is s-ewed, in that we are not allowed to venture into the regions where resistance is negative. (ne far'reaching consequence is that we cannot devise electrical networ-s which will give an output /efore receiving an input, i.e., we cannot /uild circuits which violate causality. Bhere are however non'linear passive electronic devices which have a negative resistance characteristic Hsuch as the +sa-i diode or tunnel diode12 13I, /ut this is only in the sense that there is a region in the graph of 6 vs. ! where the current goes down as the voltage is increased. Bhus negative resistance devices can go from a particular level of power dissipation to a lower level as the applied voltage is increased, /ut they can never achieve a state of negative power dissipation. 6n order to generalise $oule.s law completely, we must write it in a way which allows the current phasor 6 to adopt an ar/itrary phase /ut which gives an e<plicitly scalar result. Bhe o/vious candidate e<pression is0 , L c6cX 4 *7.1 6t transpires that this is the definitive statement of $oule.s law, as we shall demonstrateG /ut first it is interesting to note a parallel /etween it and energy laws in the wider conte<t. Bhe square magnitude theorem H*4.7I tells us that c6cX L 6 6 , and so we can write e<pression H*7.1I as ,L66 4 or more to the point0 ,L6 46 *7.* *athematical structures of this type occur everywhere in physics. ;uantum'mechanical energy equations, for e<ample, are of the same formG and in that conte<t the vector which occupies the position if 6 is -nown as the eigenvector or .state'vector. Hand also the wavefunctionI. 6f 6 descri/es the state of the system, the implication is that all we have to do is define 6 in order to determine the energy. 6f the analogy holds, then the e<pression a/ove is universally true upon provision of a definition for 6. Bhis is not difficult to demonstrate, /ecause all electrical power transmission pro/lems /oil down to the matter of delivering power to an impedance.
12 The 3rt of "lectronics, ,aul JorowitA QW1J1AR and Winfield Jill, %nd edition 17:7, Cam/ridge Eniversity ,ress. 6)B= "'#%1'25"7#'5. Bunnel H+sa-iI diode p13'1#, 9 p1"8". Bac- diode p:71, :72. 13 Physical "lectronics, C > Jemenway, 4 W Jenry, * Caulton, Wiley 9 )ons, =ew bor-, %nd edn. 1785. >i/rary of Congress cat. card no. 85'%22%5. )ection 13.80 Bhe tunnel diode, p%7"'%73.
83 Consider the system shown on the right. 6n this case, 8 and 6 are not necessarily in phase, /ut we can easily o/tain an e<pression for 6 in terms of 8, and since we now appear to have a version of $oule.s law which allows 6 to point in any direction, there is no need to impose a restriction on any of the phasors involved. Bhus we can write0 6 L 8 M H4TG\I =ow, putting the reciprocal impedance into the aTG/ form /y multiplying numerator and denominator /y the comple< conFugate of the denominator we o/tain0 6 L 8 H4 'G\I M H4X T \XI and, using the conFugate product theorem H*4.>I0 6 L 8 H4 TG\I M H4X T \XI =ow we can insert these definitions into equation H*7.*I0 ,L6 46, thus0 , L 8 8 4 H4 'G\IH4 TG\IMQH4X T \XIXR and using the square magnitude theorem H*4.7I we o/tain0 , L c8cX 4 M H4X T \XI *7.3 Bhus without any convoluted discussion a/out phases or reference phasors, we have o/tained in a few lines of alge/ra a general e<pression for the power dissipated in an impedance in terms of the applied voltage. =ow let us chec- that this is consistent with the conventional approach0 Jere we use the power factor H8 6 scalar productI rule H1-.1I. , L 8B6 L c8c c6c Cosd =ow, using the diagram on the right, we can see that Cosd HadFacent M hypotenuseI is 4MNH4X T \XI. Jence0 , L c8c c6c 4 M NH4X T \XI . . . *7.4 1rom (hm.s law we -now that0 6 L 8 M H4TG\I and using the magnitude ratio theorem H*4.1I we o/tain0 c6c L c8c M cH4TG\Ic i.e.0 c6c L c8c M NH4X T \XI =ow, su/stituting this into e<pression H*7.4I we have0 , L c8cX 4 M H4X T \XI Which is the same as equation H*7.3I and so demonstrates that the power'factor rule is already em/edded in $oule.s law when we write the latter as a properly /alanced and un-restricted vector equation. =otice also, that however we manipulate the power law, the average direction of power flow is always dictated /y the sign of the resistance. We can, incidentally, also o/tain equation H*7.3I /y using the series to parallel transformation discussed in section [email protected]
6f we write the impedance in parallel form, then the power is simply given /y the square of the voltage magnitude divided /y the equivalent parallel resistance. Bhus, if0 : L 4TG\ L 4K MM G\K Bhen0 , L c8cX M 4K where0 4K L H4X T \XI M 4 i.e.0 , L c8cX 4 M H4X T \XI
8# Bhe universal steady'state electrical power laws can /e summarised as follows0 , L c6cX 4 , L c8cX 4 M H4X T \XI , L 8B6 L c8c c6c Cosd where c6c is the reading o/tained from an ammeter, and c8c is the reading o/tained from a voltmeter. 6f the impedance has no reactive component, or if the frequency approaches "JA HDCI, the general formulae a/ove revert to their standard te<t/oo- forms0 , L 6X 4 , L !X M 4 ,L!6 where ! and 6 are the readings from AC or DC instruments and can /e positive or negativeG /ut if ! is negative then 6 must /e negative Hpresuming that the resistance to which the power is /eing delivered is positiveI.
We are often interested the way in which the gain or loss of a networ- or circuit varies over a particular /and of frequencies. We will introduce this type of analysis shortly in connection with resonant networ-s, /ut /efore doing so it is necessary to define the term ./andwidth.. *ost readers will /e aware that an amplifier will generally show a fall'off of gain at low frequencies, this often /eing due to the increasing magnitudes of the reactances of coupling capacitors in series with the signal pathG and it will also show a fall'off of gain at high'frequencies, this /eing due to a variety of factors including the falling magnitudes of the reactances of any stray capacitances in parallel with the signal path. Consequently amplifiers, and indeed many other types of circuit, usually show a hump'li-e frequency responseG and will only pass signals usefully over a particular frequency range. Bhe pro/lem in defining /andwidth therefore lies in the definition of what we mean /y .useful. and, since this will vary according to the particular application, there is really no resolution to the issue. We must therefore eschew vague concepts li-e .usefulness. in favour of a definition on which everyone can agreeG and so it is universally accepted that /andwidth, unless stated otherwise, is defined in terms of what are -nown as the .half'power points., i.e., the upper and lower frequency points at which the power delivered /y the system Hfor a constant inputI has fallen to half of that which is delivered at the frequency at which the ma<imum response occurs. Bhe half'power points are chosen, as we shall see, /ecause they have special mathematical significanceG and for simple networ-s at least, -nowledge of where they lie provides a complete definition of the frequency'response function of the system. =ow, if we call the power delivered at the frequency of ma<imum response ,/aD, then the power delivered at the half'power points is0 , L ,/aD M % and , M ,/aD L m We can e<press this ratio in deciBels using the general definition0 4atio in dB L 1">og1"H , M ,ref I Hwhere ,ref is the reference power level against which power , is /eing comparedI. i.e. 1">og1"HmI L '2."1"%777#5 Jence the half'power points are also -nown as the . '2dB . points, and the frequency interval /etween the lower point and the upper point is also often called the . '2dB /andwidth .. 6t is a good idea to /e specific in this way, /ecause when the term ./andwidth. is used without qualification, there is always the fear that it may involve some non'standard definition.
*?. deci)els 9 logarith/s
Javing Fust found cause to mention the dB notation, we should perhaps clarify some of the aspects of its use which can /e a source of confusion. 1irstly, a deciBel is a tenth of a Bel, hence the small d and the capital B in .dB.. Bhe Bel, named after Ale<ander @raham Bell, is an internationally accepted unit equal to 1" Ktransmission unitsK HBEIG the BE Hnow the dBI /eing a logarithmic relative signal measurement method introduced /y AB9B in 17%2 1#. A ratio in Bels is the logarithm of a power ratio defined in the simplest possi/le way using /ase 1" Hi.e. commonI logarithms, hence0 =MB L >og1"H, M ,ref I Bhus a ratio in deciBels is0 =MdB L 1">og1"H, M ,ref I Bhe practice of e<pressing audio power ratios on a logarithmic scale was developed /ecause human hearing has an automatic gain'control system. Bhis ma-es our hearing response logarithmic with respect to loudness, and thus ena/les us to e<tract information from sound over a vast range of sound pressures Hit also ma-es our hearing asymmetric0 if a loudspea-er produces a loud sound which is asymmetric a/out the am/ient air'pressure a<is, there will /e a change in perceived quality if the amplifier connections are reversedI. Bhe use of logarithmic scales is also favoured /ecause it allows us to represent the vast gains of electronic amplifiers, and the vast ranges of power encountered in communication systems, using convenient num/ers. 6t carries over seamlessly into the field of radio, firstly /ecause modulated radio signals are converted into sound, and secondly, radio receivers also use automatic gain control systems and so also have a roughly logarithmic response. Common logarithms are used /ecause the notation hails from the time when log ta/les were used for multiplication. Bhe deciBel is preferred over the Bel /ecause it transpires that 1dB H%#.7WI is close to the minimum change in audio power which can /e detected /y the human ear Hthis /eing 1" ' %"W, i.e., ".3 ' ".:dB, depending on the waveform and the listenerI. Jence there is rarely a need to e<press power ratios in dB to a greater accuracy than to the nearest whole num/er. An attempt to introduce a log'ratio notation /ased on natural H=aperianI logarithms resulted in an alternative unit, the =eper Hunit sym/ol =pI, i.e.0 =M=p L HmI>oge H, M ,refI giving 1 =p L :.8:8dB /ut its use never /ecame widespread in +nglish'spea-ing countries. Bhe use of log'ta/les for multiplication has of course died'out from the school curriculum, and this leaves modern students unprepared for the introduction of deciBels and other logarithmic functions. A little revision of the su/Fect will therefore not go amiss0 Bhe technique of multiplication using logarithms was introduced in 1813 /y the )cottish mathematician and astronomer $hone =eper Hspelt variously /ut nowadays usually written as .$ohn =apier., this /eing the only version which =eper himself would not have recognisedI. 6t arises from the o/servation that if two num/ers are each e<pressed as a /ase'num/er raised to a power, then the num/ers can /e multiplied simply /y adding the powers, i.e., Ba S Bb L BaOb Bhis is o/vious when the powers Halso -nown as e<ponentsI are whole num/ers He.g. 1"2 S 1"# L 1": I, /ut it wor-s Fust the same when they are not. Also, we can perform division Fust as easily /y noting that0 Ba n Bb L BaJb Consequently, in the 15th and 1:th Centuries, long /efore the advent of afforda/le calculating machines, great effort was made to produce ta/les allowing difficult multiplications and divisions to /e performed /y loo-ing up the e<ponent which, when used to raise a common /ase, represents a particular num/er. Bhese e<ponents are -nown as logarithms, and are defined as follows0
1# 3d/iralty Handboo% of #ireless Telegraphy, B.4.%2" H!ol. 66I. J*)( >ondon, [email protected]
?., Appendi< A0 Bhe DeciBel and the =eper HAppendi< availa/le from http0MMwww.g2ynh.infoI
85 6f n L Ba then a L >ogBHnI HKif n equals B to the power of a, then a is the log to the /ase B of nKI 6f a logarithm is written without the /ase su/script, then /ase 1" is usually implied, i.e., .>og. means .>og1". Halthough some older documents deviate from this conventionI. =aperian HnaturalI logarithms, which crop'up frequently in physics, use +uler.s num/er e as the /ase Heo%.51:%:I, and can /e written either K>ogeK or KlnK, the latter pronounced KlineK and /eing short for .log'=aperian.. Jence, wor-ing in /ase 1"0 if mL1"a, and nL1"b, then mSnL1"aOb. All we have to do to perform the multiplication is loo-'up the logarithms a and /, add them together, then loo- up the quantity aT/ in a ta/le of anti'logarithms to find the required mSn. Bhe anti'log of a num/er < is simply 1"D. As an alternative to using ta/les, the same operations can /e achieved /y using two identical engraved logarithmic scales and sliding one relative to the other, the device for so doing Hinvented /y William (ughtred in 18%%I /eing -nown as a slide rule18. We no longer need to use logarithms for everyday multiplication, /ut we do need to memorise some of their /asic properties in order to use logarithmic units with confidence. Bhe first and most fundamental point is that any num/er raised to the power of Aero is one, i.e., B" L 1, always, regardless of B. Jence0 >ogH1I L ", always, regardless of /ase. =ow, if we represent power gain in deciBels, i.e., =MdB L 1">og1"H, M ,ref I then if ,L,ref G =MdB L 1">og1"H1I L " i.e., a system which neither amplifies nor attenuates a signal has a gain of "dB. Also we find that if , is greater than ,ref, then =MdB is greater than Aero, and vice versa. Jence a positive quantity in dB represents gain, and a negative quantity represents loss Hi.e., negative gainI. 1inally, the additive property of logarithms allows that if we su/Fect a signal to a num/er of processes, and note the gains in dB Hpositive or negativeI for each of those processes, we can find the overall gain of the system simply /y adding all of the individual stage gains together. )o much for the /asics, /ut now we arrive at the point which causes greatest difficulty0 A quantity in dB implies a logarithmic power ratio. 6t can however also /e ta-en to represent a voltage ratio, or a current ratio, /ut the definition must be modified in that case. Bhe reason why the definition can /e e<tended to em/race current and voltage ratios is that power is a function of the voltage across, and also the current through, an impedance. Jence we can su/stitute for power using the general power laws derived earlier, i.e., ,Lc8cX4MH4XT\XI and ,Lc6cX4. Bo o/tain the voltage ratio formula we write0 =MdB L 1">og1"_ Qc8cX4MH4XT\XIR M Qc8ref cX4MH4XT\XIR ` which reduces to0 =MdB L 1">og1"Q H c8c M c8ref c IX R and if we adopt the convention that the impedance against which the two voltages are compared is a resistance0 =MdB L 1">og1"Q H! M !ref IX R A similar argument applies for the current ratio formula0 =MdB L 1">og1"Q H6 M 6ref IX R =ow everything would /e fine of we left the quantity inside the logarithm /rac-ets as the square of a voltage or current ratio, /ut everyone who teaches the su/Fect will insist on performing a .simplification., which is to note that a num/er can /e squared /y dou/ling its logarithm. Jence we get rid of the power of % /y writing0 =MdB L %">og1"H! M !ref I and
18 C#hen 2lide 0+les 0+ledC Cliff )toll, )cientific American, *ay %""8, p8:'5#.
8: =MdB L %">og1"H6 M 6refI Which is all very clever, /ut leaves people struggling to decide whether they should use 1">ogHI or %">ogHI, and so leads to lots of mista-es. )o remem/er0 a ratio in Bels is the >og of a power ratio, and a deciBel is a tenth of a Bel Hthat.s where the 1" comes fromI. By $oule.s law, the square of a voltage or current magnitude ratio is also analogous to a power ratio, and the squaring can /e o/tained /y dou/ling the logarithm Hthat.s where the %" comes fromI. 6n using deciBels, the /asic approach is to consider the power levels at two points in a circuit or power transmission system and there/y define the gain. 6t is also useful however, to e<press power in relation to some e<ternal reference or standard, and this leads to an e<tension of the notation, some commonly encountered variants /eing as follows0 4eference voltage 4eference current Enit Definition L NH,4I L NH,M4I dBm dBu dBW dB! dB relative to 1mW in #"h dB relative to 1mW in 8""h dB relative to 1W dB relative to 1! %%2.8m! 553.8m! ' 1! 3.35%mA 1.%71mA '
in old audio pu/lications and service manuals, . dBm . may /e used to mean . dB relative to 1mW in 8""h ..
By e<tending the definition in this way, the dB notation may /e used to e<press an a/solute power Hrather than a relative powerIG and if a reference resistance is specified, an a/solute voltage or current as well. 1or e<ample, if the line output level from an audio recorder is specified as '1"dBu, then the output voltage is o/tained /y rearranging the e<pression0 '1" L %">ogH!o+tM!ref I where !ref L NH".""1 S 8""I L 553.8m! Jence0 !o+t L !ref S 1"JP L !ref M HN1"I L %33.7m! 4*). Bhe dBW notation was /rought into +uropean Amateur 4adio documents some years ago, this /eing the preference in the field of /roadcast and professional radio engineering. Bhus a 3""W transmitter Hfor e<ampleI /ecomes a 1">ogH3""IL%8dBW transmitterG and it is possi/le to determine the effective radiated power H+4,I of a radio installation /y adding the transmitter power in dBW to the HnegativeI gain in dB of the antenna feeder and the gain in dB of the antenna. Bhis is all very well of course, /ut it does /eg the question0 .why, for a group of spectrum users generally only equipped to measure voltage and resistance to a reasona/le accuracy, is it necessary to state power restrictions in a way which requires a -nowledge of e<ponential functions in order to worout what they mean?. 6t would seem equally logical to state road speed restrictions in dBmph or dB-mMh, and so for the sa-e of any /ureaucrats who might read this, we will also address the question0 .do speed ratios require the 1">ogHI or the %">ogHI formula?. Bhis question, perhaps surprisingly, is not meaningless, and can /e answered /y noting that power is equivalent to energy delivered per unit'of'time. A power ratio is thus an energy per Qunit'of'timeR compared to a reference energy per Qunit'of'timeR, and since the . Qunit'of'timeRs . will cancel Hprovided that they are the same ' seconds are very popularI, a power ratio is also an energy ratio. Jence0 =MdB L 1">og1"H+ M +ref I =ewton.s laws of motion tell us that the -inetic energy of a moving /ody is given /y +LmvXM% Hwhere m is the mass and v is the velocityI, so energy is proportional to velocity squared as well as to voltage squared and current squared. Jence, a speed in dBmph is given /y %">ogHvI, so 2"mph /ecomes %7.#dBmph and 5"mph /ecomes 28.7dBmph.
87 =ow, having upgraded all of our road signs to /e in -eeping with the preferred notation for @overnment standards documents, we are only left with the pro/lem of how to measure money in deciBels. Jere we may note that currency names are often derived from weights Hof silver, /ut there has /een some devaluation since 4oman timesI, and that =ewton.s and +instein.s laws tell us that mass is proportional to energy. Bhus we can deduce that the 1">ogHI formula is the correct one in this case. We might have solved this conundrum without recourse to physics however, /y recalling the famous old saying0 Kmoney is powerK.
*@. )andwidth of a series resonator
When used to ma-e filters for the 41 and 61 amplifiers of radio receivers, high'; resonant circuits provide /etter selectivity than low'; resonant circuits. )ince good selectivity is synonymous with narrow /andwidth, there is evidently a relationship /etween /andwidth and ;, and as we shall see, this relationship is a particularly simple one if we define the /andwidth as the interval /etween the half'power H'2."1dBI points. Bhe question we must address ne<t therefore is0 Kwhat is this power to which we must relate the /andwidth?K Bhe answer in the case of an amplifier driving a resistive load is o/vious, it is the power dissipated in the load. 6n the case of a resonant circuit however, there may or may not /e a load in the normal sense, and we are left with the uncomforta/le conclusion that /andwidth must /e defined in terms of the power which would /e dissipated in the load, if there were a load. We can crac- this riddle for the series resonant case /y considering the circuit shown /elow. Bhis is the circuit of a simple /and'pass filter. 6t has an input voltage, an output voltage, and a load resistanceG and the /andwidth is very clearly the frequency interval /etween the points where the power in the load is half of that which occurs at the frequency of ma<imum response Hfor constant c8incI. 6t is also however, a series resonant circuit, and the ; at resonance can /e defined as0 ;" L \"> M 4 L '\"C M 4 where 4 is the total resistance, i.e., 4L4>T4CT4>oad G and the su/script . " . has /een added to the reactances as a reminder that ;" is defined in terms of their values at the resonant frequency f". =ow, we can easily write an e<pression for the current which flows from the generator /ecause the impedance connected across the generator is simply0 : L 4 T GH\>T\CI and so, ta-ing the current as a reference phasor, and using (hm.s law and the magnitude ratio theorem H*4.1I, we o/tain0 6 L c6c L c8inc M c:c We can also state that the power delivered to the load is0 ,>oad L 6X 4>oad and that ma<imum power will occur at the resonant frequency /ecause the total reactance will then /e Aero and the magnitude of the total impedance c:c will /e at a minimum. Jence we will call the ma<imum load power ,">oad , and the power at the /andwidth limits will /e0 ,>oad L ,">oad M % i.e., 6X 4>oad L 6"X 4>oad M % where 6" is the current at resonance. Jence the /andwidth limits lie at the points where 6X L 6"X M % i.e., where 6 L 6" M N%
5" )o the load resistance, having served to allow us to define the /andwidth, has promptly vanishedG and the /andwidth /ecomes the interval /etween the points where the current has fallen to 1MN% of its value at resonance. 1urthermore, we can o/serve that we will always o/tain this result regardless of which resistance we define as the load. 4> , 4C , and 4>oad are only sym/ols, and since the corresponding resistances are connected in series, we can swap their designations at will. We can also consider any com/ination of these resistances to /e the load, including the total resistance 4, and this will always cancel and tell us that the half'power points occur when 6L6"MN%. Bhus to define the /andwidth of a series resonant circuit, we do not need to designate any resistance as a load, we need only to consider the current. )o it transpires that we can choose any resistance in a series networ- and analyse the power dissipated in it to determine the /andwidthG and since we are interested here in the relationship /etween /andwidth and ;, the o/vious resistance to choose is 4, the total resistance. We can always isolate a portion of 4 to determine the power delivered to it or the voltage across it if we so wish, that is a trivial matter of proportionsG /ut for a general analysis, the pro/lem simplifies to that of understanding the /ehaviour of the simple series >C4 networ- shown /elow. Bhe first part of the analysis is to determine the frequency response function for this circuit and plot it as a graph to see what it loo-s li-e. A good function to plot for this purpose is the ratio ,M," vs. frequency, /ecause this ratio has a value of 1 at f" and is also in the correct form for conversion into deciBels. Bhe power ratio is equal to the square of the current ratio0 ,M," L 6XM6"X L H6M6"IX, /ecause ,L6X4 and ,"L6"X4. Jence we will start /y o/taining an e<pression for the current ratio. Bhe general e<pression for the current is0 6 L c6c L c8c M c:c where :L4TGH\>T\CI At the resonant frequency however, the impedance is purely resistive, so0 6" L c8c M 4 Jence0 6 M 6" L H c!c M c:c I M H c!c M 4 I L 4 M c :c L 4 M _ NH4X T Q\>T\CRXI ` which, /y writing the reactances e<plicitly, gives0 6 M 6" L 4 M _ NH4X T Q%]f> '1MH%]fCIRXI ` *@.1 and since ,M," L H6M6"IX0 , M ," L 4X M H4X T Q%]f> '1MH%]fCIRXI *@.* @raphs of /oth of these functions are shown /elow, the procedure used for generating them /eing to choose a value for f" and an >MC ratio, and then calculate Husing the (pen (ffice Calc spreadsheet programI a set of points at closely spaced intervals for various different values of ;" Hsee accompanying file serLres.odsI. Bhe initial choices are ar/itrary, since Has we are a/out to showI the shape of the curve o/tained depends entirely on ;". 6n this case, the author chose f"L1"*JA and >MCL1"3, i.e., CL>M1"3. !alues for > and C were then o/tained /y solving the resonance formula for >, i.e., 1"5 L 1MQ%]NH>CIR L 1MQ%]NH>XM1"3IR L 1""MQ%]>R > L 1""MQ%]S1"5R L 1.#71#3732OJ C L >M1"3 L 1#7.1#3732p1 )ince NH>MCIL1""h is also the value of \> and '\C at resonance, resonant ; values of 1"", 1" and 1 correspond to total resistances H4I of 1h, 1"h and 1""h respectively. =otice in the graphs /elow how the squaring pushes the curve of ,M," downwards in comparison to 6M6". =otice also that the half power level is 1MN%L".5"51 for 6M6" and m for ,M,", and that the deciBel scales on the right differ accordingly.
)o, having shown how the shape of the frequency response function varies with resonant ;, we will now derive an e<pression for the relationship /etween ; and /andwidth, the /andwidth /eing defined as the interval /etween the upper and lower half'power points. Bhe procedure is to write a general e<pression for the current and solve it for the frequencies at which 6L6"MN%. =otice that the word KfrequenciesK is plural0 the e<pression will /e a quadratic equation. As we have already determined, 6Lc8cMc:c, and 6" Lc8cM4. Jence, at the '2dB /andwidth limits0 6 L c8cMc:c L 6"MN% L c8cMH4N%I Bhus the /andwidth limits occur at the frequencies where0 c:c L 4N% i.e., NH4X T \XI L 4N% 4X T \X L %4X \X L 4X which, ta-ing the square root of /oth sides and noting that there are two possi/ilities from so doing, gives0
5% \ L Y4 =ow, writing \ e<plicitly we o/tain the e<pression0 Y4 L %]f> '1MH%]fCI which we must solve for f. We may proceed /y putting the right hand side onto a common denominator Hi.e., /y multiplying top and /ottom of the %]f> term /y %]fCI0 Y4 L QH%]fIX>C '1RMH%]fCI i.e., Y%]fC4 L H%]fIX>C '1 Bhis rearranges to0 H%]fIX>C Y%]fC4 '1 L " Which is a quadratic equation in the form afXT/fTcL", with aL3]X>C, /LY%]C4 and cL'1. =otice however, that this particular equation will have four solutions, rather than the usual two, /ecause the / term has a .Y. sym/ol attached to it. Bhe reason for that is that there are /oth positive and negative frequency solutions for each of the /and'edges. Bo o/tain all four of these frequencies we apply the general solution for quadratic equations H1*.3I0 f L Q'/ YNH/X ' 3acI R M %a f L _ Y%]C4 YNQH%]C4IX T3S3]X>CR `MH%S3]X>CI and using the su/stitution CLCXMC to o/tain a cancellation of C from all /ut one term0 f L _YC4 YNQHC4IX T3>CXMCR `MH3]>CI f L QY4 YNH4X T3>MCI RMH3]>I 6n order to determine which are the positive frequency solutions among these four possi/ilities, o/serve that TNH4XT3>MCI is always larger than 4. Jence the upper HpositiveI /andwidth limit is0 fT L _QTNH4X T3>MCIR T 4`MH3]>I and the lower HpositiveI /andwidth limit is0 f' L _QTNH4X T3>MCIR ' 4`MH3]>I and the /andwidth is0 fw L fT ' f' L _QNH4X T3>MCIR T 4 ' QNH4X T3>MCIR T 4`MH3]>I i.e., fw L 4MH%]>I . . . . . H*@.3I =ow recall that the resonant ; can /e defined as ;"L\>M4L%]f">M4. Jence0 ;"Mf" L %]>M4 Jence0 fw L f"M;" *@.4 Bhis is the classic e<pression for the /andwidth of an >C resonator, and is e<act when the resistance in the circuit remains constant with frequency. We have, of course, already o/served that the loss resistances of inductors and capacitors vary with frequency, /ut it transpires that this will ma-e practically no difference to the accuracy of the e<pression under most circumstances. 4esistance ma-es only a small contri/ution to the overall shape of the /andwidth function /ecause it only ma-es a significant contri/ution to the magnitude of the impedance Hand hence to the currentI when the frequency is close to resonance. 1ar from resonance, the impedance magnitude is dominated /y the reactive component unless the ; of the resonator is very low. Bhe physical laws governing the various processes which contri/ute to the loss resistance moreover are smoothly varying functions of frequency Hunless some additional system resonance is encountered in the region of interestI, and so the loss resistance component will not normally vary significantly over a small frequency interval. Consequently, for a reasona/ly high ;, the relationship0 K BandwidthLf"M; K is sufficiently accurate to /e presumed e<act for all normal engineering purposes.
3-. (ogarith/ic fre,+ency
(ne additional matter which might /e of interest is that, although some authors refer to f" as the Kcentre frequencyK, the frequency interval /etween the half'power points is not symmetrical a/out f". We can find the mid'point or median frequency /y ta-ing the average of the upper and lower /and limits, i.e., f/ L HfT T f'IM% L QNH4X T3>MCI T 4 TNH4X T3>MCI ' 4RMH%S3]>I L Q%NH4X T3>MCIR MH%S3]>I f/ L QNH4X T3>MCIR MH3]>I . . . . . . H3-.1I Bhis quantity is only equal to f" when 4k", i.e., Hnoting that HN>IM>L1MN>I 0 f/ k QNH3>MCIR MH3]>I L 1MH%]N>CI L f" Bhis limiting condition arises /ecause the /andwidth of the resonant circuit is infinitely narrow when 4k", i.e., the upper and lower /and'limits /ecome coincident with f" in the limit that 4k". 6t is therefore an essential property of a correct e<pression for the mid'/and frequency, /ut it does give us a simplification for equation H3-.1I. )quaring H3-.1I gives0 f/X L H4X T3>MCIRMH3]>IX L H4M3]>IX T f"X /ut from e<pressions H*@.3I and H*@.4I given earlier, f"M;L4MH%]>I, hence0 f/X L Hf"M%;IX T f"X L f"XH1 T Q1M%;RXI f/ L Y f" NH1 T Q1M%;RXI 3-.* Bhe HpositiveI mid point frequency is always very slightly a/ove the resonant frequency for a practical resonator, /ut /ecomes coincident with f" when ;ka Hi.e., 4k"I. Bhis s-ewing of the /andwidth function can /e said to arise /ecause f" is always closer to Aero than it is to infinity Hthe function spreads out on the high'frequency side /ecause there is more roomI. Bhe difference /etween the mid'/and frequency and the resonant frequency is however, very small, /eing ".1%#W H1%.#KJA at 1"*JAI for a ; of 1", and ".""1%#W H1%# JA at 1"*JAI for a ; of 1"". Jence the /andwidth function can /e considered to /e appro<imately symmetric for moderate ;. Although the /andwidth function is not symmetric a/out its pea- if we choose frequency as the horiAontal H<I a<is, it can /e made symmetrical if we instead plot it against an appropriately chosen function of frequency. 6n particular, we need a frequency function such that any resonance pea- is always Fust as far from Aero'frequency as it is from infinite frequency, i.e., we need an infinity to the left of the resonance, and an infinity to the right, and /oth infinities must /e of the same type. )uch a requirement is satisfied /y, and indeed is one of the principal properties of, the logarithmic function0 < L >ogHfI the choice of /ase /eing ar/itrary. Jence the pea- can /e said to /e symmetric a/out its logarithmic centre frequency <" L >ogHf"I )ince frequency can /e scaled ar/itrarily without affecting the shape of the /andwidth function Hunits of JA are not mandatoryIG this matter can /e .proved. numerically /y plotting ,M," against >ogHfI with f"L1 and noting that the function is symmetric a/out <"L>ogH1IL" for any value of ;". 6t should also /e noted that there is an infinity of scales from microscopic to macroscopic, and it is often more natural to thin- in logarithmic dimensions than in linear ones. 6n the case of frequency, this can /e seen /y considering the classic representation of the electromagnetic spectrum as illustrated /elow.
Bhere is no theoretical minimum frequency on the logarithmic scaleG although the lowest electromagnetic frequency which can /e encountered in practice is the reciprocal of the age of the universe, a/out 1MH12.5%S1"7S28#.%3%177S%3S8"XI L %.21S1"'1: JA /y current rec-oning. ,ractical DC electrical systems come nowhere close to that /ecause even the /est stop wor-ing after a few years. Jence Aero frequency is impossi/leG we employ the concept merely as a mathematical convenience for the purpose of circuit analysis. =ote incidentally, that some writers have /een moved to claim that there is a flaw in *a<well.s equations /ecause electrical formulae tend to produce infinities when frequency is set to Aero. Bhis is a fallacy of courseG the infinities /eing merely a reflection of the fact that Aero frequency is not a property of the Eniverse. A practical consequence however, is that annoying Kdivide'/y'AeroK errors occur when putting Aero frequency into Hfor e<ampleI frequency'response calculations. Bhe solution, when calculating a frequency response which needs to appear as though it starts from "JA, is to input a very low frequency instead of Aero. 1or radio'frequency calculations, starting from 1JA, instead of "JA, will usually do the tric-.
31. 3 proper definition for resonant A
=ow that we have esta/lished that ;" is an important circuit parameter, we will ta-e the opportunity to have another loo- at at its definition. Bhe point is that there is something horri/ly unsatisfying a/out writing0 ;" L \">M4 '0 ;" L '\"CM4 6t seems more logical that the definition should simultaneously involve /oth inductance and capacitance. 6f so, then why not write0 ;" L Q TNH'\"C \">I R M 4 which is the same as multiplying the two standard definitions and ta-ing the positive square root Hi.e., ta-ing the geometric mean of the two definitionsI? (f course0 '\"C \"> L %]f"> MH%]f"CI L >MC Hi.e., the >MC ratioI hence0 ;" L Q NH>MCI R M 4 or 31.1 ;" L 4" M 4 Jere is a definition of resonant ; which properly involves all of the componentsG and as an added /onus in calculation, does not involve the resonant frequency.
3*. )andwidth in ter/s of A
A /andwidth function for the series resonator was derived earlier and given as equation H*@.*I0 , M ," L 4X M H4X T Q%]f> '1MH%]fCIRXI =ow that we have a sensi/le definition for ;" however, we can see that it can /e used as a su/stitution for 4 in the e<pression a/ove, i.e.Hfrom 31.1I0 4 L Q NH>MCI R M ;" Jence0 QH>MCI M ;"XR ,M," L QH>MCI M ;"XR T Q%]f> '1MH%]fCIRX =ow, if we forci/ly factorise the quantity >MC from the right hand term in the denominator we o/tain0 QH>MCI M ;"XR ,M," L QH>MCI M ;"XR T H>MCI_ QNHCM>IRQ%]f> '1MH%]fCIR`X which, noting that >MN>LN> and HNCIMCL1MNC, simplifies to0 Q1 M ;"XR ,M," L Q1 M ;"XR T _ f%]NH>CI '1MQf%]NH>CIR `X We cans su/stitute for the quantity %]NH>CI /y noting that the standard series resonance formula can /e rearranged thus0 %]NH>CI L 1Mf" Jence0 Q1 M ;"XR ,M," L H3*.1I Q1 M ;"XR T Q Hf M f"I ' Hf" M fI RX Bhis puts the /andwidth function into a form most similar to a curve -nown as the >orentAian line' shape function Hne<t sectionI, /ut a further simplification is possi/le /y multiplying /oth numerator and denominator /y ;"X 0 1 ,M," L H3*.*I 1 T _ ;"Q Hf M f"I ' Hf" M fI R `X which demonstrates in the clearest possi/le way that the /andwidth of an >C resonator is dictated entirely /y ;" and f".
33. (orentzian lineJshape f+nction
Bhe electrical resonance curve is closely related to a simple mathematical function -nown as the orent!ian Hor CauchyI line'shape function, which has the general form0 h wX yL H33.1I wX T H< ' <"IX where h is the pea- height and w is called the half'width. Bhe e<pression can also /e written0 y 1 L H33.*I h 1 T QH< ' <"IMwRX which is the form most similar to equation H3*.*I. Bhe >orentAian is regarded as the characteristic signature of natural electromagnetic resonance processes. 6n particular, the pea-s in molecular and atomic spectra in the microwave, optical, <'ray and gamma'ray regions are all of this form when displayed on a linear amplitude Hy'a<isI scale. Bhe curve is called a line'shape function /ecause the narrow spi-es which occur when dense spectra are drawn /y a chart'recorder or otherwise displayed are traditionally -nown as lines. 6t is only when the frequency scale is e<panded that the individual pea-s resolve into >orentAians. 6n comparing the >orentAian to the electrical resonance curve, we may first note that the >orentAian is always e<actly symmetric a/out <", and that <" can /e set to Aero. We have noted /efore Hsection 3-I that the electrical resonance curve is s-ewed when plotted against linear frequency, /ut /ecomes symmetric to a good appro<imation when the ; is high. We also noted that the resonance curve can /e made perfectly symmetric /y plotting it on a logarithmic frequency scaleG in which case, since the logarithm of unit frequency is Aero, the curve can also /e symmetric a/out >ogHfIL". 6n fact, natural resonance processes have such high ; that they appear symmetric on linear, logarithmic, and even reciprocal HwavelengthI scalesG /ut to find the relationship /etween the >orentAian and the electrical curve, it is o/vious that we must identify the <'a<is as corresponding to logarithmic frequency, i.e., <L>ogaHfI , where the /ase a can /e chosen ar/itrarily. Jere we will use =aperian logarithms /ecause it will allow us to use the series e<pansion of e to solve the pro/lem. Jence we choose0 < L >ogeHfI which means that0 f L eD and f" L e D )u/stituting these identities into the electrical resonance curve H3*.*I we o/tain0 1 ,M," L 1 T _ ;"Q HeD M eD I ' HeD M eDI R `X
/ut, from the rules of logarithms discussed in section *?0 eD M eD L eDJD and eD M eD L eD JD L eJQDJD R Jence0 1 ,M," L 1 T Q ;" HeDJD ' eJQDJD R I RX
55 Bhe quantity eDJD ' eJQDJD R is related to a function -nown as the hyper/olic sine H)inh, pronounced KshineKI, which is defined as0 )inhH<I L HeD ' eJDIM% Jence0 1 ,M," L H33.3I 1 T Q % ;" )inhH< ' <"I RX
Bhe function eD can /e e<panded as an infinite series0 <% eD L 1 T < T %p T 2p <2 T 3p <3 T #p <# T .............
where an e<clamation mar- indicates a factorial number, the factorials /eing defined as0 1actorial "p 1p %p 2p 3p np HnT1Ip !alue 1 1 %S1 2S%S1 3S2S%S1 nHn'1IHn'%IS . . . . . . S1 HnT1ISnp
6t follows that the series for eJD is0 <% e L1'<T %p %<2 %)inhH<I L %< T 2p or <2 )inhH<I L < T 2p T #p <# T 5p <5 T 7p <7 T......... T #p
<2 ' 2p T
<3 ' 3p %<#
<# T ............. #p %<5 T 5p T 7p %<7 T.........
and that /y su/tracting one series from the other we can o/tain a series for eD'eJDL%)inhH<I
=ow notice that when the magnitude of < is somewhat less than 1, the magnitudes of the terms in which < is raised to a high power /ecome very small, and so we can ma-e the appro<imation0 )inhH<I o < when c<c f 1, and )inhH<I k < when c<c ff 1 H . o . means .appro<imately equal to.G . ff . means .much less than. I. )u/stituting this into equation H33.3I we get0 1 ,M," o 1 T Q % ;" H< ' <"I RX which is a >orentAian with wL1MH%;"I. Jence the electrical resonance curve is >orentAian when c< ' <"c ff 1. Bhe electrical resonance curve is, of course, an electromagnetic resonance curveG and li-e any spectral line, is >orentAian when the ; of the resonance is reasona/ly large.
34. 1aDi/+/ power transfer
)o far we have considered generators to /e sources of constant 4*) voltage. 6n reality, in the a/sence of a control system to -eep it constant, the output voltage of a generator will droop as the output current is increased. Bhis means that the generator has an internal impedance, which is somehow distri/uted throughout its wiring and component parts, /ut which will /e seen from outside as though there is a single impedance in series with an otherwise perfect generator. Bhis impedance is -nown as the source impedance or the generator.s output impedance, and must often /e ta-en into account when carrying out circuit analysis. 6n particular, it is necessary to include the source impedance e<plicitly when determining the characteristics of the parallel resonator /andpass filterG /ut there are various connotations relating to power transmission in general which must /e addressed. Bhe /asic matter is that of the effect that the load impedance has on the amount of power delivered to the load, and is encapsulated in a set of relationships -nown as the maximum power-transfer theorem. 1or the special case of a generator with a purely resistive output impedance and a purely resistive load, we can o/tain the ma<imum power'transfer condition using a graphical method. Bhe circuit to /e considered is shown /elow, where 4g is the generator output resistance, and 4 is the load. ! is the off'load generator voltage, i.e., it is the voltage which will /e seen at the generator terminals when the load 4 is disconnected. 6t should also /e o/vious /y inspection that no power is delivered when 4L" Hshort'circuitI, and also that no power is delivered when 4 is disconnected Hi.e., when 4kaI. Jence we e<pect a pea- in power output at some intermediate value of 4, and we can o/tain this value in relation to 4g /y determining the relationship /etween , and 4 and plotting it as a graph. 6n the circuit on the right, the power delivered to the load is0 , L 6X 4 Where0 6 L ! M H4 T 4gI Jence0 , L !X 4 M H4 T 4gIX 34.1 Bhis function is plotted /elow, for constant !, and shows that ma<imum power output occurs when 4L4g.
Bhe result is, of course, well -nown, /ut it is /y no means the whole story, and its interpretation is su/Fect to various common misconceptions. We can settle all of these issues /y deriving the complete ma<imum power transfer condition Hsee /o< /elowI. Bhis requires the use of calculus, which will not /e e<plained here, /ut those unfamiliar with the technique may still avail themselves of the result.
The /aDi/+/ power transfer theore/: 6n the circuit shown on the right, the power , delivered to the load : is0 , L c6cX 4 where c6c L c8c M cH: T :gIc Jence0 , L c8cX 4 M cH: T :gIcX L c8cX 4 M cH4T4g TGQ\T\gRIcX , L c8cX 4 M QH4 T 4gIX T H\ T \gIXR Bhere are two ma<imum power transfer conditions to /e o/tained here, one /eing the value of load reactance, and the other /eing the value of load resistance. 1or changes in either of these varia/les, there will /e a pea- in the graph of power versus the varia/le, and the pea- will of course occur at the point where the gradient of the curve is Aero. Jence, for the reactance condition, ma<imum power transfer occurs when q,Mq\L", and for the resistance condition, ma<imum power transmission occurs where q,Mq4L" Hwhere q is -nown as Kpartial dK or Kcurly dK and indicates a partial differentialG i.e., differentiation of one varia/le with respect to another is carried out with all other varia/les held constantI. 6n order to carry out these differentiations on the e<pression a/ove, we can use the quotient rule0 6f y L =MD then dyMd< L HDd=Md< ' =dDMd<IMDX Jence if we let = L c8cX 4 and D L H4T4gIXTH\T\gIX L 4XT4gXT%44gT\XT\gXT%\\g then0 q=Mq\L", qDMq\L%\T%\g, q=Mq4Lc8cX, and qDMq4L%4T%4g. Jence0 q,Mq\ L Q" ' c8cX4H%\T%\gIR M DX L '%c8cX4H\T\gI M DX therefore0 q,Mq\L" when \L'\g Hma<imum power transfer occurs when the power factor is 1I and q,Mq4 L _ c8cXQH4T4gIXTH\T\gIXR ' c8cX4H%4T%4gI `M DX L c8cXQ H\T\gIX T 4X T 4gX T %44g '%4X '%44g R M DX L c8cXQ 4gX T H\T\gIX '4X R M DX therefore0 q,Mq4L" when 4gXTH\T\gIX'4XL" , i.e., q,Mq4L" when 4LNQ4gXTH\T\gIXR =otice that this latter ma<imum power transfer condition is a magnitude0 it is the same asG 4 L c4g TGH\T\gIc i.e., ma<imum power transfer occurs when the load resistance is equal to the magnitude of the impedance formed /y the source resistance and the total reactance. Bhis also means that if the source impedance is purely resistive, then ma<imum power transfer occurs when the magnitude of the load impedance is equal to the source resistance. (/serve also that when the unity power' factor condition \L'\g is satisfied, the H\T\gI term disappears and ma<imum power transfer occurs when 4L4g . Bhus the overall ma<imum power transfer condition occurs when 4L4g and \L'\g, i.e, when : L :g Bhe condition o/tained when the load impedance is the comple< conFugate of the source impedance is -nown as a conFugate match.
:" We can address the most common misconception regarding impedance matching /y stating that, although unity power'factor H\L'\gI is always desira/le, is it is not necessary and not always desira/le that the load resistance should /e equal to the source resistance. Bhe reason can /e understood /y considering the poor generator, which must dissipate power in its internal resistance, and will therefore get hot. 6f we assume that power'factor correction will normally /e carried out, then there is no need to consider the reactances in the system, and we can analyse the power dissipated in the generator using the case where /oth the source impedance and the load are purely resistive. Bhus0 ,g L 6X 4g where the current is, as defined earlier0 6L!MH4T4gI. Jence0 ,g L !X 4g M H4T4gIX 34.* We will plot this function shortlyG /ut when doing so it will /e interesting to use the comparison /etween the load power and the power wasted in the generator as a measure of the power transmission efficiency. We can define efficiency as0 Bransmission efficiency L ,ower delivered M Botal power generated and here we will give it the sym/ol r [email protected]
lower case .eta.I. Bhus0 r L , M H, T ,gI =ow, su/stituting the definitions of , H34.1I and ,g H34.*I into this e<pression we get0 r L !X4MH4 T4gIX M _ Q!X4MH4 T4gIXR T Q!X4gMH4 T4gIXR ` i.e., r L 4 M H4 T 4gI )hown plotted /elow for comparison are0 ,, the power delivered to the loadG ,g , the power dissipated as heat in the generatorG ,T,g , the total power generatedG and r , the ratio of power delivered to power generated.
Bhe ta/ulated results /elow show the various power levels as a proportion of the ma<imum delivera/le power ,/aD , and are applica/le to any power'factor corrected generator'load system.
(oad 4 M 4g " 1M18 1M: 1M3 1M% 1 % 3 : 18
Total Power H,T,gI M ,/aD .................... 3."" ................... 2.58 .................. 2.#8 ................ 2.%" ............. %.85 .......... %."" ...... 1.22 .... ".:" .. ".33 . ".%8
Power (oss ,g M ,/aD .................... 3."" .................. 2.#3 ................ 2.18 ............. %.#8 ......... 1.5: ...... 1."" .. ".33 . ".18 "."# "."1
(oad Power , M ,/aD "."" . ".%% .. ".3" ... ".83 .... ".:7 ..... 1."" .... ".:7 ... ".83 .. ".3" . ".%%
(oad power M dB 'a '8.## '3."2 '1.73 '".#1 "."" '".#1 '1.73 '3."2 '8.##
"fficiency 4 M H4T4gI "."" . "."8 . ".11 .. ".%" ... ".22 ..... ".#" ....... ".85 ........ ".:" ......... ".:7 ......... ".73
=otice in the graph a/ove that as the load resistance 4 is increased and /ecomes greater than source resistance 4g , the power delivered to the load tails off gently. Bhe reason for this /ehaviour is that, as the current drawn from the generator reduces, the output voltage increasesG and so the system possesses a self'regulating property when lightly loaded. When the load is twice the source resistance, the power delivered is still :7W of the ma<imum possi/le, a droop in output of only ".#1dB. Bhe maFor advantage of light loading however is seen in the transfer efficiency. When a conFugate match is achieved, the efficiency is only #"W, /ut it rises to 85W H%M2I when 4L%4g , and :"W H3M#I when 4L34g . Bhis means that light loading, when compared to conFugate matching, gives a reduction in generator dissipation and power input for a given power output. 6n radio practice, of course, the generator is a radio transmitterG and if the transmitter is designed for light loading it can have smaller heat'sin-s and reduced /attery or mains power consumption in comparison to a transmitter designed for conFugate matching. Consequently, the figure often referred to as the Koutput impedanceK of a radio transmitter Hoften #"hI is usually nothing of the sort, it is instead Hand should /e calledI the preferred load-resistance, or alternatively the design load resistance. Bhe preferred load resistance of a /road/and transistor power amplifier is usually higher than the output impedance, and attempting to provide such an amplifier with a conFugate match will result in e<cessive internal dissipation, overheating, and possi/ly catastrophic failure. 1ortunately, most modern amplifiers are provided with protection circuitry to prevent over' dissipation, and this circuitry gives the transmitter a loading characteristic which ma-es it appear that the source resistance is higher that it really is. Bhis loading characteristic will /e different from the power transfer curve derived a/ove /ecause it is caused /y the action of non'linear circuit elements Hlevel detectors etc.I, and so the load resistance which corresponds to the middle of the permitted operation window is -nown as the pseudo output-impedance Hor, if you li-e, the pseudo source-resistanceI. Bhe diagram /elow shows what the power transfer curve might loo- li-e with the operation window centred on twice the source resistance Hunprotected transfer'function shown dottedI.
=otice that the protection circuitry also operates when the load resistance is higher than the preferred value. Bhis is not usually necessary for the protection push'pull transistor power amplifiers Hthe most common type of output stage in modern practiceIG /ut it helps to ensure that any harmonic suppression filter after the amplifier will function correctly, and it occurs /ecause the load impedance is traditionally detected using a /ridge circuit Hoften called a reflectometer or )W4 /ridge, /ut really an impedance /ridgeI /alanced for a particular value of resistance. An interesting discussion of the conditions which provo-e transistor failure is given /y Bo/ ,earson15. 6f the protection circuitry is correctly designed and adFusted, the pseudo output'impedance should /e the same as the preferred load'resistance. When determining the effect of source impedance on the ; of antenna systems and /andpass filters however, it is the true output-impedance, not the preferred load'resistance which must /e used. Enfortunately, this quantity is often impossi/le to o/tain from the manufacturer.s dataG /ut, as we shall see shortly, it can /e measured with the aid of two dummy load resistors of different value.
35. The potential di!ider.
Bhe potential divider is the simplest three-terminal electrical networ-. We have made some use of its properties already, /ut there comes a point when it is useful to characterise it formally. Jere we will do so for the general case, which is that of defining the voltage at the intersection of two impedances. 4eferring to the diagram0 8o+t L 6 :1 where, since 8inL 6H:1T:%I 0 6 L 8in M H:1 T :%I hence 8o+t L 8in :1 M H:1 T :%I 35.1 and if we multiply the right'hand side /y :%M:%0 8o+t L 8in H:1 MM :%I M :% 35.* =ote that :% is the sum of the source impedance and any additional impedance placed in series with the generator. 8in is the off'load generator voltage.
15 CHow )ig is a )ad 2#0&C Bo/ ,earson, @31JE, 4ad Com, *arch 1772, p83'8#, April 1772, p8%'82. Bhe greatest danger for push'pull transistor amplifiers lies in low load resistance which, for a given value of )W4, is more pernicious than residual reactance. )W4 is a poor matching criterion /ecause it does not indicate whether the magnitude of the load impedance is too high HharmlessI or too low HharmfulI.
:2 6f the impedances are pure resistances, the formula a/ove reverts to0 !o+t L !in 41 M H41 T 4%I 35.3 where 41 is the resistance across which !out is said to appear. Alternatively, multiplying /y 4%M4%0 !o+t L !in H41 MM 4%I M 4% 35.4
37. '+tp+t i/pedance of potential di!ider
Bhe output impedance of a networ- is defined as that impedance which, when placed in series with a hypothetical perfect generator, accounts for the drop in output voltage which occurs when a load is connected. )hown /elow is a representation of a potential divider networ- loaded with an impedance :>. 6f the load is removed, the output voltage is 8o , /ut when the load is connected, the output drops to a new voltage 8o. Hthe single inverted comma is pronounced KprimeKI. Bhis situation is modelled on the right as a perfect generator with an output 8o in series with an impedance :o , the latter /eing the output impedance we wish to define. Esing the definitions given in the diagram0 :o L H 8o ' 8o. I M 6> where0 6> L 8o. M :> Jence, com/ining these two equations0 :o L :> H 8o ' 8o. I M 8o. L :> Q H 8o M 8o. I ' 1 R . . . . . . H37.3I 1rom equation H35.*I given a/ove0 8o L 8 H:1 MM :%I M :% and /y considering :> as part of the potential divider itself0 8o. L 8 H:1 MM :> MM :%I M :% Jence 8o M 8o. L H:1 MM :%I M H:1 MM :> MM :%I L Q H1M:1I T H1M:>I T H1M:%I R M Q H1M:1I T H1M:%I R L 1 T H1M:>I M Q H1M:1I T H1M:%I R 8o M 8o. L 1 T H:1 MM :%I M :> )u/stituting this into H37.3I gives0 8o L D> _ 1 T Q H:1 MM :%I M D> R ' 1 ` i.e.0 :o L :1 MM :% Bhe output impedance of a potential divider is the parallel com/ination of the component impedances. =ote that the output impedance of the main generator is part of :%. 6f however, as is often the case, this output impedance is small in comparison to the total :% , then it can /e neglected.
3>. ThS!enin;s Theore/.
Bhe current in any impedance :, connected to a networ- consisting of any num/er of impedances and generators, is the same as though the impedance were connected to a single generator having an output impedance equal to the impedance seen loo-ing /ac- into the networ- when all generators are replaced /y their output impedances, and an output voltage equal to the voltage which appears at the terminals when : is disconnected. BhCvenin.s theorem Hpronounced KBae'ven'inKI arises from the o/servation that the output impedance of a generator is effectively in parallel with its load, and so the output impedance of an active networ- is its impedance when all of its generators are replaced /y their internal impedances. Bhis is entirely logical when we thin- of the generator as an ideal generator in series with an impedance, /ecause the ideal generator part of the model is a short'circuit with regard to any power reflected /ac- into the networ-. BhCvenin.s theorem simply ta-es this o/servation to its logical conclusion /y allowing that the networ- can /e represented as a single generator, which may /e characterised completely /y -nowing its output impedance and its off'load voltage Hand its output spectrum, /ut in a linear networ- we can treat each frequency component separately and so need only consider sine wavesI. =otice that in the preceding section we could have o/tained the output impedance of the potential divider directly /y using BhCvenin.s theoremG i.e., with the generator replaced /y a short' circuit, it is o/vious that the impedance loo-ing /ac- into the networ- is :1MM:%. Bhis is an e<tremely useful tric-G /ut even more useful analytically is the technique of replacing a complicated networ- with a single generator and a series impedance Hi.e., the output impedanceI. Bhe replacement networ- is -nown as the .BhCvenin equivalent..
3?. 1eas+ring so+rce resistance
6n the test setup shown /elowG the output voltage of a generator is measured with two different load resistances, all other varia/les /eing -ept constant, and the circuit /eing constructed in such a way as to minimise stray capacitance and inductance Hi.e., using very short wiresI. 6t is assumed that the source impedance is purely resistive, this /eing reasona/le in the case of a transistor 41 amplifier, /ut very unreasona/le in the case of a tuned valve Htu/eI 41 amplifier. 6n order to avoid interference from any protection circuitry, the test should /e carried out at a low power level Hf1"W of ma<imum outputI. Bhe voltmeter should /e capa/le of measurement at the generator frequency and should have a high input resistance. An oscilloscope with a high' impedance pro/e is suita/le, /ut ordinary multimeters do not wor- at radio frequencies. (nly the voltage ratio needs to /e determined accurately, the a/solute voltages are immaterial. >et the output voltages /e !1 when 41 is connected, and !% when 4% is connected. Bhe source and load resistances form a potential divider. Jence Husing equation 35.3I0 !1 L ! 41 M H4g T 41I and !% L ! 4% M H4g T 4%I 4earranging /oth of these e<pressions to get ! on its own and then equating them gives0 ! L !1H4g T 41IM41 L !%H4g T 4%IM4% 4%!1H4g T 41I L 41!%H4g T 4%I 4g H4%!1 ' 41!%I L 414% H!% ' !1I
:# 4g L 414%H!% ' !1I M H4%!1 ' 41!%I 6f HsayI !1 is factored out of the numerator and denominator, a form is o/tained which ma-es it clear that only the voltage ratio is needed0 4g L 41 4%H Q!%M!1R ' 1I M H4% ' 41!%M!1I 3?.1 A respecta/le difference /etween the two load resistors is necessary in order to minimise the effect of measurement errors, /ut too large a deviation from the preferred load resistance is li-ely to provo-e a transistor power'amplifier.s protection circuitry. 1or a transmitter designed to operate into a #"h loadG %# and 1""h dummy'load resistors correspond to the upper and lower %01 )W4 points and should give an easily discerni/le output voltage difference. A %#h resistor can /e had /y connecting two #"h dummy load resistors in parallel with a coa<ial B'piece. 1""h coa<ial resistors are less readily availa/le, /ut an old'fashioned 5#h load will do instead. 6n the calculation, the actual resistance of the load measured with an accurate resistance meter should /e used Hrather than the nominal value stamped on the resistorI. "Da/ple0 Bhe output voltage of a Kenwood B)32") 1""W J1 transmitter was measured with two different dummy loads. Bhe measurement frequency was 1.7*JA, and the test power level was very appro<imately 1W. (ne load was a 5#h nominal coa<ial resistor measuring 5#.1Y".5h, the other was the com/ination of this resistor and a #"h nominal coa<ial resistor in parallel with it, the com/ination measuring %7.8Y".2h. Bhe voltage ratio was measured using an oscilloscope with a 1"*h S1" pro/e. Bhe resistors and the pro/e were attached directly to the antenna soc-et using coa<ial B'pieces Hno ca/lesI. Bhe measurement was made /y attaching and removing the #"h resistor from the B'piece with the transmitter running and noting the change in the pea- to peae<cursion of the output waveform. Esing the following designations0 4%L5#.1h, 41L%7.8h, the voltage ratio !%M!1 was 1.283 Y"."3. Esing equation H3?.1I, the source resistance 4g was calculated to /e %2.2h. An error analysis Hsee ne<t sectionI, gave an estimated standard deviation of 2.3hG i.e., 4gL%2.2Y2.3h. =ote incidentally, that this determination assumes that the output impedance does not change with power output level. @iven that power transistors are non'linear devices, this may not /e the case.
While it would /e inappropriate here to delve too deeply into the su/Fect of scientific data analysisG the reader should nevertheless /e aware that all physical measurements are meaningless unless they have some -ind of error'window or confidence interval associated with them. Bhis is not a serious pro/lem when ta-ing a reading with say, a multimeter, /ecause Hassuming that the instrument has /een cali/ratedI, the manual will say what the measurement accuracy is. A digital multimeter, for e<ample, might have a quoted accuracy of Y".:W Y1digit Hi.e., Y1 in the last decimal placeI for its resistance ranges, so if we o/tain a resistance reading from this instrument of HsayI 5#.1h, the actual measurement will have a confidence interval of Y".8 Y".1, i.e., the reading should /e recorded as 5#.1Y".5h. )cientists and engineers normally equate error /oundaries stated in this way with the estimated standard deviation H+)DI of the measurementG where, on the assumption that errors are scattered randomly according to a .normal. or @aussian distri/ution, a standard deviation represents a region where we have a 8:W confidence that the true result will lie. Bhe standard deviation is usually given the sym/ol s [email protected]
lower case .sigma.I, and so if we o/tain a measurement <Ys, we have 8:W confidence that the true answer lies /etween <'s and <Ts. 1rom the properties of the @aussian error distri/ution also, we have a 7#.#W confidence that the true answer lies /etween <'%s and <T%s, and a 77.5W confidence that the true answer lies /etween <'2s
:8 and <T2s Qref.1:R. Bhe use of standard deviations rather than ./ric- wall. tolerances reflects the reality that there is always a finite pro/a/ility that the true result will lie outside the stated error range. We can only ever have absolute confidence that the magnitude of the true answer lies somewhere /etween Aero and infinity, /ut we e<pect only 2 measurements in every 1""" to fall outside <Y2s. 6t is always advisa/le to try to write down an +)D for every measurement made. Bhis is a reasona/ly straightforward matter where direct measurements are involved, /ut a difficulty arises in situations where several measurements are made and then put into a formula in order to o/tain the required result. Bhe pro/lem is that of wor-ing out how much influence the deviation of a particular varia/le has on the overall result, and how to add the various deviations together in order to arrive at the overall +)D. 6t is therefore fortuitous that we have /een engaged in the study of vectors, /ecause it turns out that this is a pro/lem of vector addition and magnitudes. 6f two or more measurements are made in such a way that the outcome of one has no influence on the outcome of any of the others, the measurement errors are said to /e uncorrelated. An e<ample of uncorrelated errors is that of readings ta-en from two separate instruments, where an error or inaccuracy in the reading of one instrument is not related to any error or inaccuracy in the reading of the other. (n the other hand, the errors in two measurements made using the same instrument may /e correlated, in the sense that if the instrument always reads too high or too low, it will introduce errors in the same direction in /oth cases. 6f measurement errors are correlated, then it means that there is some systematic Hdesign, interpretation, or cali/rationI defect in the measuring processG /ut if we /elieve that the measurements have /een made to the /est of our a/ilities with the equipment availa/le, then it is usually sensi/le to assume that any measurement errors are uncorrelated. =ow, if the errors in two or more measurements are uncorrelated, this means that a deviation from the true value in one measured quantity can occur without influencing the deviations in any of the other quantities. 6f we determine a quantity /y applying a formula to a set of measurements, each measurement will contri/ute a random error to the result, /ut there is Fust as much chance that the error due to one measurement will partly cancel the error due to another as there is that a pair of error contri/utions will /oth increase or decrease the result. Bherefore it will /e unduly pessimistic to add the uncertainty contri/utions of the individual measurements directly. 6nstead, we should allow for the independence of the uncertainty contri/utions /y regarding each one as a vector pointing in a direction which is at right angles HorthogonalI to all of the others. 6n effect, /y virtue of its randomness, each uncertainty contri/ution e<ists in its own dimension, and we may identify its magnitude as its length in that dimension. 6t follows that the overall uncertainty is the length Hi.e., the magnitudeI of the vector which results from the addition of a set of orthogonal uncertainty vectors. Bhis situation is represented in the diagram /elow, where E1, E%, and E2 are the uncertainty contri/utions to the determined value of an un-nown, and E is the overall uncertainty in the result. We can easily find E /y successive application of ,ythagoras. theorem, as follows0 >et the magnitude of the vector sum of $1 and $% /e E1%0 E1% L NH E1X T E%X I Bhen E is the magnitude of the vector sum of $1% and $20 E L NH E1%X T E2X I /ut E1%X L E1X T E%X Jence0 E L NH E1X T E%X T E2X I
ata 0ed+ction and "rror 3nalysis for the Physical 2ciences, ,hilip 4 Bevington. *[email protected]
'Jill, 1787. >i/rary of Congress cat. card t 87'1873%. %'%0 )ample mean and standard deviation. Area under the @aussian distri/ution0 Ba/le C'%, p2":.
:5 Bhis process can /e e<tended to find the magnitude of a vector in an ar/itrary num/er of dimensions Hwe can.t ma-e perspective drawings in more than three dimensions, /ut there is no restriction on the num/er of dimensions that a vector can haveI. Jence0 E L NH E1X T E%X T E2X T . . . . T EnX I =ow note that this formula says0 Kto find the overall uncertaintyG calculate the sum of the squares of the uncertainty contributions and ta-e the square root.K Bhe uncertainty contri/utions are not the same as the uncertainties in the measurements made. 6magine that an un-nown quantity < is given /y a formula f, which is a mathematical function involving measura/le quantities Hvaria/lesI m1, m%, m2, etc. We can e<press this situation /y writing0 < L fHm1, m%, m2, ...I and we can determine < /y plugging m1, m%, m2, etc. into the formula. We can also determine the uncertainty contri/ution due to any one of the varia/les /y changing it and noting the change which occurs in <. Bhe o/vious amount /y which to change the varia/le is its standard deviation, hence0 <Ts<1 L fHm1Ts1, m%, m2, ...I Jere we have assumed that a positive change in m1 will cause a positive change in <. Bhis might not /e the case, /ut since we intend to add the contri/utions from changes in each of the varia/les as orthogonal vectors, it ma-es no difference either way. =ow, restoring m1 to its original value we determine the uncertainty contri/ution due to m%. <Ts<% L fHm1, m%Ts%, m2, ...I and so on. 6f we wor- through all of the varia/les in this way and determine their error contri/utions, we can o/tain an estimate of the standard deviation of < /y summing the squares of the contri/utions and ta-ing the square root0 s L NH s<1X T s<%X T s<2X T . . . . T s<nX I =ote that there are a num/er of assumptions inherent in this procedure0 firstly, as discussed /efore, that the uncertainties are uncorrelatedG and secondly that we have assumed that the function f is linear for changes in any of the varia/les. Bhe latter condition is normally true to a good appro<imation for small changes, and the effect of any non'linearity is mitigated /y the fact that the o/Fect of the e<ercise is to o/tain an estimate. "Da/ple: Bhe output resistance 4g of an 41 amplifier was determined /y loading the output with two different resistances and noting the change in the output voltage with all other conditions held constant. Bhe applica/le formula is equation H3?.1I0 4g L 41 4% H=! ' 1I M H4% ' 41 =!I Where =! is the ratio of the output voltages0 =! L !%M!1 Bhe voltage measurements were made using an oscilloscope, and it was considered that each measurement had an uncertainty of a/out %W. 6t was also considered that these uncertainties were uncorrelated /ecause they were incurred /y different operationsG one operation /eing to set the transmitter carrier level and oscilloscope b'shift until the waveform Fust touched the top and /ottom of the measuring graticule with the higher value resistor connected, the other /eing to read the height on the graticule with the lower value resistor connected. Bhe overall uncertainty of the voltage ratio measurement was therefore ta-en to /e the square root of the sum of the squares of the two voltage measurementsG i.e., NH%XT%XIL%.:W, which was rounded to 2W in view of the appro<imate nature of the estimate. Bhe actual voltage ratio was 1.283, and 2W of 1.283 is "."3. Jence0 =! L 1.283 Y"."3 Bhe resistances were measured using a multimeter -nown to read correctly within Y".1h against a standard resistance of 1""."h. Bhe stated accuracy of the instrument was Y".:W Y1 digit. Bhe
:: measured resistances were 41L%7.8h and 4%L5#.1h. Jence0 41L%7.8Y".23h 4%L5#.1Y".5h Bhe output impedance 4g was calculated from the formula H3?.1I using a spreadsheet program and determined to /e %2.2h. Bhe output impedance was also calculated with each of the measured values individually incremented and decremented /y an amount equal to its estimated standard deviation, and the resulting deviation in 4g was noted. Bhe spreadsheet H0gL/eas.odsI is shown /elow0
=ote that the formula is somewhat non'linear in its /ehaviour /ecause the deviations caused /y incrementing and decrementing a varia/le are not e<actly equal and opposite. Bhe correct way to allow for this effect is to ta-e the average of the deviation magnitude H4*)I for each case. Bherefore, the estimated standard deviation in 4g is0 s L NH ".#57X T ".%#2X T 2.2#7X I L 2.31: Jence0 4g L %2.2 Y2.3 h =otice that the maFor contri/utor to the uncertainty in 4g in this case is the uncertainty in =!. We cannot ignore the effect of the resistance uncertainties however, /ecause if we repeat the e<periment with more closely spaced values for 41 and 4% , we will find that their contri/utions to the uncertainty increase dramatically. 3nalytical approach to error analysis: While the error analysis technique Fust descri/ed is perfectly respecta/leG those who write computer programs will generally prefer an analytical approach. Bhe derivation of an error function from a formula requires the use of calculus. Bhose who are unfamiliar with calculus may proceed to the ne<t section without losing trac- of the narrative. Bhe analytical form of an error function is o/tained from the o/servation that an error in a varia/le is transmitted through a formula according to the rate of change of the formula with respect to the varia/le. Bhus the error contri/ution from a varia/le is the partial derivative of the formula with respect to the varia/le multiplied /y the deviation in the varia/le. Jence if < L fHm1, m%, m2, ...I and the +)Ds of the measured quantities are s1, s%, s2, etc.G the contri/ution which the varia/le m1 ma-es to the +)D of < is given /y0 sD1 L H qfMqm1I s1 and so on Hstrictly we should ta-e the modulus of the derivative /ecause standard deviations are /y definition positive, /ut it does not matter in this case /ecause orthogonal addition involves squaring
:7 of the error contri/utionsI. Jence the analytical form of the error function is0 s L N_ QHqfMqm1Is1RX T QHqfMqm%Is%RX T QHqfMqm2Is2RX T .... ` "Da/ple0 Bhe output impedance of a generator is o/tained from the formula0 4g L 41 4% H=! ' 1I M H4% ' 41 =!I Differentiation of this function requires the use of the quotient rule0 6f y L =MD then dyMd< L HDd=Md< ' =dDMd<IMDX where, in this case, the numerator is0 = L 414%H=! ' 1I L 414%=! ' 414% and the denominator is0 D L 4% ' 41=! Differentiating the numerator with respect to each of the varia/les gives0 q=Mq41 L 4%H=! ' 1I , q=Mq4% L 41H=! ' 1I , q=Mq=! L 414% and differentiating the denominator with respect to each of the varia/les gives0 qDMq41 L '=! , qDMq4% L 1 , qDMq=! L '41 Esing these results we o/tain0 q4gMq41 L Q DHq=Mq41I ' =HqDMq41I R M DX L Q H4% ' 41=!I 4%H=! ' 1I ' 414%H=! ' 1IH'=!I R M H4% ' 41=!IX L Q H4% ' 41=!I 4%H=! ' 1I T 414%H=! ' 1I=! R M H4% ' 41=!IX L Q 4%X=! ' 4%X ' 414%=!X T 414%=! T 414%=!X ' 414%=! R M H4% ' 41=!IX q4gMq41 L 4%XH=! ' 1I M H4% ' 41=!IX q4gMq4% L Q DHq=Mq4%I ' =HqDMq4%I R M DX L Q H4% ' 41=!I41H=! ' 1I ' 414%H=! ' 1I R M H4% ' 41=!IX L Q 414%=! ' 414% ' 41X=!X T 41X=! ' 414%=! T 414% R M H4% ' 41=!IX q4gMq4% L 41X=!H1 ' =! I M H4% ' 41=!IX q4gMq=! L Q DHq=Mq=!I ' =HqDMq=!I R M DX L Q H4% ' 41=!I414% ' 414%H=! ' 1IH'41I R M H4% ' 41=!IX L Q 414%X ' 41X4%=! T 41X4%=! ' 41X4% R M H4% ' 41=!IX q4gMq=! L 414%H4% ' 41I M H4% ' 41=!IX Bhe error function in this case is0 s L N_ QHq4gMq41Is41RX T QHq4gMq4%Is4%RX T QHq4gMq=!Is=vRX ` Bhe derivatives all share a common denominator DX, and so on writing the e<pression in full, a factor H1MDXIX can /e removed from the square root /rac-et. Jence0 s L Q1MH4% ' 41=!IXR N_ Q4%XH=!'1Is41RX T Q41X=!H1'=! Is4%RX T Q414%H4%'41Is=vRX ` 6n the previous section, we determined 4g L %2.2h from the following measurements0 41L%7.8Y".23h , 4%L5#.1Y".5h , =! L 1.283Y"."3 Bhese give0 DX L H4% ' 41=!IX L 1%"#.:852 q4gMq41 L 4%XH=! ' 1I M DX L %"#%.7828 M 1%"#.:852 L 1.5"%# q4gMq4% L 41X=!H1 ' =! I M DX L '32#."1"" M 1%"#.:852 L '".28"5 q4gMq=! L 414%H4% ' 41I M DX L 1"1133.8: M 1%"#.:852 L :2.:551 s L N_ QHq4gMq41I S s41RX T QHq4gMq4%I S s4%RX T QHq4gMq=!I S s=vRX ` L N_ Q1.5"%# S ".23RX T Q".28"5 S ".5RX T QH:2.:551 S "."3RX ` L N_ ".#5:7X T ".%#%#X T 2.2##1X `
7" =ote that the error contri/utions in the e<pression a/ove are very close to the averages of the deviations calculated /y the incremental HspreadsheetI method used previously. 1inally we have0 s L 2.313 4g L %2.2 Y2.3h A spreadsheet version of this calculation Hwhich can /e used as a templateI is given on sheet % of the accompanying file0 0gL/eas.ods .
4-. 3ntenna syste/ A
6n previous sections we showed that conFugate matching is not necessarily a good idea, and that radio transmitter manufacturers do not necessarily design power amplifiers to wor- into a conFugate load. Besides the o/vious advantages in terms of output regulation and efficiency howeverG there is a further reason to load a radio transmitter lightly in cases where the input impedance of an antenna system changes rapidly with frequency or is su/Fect to varia/le environmental factors. 4ecall from section 34 that the true ma<imum power transfer condition occurs when0 4LNQ4gXTH\T\gIXR Consequently, if the antenna system is su/Fect to distur/ances which can cause a reactive component to appear after matching has /een carried out, then the /est average ma<imum power output will /e o/tained when the load resistance is somewhat higher than the source resistance. ,ossi/le causes of transient residual reactance are many and various, including0 changing physical environment Hof mo/ile and porta/le transmittersI, wind, rain, component heating, /irds, etc., and Has is easily forgottenI modulation. 6n section @, we discussed an electrically'short inductively'loaded vertical antenna system. Bhe equivalent circuit of this antenna is shown on the rightG with one e<tra resistance added, that /eing the HtrueI source resistance 4g . With this additional piece of information, it /ecomes possi/le to calculate the ;, and hence the /andwidth, of this system. 6f we ta-e the same e<ample component values as were used /efore, we have0 :> L 4> T G\> L 5.# TG2""" :a L 4a T 4r TG\a L %.# 'G2""" giving an input resistance of 1"h for a whip length of a/out "."5V. Bhe whole system is of course, a series resonator, and we can define the circuit ; as0 ;" L c\>cM4total L c\acM4total )ince /andwidth is proportional to f", pro/lems of e<cessive ; are li-ely to occur at low operating frequencies, so let us see what happens if this antenna is /uilt to operate on HsayI 1.7*JA, with the generator source resistance adFusted to /e #h. Bhis will ma-e the total series resistance 1#h, and with \>L2"""h, the ; will /e 2"""M1#L%"". Bhe '2dB /andwidth of the antenna will therefore /e f"M;L7.#KJA. Bhis is wide enough to accommodate a communications ))B signal H%.5KJA /andwidthI /ut there is very little latitude for incidental detuning, and the antenna will e<hi/it a small variation of input impedance depending on the modulation frequency. >ight loading of the generator will help to offset these pro/lems, /ecause it will create a situation where transient detuning forces the generator'load system closer to its ma<imum power transfer point Halthough detuning won.t increase the amount of power delivered, light loading will give /etter regulation than a conFugate matchI. =ote incidentally, that the antenna discussed a/ove is not physically small when designed for operation in the 18"m /and. Bhe wavelength at 1.7*JA is cMf L1#5.:mG and so a "."5V rod will /e 11m long, and consequently far too large for mo/ile use. Bo ma-e a mo/ile antenna, the rod must
71 /e shortenedG and this will reduce the antenna capacitance Hand hence increase the reactanceI, and sadly for efficiency, will cause the radiation resistance to fall. Bhe larger antenna reactance will necessitate a larger loading reactance, and although this will /ring more resistance with it, the increase in reactance will /e greater than the increase in total resistance and the ; will rise. A point can /e reached where serious curtailment of the modulation /andwidth occurs, although, for this system, it it not predicta/le using lumped'component theory. Bhe coil can /e regarded as a lumped component provided that the whip is long enough to ensure that most of the radiation occurs from the whip rather than from the coil. 6f that condition applies, then the ; of the antenna system can never /e larger than the ; of the loading coil /ecause the total series resistance will always /e that of the coil plus a little e<tra. Bhe ma<imum tolera/le ; Hcausing some, /ut not serious, audio degradationI occurs when the antenna system /andwidth is the same as the audio /andwidth, and for ))B on 1.7*JA this figure is 17""M%.5L5"3. 6t is e<tremely difficult to ma-e a lumped inductor with a ; of greater than a/out 3"", so on 18"m the ; limit can /e avoided /y controlling the length of the coil. 6f, on the other hand, the whip is of length compara/le to or shorter than the coil, then most of the radiation occurs from the coil. 6n that case, the lumped component description fails completely, and the system is /est descri/ed as a quarter'wave transmission'line resonator. 6n the transmission'line regime, the ; and hence the voltage magnification can /ecome enormous, and the usa/le input power is limited /y the tendency for the air around the top of the coil to ionise and /ecome electrically conductive. Coils operated at or slightly /elow the quarter'wave transmission line resonance frequency are used for artificial lightning e<periments, in which conte<t they are -nown as .Besla coils.. 6n particular, the voltage'magnifier coil connected in series with the output from a step'up transformer is -nown as the .+<tra Coil.. Bhe transmission'line properties of coils will /e discussed in a separate article. While on the su/Fect of *1 and J1 mo/ile antennasG when forced to use a very short whip, it is possi/le to increase the antenna capacitance artificially Hand hence reduce the reactanceI /y adding a capacitance hat to the antenna Hsome prongs stic-ing'out sideways symmetricallyG or, if there.s a ris- that you might po-e someone.s eye out, an aluminium dis- I. 4educing the antenna reactance in this way reduces the amount of loading inductance required, and hence allows the coil to /e wound with thic-er wire for a given siAe Hless resistanceI. ,lacing the hat at the top of the antenna moreover, increases the current in the vertical section, and actually increases the radiation resistance slightly Hevery little helpsI.
41. )asic i/pedance transfor/er
6n the previous section it was implied that the generator output impedance could /e adFusted, /ut we have yet to offer any method for doing so. Bhere are numerous options in this respect, /ut for the present purpose it will /e sufficient to have Fust one0 the transformer. Bransformers are discussed in detail in a separate article17G /ut here we will avail ourselves of the properties of straightforward H/ut unfortunately mythicalI circuit models -nown as the perfect transformer and the ideal transformer. An ideal transformer has no losses, and its voltage ratio is the same as its turns ratio. 6n truth, well'designed transformers can have power'transfer efficiencies of more than 7:W within a certain /and of frequencies, and so the myth of the ideal transformer is not so far from reality. A perfect transformer is an ideal transformer which has large winding reactances, so that the off'load input current is negligi/leG which means that its current ratio is the inverse of its voltage ratio. Jere
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7% we will assume that a tightly'coupled transformer is ideal when operating within its pass'/and, on the understanding that it requires a more advanced analysis to determine what the pass'/and is. )uch a transformer is also appro<imately perfect when used as part of a low'impedance electrical networ-. A transformer loaded with an impedance : is represented on the right. Jere =, is the num/er of turns in the primary Hgenerator sideI winding, and =) is the num/er of turns in the secondary Hload sideI winding. Bhe dots ne<t to the windings indicate either the start or the finish Hit doesn.t matter how this is designated, as long as it is done consistentlyI, and it it assumed that /oth coils are wound in the same sense Hcloc-wise or anticloc-wise when loo-ing at a particular end of the coilI. Bhe dotted line /etween the coils indicates that the transformer is wound on a magnetic core, the purpose of which Hin this instanceI is to produce a very tight magnetic coupling /etween the windings. 6f all of the magnetic field from the primary winding is captured /y the core and lin-ed to the secondary winding Hi.e., if there is no magnetic lea-ageI, and if the coils and the core have no heating losses, then all of the power delivered /y the generator is transferred to the load. Also, if the inductive reactance of the windings is much larger than the magnitudes of the impedances seen on either side, and the capacitance of /oth windings is very small, then the secondary voltage will /e in phase with the primary voltage, and the secondary current will /e in anti'phase with the primary current Hi.e., as a current appears to flow into the primary, a current appears to flow out of the secondaryI. 6f the num/er of turns in the secondary winding is greater than the num/er of turns in the primary, then 82 will /e larger than 8, , and vice versaG and the voltage transformation will /e in proportion to the turns ratio, i.e., 8) L 8, =) M =, . . . . H41.1I 6t follows that if the power produced /y the generator is transferred to the load without loss, then the 86 product will /e conservedG which means that if the voltage is stepped up, then the current will /e stepped down to -eep 86 very nearly constant Hand vice versaI. Bhis implies that the transformer performs on the current the inverse of the transformation it performs on the voltage, i.e. Hinterpreting the currents in the sense of the arrows in the diagram a/oveI0 6s L 6, =, M =) . . . . H41.*I =ow, /y definition, the impedance loo-ing into the transformer primary is0 :. L 8 , M 6 , which gives, using H41.1I and H41.*I as su/stitutions0 :. L 8) H=,M=)I M Q 6) H=)M=,I R and since : L 8)M6) 0 :. L : H=,M=)IX 41.3 Bhus, to a reasona/ly good appro<imationG a tightly'coupled transformer having relatively large winding reactances scales an impedance according to the square of the turns ratio. =ow let us consider the pro/lem in reverse, and see what a transformer does to the output impedance of a generator. Jere we will call the apparent source impedance as seen from the secondary side of the transformer :g., with :g as the actual generator output impedance. Bhe relationship /etween :g. and :g is perhaps guessa/leG /ut to derive it mathematically requires a tric-, which is that of defining an equivalent circuit with all of the source resistance moved to the secondary side of the transformer. A suita/le approach to the derivation is then to write e<pressions for the voltage 8 across the load using /oth the original and the equivalent circuits and then equate the two e<pressions.
1or the left'hand circuit, let us define :. as the load impedance seen /y the generator, its relationship to to the load : /eing given /y equation H41.3I a/ove0 :. L : H=,M=)IX Bhe voltage 8. is then the output of a potential divider formed /y :g in series with :., i.e.0 8. L 8g :. M H:g T :.I and 8. is related to the load voltage 8 /y the turns ratio, ie.0 8 L 8. =) M =, Jence0 8 L H=)M=,I 8g :. M H:g T :.I 8 L H=)M=,I 8g Q1 T H:. M :gI R . . . . H41.4I 1or the right hand circuit, 8 is the output of a potential divider formed /y :g. and : 0 8 L 8g. : M H:g. T :I 8 L 8g. Q1 T H: M :g.I R where0 8g. L H=)M=,I 8g Jence0 8 L H=)M=,I 8g Q1 T H: M :g.I R +quating this to e<pression H41.4I gives0 1 T H:. M :gI L 1 T H: M :g.I i.e., :g. L :g : M :. /ut, /y rearrangement of equation H41.4I, : M :. L H=)M=,IX, hence0 :g. L :g H=)M=,IX 41.5 Bhus a tightly'coupled output transformer scales the source impedance according to the square of the turns ratio, a generator with a low output impedance /eing converted into a generator with a high output impedance /y means of a step'up H=)e=,I transformer and vice versa. Bhe /road/and output transformer of a fairly typical 1""W short'wave radio transmitter HKenwood B)32"sI is shown on the right. Bhe transformer core is a /loc- of ferrite with two hollow channels passing through it H-nown colloquially as a Kpig noseKI. Bhe primary winding consists of two short lengths of copper or /rass tu/ing passing through the core and connected together at one end /y a strip of copper'laminate /oard. Bhe secondary winding is a length of ,B1+'coated multi'strand silver'plated copper wire threaded through the copper tu/es Hthe reason for the choice of materials is e<plained in another article%"I. Bo ma-e a complete turn around the core, a conductor must pass through one hole and /ac- through the other. As shown /elow diagrammatically, the copper tu/es form a centre'
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73 tapped single turn, with the DC power supply HBTI connected to the centre tap, and the other ends connected to the collectors of the 41 power transistors Ha matched pair of %)C%%7"sI. Bhe transformer in the photograph has four turns and so increases the amplifier output impedance /y a factor of 18. Bhere is something more to the use of an output transformer than impedance transformation however, the principal issue /eing that the transmitter discussed a/ove uses a 12.:! power supply and yet must deliver 1""W into a #"h load. Bhe required output power and target load impedance defines the output voltage as !LNH,4ILNH1""S#"IL5".5! 4*), i.e., 5".5S%N%L%""! pea-'to'pea- Hp'pI. A simplified version of the power amplifier circuit is shown /elow, and we can deduce the minimum allowa/le transformer step'up ratio /y e<amining it.
Bhis is a so'called push-pull amplifier circuit, in which one transistor provides the positive half' cycle of the output waveform, and the other transistor provides the negative half'cycle. When a /ipolar transistor is turned hard on, its collector voltage does not go to Aero, /ut stops at some saturation voltage, which is usually around 1!. Also, it is not a good idea to drive the transistors close to saturation /ecause this will lead to considera/le distortion of the output waveform. Bherefore we must assume that the output stage can produce positive and negative half cycles of no more than a/out 1%.#! across half of the primary winding, i.e., %#! per transistor across the whole winding , hence #"! p'p. Bo o/tain %""! p'p H5".5! 4*)I therefore, a voltage step'up ratio of 103 is required. Bhe fact that this transformation increases the source impedance /y a factor of 18 is a secondary considerationG and is of no great concern unless the source impedance /egins to approach the design load resistance, the latter situation /eing associated with low transfer efficiency and poor load regulation as discussed earlier. 6t follows, that to -eep the output impedance as low as possi/le, a step'up ratio Fust sufficient to provide the required output voltage is optimal. Bhe actual output impedance H4g.I of the B)32"s transmitter Hmeasured at the antenna soc-et, see the e<ample at the end of section 3?I is a/out %2h Hmeasured %2.2Y2.3h at 1.7*JAI for a design load resistance of #"h. Bhe output impedance of the power amplifier H4gI is therefore appro<imately %2M18L1.3h. Dye and @ran/erg%1 give an appro<imate formula for calculating the output impedance of a transistor power amplifier /elow 1""*JA as0 4g L H!cc ' !satIX M ,o+tQ/aDR where !cc is the supply voltage, and ,o+tQ/aDR is the ma<imum power availa/le from the amplifier. 6f we assume a saturation voltage !sat of a/out 1!, this gives0 4g L 1%.:X M 1"" L 1.83h. *ultiplying this /y 18 gives 4g.L%8.%h, which is within 1s of the measurement without ta-ing any of the circuitry /etween the power amplifier and the antenna soc-et into account.
%1 0adio Fre,+ency Transistors, =orm Dye and Jelge @ran/erg. *otorola inc. M Butterworth Jeinemann, =ewton *A. 1772. 6)B= "'5#"8'7"#7'2. (utput impedance of a power amplifier0 p11:.
7# =otice incidentally, that the power amplifier is shown as feeding into a low'pass filter H>,1I /efore connection to the antenna system. )uch a filter is always necessary with a /road'/and transistor power amplifier, /ecause such amplifiers produce relatively high levels of harmonics. Bhe push'pull configuration actually cancels even harmonics, /ut there are still high levels of odd harmonics H2rd, #th, 5th etc.I which must /e removed Hin engineering, the first harmonic is the same as the fundamentalI. 6n section 34 we noted that the power amplifier protection circuitry operates when the load impedance is too high, as well as when it is too low. Bhis is not usually necessary for the protection of the amplifier, /ut the >,1 may not provide the required degree of harmonic attenuation when incorrectly terminatedG and so the protection circuitry helps to -eep spurious emissions within accepta/le limits if the load impedance is too high.
4*. 3+to transfor/ers
.Auto'transformer. Hself'transformerI is Fust another name for a tapped inductor. Bhe transformation rules for tightly'coupled auto'transformers are identical to those for tightly'coupled transformers with separate windings. Bhe significant functional difference /etween the two types of transformer is that an auto'transformer does not provide DC isolation /etween source and load. A more su/tle difference is that a transformer with separate windings, /y Fudicious use of electrostatic shielding, can /e made in such a way that the coupling /etween the primary and secondary windings is almost entirely magnetic. An auto'transformer will always e<hi/it some stray capacitive and resistive Hpotential dividerI coupling, and so if its inductance is part of a filter circuit, the filter may e<hi/it poor attenuation of signals outside its pass/and. Bhe step'up and step'down auto'transformer configurations are shown /elow. Also shown is the somewhat redundant 101 auto'transformer, otherwise -nown as an inductorG the point in including it /eing to draw attention to the inductive reactance which every transformer places in parallel with its load.
6t should /e o/vious /y inspection of the .101. auto'transformer circuit, that the voltage ' current relationship for the load seen /y the generator is given /y0 8M6 L G\> MM : 6f the coil has losses moreover, we can represent these as a resistance H4> sayI in series with the coil0 8M6 L H 4> T G\> I MM : We can also transform the impedance of the coil into its parallel form Hsee section [email protected]
, in which case the load on the generator /ecomes0 8M6 L 4>p MM G\>p MM : Bhe implication is that, unless the magnitude if the inductive reactance is very much larger than the magnitude of the load impedance, the transformer will not preserve the load phase relationship. 6f the load is reactive, the parallel loss component will also alter the load phase relationship slightly. 6n the previous section, we introduced the idea that an impedance located on one side of a transformer can /e transferred to the other side in an equivalent circuit /y the act of multiplying it /y the turns'ratio squared. )o we might represent the inductance of a transformer as a separate inductance > in parallel with the primary side of an ideal transformer Hof otherwise infinite inductanceI, or we might represent it as an inductance >. in parallel with the secondary side. Bhe
78 transformation rule H33.3I tells us that0 G%]f > L GH=)M=,IX %]f >. i.e., > L H=)M=,IX >. Bhis is a remar-a/le result /ecause, not only does it give us the /asis for constructing equivalent circuits to serve as models for real transformers, it also tells us something a/out inductors. Bhe e<pression can only /e true if the inductance of the coil is proportional to the square of the num/er of turns in it. We can see why /y considering the two 10= auto'transformer equivalent circuits shown /elow0 6n the left'hand circuit, the inductance of the transformer is referred to the primary side, and for reasons of convention is given the sym/ol A>. 6n the right'hand circuit, the inductance is referred to the secondary side and is given the sym/ol >. 1rom the foregoing discussion, we can immediately write the relationship /etween > and A>0 > L =X A> We can also interpret > as the inductance of the whole coil, and A> as the inductance of one turn of the coil. A> is -nown as the inductance factor, and depends on the physical dimensions of the coil and the nature of any magnetic core material. 6t may /e interpreted either as the inductance of a one'turn coil, or as the inductance of an auto'transformer referred across a one'turn tap. A> has the units of inductance HJenrysI, /ut is more informatively given units of inductance M turnX HKJenrys per turn' squaredKI. Contin+o+sly !ariable a+toJtransfor/er: (ne of the draw/ac-s of ferrite or iron'cored transformers as impedance matching devices is that the the transformation ratio can only /e altered in a stepwise fashion, /y changing windings or tappings one turn at a time Hor half a turn if the core has two holesI. 6f the turns in the coils are few, as tends to /e the case in radio'frequency applications, then the steps availa/le can /e very coarse indeed. 6t is however possi/le to ma-e a continuously varia/le inductor or auto'transformer /y rotating a coil a/out its a<is and tapping into it with a rolling contact, the coil end'connections /eing made /y slipping contacts H-nown, for historical reasons, as K/rushesKI. )uch a device is -nown colloquially as a Kroller coasterK, and an e<ample is shown in the photograph on the right. Bhis is the motor'driven varia/le impedance transformer from a 17#5 vintage Collins 1:">'2A automatic J1 antenna tuner. Bhe tuner is designed to match end'fed wire H*arconiI antennas of 13 to 3" metres in length over a frequency range of % to %#*JA, and is for use with transmitters with an output of up to 1#"W and a preferred load impedance of #%h. An interesting feature of the transformer is that it achieves a continuous transition from step'down to step'up /y having an overwind Hsee diagram rightI, i.e., the /rush contact at one end of the coil goes to a centre'tap, and the end of the coil is left unconnected. Bhe coil has %: turns, and the input tap is at 13 turns, so a ma<imum impedance step'up of appro<imately 301 is o/taina/le.
75 Bhe disadvantage of the Collins transformer is that the coil does not have a magnetic core. Bhe stray magnetic fields will therefore induce currents Heddy currentsI in the surrounding metalworand give rise to resistive losses. Bhe open magnetic circuit also implies that the impedance transformation o/tained will not /e e<actly proportional to the square of the turns ratio, and due to the a/sence of a magnetic core the inductance might appear on first consideration to /e rather low. Bhe inductance for the whole coil, estimated using Wheeler.s 1ormula%% is a/out %"OJ, giving only a/out #OJ when referred to the primary side. Bhis will give rise to significant phase shift at lower frequencies, the inductive reactance seen /y the transmitter at %*JA /eing something around %]S%S1"8S#S1"'8L82h. 6t transpires however, that the choice of primary reactance a/out equal to the target input impedance at the lowest operating frequency is sensi/le, /ecause in addition to the impedance transformer, the antenna tuner also has power'factor correction components. 6n the process of adFusting a reactance in series with the antenna to achieve a resistive input impedance, any phase shift due to the transformer is automatically ta-en into account. Consequently, it is possi/le to -eep the inductance small, which helps in the avoidance of self'resonance pro/lems at the high'end of the operating frequency range.
43. Prototype :J/atching networ%
An antenna matching system /ased loosely on the Collins 1:">'2 is depicted in the diagram /elow. Bhe only maFor difference is that the transition from step'down to step'up is accomplished /y means of a change'over relay. Bhis increases the transformation range in comparison to the overwind method, /ut also increases the comple<ity of the control system.
Bhis is the prototype of all antenna tuners in the sense that it approaches the impedance matching pro/lem in the simplest possi/le way. Bhe o/Fect of the e<ercise in every case is to transform the impedance in its two dimensions0 magnitude and phase, and the most direct approach is to do so using one device which only affects the magnitude and one device which only affects the phase. Bhe magnitude'correcting engine is the varia/le auto'transformer, and the phase'correcting engine is a series reactanceG relays /eing provided to insert a series coil in the event that the antenna is capacitive, or a series capacitor in the event that the antenna is inductive. )uch a matching unit can, of course, /e controlled manually, /y the e<pedient of providing it with control -no/s and switches instead of motors and relays. Bhis approach replaces the automatic control system with a human /eing, /ut ma-es no allowance for the fact that humans in general have little aptitude for the tas-. Jere we monitor the load magnitude and phase using /ridge circuits, which are the su/Fect of a separate article. Bhe /ridges produce error signals, which tell their respective control systems which way to go in event that the error e<ceeds a certain preset
%% )olenoids. D W Knight Qwww.g2ynh.infoMR
7: threshold. =ot shown on the diagram, /ut necessary to ma-e the system wor-, are limit switches, two for each varia/le device. Bhese tell the control system when a motorised device has hit one of its end'stops0 so that the change'over relay can /e switched and the motor direction reversed in the case of the impedance transformerG so that the switch'over from coil to capacitor or vice versa can /e made in the case of the series reactance networ-G and as protection against motor /urn'out in the event that the load is outside the matching range. Bhe control systems for magnitude and phase are shown as /eing completely separateG which they are e<cept in respect of common signals, such as the request for a tuning carrier or an instruction to reduce power, which they might send to the transmitter on detecting a matching error. Bhe independence of the two systems is possi/le /ecause the two chosen matching criteria are independent, i.e., the two matching processes can proceed simultaneously without altering the outcome. Bhe system can even adFust itself when presented with a speech ))B signal, /ut will reach a solution fastest when the error signals are continuously availa/le. (ne desira/le property of this matching system, and of any properly designed matching system, is that it corrects for the defects of its own components. 6n this case, when the phase control system adds series inductance Hfor e<ampleI, the increasing resistance of the coil will increase the impedance magnitude seen at the input, /ut the magnitude control system will simply /ac-'off to compensate. )imilarly, the inductance of the impedance transformer will cause a positive phase shift, /ut phase control system will /ac-'off in the capacitive direction to compensate. While the simple magnitude'phase matching system is entirely practical however, it has never /een particularly popular. Bhe reason is that it is difficult to design an efficient and resonance'free varia/le /road/and transformer. Bhe required transformations can Fust as well /e o/tained using only varia/le capacitors and inductors, and this su/Fect is e<amined in detail in a separate article%2.
44. 3d/ittance, cond+ctance, s+sceptance
Bhe linear circuit analysis technique demonstrated so far consists of /rea-ing the circuit down into two'terminal networ-s and treating those networ-s as impedances. Bhis approach has allowed us to attac- a wide range of pro/lemsG /ut it results in e<tremely messy alge/ra when impedances in parallel are involved. Eltimately, we need a way of dealing with ar/itrarily large num/ers of impedances in parallel, Fust as we can already deal with any num/er of impedances in seriesG and it transpires that this can /e achieved /y defining the properties of our component two'terminal networ-s not in terms of impedance, /ut in terms of the reciprocal of impedance, this /eing called admittance. By so doing, we move the pro/lem out of what we so'far thin- of as its natural space Himpedance spaceI and into what is -nown as its reciprocal spaceG and the re'definition, trivial though it is in the case of phasors, is -nown as a reciprocal-space transformation. Bhe reciprocal space transformation is another mathematical invention of $ames Cler- *a<well. 6ts most far'reaching application is in the field of \'ray crystallography, it /eing the means /y which the \'ray diffraction patterns of crystals are traced /ac- to the internal arrangement of atoms. Jere however, we need only a simplified version, /ecause the pro/lems we wish to solve are strictly two'dimensional. Bhe reciprocal of impedance space is -nown as admittance space. A pair of two'dimensional reciprocal spaces has the property that straight lines in one appear as circles in the other Ha correspondence which is used e<tensively in the article K6mpedance *atchingK, cited earlierIG /ut the real power of the transformation lies in the fact that phasor pro/lems requiring the dou/le'slash product in one space, /ecome pro/lems of addition in the other.
%2 6mpedance matching. D W Knight. Qwww.g2ynh.infoMR
77 Converting an impedance into an admittance is simply a matter of ta-ing the reciprocal. Admittance is usually given the sym/ol T Hand here we put it in /old /ecause it is comple<I, hence0 TL1M:. =ow, if :L4TG\ this gives0 T L 1MH 4 TG\ I which can /e put into the aTG/ form /y multiplying the numerator and denominator /y the comple< conFugate of the denominator, i.e.0 4 ' G\ TL H4 T G\I H 4 ' G\I hence0 4 ' G\ TL H4X T \XI Bhis e<pression can /e written0 T L @ T GB where the real part of the admittance, @, is called the conductance, and the imaginary part, B, is called the susceptance Hof the networ- under considerationI. 1rom the a/ove, we can e<tract definitions for conductance and susceptance which are0 Cond+ctance, 2+sceptance, @ L 4 M H4X T \XI B L '\ M H4X T \XI
=ow o/serve that when the impedance of a networ- is purely resistive, the conductance is 1M4, and @L1M4 is the definition of conductance in DC electrical theory. When an impedance is purely reactive, the susceptance BL '1M\ Hsusceptance has no DC counterpartI. Admittance, conductance, and susceptance, of course, have unitsG and the modern unit in this case is the "iemens, which is given the dimension sym/ol capital ) Has opposed to the second, which has a small sI. 6n old te<t/oo-s and papers, the unit of admittance is often given as the .*ho. H(hm spelt /ac-wardsI, /ut in either case, the actual dimensions are in reciprocal (hms, i.e., Mh or h'1. A pure resistance of #"h therefore corresponds to a conductance of 1M#" )iemens, i.e., %" milli')iemens or %"m). A pure reactance \L1""h corresponds to a susceptance BL1"m), and so on. K)iemensK, incidentally, is a name, li-e K$onesK. Bhe singular of $ones is not $one, and so )iemens -eeps its final s in /oth singular and plural forms Hone )iemens, several )iemensI. Bhe plural K)iemensesK is not recommended H/ut is a lot less em/arrassing than the quasi'singular K)iemenKI. Bhe dou/le'slash product was previously defined as H1>.5I0 a MM b L abMHaTbI We can demonstrate that addition is the reciprocal'space counterpart of the dou/le slash operator /y transforming the parallel impedance formulaG i.e., if0 : L :1 :% M H:1 T :%I then T L H:1 T :%I M :1 :%. 6f we let T1L1M:1 and T%L1M:%, then T L T1 T% Q H1MT1I T H1MT%I R Which rearranges to0
1"" T L T1 T T% i.e., when two networks are placed in parallel) their admittances are added. 4ecall that the formula for resistances in parallel, 4L414%MH41T4%I is a rearrangement of the e<pression0 1M4 L H1M41I T H1M4%I 6t should now /e apparent, that what the formula really says is0 @ L @1 T @ % Bhe formula for impedances in parallel is of course a rearrangement of0 1M: L H1M:1I T H1M:%I and this e<pression can /e e<tended to cover any num/er of impedances in parallel /y adding more terms, i.e.0 1M: L H1M:1I T H1M:%I T H1M:2I T . . . . . T H1M:nI Bhis is a sum of admittances, and may /e re'written as0 T L T1 T T% T T2 T . . . . . T Tn We can e<press this result using the dou/le slash notation0 1MH :1 MM :% MM :2 MM . . . MM :n I L T1 T T% T T2 T . . . . . T Tn where T% L 1M:% H- /eing any su/scriptI. Bhe admittance representation of an electrical circuit is no less authoritative than the impedance representation, and is no more difficult to use. Admittances are phasors, and all of the phasor techniques we have developed in this chapter will wor- on them. 6t is however, helpful to remem/er that a HnumericallyI large admittance corresponds to a small impedance and vice versa. 0eciprocalJspace co+nterparts 6/pedance space 6mpedance : L 4TG\ L 1MT 4esistance 4 L @[email protected]
4eactance \ L '[email protected]
,ure resistance 4 L [email protected]
,ure reactance \ L '1MB 6nductive reactance \> L %]f> Capacitive reactance \C L '1MH%]fCI MM operator T operator )traight line Circle
3d/ittance space Admittance T L @TGB L 1M: Conductance @ L 4MH4XT\XI )usceptance B L '\MH4XT\XI ,ure conductance @ L 1M4 ,ure susceptance B L '1M\ 6nductive susceptance B> L '1MH%]f>I Capacitive susceptance BC L %]fC T operator MM operator Circle )traight line
1"1 (ne further issue of which the reader will need to /e aware is that two different definitions of admittance appear in the electrical and electronic literature. )ome authors He.g., Jartshorn%3I use [email protected]
, as is done hereG and others He.g., >angford')mith%#I use [email protected]
'GB. Bhe .alternative. definition gives BL\MH4XT\XI, and thus B>L1MH%]f>I, and BCL '%]fC. 6n the ne<t section, we analyse the parallel resonator /andpass filter and determine the relationship /etween resonant frequency, /andwidth and ;. 6n the author.s first attempt at the derivation, the definition [email protected]
'GB was used, and the formula which resulted had it that either ; is negative or f" is negative. Bhe change to [email protected]
fi<ed the pro/lem and so, since ; is /y definition positive when loss resistance is positive, the other definition is wrong according to the convention that frequency is positive. We may also note a reflection symmetry in the correct choice, in that we have \>L%]f> in impedance space, and BCL%]fC in admittance space, etc.G i.e., inductive reactance and capacitive susceptance are positive, capacitive reactance and inductive susceptance are negative.
45. Parallel resonator )PF
Bhe reader may have noticed that, having determined the relationship /etween /andwidth and ; for a series resonator, we did not immediately do the same for a parallel resonator, /ut instead digressed into the su/Fects of source impedance and impedance transformation. Bhere was a very good reason for doing so, as we shall soon see, which is that there is no satisfactory design procedure for parallel'resonant /andpass filters if the source and load impedances cannot /e controlled. Bhis situation prevails /ecause, in order to use the resonator as a filter, we need methods for inFecting energy into it and e<tracting energy from it, and the impedances presented /y these input and output networ-s affect the ;. Bhe prototype /and'pass filter is shown on the right. Bhe generator and load coupling scheme used is not the only one possi/le, /ut all other schemes are equivalent to this one after suita/le transformation. Jere we inFect energy via a source resistance 4), which is the sum of the generator output resistance and any additional resistance placed in series with it. 4, is the parallel com/ination of the resonator dynamic resistance and any load resistance which might /e placed across it. =otice that we have provided the model with source and load resistances rather than impedances. We are at li/erty to do so without affecting the generality of the analysis, /ecause any reactive components in the source and load impedances will turn out to /e effectively in parallel with the resonator. Bhis means that these additional reactances will modify the effective values of \Cp and \>p Hi.e., they will change the resonant frequencyI, /ut they will not affect the general circuit /ehaviour provided that they do not e<hi/it any self'resonances in the analysis frequency range. 6f we define 8" as the output voltage at resonance, then the /andwidth function is c8M8"c plotted against frequency. We can write e<pressions for 8 and 8" /y treating the circuit as a potential divider, thus, noting that \Cp MM \>p ka at resonance Hi.e., when f L f"I we get0 8" L 8g 4, M H 4)T 4, I and if we choose the generator voltage as our phase reference we can drop dimensions0
%3 4adio'1requency *easurements /y Bridge and 4esonance *ethods, >. Jartshorn Q,rincipal )cientific (fficer, British =ational ,hysical >a/oratoryR, Chapman 9 Jall, 173" H!ol. \ of K*onographs on +lectrical +ngineeringK, ed. J , boungI. 2rd imp. 173%. Ch. 6, section 20 Defines Admittance as [email protected]
, hence B>L'1M^> and BCL^C. %# 0adio esigner;s Handboo%, +d. 1ritA >angford')mith. 3th edition. 3th impression Hwith addendaI, 6liffe ,u/l. 17#5 QA later reprint e<ists H1785I 6)B= " 5#"8 282#1R. )ection 3.8HvI, p1#20 Defines inductive susceptance as positive and capacitive susceptance as negative, hence [email protected]
'GB. Bhis is contrary to the more convincing derivation given /y Jartshorn Ha/oveI.
1"% !" L !g 4, M H 4) T 4, I We will also avail ourselves of a useful property of the potential divider formula H35.4I, which is that if we multiply it /y a unit quantity consisting of the source resistance divided /y itself Hi.e., 4)M4) I, it /ecomes a dou/le'slash product0 !" L !g H 4, MM 4) I M 4) )imilarly, for the output voltage in general0 8 L !g H 4, MM G\Cp MM G\>p I M Q 4) T H 4, MM G\Cp MM G\>p I R and using the associative rule H1>.4I0 8 L !g H 4) MM 4, MM G\Cp MM G\>p I M 4) )o we can write the ratio 8M!" as0 8M!" L H 4) MM 4, MM G\Cp MM G\>p I M H 4, MM 4) I Bhe /andwidth function is the magnitude of this e<pressionG /ut with all of the components represented as impedances, anyone attempting to e<pand and simplify it, or isolate part of it as the load, will have a hard time -eeping trac- of all of the intermediate terms. We will therefore convert it into an admittance pro/lem, using the relationship0 1MH :1 MM :% MM :2 MM . . . MM :n I L T1 T T% T T2 T . . . . . T Tn Jence0 1 M H @) T @, T GBCp T GB>p I 8M!" L 1 M H @, T @) I Where @ stands for conductance and B for susceptance, and @)L1M4) , @,L1M4, , BCpL'1M\Cp and B>pL'1M\>p. Bhe e<pression a/ove can /e re'written0 @) T @ , 8M!" L @) T @, T GBCp T GB>p and the magnitude is0 NQ [email protected]
) T @,IX R c8c M !" L NQ [email protected]
) T @,IX T HBCp T B>pIX R i.e., @) T @ , c8c M !" L NQ [email protected]
) T @,IX T HBCp T B>pIX R Bhis can /e plotted against frequency /y su/stituting BCpL%]fC, and B>pL'1MH%]f>,I, /ut we will not /other to do so here /ecause it is identical in appearance to the graph of c6cM6" for a series resonator given in section *@. We will instead go on to determine the half'power points /y noting that, whatever proportion of the parallel resistance 4, is designated as the load, power will always /e delivered to it in proportion to c8cX, so the half'power points occur when c8cL!"MN%. Jence, at the half'power points we have0
1"2 @) T @ , L 1MN% NQ [email protected]
) T @,IX T HBCp T B>pIX R Which, upon squaring gives0 [email protected]
) T @,IX L m [email protected]
) T @,IX T HBCp T B>pIX and upon inversion gives0 HBCp T B>pIX T1L% [email protected]
) T @,IX i.e.0 HBCp T B>pIX M [email protected]
) T @,IX L 1 and ta-ing the square root0 HBCp T B>pI M [email protected]
) T @,I L Y1 Bhus0 B>p T BCp L [email protected]
) T @,I and if we define the sum @)[email protected]
, as @; Hi.e., the conductance which determines the ;I0 B>p T BCp L [email protected]
; =ow, using the su/stitutions BCpL%]fCp and B>pL'1MH%]f>pI, we o/tain0 Q '1MH%]f>pI R T %]fCp L [email protected]
; and /y factoring out 1MH%]f>pI from the left hand side and re'arranging0 Y%]f>[email protected]
; L '1 T H%]fIX>pCp i.e., Q3]X>pCpRfX YQ%]>[email protected]
;Rf '1 L " Bhis is a quadratic equation in f with aL3]X>pCp , /LY%]>[email protected]
; , and cL'1. 6t has four solutions as was the case for the series resonator Hsection *@I, these /eing the upper and lower /andwidth limits for positive and negative frequencies. Bo solve it we apply the standard formula0 f L Q'/ YNH/X ' 3acI R M %a Jence0 f L _ Y%]>[email protected]
; YNQH%]>[email protected]
;IX T 3S3]X>pCpR ` M H%S3]X>pCpI and using the su/stitution >pL>pXM>p to o/tain cancellation of >p from all /ut one term0 f L _ Y>[email protected]
; YNQH>[email protected]
;IX T 3H>pXM>pICpR ` M H3]>pCpI i.e.0 f L _ [email protected]
; [email protected]
;X T 3CpM>pI ` M H3]CpI =ow, since [email protected]
;X T 3CpM>pI will always /e larger than @;, we can identify the positive frequency upper /andwith limit as0 fT L _ [email protected]
;X T 3CpM>pIR T @; ` M H3]CpI and the positive frequency lower /andwidth limit as0 f' L _ [email protected]
;X T 3CpM>pIR ' @; ` M H3]CpI and the /andwidth is0 fw L fT ' f' L @;MH%]CpI Bhis is the admittance counterpart of the result o/tained at this stage in the derivation of the ; of a
1"3 series resonator Hequation *@.3I and so we will deduce that the /andwidth of the parallel resonator B,1 is f"M;", and use this deduction to find a definition for ;. f"M;" L @;MH%]CpI %]f"Cp L BCp" L ;" @; ;" L BCp" M @; =ow let 4;[email protected]
;, where 4; is Kthe resistance which determines the ; K. Also o/serve that BCp"L '1M\Cp" , and at resonance '\Cp"L\>p" . Jence0 ;" L '4; M \Cp" L 4; M \>p" or, in -eeping with the definition of resonant ; given in section 310 ;" L 4; M N(>pMCpI 45.1 4; is simply the parallel com/ination of the source resistance, the load resistance, and the dynamic resistance of the resonator, i.e.0 4; L 4) MM 4p" MM 4>oad Bhis result gives us the theoretical information we need in order to /e a/le to design parallel resonant /andpasss filters. 1irstly, we may o/serve that the source and load impedances are effectively in parallel with the resonator, which is why any minor source and load reactances can /e lumped with the resonator reactances and cause only a detuning effect Hif such reactances are very large however, they will cause a significant change in the dynamic resistance and the pro/lem is /est re'analysed from scratchI. Bhe source, load, and dynamic resistances however, are critical in determining the ;, and we need to o/tain high values for all of them in order to o/tain a high ;. We can of course adFust the source and load resistances using transformersG and as we shall see shortly, we can replace the resonator coil with a transformer so that the inductor and the transformer /ecome one and the same. Before we loo- at such coupling schemes however, we must draw attention to a particularly misleading inference of the formula, which is that high ; can /e o/tained /y ma-ing the ratio >pMCp as small as possi/le. Bhis suggestion has appeared in at least one amateur radio pu/lication, /ut it is a fallacy. 6f the reactive components are of reasona/le quality, the parallel form >MC ratio H>pMCp I is only slightly different from the series form >MC ratio, and as we showed in section *1, imaginary resonance can occur if the >MC ratio /ecomes too low. Bhe imaginary resonance condition is entirely a function of the series HlossI resistances of the coil and the capacitor. 6t is nothing to do with the source and load resistances /ecause avoidance of imaginary resonance is a matter of ensuring that the T7"Z component of the coil current at resonance is sufficiently large to cancel the '7"Z component of the capacitor current Hor vice versa, /ut in practice coils are more lossy than capacitorsI. Consequently, the design procedure for a parallel resonator B,1 is to ma-e the >MC ratio large enough to o/tain a good strong resonance Hwithout ma-ing the inductance so large that stray capacitance and coil self'capacitance prevent the target ma<imum frequency from /eing reachedI, and then to ma-e 4p" even larger H/y minimising loss resistancesI in order to o/tain a useful wor-ing ;.
47. $nloaded A of parallel resonator
Bhe e<pression for ; o/tained in the previous section is sometimes referred to as the loaded * of the resonator, /ecause it is the ; which results when the source and load impedances are ta-en into account. We may also imagine that the resonator has an unloaded *, which is that which o/tains when the source and load are disconnected. 6t is not immediately o/vious why we should wish to employ such a concept, /ecause it is impossi/le to use the resonator without coupling to it in some wayG /ut it is nevertheless useful /ecause it sets an upper limit on the ; which can /e o/tained in a practical circuit. 6t is o/viously o/tained /y su/stituting the dynamic resistance in place of 4; in
1"# equation H45.1I, i.e., ;"u L 4p" M NH>pMCpI /ut it would /e a lot more useful intuitively if we could e<press it in terms of the coil and capacitor impedances in their series H4TG\I forms. We can do so /y using the series'to'parallel transformation Hsection [email protected]
and using the definition of ; from section 31 as precedent, we e<pect a result in the form0 ;"u L QNH>MCIR M 4 . . . H47.1I the point /eing to find out what is meant /y 4 in this case. Bhe translation from parallel to series form is indicated in the set of equivalent circuits shown /elow0
Jere we identify 4p" as 4CpMM4>p, i.e.0 4p" L 4Cp 4>p M H 4Cp T 4>p I and an e<pansion in terms of the series forms of the impedances has already /een given as equation H*-.5I0 H4CX T \CXI H4>X T \>XI 4p" L 4>H4CX T \CXI T 4CH4>X T \>XI Bhe unloaded ; is defined as0 ;"u L 4p" M NH'\Cp \>pI and, from the series'to'parallel transformation Hequation [email protected]
, we have0 \Cp L H4CX T \CXI M \C . . . H47.*I and \>p L H4>X T \>XI M \> . . . H47.3I ,utting all of this together we have0 H4CX T \CXI H4>X T \>XI M Q 4>H4CX T \CXI T 4CH4>X T \>XI R ;"u L NQ 'H4CX T \CXI H4>X T \>XI M \C \> R and noting that NH'\C \> ILNH>MCI, this rearranges to0 QNH>MCIR _NQ H4CX T \CXI H4>X T \>XI R ` ;"u L Q 4>H4CX T \CXI T 4CH4>X T \>XI R )o, at this point we have e<tracted NH>MCI as required /y equation H47.1I, and the resistance /y which NH>MCI must /e divided in order to o/tain ;"u is0
[ 4 4 \ 4 4 \ ]
> % C % C C % > % >
Which, upon e<panding the numerator gives0 4>XH4CXT\CXIX 4X L H4CXT\CXIH4>XT\>XI T H4CXT\CXIH4>XT\>XI 4CXH4>XT\>XIX T H4CXT\CXIH4>XT\>XI %4>4CH4CXT\CXIH4>XT\>XI
Bhe simplification we require here comes from noting that the terms H4CXT\CXI and H4>XT\>XI occur in the e<pressions for \Cp and \>p given a/ove Hequations 47.* and 47.3I, and that at resonance '\CpL\>p. Jence0 H4CX T \CXI M H'\CI L H4>X T \>XI M \> i.e.0 H4CX T \CXI M H4>X T \>XI L '\C M \> and H4>X T \>XI M H4CX T \CXI L \> M'\C Jence0 4X L 4>XH'\CM\>I T 4CXH\>M'\CI T %4>4C which can /e factorised0 4X L _ 4>QNH'\CM\>IR T 4CQNH\>M'\CIR `X Jence0 4 L 4>QNH'\CM\>IR T 4CQNH\>M'\CIR . . . H47.5I Hstrictly Y4, /ut resistance is positive, allowing us to ignore the negative solutionI and so0 ;"u L QNH>MCIR M _ 4>QNH'\CM\>IR T 4CQNH\>M'\CIR ` Bhis is an e<act solution provided that 4> and 4C do not varyG and once again may /e assumed e<act for normal engineering purposes /ecause 4> and 4C will not vary significantly in the vicinity of the resonant frequency. =ote however that \>L'\C to an e<tremely good appro<imation when the >MC ratio is reasona/ly large, and this relationship is e<act when 4>L4C. Jence, for most practical purposes0 ;"u L QNH>MCIR M H4> T 4CI 47.7 Which means that the unloaded ; of the parallel resonator is the same as that of the series resonator, it is the square root of the >MC ratio divided /y the total series resistance. 6n other words, we can estimate the unloaded ; of the parallel resonator /y considering it to /e a series resonator connected as a loop.
4>. C+rrent /agnification
Bhere is another way to determine the unloaded ; of a parallel resonator, which stems from the o/servation, that Fust as a series resonator e<hi/its the phenomenon of voltage magnification, the parallel resonator e<hi/its current magnification. 6n effect, the parallel resonator is a series resonator connected in a different way, /ecause its characteristics at resonance are principally determined /y a large circulating current, and the current it draws from the generator is small in comparison H; times smaller than the circulating current in factI. 6n the diagram on the right, the current 6 flowing into the resonator is 6>T6C, where0 6C L 8 MH4C TG\CI L 8H4C 'G\CIMH4CXT\CXI and
1"5 6> L 8 MH4> TG\>I L 8H4> 'G\>IMH4>XT\>XI /ut at reasonance 6 is real, which means that the imaginary parts of 6> and 6C add up to Aero, and the total current at resonance /ecomes0 u x 4C 4> v v 6" L ! T v v H4CXT\CXI H4>XT\>XI w y H! and 6" are now in phase and will /e treated as realI. ,utting the e<pression onto a common denominator yields0 u x 4 H4 XT\>XI T 4>H4CXT\CXI v C > v 6" L ! 4>.1 v v H4CXT\CXI H4>XT\>XI w y where the term inside the square /rac-ets is the reciprocal of the dynamic resistance Hsee equation *-.5I, i.e., 6"L!M4p" . =ow, the current circulating in the resonator can /e determined from either /ranch as the total current flowing in the /ranch, less the current drawn from the generator. Jence the circulating current is simply the imaginary part of the current in the /ranch. Bhe resonant condition Hand the concept of circulationI also implies that the circulating current is of the same magnitude for /oth /ranches, /ut of opposite sign. Jence, if we call the circulating current 6;, then Hta-ing the imaginary parts of the e<pressions for 6C and 6> a/oveI we have0 6; L G!\>MH4>XT\>XI L 'G!\CMH4CXT\CXI and the magnitudes are0 c6;c L !\>MH4>XT\>XI L !H'\CIMH4CXT\CXI We can also create a definition involving /oth /ranches /y ta-ing the geometric mean0 c6;c L ! N_ H'\C\>I M Q H4CXT\CXI H4>XT\>XI R ` which allows us to e<tract the >MC ratio0 c6;c L ! N_ H>MCI M QH4CXT\CXI H4>XT\>XIR ` . . . . H4>.*I =ow, let us define the unloaded ; of the resonator as the ratio of the circulating current to the through current0 ;"u L c6;c M 6" Which can /e e<panded using equations H4>.1I and H4>.*I0 u x H>MCI M QH4CXT\CXI H4>XT\>XIR v v ;"u L v v Q4CH4>XT\>XIT4>H4CXT\CXIRX M QH4CXT\CXIH4>XT\>XIRX w y and rearranged0 u x H4CXT\CXIH4>XT\>XI v v ;"u L QNH>MCIR v v Q 4CH4>XT\>XIT4>H4CXT\CXI RX w y Bhe rightmost square'root /rac-et is simply the reciprocal of 4 as defined in equation H47.4IG hence0 ;"u L QNH>MCIR M 4 and we have proved that the current'magnification definition for unloaded ; is identical to that o/tained on the assumption that ; is the magnitude of the resonant frequency divided /y the /andwidth of the resonator.
1": Bhe only residual issue is that of why the e<act e<pression for 4 is Has given /y equation 47.5I0 4 L 4>QNH'\CM\>IRT4CQNH\>M'\CIR rather than simply 4L4>T4C. Bhis however can /e understood /y noting that the HrealI current flowing through the resonator will /e very slightly /iased in favour of the /ranch with the lowest resistance. Bhis difference is very small for practical resonators of moderate unloaded ;, and may normally /e ignored.
4?. Controlling loaded A
As determined earlier H45.1I, the ; of a parallel resonator can /e given as0 ;" L H 42rc MM 4p- MM 4(oad I M N(>pMCpI where, with a slight change from the previous notation, 42rc is the output resistance of the energy inFection networ- Hthe sourceI, 4p- is the dynamic resistance of the resonator, and 4(oad is the input resistance of the networ- to which energy is /eing delivered. Bhe inductance and capacitance are defined in their parallel forms so that the dynamic resistance can /e treated as a separate parallel element.
Bo /e continued . . . . .