ECE Formula Sheet

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Communication Systems  Amplitude Modulation :  DSB-SC : u (t) =  m(t) cos 2

Power P =

    



π      t

 

Conventioanal AM :

[1 + m(t)]mCos( )2π t . as long as |(m()t)| ≤ 1 demodulation is simple . () m |()|

u (t) = Practically m(t) = a

(t) .

Modulation index a = Power =



 +

  

 

(t) =

  ,

 

 

→ Square law Detector SNR =  ( ) ↓ a a → amplitude Sensitivity R          R       RC << 1/ω    ≥   Cos (2π  t + ∅ (t) ) ∅→(maxt) 2πphase    devi(m()a ttio)→n.d∆∅t →=           → max requency deviation ∆ =        = 2 (β + 1)       → 98% power → (SNR)     R = m(t) cos 2π      ∴  

SSB-AM : : 

 

Square law modulator    = 2 /   Envelope Detector  

Frequency u (t) =

 

C (i/p) < < 1 /  

C (o/P) >> 1/

 

 

Phase Modulation : Angle Modulation : 

 

 

 phase & frequency deviation constant

 

 max | m(t) |  max |m(t) |

Bandwidth : :  Effective Bandwidth

 

 

Noise in Analog Modulation :-  :-   =  

 =

 =

 

 =

 / 2

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 1 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo  www.facebook.com/onlin ok.com/onlineies eies  

 

Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

→ (SNR)  //      (SNR)   → (SNR)  //    (SNR)        η   η                                    → ω       ω →       → → R        ω → → →  =

 =

 =

 =

 =

 .

 =

 =

 =

 =

 =

 =

 =

 =

 

 

 

.

 =

Noise in Angle Modulation :-

 =

PCM :  Min. no of samples samples required for reconstruction reconstruction = 2  =

 Total bits required = v  Bandwidth =

/2 = v

 bps .  /

v

 ;

 = Bandwidth of msg signal .

 bits / sample

2 = v .

 

 SNR = 1.76 + 6.02 v 

 As Number of bits increased SNR increased by 6 dB/bit . Band width also increases.

Delta Modulation :-

 By increasing step size slope over load distortion eliminated [ Signal raised sharply ]

 By Reducing step size Grannualar distortion eliminated . [ Signal Signal varies slowly ]

Digital Communication

→ → →→

Matched filter:  impulse response a(t) =



 ( T –  T –  t) . P(t)



 i/p

 Matched filter o/p will be max at multiples of ‘T’ . So, sampling @ multiples of ‘T’ will give max SNR (2  point )  matched filter is always always causal causal a(t) = 0 for t < 0  Spectrum of o/p signal of matched filter with the matched signal as i/p ie, except for a delay factor ;  proportional to to energy spectral density density of i/p. nd

∅()  ∅ ∅ ∅ e   ∅() |∅()  |    e   = =

 =

(f) (f) = (f) *(f)  

 

 

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 2 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo  www.facebook.com/onlin ok.com/onlineies eies  

 

Institute Of Engineering Studies (IES,Bangalore) 



Formulae Sheet in ECE/TCE Department

 o/p signal of matched filter is proportional to shifted version of auto correlation fine of i/p signal

∅∅ RR∅∅ → |g(t)  g(t)  ddtt| ≤  g(t) dt  |g(t)|  g g   (t –   (t –  T)  T)  (0)  which proves 2nd point

(t) = (T) =

At t = T

Cauchy-Schwartz in equality :-

If

 dt     (t) then equality equality holds holds othe otherwise rwise ‘<’ holds

(t) = c

Raised Cosine pulses : 

(())   ()   | |          ≤      cos  ||     | |        ≤| | ≤    ⇒         α → rol o actor P(t) =

 .

 

P(f) =

  Bamdwidth of Raised cosine filter



 =

 

 Bit rate

 → signal time period →            →         1                → → →→   e/  

 

 For Binary PSK

 = Q

 = 2Q

 

 4 PSK

 = Q

 =  erfc

       =

 

.

 

FSK:For BPSK

 = Q

 = Q

=  erfc

 

 All signals have same energy (Const energy modulation )

 Energy & min min distance both can be kept constant while increasing no. of points . But Bandwidth Compramised.  PPM is called as Dual of FSK .  For DPSK

 =  

 

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→ → → →

Formulae Sheet in ECE/TCE Department

 Orthogonal signals require factor of ‘2’ more energy to achieve same



 as anti podal signals

 Orthogonal signals are 3 dB poorer than antipodal signals. The 3dB difference is due to distance b/w 2  points.  For non coherent FSK

  e/  =  

 

 FPSK & 4 QAM both have comparable performance .  32 QAM has 7 dB advantage over 32 PSK.

       .

  Bandwidth of Mary PSK =



  Bandwidth of Mary FSK =



  Bandwidth efficiency efficiency S =



  Symbol time



  Band rate =



  log  =

 

 

 =

 =

  ; S=

 ;S=

 

 

 .

 

 

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 4 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo  www.facebook.com/onlin ok.com/onlineies eies  

 

Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

Signals & Systems

→  |x(t)|  ddtt    |[]| lim→   |x(t)|   ddtt lim→   |x[n]| → →→ x(t) →x(t) →x → x x → ∞ →⇒ ⇒ → ∞ → → → → →→ → →→ → ⇒    h(t)ddtt   e e  [e] e        = =

  Energy of a signal

 

 

 

  = =

 Power of a signal signal P =

(t)          ;  +      +   iff

 

(t) orthogonal

(t) &

 power.    Shifting & Time scaling won’t effect power . Frequency content doesn’t effect  power.  if power =    neither energy nor power signal Power = 0  Energy signal Power = K  power signal

 Energy of power signal =

 ; Power of energy signal = 0

 Generally Periodic & random signals Aperiodic & deterministic deterministi c

 Power signals  Energy signals

Precedence rule for scaling & Shifting :

x(at + b)

  (1) shift x(t) by ‘b’  x(t + b) (2) Scale x(t + b) by ‘a’  x(at + b)

x( a ( t + b/a))

 (1) scale x(t) by a  x(at) (2) shift x(at) by b/a  x (a (t+b/a)).

 x(at +b) = y(t)

 x(t) = y

 

  Step response response s(t) = h(t) * u(t) =



 

S[n] =

   

 

 u(t) *

 u(t) =

 Rect (t / 2

)*

  S’ (t) = h(t)  h(t)  h[n] = s[n] s[n] –   –  s[n-1]  s[n-1]

 [

 Rect(t / 2

 -

] u(t) .

 ) = 2

,

 min (

) trapezoid (

,

)

  Rect (t / 2T) * Rect (t / 2T) = 2T tri(t / T)



Hilbert Transform Pairs : 

 e  //  dx σ  2π     x  e/  ddxx σ  2π    σ  =  

 ;

  =

 

   > 0

Laplace Transform :- 

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  (s)  e  x(t)  e

x(t) =

 

Formulae Sheet in ECE/TCE Department

 ds

 ds

X(s) =

Initial & Final value Theorems :   x(t) = 0 for t < 0 ; x(t) doesn’t doesn’t contain any impulses impulses /higher /higher order singularities singularities @ t =0 then then 

x(

∞ lliimm→→(()) )=

x( ) =

 

 

Properties of ROC :-

ω

1. X(s) ROC has strips parallel to j  axis 2. For rational laplace transform ROC has no poles

→ →→

3. x(t)

 finite duration & absolutely integrable then ROC  entire s-plane

4. x(t)

 Right sided then ROC right side of right most pole excluding pole s =

5. x(t) 6. x(t)

 left sided

ROC left side of left most pole excluding s= -

 two sided

ROC is a strip

∞ ∞ ∞  

7. if x(t) causal

ROC is right side of right most pole including s =

8. if x(t) stable

ROC includes j -axis

Z-transform

x[n] = X(z) =

ω

:-

  x()   x[n]    

 

 

 

 dz

 

 

Initial Value theorem : If x[n] = 0 for n < 0 then x[0] =

lim→ [] lim→(1)

Final Value theorem :  = =

lim→ ()

 

 X(z)  X(z)

Properties of ROC :- 

1.ROC is a ring or disc centered @ origin 2. DTFT of x[n] converter if and only if ROC includes unit ci circle rcle 3. ROC cannot contain any poles

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 6 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo  www.facebook.com/onlin ok.com/onlineies eies  

 

Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

→→

4. if x[n] is of finite duration then ROC is enter Z-plane except poss possibly ibly 0 or 5. if x[n] right sided then ROC  outside of outermost outermost pole excluding excluding z = 0 6. if x[n] left sided then ROC  inside of innermost pole including z =0 7. if x[n] & sided then ROC is ring 8. ROC must be connected region 9.For causal LTI system ROC is outside of outer most pole including   10.For 10. For Anti Causal system ROC is inside of inner most pole including ‘0’  ‘0’  



 



11. System said to be stable if ROC includes unit circle . 12. Stable & Causal if all poles inside unit circle 13. Stable & Anti causal if all poles outside unit circle. Phase Delay & Group Delay :-  When a modulated signal is fixed through a communication channel , there are two different delays to be considered.

∅ ω  ∅ ω ↓ω ↓   ∅() 

(i) Phase delay:  Signal fixed @ o/p lags the fixed signal by (

 ∅()  = -

  where (

) phase

) = K H(j )

  Frequency response of channel   for narrow Band signal signal

 =

Group delay

  Signal delay / Envelope delay

Probability & Random Process:-

→→ →→

 P (A/B) =

(()) (())

 ⇒ 

 

 Two events A & B said to be mutually exclusive /Disjoint if P(A  B) =0  Two events A & B said to be independent independent if P (A/B) = P(A)  P(A  B) = P(A) P(B)  P(Ai / B) =

 =

  ()       ()  

 

 ≤  ∞∞ ≤ ∞  x≤ ≤ x  x  x ≤ 

CDF :-  Cumulative Distribution function

(x) = P { X

 x   x }

Properties of CDF :

   ( ) = P { X    } = 1    (- ) = 0    (  X    ) =  ( ) -  ( )   Its Non decreasing function   P{ X > x} = 1 –  1  –  P  P { X  x} = 1-  (x)





 



PDF :- 

Pdf =

        (x) (x) =

 

 

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Institute Of Engineering Studies (IES,Bangalore) 

       = x δ x       ≥  ∞                x ≤ x     (x  )dxdx    x         σ  () x  →  g(x)    (  x  )  ≤            < 1       <<     (a) x            e()/ σ       x  e() /  (x) =

Pmf =

} (x =

Formulae Sheet in ECE/TCE Department

)

Properties:-  (x)  0  

 

(x) =

 

( )=





  P{

 (x) * u(x) =

(x) dx =1 so, area under PDF = 1

<X



(x) dx

 

 

 } =

Mean & Variance :-

Mean

(x) dx

 = E {x} =

Variance

 = E {

 } = E {

 E{g(x)} =

}-

 

 dx

Uniform Random Variables :

Random variable X ~ u(a, b) if its pdf of form as shown below (x) =

(x) =

Mean =

 

Variance =

 / 12

E{

 } =

 

Gaussian Random Variable :- 

 

(x) =

X ~ N (

Mean =

 

)

 

 dx =  

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Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

  x  e()/ σ       λ e        ee||  ∞ ≤≤ ≤  ∞   (x  ) ∞     (xx y) ∞      ∞ ∞                  ()) /          (    )  /         ≤x   ( )  dx =

 

Variance =

 

Exponential Distribution :- 

 (x) =

 

 u(x)

 ) u(x)

(x) = ( 1-

Laplacian Distribution :- 

(x) =  

 

Multiple Random Variables :- 

       

 (x , y) = P { X  x , Y  y }  (  , y) = P { Y < y } =  (x , ) = P { X  x } =  (x) ;  (- , y) =  (x, - ) =  (- , - ) = 0







 dy ;

  = =



 

=



 

(y/x) =





(y)

 (x, y) dx

 (y) =

 =



 

 

Independence :-  X & Y are said to be independent if      (x, y) = (x) . (y) P{X

≤ ≤   ≤ ≤

⇒                              ⇒ → t t  t ) t  t t  → → R t t  t t 

(x , y) = (x) (y)  x, Y  y} = P { X  x} . P{Y

 y}

Correlation:  (x, y). xy. dx dy Corr{ XY} = E {XY} = If E { XY} = 0 then X & Y are orthogonal orthogonal .

Uncorrelated :-  Covariancee = Cov {XY} = E { (X Covarianc ( X - ) (Y- } = E {xy} –  {xy} –  E  E {x} E{y}. If covariance = 0  E{xy} = E{x} E{y}

  Independence



 uncorrelated but converse is not true.

Random Process:-  Take 2 random process X(t) & Y(t) and sampled @

X(

 , X( ) , Y( ) , Y ( )  ,

 Auto correlation

( ,

,

 

 random variables

) = E {X( ) X( ) }

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Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

→→ CRtttt tt  tt t  t R t t  t  t → C t t t  t t  t R t t  t  t → C t t ⇒ R t t  t  t → → RR t ttxt⇒  t  t → RR R≤ R → RR  ≤RR  R R S≤ωR RR () e    (ω)edω R   S ω RS ω |(ω)|(ω)  ddωω R  δ  →  Auto covariance  cross correlation correlation  cross covariance

   

( , ) = E { X( ) - ( )) (X( ) ( , ) = E { X( ) Y( ) } ( , ) = E{ X( ) - ( )) (Y( ) -

( )}=

( )}=

( ,

( ,

)-

)-

( )

( )

( )

( )

 ( , ) = 0   ( , )= ( ) ( )  Un correlated  ( , ) = 0 Orthogonal cross correlation correlation = 0  (x, y ! , ) =  (x! ) (y ! )  independent

Properties of Auto correlation :- 

  (0) = E {  }   ( ) = (- )  even   | ( ) |   (0)







Cross Correlation

( )= (- )   ( )   (0) . (0)   ( )|   (0) + (0)   2|







Power spectral Density :- 

  P.S.D



( )=

 



(j ) =

 

(j ) =

(j )

 

 

 

  Power = (0) =   ( ) = k ( )  white process  



 d  

 



Properties :  (j ) even    (j )  0  

S ω ≥

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Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

Control Systems  nd

Time Response of 2  order system :- 

Step i/P :

 



            → 

C(t) = 1-

 (sin

 

   

     t ± 

 

sin ω 1 tan    tan   e lim  sin  tan      →→ ω → Damping actor        tan    ω e(1+ ω   e(t) =



 



 =

 )

 

 

 

 

     Damping ratio ;

 < 1( nder damped damped ) :C(t) = 1- =

  Sin

 = 0 (un damped) :c(t) = 1- cos

t

 = 1 (Critically damped ) :C(t) = 1 -

t)

 > 1 (over damped) :-

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Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

                    t∅       ∅ t a n     t   e/ t . t   ()   /   ()  Dampi n g  a ct o r t ≈ t t    ω ω 12 ω ω (12  +    + 2)//  

C(t) = 1 -

 

T=

 >

 >

 >

 

Time Domain Specifications :-

  Rise time

 =

  Peak time

 =





 

 

  Max over shoot %  =   Settling time  = 3T

 × 100 5% tolerance 2% tolerance





= 4T

  Delay time

 =



 

 =

 

 

 =

 



 



  Time period of oscillations T =    No of osci oscillations llations =  =

 



 

     1.5    = 2.2 T  =   Resonant peak

 =

 ω ω ω



 ;



  Bandwidth

 

 

 =



 

 

 

 <

 <

 

Static error coefficients :- 

  Step i/p :



e lim→ () lim→  () lim→ ()  e e  lim→ lim(→)((())  () t e   lim→ s  ()(() →→ee ∞ → e/  =

  = =

  = =

 =

 (positional error)

 =

  Ramp i/p (t) :



 =

  Parabolic i/p ( /2) :



Type < i/p Type = i/p Type > i/p

     

  Sensitivity S =



 

 = 1/

 =

 

 =

 

 

 

 

 =    finite  = 0

/

 sensitivity of A w.r.to K.

  Sensitivity of over all T/F w.r.t forward path T/F G(s) :



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Institute Of Engineering Studies (IES,Bangalore) 

Open loop:

S =1

Closed loop :

S=

Formulae Sheet in ECE/TCE Department

 (()(()

 

  Minimum ‘S’ value preferable



  Sensitivity of over all T/F T/F w.r.t feedback T/F H(s) : S =



Stability RH Criterion :- 

()(() (()(()

 

       

Take characteristic characteristic equation 1+ G(s) H(s) = 0 All coefficients should have same sign There should not be missing ‘s’ term . Term missed means presence of at least one +ve real part root If char. Equation contains either only odd/even terms indicates roots have no real part & posses only imag parts there fore sustained oscillati oscillations ons in response.   Row of all zeroes occur if (a) Equation has at least one pair of real roots with equal image but opposite sign (b) has one or more pair of imaginary roots (c) has pair of complex conjugate conjugate roots forming symmetry about origin.





 



Electromagnetic Fields 

→→ →→

Vector Calculus:-   A. (B × C) = C. (A × B) = B. (C × A)  A×(B×C) = B(A.C) –  B(A.C) –  C(A.B)  C(A.B)  Bac –   Bac  –  Cab  Cab rule

→  a (|.|)   a (.|.|)  

 Scalar component of A along B is

 = A Cos

 = A .

 Vector component of A along B is

 = A Cos

 .

    





 =

 

 

..ds  ((..)) →→ ergencevatliovsse/pot  .ential. → source   . <  ⇒ sink .==→ →sol.ir)eonoitatdioalnal/Di/vconser   |(|)   R        |()   ∞ ρ  ρa a→    a 

Laplacian of scalars : =     =     A =  (   

 =

 Divergence theorem  Stokes theorem  

 

 = 0

 

 Harmonic .

Electrostatics :-

  Force on charge ‘Q’ located @ r F =



 

  E @ point ‘r’ due to charge located @



  E due to



 =

 line charge @ distance ‘  ‘ E =

  E due to surface charge



 is E =

 

 ;

 .

 

  ..

 =

 .

 

 

 

 (depends on distance)

   unit normal to surface (independent (independent of distance)

  For parallel plate capacitor @ point ‘P’ b/w 2 plates of 2 opposite charges is



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Institute Of Engineering Studies (IES,Bangalore) 

 a   (  a            Ψ s .

E=

 

-

Formulae Sheet in ECE/TCE Department

)

  ‘E’ due to volume charge c harge E =

→ → →Ψ ρ= .⇒D D .ds   ρ. dvdv →    a a .d         |   |   →  →  ∝  ∝ ∝       D. dv     dv ρ ρ e/   σ   

 Electric flux density D = Flux  =  

   .

 

D

 independent of medium

Gauss    TotalLaw flux:-coming out of any closed surface is equal to total charge enclosed by surface .  =       = =  =    

 Electric potential

  (independent (independent of path)

 =   = -

 = -

 =

 . dr

 

 -

 (for point charge )

r

  Potential @ any point (distance = r), where Q is located same where , whose position is vector @  



V=

 

 V(r) =

  + C . [ if ‘C’ taken as ref ref potential potential ] 

   × E = 0, E = - V  For monopole E

V

    ; Dipole E

   ;

V

   .

   

  Electric lines of force/ flux /direction of E always normal to equipotenti equipotential al lines .  =    =     Energy Density  =    





  Continuity Equation

.J = -



 



 =

 

  where

 .

 = Relaxation / regeneration time =

/  (less for good conductor )

Boundary Conditions :   = Tangential component of ‘E’ are continuous across dielectric-dielectric dielectric -dielectric Boundary .   Tangential Components of ‘D’ are dis continues across Boun dary .  

 



 =

 ;

 =

/

 .

D D ρ              →        .d   .ds

   Normal components components are are of ‘D’ are are continues continues , where as ‘E’ are dis continues. continues.      =  ;  =  ;  =  =  





 



 =

 

 =

 

=

=

t

 

 

 

Maxwell’s Equations :- 

 faraday law

 =

 Transformer emf =

 =  =

 = -

 

 

     ds ⇒

   × E = -

 

 

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Formulae Sheet in ECE/TCE Department

s

→ →

      × 

 Motional  Motion al emf emf =    × H = J +



 =  × (  × B).

 

Electromagnetic wave propagation :-   × H = J +   D= E    ×  E = -   B= H      

                      

. D  =     ρ          J =σ .  = ⊥    /   ρ ρ   ω(σ+ω) α β       βα ω       1 +  +1   e ω β   η η    /  η  η  /  σ ω η α β α → σ α → N| log  β →  σ ω σ  →ωω1→β λ πβ          α β ω      ω =       λ = 2π/β  η =   / ∠  

 = -



 =

  ; E.H = 0

For loss less medium

 -

 

 

 =

 

 

  E(z, t) =

 

 

 





 =

  | |=

 cos( t –   z) ;

|  | <  



   

E  H in UPW

E=0

 =

 =  =

  =  + j .

 =

 =

 /  .

 

tan 2

 = /

.

  =  + j      attenuation constant  Neper /m .   For loss less medium  = 0;  = 0.      phase shift/length ;  =  /  ;  = 2 /  .



|

 = 20  =

 = 8.686 dB





 =  =  /  = tan    loss tanjent  = 2       If tan  is very small (  < < )  good (lossless) dielectric   If tan  is very large (  >> )  good conductor   Complex permittivity  =    =   - j   ..









  Tan  =

 =



 .

Plane wave in loss less dielectric :- ( σ ≈ 0) 

 



 = 0 ;  =

 

 

 



  E & H are in phase in lossless dielectric 



Free space :- (  = 0,  =

   ,  =

 )

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Formulae Sheet in ECE/TCE Department

α β ω       λ = 2π/β  η = =   / <  = 12π ∠ σ ω σ/ω → ∞ ⇒ σ = ∞  =    =  αSkiη =nβdept  2 πh  eδσ/= 1/α 2ω/σ   λ π β η    ∠     R  RR .    (  )   ds     +  –  σ       |   δ | e  a    (s)   a a a →→a a a  

Good Conductor :-  >>

 =  =



     

 

 ,

 ; u = 1/  = 0 ,  =   Here also E & H in phase .



 

 ; u =

  ;  = 2  /   ;

 

 

 

 =

 



 



 

 =

  Skin resistance



 

 =

 

 =





 

 =

 =

 

 [

 

 

.

Poynting Vector :- 

 

 =-



S

 

] dv  

 dv 

v

(z) =      cos         Total time avge power crossing given area



 =



  ds ds

S



Direction of propagation :- (  ×   =  

 ×   =    Both E & H are normal to direction of propagation  Means they form EM wave that has no E or H component along direction of propagation .

Reflection of plane wave :-  (a) Normal incidence 

Reflection coefficient

Γ    =

 =

 

 Τ Γ there is a standing wave in medium   wave in medium ‘2’.    (nπ/β) () n =  1 2….    coefficient

 =

 =

 

Medium-I Dielectric , Medium-2 Conductor : >  :-

 

Max values of |

| occurs

 = -

 =

 =

 =

 

 

 

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Formulae Sheet in ECE/TCE Department

   β  ()  ⇒  ()     β  π ⇒        || || ||t  |Γ| ⇒||≤δ ≤ |∞ | Γ    r  σ r  Γ  ( R+  R+ ωe))(+ (+ωωC)β α β e ω β           <

 

 =

 

 occurs @

 :-

 = n    

 =

   

 =

 =

 =

() 

 

 

 min occurs when there is | |max

Since | | < 1

  1

 

 ; |  | =

  =

 =

S=

       

Transmission Lines :-  Supports only TEM mode   LC =  ; G/C =  /  .  

 

   = 0 ;

 -



   =   V(z, t) =

  = =  + j    cos ( t- z) +  



 



 

 =



   = 0

 -

 =

 =

 =

 cos ( t + z)

 

Lossless Line : (R = 0 =G; σ = 0)

→ γ  α/C  β ω C     α =  β = w w  C     λ = 1/   C     C    → α = =  R    β = ω  ω   = ωCωC    = ω  C    →         λ = 1/  C             //  γ β ⇒ β β                      Γ   Γ   ||||        ||||  /S     =  =

 + j  = j  

 

 , u = 1/

 

 

Distortion less :(R/L = G/C)  

 =

 

 =

 

 

 

 ; u =

 =

 ; u  = 1/C , u

 = 1/L

i/p impedance :-

 

 =

 =

  for lossless lossless line

 = j  

 tan hj l = j tan l

 

 

  VSWR =



 =

 

  CSWR = -     Transmissio Transmission n coefficient S = 1 +     SWR =  =  =  =  







(

 

 =

 =S

 

 

 =

 =

 





Shorted line :-

 = -1 , S =

 

 >

 =

=

 

) (

 <

)

 = j

 tan

l  

Γ ∞    β

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Γ 

 

 = -1 , S =



 



∞    β  

 =

 = j

 tan l.

 may be inductive or capacitive based on length ‘0’

  λλ/2 → capacitive( Γ  ∞ β  λ λ           Γ  If

Formulae Sheet in ECE/TCE Department

<

l

 / 4

 inductive (

 +ve)



 <  <

Open circuited line :-   = -j  cot l    =  = 1 s =  

l   < <

 / 4 capacitive

 < l   < <

 

 -ve)

/2 inductive

 

 =

Matched line : (  = )  = 0 ; s =1  =    No reflection reflection . Total Total wave

 . So, max power transfer possible .

Behaviour of Transmission Line for Different lengths :- 

λ λ λλ →  ⇒→    = )          e h k k ∴  γ    +    ω ω      → k  +ω   →     +      γγ α αβ β k  +   →   γ = β α =  β =  k                       +  u =  /4

l

l

 

 

=  /2 :

 =

 

Wave Guides :TM modes : (

 =

 sin

 =

 +

 

 impedance reflector @ l   = =  /2

 

 x  x 

 

 impedance inverter @ l   = =  /4

sin

 y

 

   =

 

where k =

 

 

m  no. of half cycle variation in X-direction

n  no. of half cycle variation in Y- direction .  =   Cut off frequency  

 

 <

   Evanscent mode ;

 

 >

   Propegation mode





 = 0;

 = 0 =  

 =  ;

 = 0

 

 

 



 =

 

 

 = phase velocity =

 is lossless dielectric medium

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Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

λ u        ( )() β β  1          β ω β  uη ωβ λ  πβ  u   →1          η η  1           η →   →         e → η η η  1        →      e  η             η  



 



/  =

 =

 =

 



 

 =

 

 

/  

 =



 

 = -

 =

 phase velocity & wave length in side wave guide

 

 

   impedance of UPW in medium

  cos

 

 

 =

 

 >

 

 Dominant mode

 

 

 /

 =

= phase constant in dielectric medium.

 = 0) 

 cos

 =

/f

 =

 

TE Modes :- (

/ W

 = 2 /   =

 =

 

  = =

 

Antennas :- 

Hertzian Dipole :- 

 =

 =

 

 sin  

 

Half wave Dipole :-  

 

 =

 ;

 =

EDC & Analog 

//... ....   Ne( )/   Nn e(n   N)/N e/    ≤ 1

  Energy gap



 



       

 





 =

 - KT ln

  Energy gap depending on temperature  Energy

 =

 + KT ln

 No. of electrons electrons n =    No. of holes holes p =   Mass action law  =  = Drift velocity  = E (for si

 

 

       

(KT in ev)

   cm/sec)

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 19 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com    www.facebook.com/onlin ok.com/onlineies eies   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo

 

Institute Of Engineering Studies (IES,Bangalore) 

σ ρ.  σR R ρ ρ → N  N n  N n      xN xN J D  J D     ≈   np NN np nn NN      

  Hall voltage

 =

 . Hall coefficient

 = 1/  .

 .    Conductivity  =  ;  =   Max value of electric field @ junction

 = -  







  Charge storage @ junction



Formulae Sheet in ECE/TCE Department

   charge density = q

.

.

 . 

 

  = qA

 = qA

 = -

 = -  

 = ne …  … 

EDC

  Diffusion current densities  = - q      = - q   Drift current current Densities Densities = q(p + n )E   ,  decrease with increasing doping concentration .

 



 



 =

 = KT/q

 25 mv @ 300 K

  Carrier concentration in N-type silicon   Carrier concentration in P-type silicon

 

  Junction built in voltage



 =

 

 

 ln  =

 =

  = =

  = =

/

  = =





 +

 

 

 =

 . A .

 =

 

 

  (for forward forward Bias) Bias)  ;

   /     = Aq

 

 = Aq

 

 

 

 

 

 n  D +n    D  1    C    ⇒ C ∝ → 2()/         Q=

 +

 =

I

  Diffusion capacitance



 

  = =

 ;

  Saturation Current  = Aq     Minority carrier life time  =   Minority carrier charge storage



   

 

  Forward current I =



 /  /

 

  Depletion capacitance



 =  =

 

  Charge stored in depletion region



 ;  ;

 =12.9    qx .x     +   ( +  )      C  C / CC 2CC1+1 +   +   n  /  1

* 

 =  =

  

  Width of Depletion region



   

 =

 

 /

 ;

 =

 

 /

 =  ,  =    = mean transist time  I = .g

 

   I.

   carrier life time , g = conductance = I /  =       =  =  (open condition)   Junction Barrier Voltage   = =  - V (forward Bias) =  + V (Reverse Bias)

 

  Probability of of filled states states above ‘E’ f(E) =



 

( )/ 

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 20 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com    www.facebook.com/onlin ok.com/onlineies eies   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo

 

Institute Of Engineering Studies (IES,Bangalore) 

  Drift velocity of

e  ≤ 1  ⇒   



  Poisson equation



 

     

 

 =

     = E =

 =  = –   –      +

 =

 

 

 

 Collector current when base open

   Collector current when

 = 0

 >

    - 2.5 mv

 C ;

   

  or

  D.C current gain



  (



=

 )

 =

 =

    when

 >

C

 

 =

 =

Conversion formula :-  CC ↔ CE  =  ;  = 1 ;  

 = - 0.25 mv

 

  Over drive factor =



 .

 

 =

  Small signal current gain



 

 )

 (1-

  Large signal Current gain





 Active region

Common Emitter :-   = (1+ )  +   

 cm/sec

    α α → e/  β → β β =   →  → / →  /   β  h β  β  h ≈ β          h  h  C R C       (     )         h  ∵  β β →  h h h h h h h h  h    h h  h 

Transistor :-   =  +      =  –   –         

 =

   

Formulae Sheet in ECE/TCE Department

 = - (1+

 

 =

 

 

);

 =

 =  =

 

 

 

CB ↔ CE

 



 =

 ;

 =

 -

 ;

 =

 ;

 =

 

CE parameters in terms of CB can be obtained by interchanging B & E . Specifications of An amplifier :-

 



      h h   . . .      h      . .    = =

 =

 

  = =

  + +

 

 =

 -

 

 

 

 =

 =

 =

 =

 =

 

 

Choice of Transistor Configuration :For intermediate stages CC can’t be used as  < 1   CE can be used as intermediate stage   CC can be used as o/p stage as it has low o/p impedance   CC/CB can be used as i/p stage because of i/p considerations.  

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 21 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com    www.facebook.com/onlin ok.com/onlineies eies   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo

 

Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

Stability & Biasing :- ( Should be as min as possible)

∆∆ S ∆∆  S ∆∆            ∆ ∆ S ∆  S ∆ββ  β        ≈      βR R R    R   

  For S =



 

 = S.

 +

  = =

 

 +

  For fixed bias S =



 

 

 =1+  

  Self bias S =



 



 =

 

 =

  ;

 

 

  Collector to Base bias S =



  = =

 

 1+

0 < s < 1+   =

  > 10

 

 

 

 

  For thermal stability [

 - 2  (



 +

g   rr hh rg rg hr hh g        (h   )     ( )  C C g R              ≈       ()       ( )  1 + h       1 + β        =  =  =  =

 <

  R R  

Hybrid –  Hybrid  – pi( pi(π)- Model :-

 = |  | /

  

 . S] < 1/   ;

)] [ 0.07

 

 

 

 /

    /    - (1+

 )

 

For CE :-

 



 



 =  =

 =

 

    = =

 =

;

 

 +

C=

 (1 +

 

 )

 = S.C current gain Bandwidth product  = Upper cutoff cutoff frequency frequency

For CC :-

 



 =

 

 

 =

 

 =

For CB:-

 



 =

=(

)

 = (

)

 

GATE/IES/JTO/PSUs in  in Bangalore @ Malleshwaram & 22 No.1 Training center for GATE/IES/JTO/PSUs Jayanagar,, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Jayanagar Email : [email protected]  [email protected]  Site: www.onlineIES Site: www.onlineIES.com .com    www.facebook.com/onlin ok.com/onlineies eies   Google+: http://bit.ly/gplus_iesgate FB: http://bit.ly/gplus_iesgate FB: www.facebo

 

Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

                                α    e/ α   α    e/              t2 /  .1 .       /             .   t  t  + t +   

 =



Ebress moll model : (1 +  = -

 = -

 +

 

 >

 >

   

)

 (1-

)

 

 =

Multistage Amplifiers :-

 



 

* =

  ;

  Rise time

 =



 



 



 



 =

 

 

 = 1.1

       

 = 1.1

 = 1.1

 

 =

        +       + 

 

 

Differential Amplifier :-

 



 



 h| | h R h R ≈ βR g   g α → α  R ↑ →  ↑   ↑  CRR ↑  (β )    Ωβ  ≈  R R ()    g β g   = =

 =

 =

  CMRR =



)2

 + (1 +

 

 



 



 

 +

 = (1 +

 )

   DC value of

   

 

 

   1 ( < 1)

);

  = =

 =

   ,

;

Darlington  = (1Pair + ):-(1 +   

 

   2

 

 of BJT/4

 =

 

 = 2

[ if

 &

 have same type ] =

 

 

 

Tuned Amplifiers : (Parallel Resonant ckts used ) :  

 



              =

 

Q

 ‘Q’ factor of resonant ckt which is very high high  



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Formulae Sheet in ECE/TCE Department

   ∆                  ∆   

  B.W =  /Q



 



 



 =

 -

 =

 +

 

  For double tuned amplifier 2 tank circuits with same



 used .

 =

 .

                    

MOSFET (Enhancement) [ Channel will be induced by applying voltage] 

 → ≥  i → i ∝     ↑ → ≈→    i       →   C        i         → r    (     ) →

p-substrate    NMOSFET formed in p-substrate      channel will be induced &   If      +ve for NMOS      (  -  ) for small  



  

 



 

 =

  channel width

 -

  For every



 >

 [ (

 0

 there will be

 -

 )

 -  

 pinch off further increase no effect  

   triode region (

 ]

 =

 



 



 source )

      channel width @ drain reduces .

 =



 (Drain

 =

 [

 =

 ]

 

 <

 -

)

 

 saturation

 Drain to source resistance in triode region

PMOS : are –   – v vee ,  are  , Device operates in similar manner except    enters @ source terminal & leaves through Drain .  

   i ≤  →      ≥   → i    (      )     C  ≤   → →      

 =

 

   

 induced channel  [

 -  

 ]

 =

 -

 

   Continuous channel  

     -    Pinched off channel .    NMOS Devices Devices can be made made smaller smaller & thus operate operate faster faster . Require low power supply supply .   Saturation region  Amplifier   For switching operation Cutoff & triode regions are used

 



 



NMOS

PMOS

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 ≥   ≤               ≥      →→ ≤ →→  i  ∴               

     

 -

 >

   

 -

 

→ → →

Formulae Sheet in ECE/TCE Department

 induced channel

 -

 <

 

   

 -

 

 Continuous channel(Triode region)  Pinchoff (Saturation) (Saturation)

i    ⇒ 

Depletion Type MOSFET :- [ channel is physically implanted .

  For n-channel



 



 -

 

 +ve   -ve

 

 characteristics are same except that

 =

 = 0 ]

 enhances channel .  depletes channel

 is –   is – ve ve for n-channel

  Value of Drain current obtained in saturation when



 flows with

 = 0

 

 .

 .

MOSFET as Amplifier :-

              i     (      ) ⇒ g    (      )  g R       ()  ≤  ⇒ i →          →  ≤  ≤ i   ≤=   21  ≤  ≤  ≥            ⇒         →    ||      ||       ≥  

 -    >   For saturation   To reduce non linear distortion

 

 



 



 =

 =-

 

 

 -

 )  )

 =

   

 

 

  Unity gain frequency



 < < 2(

 =

 

JFET :-

   

 

       

     = 0  Cut off         0,

 

 Triode

 



   

   0 ,

   

 

 -

   Saturation

Zener Regulators :-

  For satisfactory operation



   

 +

 

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R    r R  =

 

  Load regulation regulation = - (  ||



  Line Regulation =



Formulae Sheet in ECE/TCE Department

  For finding min



 )

 .

R       take

 &

 ,

 (knee values (min)) calculate according to that .

Operational Amplifier:- (VCVS)

         



Fabricated with VLSI by using epitaxial method High i/p impedance , Low o/p impedance , High gain , Bandwidth , slew rate .   FET is having high i/p impedance compared to op-amp .  Gain Bandwidth product is constant .  



Closed loop voltage gain







 



 



  β →

 =

 

 

 

 LPF acts as integrator ;

 =

 

 =

 

 =

 .

  dt ;

 = A.

  Max operating frequency



   feed back factor

  dt ;

 

 =

  Slew rate SR =



 

 ⇒  ⇒   dtdt →           ∆∆ ∆∆ ∆∆ ∆∆       . ∆ ∆   

  For Op-amp integrator



 =

Differentiator Differentiator

  (HPF)   = -  

 

 

 =

 =

 .

  In voltage follower Voltage series feedback



  In non inverting mode voltage series feedback



  In inverting mode voltage shunt feed back



 



 



 η      η    = -  

 ln

 = -

 

= -  

 ln

 

 

  Error in differential % error =



     

× 100 %

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Power Amplifiers :-

  Fundamental power delivered to load



     R+ +. R.    

 =

  Total Harmonic power delivered to load



Formulae Sheet in ECE/TCE Department

 =

=

 =

 

 

   1D+  +  + … …   

 

= [ 1+

]

 

 

 +D   +. . + D D      η

Where D =

 =

 

 

D = total harmonic Distortion .

Class A operation :o/p  flows for entire     ‘Q’ point located @ centre of DC load line i.e.,  =  / 2 ;  = 25 %    Min Distortion , min noise interference , eliminates thermal run way     Lowest power conversion efficiency & introduce power drain    =     -     if  = 0, it will consume more power   

 

   i  i  18 η   i    →   18 

 is dissipated in single transistors only (single ended) 

Class B:- 

 flows for    ; ‘Q’ located @ cutoff ;  = 78.5% ; eliminates power drain   Higher Distortion , more noise interference , introduce cross over distortion  = 0 [ transistors are connected connected in that way ]   Double ended . i.e ., 2 transistors .    power dissipated by 2 transistors tr ansistors .   =    = 0.4    

 





 i   

 =

 

Class AB operation :-

 flows for more than    & less than     ‘Q’ located in active active region region but near to cutoff cutoff ;  = 60%   Distortion & Noise interference less compared compared to class ‘B’ but more in compared to class ‘A’ Eliminates cross over Distortion   Eliminates











ηη         h  ≥       

Class ‘C’ operation :  flows for < 180 ; ‘Q’ located just b below elow cutoff ;  = 87.5%   Very rich in Distortion ; noise interferenc interferencee is high .   Oscillators :- 

RC-phase shift oscillator oscillator f =   For RC-phase



f=

 

 



   4k + 23 +  

where k =

R/R

 

 > 29

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  For op-amp RC oscillator f =



       |   ≥  

|

 29

Formulae Sheet in ECE/TCE Department

⇒ R ≥    R  

   29

 

Wein Bridge Oscillator :-

f=

   h ≥

   3

 

   3 A  3

≥

⇒ R  ≥    R  

   2

 

Hartley Oscillator :-

f=

    h | ≥       ( ) ↓|    ≥       h | ≥               | ≥     | ≥  |

|

 

   

 

   

|A  

 

 

 

Colpits Oscillator :-

f=

|

|

 

   

 

 

|A  

   

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Institute Of Engineering Studies (IES,Bangalore) 

Formulae Sheet in ECE/TCE Department

MatheMatics  Matrix :-     If |A| = 0  Singular matrix ; |A|  0 Non singular matrix Scalar Matrix is a Diagonal matrix with all diagonal elements are equal   Unitary Matrix is a scalar matrix with Diagonal e element lement as ‘1’ (  =  =  )   If the product of 2 matrices are zero matrix then at least one of the matrix has det zero   Orthogonal Matrix if A  = .A = I    =     A=  Symmetric   A=   Skew symmetric

→   →→



  ⇒  

 () 

Properties :- (if A & B are symmetrical ) A + B symmetric   KA is symmetric   AB + BA symmetric   AB is symmetric symmetric iff AB = BA BA      symmetric ; A skew symmetric.   For any ‘A’  A + Diagonal elements of skew symmetric matrix are zero   If A skew symmetric    symmetric matrix ;    skew symmetric   If ‘A’ is null matrix then then Rank of A = 0. 0.    

→  →



   →

Consistency of Equations :r(A, B)  r(A) is consistent consistent   r(A, B) = r(A) consistent consistent &   if r(A) = no. of unknowns then unique solution r(A) < no. of unknowns then  solutions .



Hermition , Skew Hermition , Unitary & Orthogonal Matrices :- 

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Formulae Sheet in ECE/TCE Department

  →→

   =    then Hermition    =    then Hermition   Diagonal elements of Skew Hermition Matrix must be purely imaginary or zero   Diagonal elements of Hermition matrix always real .   A real Hermition matrix is a symmetric matrix.   |KA| =  |A|

 

 









λλ   λλis λig……en valλ ue o  then 1/ λ →  ||   →  λ λ …….. λ   λ   → λ →λλ…………. λ . λ (    ) → (λ  k) (λ  k) λ λ λ  | | ||()

Eigen Values & Vectors :-  Char. Equation |A –  |A –   I| = 0.   Roots of characteristic equation are called eigen values . Each eigen value correspo nds to non zero solution X such that (A –  (A –   I)X = 0 . X is called Eigen vector . Sum of Eigen values is sum of Diagonal elements (trace)   Product of Eigen values equal to Determinent of Matrix .   Eigen values of  & A are same  

 



 



 

 &

 are Eigen values of A then

 

,

 .

 

 ,

A + KI

 

 

 , K

 

is Eigen value of adj A.

     

 + k ,

 + k , ……..  , ………

 + k

 

  Eigen values of orthogonal matrix have absolute value of ‘1’ .  .    Eigen values of symmetric matrix also purely real .   Eigen values of skew symmetric matrix are purely imaginary or zero .    ,  , ……  distinct eigen values of A then corresponding eigen vectors

 

 

linearly independent set .   ;   adj (adj A) =



| adj (adj A) | =

 

      ,

, .. …

 for

Complex Algebra :-

  Cauchy Rieman equations



 



 



             

          ()/(a)   (     ( )   (  () ⇓   (  ()   a ( ) a   ( )   Neccessary & Sufficient Conditions Conditio ns for f(z) to be analytic

 dz =

f(z) = f( ) +

)

 [

 +

(a) ] if f(z) is ana analytic lytic in region region ‘C’ & Z =a is single single point )

 + …… +

)

+ ………. Taylor Series  Series 

 

if

 = 0 then it is called Mclauren Series f(z) =

  ; when

 =

 

  If f(z) analytic in closed curve ‘C’ except @ finite no. of poles poles then



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 (    )  dd π lΦim→( ()  → ( )    )      a) lim

Formulae Sheet in ECE/TCE Department

 = 2 i (sum of Residues @ singular points within ‘C’ )  ) 

Res f(a) =

 

=

(a) /

(a) (a)

=

 ((

 f(z) )

Calculus :-

Rolle’s theorem ::-

If f(x) is (a) Continuous in [a, b] (b) Differentiable in (a, b) (c) f(a) = f(b) then there there exists exists aatt least least one value C  (a, b) such that

    

(     )()

 = 0 .

  (  c  )



Langrange’s Mean Value Theorem ::- 



If f(x) is continuous in [a, b] and differentiable differentiable in (a, b) then there exists exi sts atleast one value ‘C’ in (a, b) such that

(c) = (c)

 

Cauchy’s Mean value theorem ::- 

If f(x) & g(x) are two function such that (a) f(x) & g(x) continuous in [a, b] (b) f(x) & g(x) differentiable differentiable in (a, b)

g ≠ ∀  (  c  ) g(c) ( (   ))((())    ( ) ( ) (  x   ). dxdx    x     . dx d x    x     . dx d x        (  x   )ddxx  (  a    x)ddxx

(c)

(x) (x)

 0

 x in (a, (a, b)

Then there exist atleast one value C in (a, b) such that   //

 =

 

Properties of Definite integrals :-

  a<c<b  



 +

  =



 =

 

 

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 (  x   ). dxdx  (  x   )ddxx

 

 = 2

f(x) is even

= 0

 



 



 



 



 



 



f(x) is odd

 (  x   ). dxdx  (x   )dxdx  (  x   ). ddxx  (  x   )ddxx a   +x).dx  (  x   ). dxdx  ( a+x)  x  (x   ). dxdx   (  x   ).dx / sinx / cosx  ()())())())())……….…)…….……………… ) )() ) )………. …………….…    () ) ) () )………() ………(…()() ) )……… )…….… ….(  )). / sinx cosx  ())()())……..()) () π  

 = 2

if f(x) = f(2a- x)

= 0

if f(x) = - f(2a –  f(2a –  x)  x)

 

 = n

if f(x) = f(x + a)  

 =

 

 =  

if f(a - x) = f(x)

=

 =

  if ‘n’ odd  odd  

=

 



Formulae Sheet in ECE/TCE Department

 .

 .

  if ‘n’ even  even 

 . dx =

 

Where K =  / 2 when both m & n are even otherwise k = 1 Maxima & Minima :-

A function f(x) has maximum maximum @ x = a if A function f(x) has minimum minimum @ x = a if Constrained Maximum or Minimum :-

     

(a) = 0 and (a)

(a) = 0 and

     

 (a) < 0  (a)

 (a) > 0

To find maximum or minimum of u = f(x, y, z) where x, y, z are connected by Working Rule :-

Φ

 (x, y, z) = 0

λϕ

(i) Write F(x, y, z) = f(x, f (x, y, z) +   (x, y, z) (ii) Obtain

    = 0,

 = 0 ,

 = 0

(ii) Solve above equations along with

 = 0 to get stationary point .

ϕ

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Formulae Sheet in ECE/TCE Department

Laplace Transform :-

  L



 ()) s s s             = =

  ( ) t⇔  (s)( dds1)s   (u)   du ⇔ ( )  ( )  

  L{



 



 



 f(s) -

 

 

 f(t) } =  

 f(0) -

  (0) (0) ……

 (0)

 f(s)

 

 

 f(s) / s .

Inverse Transforms :-

 



 



 



 



 



 =

 t sin at

 =

  [ sin at at + at cos at at]]

 =

  [ sin at at - at cos at]

()    

 = Cos hat

 = Sin hat

Laplace Transform of periodic function : L { f(t) } = Numerical Methods :-

   (  ) 

 

Bisection Method :- 

x xx x x x x x x x x  (  )() x)

(1) Take two values of  &  suc  such h that that f( ) is +ve & f( ) is –  is – ve ve then +ve then root lies between  &  otherwise it lies between  &  . Regular falsi method :-

Same as bisection except

 =

 -

  f(

 =

 

x  x

  find f(

) if f(

 

Newton Raphson Method :- 

x x  ((  ))  =

 –   –  

 

Pi cards Method :-

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)

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Formulae Sheet in ECE/TCE Department

y y  (xy) ←   

 +

 =

 

 = f(x, y)

Taylor Series method :-

 y y   y() y y() y

 (    )        y x ( y ) (y)  x y ←   x y x + h y  x y x y()

 = f(x, y)

y=

 + (x-

)

 +

 

 + ………….

Euler’s method ::-

)

,

 + h f(

 =

 

 =

 +  [f(

,

) + f(

 =

 +  [f(

,

) + f(

:

,

,

 = f(x, y

( )   (y)    



)]

: Calculate till two consecutive value of ‘y’ agree  agree  

y() y  x + h y y y  x + h y x + 2h y  =

 + h f(

 =

)

,

 +  [f(

,

) + f(

,

)

………………   ………………

Runge’s Method ::- 

  k x x +y yy +k  k  x  y k   = h f(

,

)

 = h f(

 = h f(  =

,

+h ,

 = h ( f (

)

 +

+h ,

)

 +

   

finally compute K =  (

+4

 +

)

  )) ))

Runge Kutta Method :-

k x y  k   x +  y +     = h f(

 = h f(

,

)

,

)

       +  

finally compute K =  (

+2

 + 2

)

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k x +   y +   k x y k  = h f(

 = h f (

,

 )

 +

+h ,



)

Formulae Sheet in ECE/TCE Department

 approximation vale

y  y =

 + K .

Trapezoidal Rule :-

   (  x   ). dxdx  y yyy y y y x x x  (  x   ). ddxx  y y y y y y + y +…. + y)     (x   ). ddxx  y y y y y y y y + y ++….…. + y)  =

) + 2 (  +

 +

 [ (

f(x) takes values

,

 ,

 ,

@

 + …….

 )]

 ) + 2 (

 ….. …..  

 …….. ……..  

Simpson’s one third rule ::-

 =

 +

 [ (

)+4(

 +

 + …….

)+3(

 +

 +

]

Simpson three eighth rule : -

 =

 [ (

 +

 +

 + …….

 )+ 2 (

]

Differential Equations :-  Variable & Seperable :-

General form is

ϕ  (y) dy  ϕ(x) dx f(y) dy =

Sol:

(x) dx

 =

+C.

Homo generous equations :-

General form

 =

 

f(x, y) &

 ⇒(()    

Sol : Put y = Vx

 

 = V + x

 Homogenous of same degree

ϕ(xx y)

 & solve

Reducible to Homogeneous :-

General form

 ≠ 

 =

 

(i)       Sol : Put

x=X+h

y=Y+k

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⇒  (() )    =

(ii)

 

 =

 

Choose h, k such that

        () )  ⇒    

Sol : Let

 =

Formulae Sheet in ECE/TCE Department



 becomes homogenous then solve by Y = VX

 =  

 =

Put ax + by = t

 

 =

/b

Then by variable & seperable solve the equation . Libnetz Linear equation :-

 e. .(.)  +py = Q

General form I.F =

where P & Q are functions of “x”  “x” 

 



Sol : y(I.F) =

 dx + C .  dx

Exact Differential Equations :-

M

General form M dx + N dy = 0

If

 N

y Nx  =

→ →

  f (x, (x, y)

 f(x, y)

  then

 .dx (terms o N () e () e   (    ) e

Sol :

 

+

containing x ) dy dy = C

( y constant )

Rules for finding Particular Integral :- 

 

 =

 

 

= x =

 

 

if f (a) = 0

    x    ( ) e        

 

 

if

(a) = 0 (a)

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()

()   (   )   () x [ (D   )] x    ( ) ()  e e ()  sin (ax + b) =  sin

  sin (ax + b)

= x

 

 =

  sin (ax + b)

=

 

 

 

 f(x) =

f(-

 

a ≠ a )

f(-

Formulae Sheet in ECE/TCE Department

 0

)= 0

Same applicable for cos (ax + b)

  sin (ax (ax + b)

  f(x)

Vector Calculus :Green’s Theorem ::-

 (ϕ dx+  dy)   Ψx  ϕy  =

 dx dy

This theorem converts a line integral around a closed curve into Double integral which is special case of Stokes theorem . Series expansion :Taylor Series :-

 (   )   ( ) (xa)   (  ) (xa)   (   )    ( ) x   (  ) x  (1+x) () x  e                      →

f(x) = f(a) +

 (x-a) +

f(x) = f(0) +

 x +

 

 

 = 1+ nx +

 = 1 + x +

 + …………+

 + …………+

 

 

+ ……. (mc lower series )  ) 

+ …… | nx| < 1

 + ……..  …….. 

Sin x = x -

  + +

  - ……..

Cos x = 1 -

  + +

  - ……..

Digital Electronics 

  Fan out of a logic gate =



 or

  or     Noise margin margin :    =   Power Dissipation





 

 

 -

 =

 

 

 

   when o/p low

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    





  

     

Formulae Sheet in ECE/TCE Department

 →   

→ 

   when o/p high .

TTL , ECL & CMOS are used for MSI or SSI    Logic swing : RTL , DTL , TTL  saturated logic ECL  Un saturated logic Advantages of Active pullup ; increased speed of operation , less power consumption . For TTL floating i/p considered as logic “1” & for ECL it is logic “0” .  



“MOS” mainly used for LSI & VLSI . fan out is too high ECL is fastest gate & consumes more power . CMOS is slowest gate & less power consumption  NMOS is faster faster than CMOS . Gates with open collector o/p can be used for wired AND operation (TTL) Gates with open emitter o/p can be used for wired OR operation (ECL) ROM is nothing but combination of encoder & decoder . This is non volatile memory . SRAM : stores binary information informatio n interms of voltage uses FF. DRAM : infor stored in terms of charge on capacitor . Used Transistors & Capacitors . SRAM consumes more power & faster than DRAM . CCD , RAM are volatile memories memories . 1024 × 8 memory can be obtained by using 1024 × 2 memories  No. of memory memory ICs of capacity capacity 1k × 4 required to construct memory of capacity 8k × 8 are “16”

 11  

DAC 

  FSV =



 

 

  Resolution =



ADC 

 =

 /    =

  Accuracy = ±   LSB = ±    Analog o/p = K. digital o/p

 

 

1%

 

* LSB = Voltage range / * Resolution =

2

   

* Quantisation Quantisation error =

 

 %

PROM , PLA & PAL :-

AND Fixed

OR Programmable

PROM

Programmable fixed

PAL

Programmable Programmable

PLA

  Flash Type ADC :



Fastest ADC :-

22 → → 2 →  

 comparators    resistors  × n  Encoder

  Successive approximation ADC : n clk pulses   Counter type ADC :  - 1 clk pulses





2

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Institute Of Engineering Studies (IES,Bangalore) 

  Dual slope integrating type :



Flip Flops :-

2

Formulae Sheet in ECE/TCE Department

 clock pulses .

R    

  a(n+1) = S +



 Q  Q

=D = J  + Q = T  +  Q Excitation tables :- 

J

K

0

0 0

x

0

0 0

0

0 0

0

0

1 1

x

0

1  1

0

1 1

1 0

1

0 x 1 x

1 0

1

0 0 1 1

1

0 1 1 0

S

R

0

0 0

x

0

1 1

0 0 1   1 x

1

       



  

1

T

D

1

1

For ring counter total no.of states = n

t  t 

For twisted Ring counter = “2n” (Johnson counter counter / switch tail Ring counter ) .  .  To eliminate race around condition  <<  . In Master slave master is level triggered & slave is edge triggered

Combinational Circuits :Multiplexer : AB 00 01   I1 I0  _  2 OC 0

I0

10  I2

11 I3

4

6

3   5

7

 _  C

1

I1

C

I

1C 1

2

I3

0  A   B

         

C

(2, 5, 6, 7)

2

 i/ps ; 1 o/p & ‘n’ select lines. It can be used to implement Boolean function by selecting select lines as Boolean variables For implementing ‘n’ variable Boolean function  × 1 MUX is enough enough . For implementing “n + 1” variable Boolean  × 1 MUX + NOT gate is required . For implementing “n + 2” variable Boolean function  × 1 MUX + Combinational Ckt is required   If you want to design  ×  1 MUX using  × 1 MUX . You need    × 1 MUXes

  







2

 2 2 2 2

2 2

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Decoder :-

Formulae Sheet in ECE/TCE Department

2

  n i/p &  o/p’s   used to implement the Boolean function . It will generate required min terms @ o/p & those terms





should be “OR” ed to get the result .    Suppose it consists of more min terms then connect the max terms to NOR gate then it will give the same o/p with less no. of gates .   If you want to Design m ×   Decoder using n ×   Decoder . Then no. of n ×   Decoder





required =



 .

2 2 t 2 t 

  In Parallel (“n” bit ) total time delay =   For carry look ahead adder delay = 2





  .

 .

2

Microprocessors 

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

  Clock frequency =  crystal frequency   Hardware interrupts





  Software interrupts



TRAP (RST 4.5) RST 7.5 RST 6.5 RST 5.5

0024H both edge level  Edge triggered 003CH 0034 H level triggered 002C

INTR

Non vectored

RST 0 RST 1 2 : : 7



Formulae Sheet in ECE/TCE Department

0000H 0008H 0010H 0018H

Vectored

0038H

 



S1   S 0

 



0

0

Halt

0

1

write

1

0

Read

1

1

fetch

HOLD & HLDA used for Dir ect Memory Access . Which has highest priority over all interrupts .

Flag Registers :S

   AC X P X Z   X   CY

  Sign flag :- After arthematic operation MSB is resolved for sign flag . S = 1

 -ve result



If Z = 1  Result = 0 AC : Carry from one stage to other stage is there then AC = 1   P : P =1  even no. of one’s in result .  CY : if arthematic operation Results in carry then CY = 1  For INX & DCX no flags effected In memory mapped I/O ; I/O Devices are treated as m memory emory locations . You can connect max max of 65536 devices in this technique .    In I/O mapped I/O , I/O devices are identified by separate 8-bit address . same address can be used to identify i/p & o/p device .    Max of 256 i/p & 256 o/p devices can be connected .  



           



 













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Formulae Sheet in ECE/TCE Department

Programmable Interfacing Devices :-

                 



















→ →→ →

8155  programmable peripheral Interface with 256 bytes RAM & 16-bit counter 8255   Programmable Interface adaptor 8253  Programmable Interval timer 8251  programmable Communication interfacing Device (USART) 8257  Programmable DMA controller (4 channel) 8259  Programmable Interrupt controller 8272  Programmable floppy Disk controller CRT controller Key board & Display interfacing Device

D D D   D   D D

RLC :- Each bit shifted to adjacent left position .

 becomes

 .

CY flag modified according to

RAL :- Each bit shifted to adjacent left position . ROC :-CY flag modified according RAR :-

D

  becom becomes CY & CY becomes

D

 .

 

 becomes CY & CY becomes

 

CALL & RET Vs PUSH & POP POP :-  CALL & RET 



  When CALL executes ,  p  p automatically automatically stores stores



16 bit address of instruction instruction next to CALL on the Stack   CALL executed , SP decremented by 2   RET transfers contents of top 2 of SP to PC   RET executes “SP” incremented by 2







PUSH & POP 

* Programmer Programmer use PUSH to save the the contents contents rp on stack * PUSH executes “SP” decremented by “2” .  * same here but to specific “rp” .  * same here

Some Instruction Set information :CALL Instruction

CALL CC CM CNC CZ

→ → → → →

 18T states SRRWW

  Call on carry

9 –  18  18 states

  Call on minus

9-18

  Call on no carry   Call on Zero ; CNZ call on non zero zero

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CP CPE

→ → →

  Call on +ve

 

CPO

Formulae Sheet in ECE/TCE Department

Call on even parity

  Call on odd parity

RET : - 10 T RC

: - 6/ 12 ‘T’ states states

Jump Instructions :-

JMP JC JNC JZ JNZ JP JM JPE JPO

             



 

 





→ → → → →→ → →

 10 T

  Jump Jump on Carry

7/10 T states states

  Jump on no carry

  Jump on zero

 Jump on non zero

 Jump on Positive  Jump on Minus  Jump on even parity

 Jump on odd parity .

PCHL : Move HL to PC 6T PUSH : 12 T ; POP : 10 T SHLD : address : store HL directly to address 16 T SPHL : Move HL to SP 6T STAX :   store A in memory 7T STC : set carry 4T XCHG : exchange DE with HL “4T”  “4T” 

R

XTHL :- Exchange stack with HL 16 T

→→ →→ → →

  For “AND “ operation “AY” flag will be set & “CY” Reset  Reset    For “CMP” if A < Reg/mem : CY  1 & Z  0 (Nothing but A-B)

 

A > Reg/mem : CY

 0 & Z

A = Reg/mem : Z

 1 & CY



 0  0 .

  “DAD” Add HL + RP (10T)  fetching , busidle , busidle   DCX , INX won’t effect any flags . (6T)  (6T) 





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DCR, INR effects all flags except carry flag . “Cy” wont be modified “LHLD” load “HL” pair directly “ RST “  12T states SPHL , RZ, RNZ …., PUSH, PCHL, INX , DCX, CALL  fetching has 6T states PUSH –  PUSH  –  12   12 T ; POP –  POP –  10T  10T 

   

 



Formulae Sheet in ECE/TCE Department





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