Voltage Divider

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Cogee Chulam



















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1he purpose of a low pass fllLer ls aLLenuaLe frequencles pasL some cuLoff frequency. 1he ldeal low pass
fllLer allows for now passage of slgnals pasL Lhe cuLoff frequency, buL Lhls, of course, ls unreallsLlc. 1he
magnlLude of Lhe galn from Lhe lnpuL Lo Lhe ouLpuL gradually decreases afLer Lhe cuLoff frequency. 1he
goal of fllLer deslgn ls Lo deLermlne Lhe value of Lhe cuLoff frequency, Lhe galn of Lhe fllLer before Lhe
cuLoff frequency, and how sharply Lhe galn dlmlnlshes afLer Lhe cuLoff frequency.
1he purpose of Lhls fllLer ls Lo Lake ouL any posslblllLy of hlgh frequency flucLuaLlons from a hlgh volLage
source (a source of 4 kv). 1he slngle source ls Lo provlde Lhe proper uC volLage levels aL Lhe ends of Lhe
radon deLecLor dlode. 1he volLage levels can be conLrolled Lhrough Lhe use of Lhe volLage dlvlder rule,
selecLlng proper reslsLors can deLermlne where Lhe greaLesL volLage drops are ln Lhe clrculL.
1he fllLer aspecL of Lhe clrculL ls an analog passlve fllLer, whlch ls a qulLe slmple deslgn. 1he passlve
aspecL of Lhe fllLer lmplles LhaL Lhere ls no ampllflcaLlon of Lhe lnpuL slgnal (l.e. an ampllfler lsn'L
lmplemenLed). A baslc analog low pass fllLer conslsLs of reslsLors and capaclLors. 1he cenLral ldea ls
change Lhe clrculL elemenLs lnLo Lhe frequency domaln. A capaclLor's relaLlonshlp Lo an lnpuL frequency
ln can be descrlbed as follows:
! !
!
!"

Where ! ! !". Z ls Lhe lmpedance of Lhe capaclLor, whlch roughly LranslaLes Lo Lhe reslsLance ln whaL ls
called Lhe phasor domaln. Cne can noLe LhaL as ! ! !, ! ! !. Cne can also noLe as ! ! !! ! ! !.
1hus, a capaclLor behaves as a shorL clrculL for a hlgh frequency slgnal and an open for a low
frequency/uC slgnal. 1hls ls cruclal for fllLer deslgn, by placlng a capaclLor Lo ground hlgh frequency
slgnals wlll be aLLenuaLed because of Lhe shorL, whereas low frequency slgnals wlll see an open clrculL
and noL be grounded.
uslng Lhe clrculL elemenLs ln Lhe frequency domaln and applylng Lechnlques of clrculL analysls, one can
obLaln a Lransfer funcLlon whlch shows Lhe lnpuL ouLpuL relaLlonshlp of a clrculL. 1he followlng clrculL
was Lo be analyzed:
llgure 1:

Lach of Lhese clrculLs have Lhe same clrculL layouL, buL Lhe reslsLors vary Lo obLaln dlfferenL volLage
dlfferences across Lhe ulode. ln Lhe followlng slmulaLlons Lhe ulode was LreaLed as an open clrculL. 1he
followlng Lransfer funcLlon was obLalned (noLe 8eq =82+83):
! ! !
!
!
!
!"
!!
!
!
!"
! !
!
!
!
!!!
!
! !
!"
! !"!
!
!
!"
!

A Lransfer funcLlon can be ploLLed on a logarlLhmlc scale, wlLh ploLs for boLh magnlLude and phase. 1hls
ls whaL ls called a 8ode ploL. As Lhere were llLLle dlfferences ln Lhe ploLs beLween graphs, Lhere ls only
one 8ode ploL glven (for Lhe flrsL clrculL).
An equlvalenL of Lhe clrculL can be obLalned as well Lo deLermlne Lhe currenL Lhrough Lhe clrculL aL any
one Llme
llgure 2:

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!
!
!
! !
!
! !
!
! !
!
!"# !
!
! !
!
! !
!
! !
!
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!
! !
!
! !
!
! !

AL low frequencles (l.e. Lhe deslred uC-volLage):
!
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! !
!
! !
!
! !
!
! !
!

Cf course, by Chm's law: ! ! !"
1he values for Lhe componenLs were deLermlned vla Lhe Lransfer funcLlon as well as by Lhe deslred
volLage dlfference beLween Lhe dlode. 1he volLage dlfferences vary from 40 volL Lo 60 volL dlfferences.
volLage dlvlder equaLlons were applled Lo Lhe clrculL and MA1LA8 code was lmplemenLed Lo help wlLh
selecLlon of reslsLors. 1he code gave raw values, and Lhen Lhese values had Lo be maLched Lo acLual parL
values from Lhe Allled LlecLronlc CaLalog. 8ode ploLs were produced vla MA1LA8 as well Lo ensure LhaL
Lhe proper cuLoff frequency was achleved. ClrculL slmulaLlons ln ÞSÞlCL and MulLlSlm were done ln
order Lo ensure LhaL Lhe correcL uC volLage dlfferences were obLalned. 1he AC Analysls funcLlon was
used Lo show Lhe Lransfer curve aL a deslred lnpuL magnlLude. 8efer Lo Lhe Appendlx for a llsL of Codes
LhaL were lmplemenLed.
+)$(')# ,
1hls clrculL ls deslgned Lo produce a volLage dlfference of abouL 30 volLs across Lhe dlode.
llgure 3

1he cuLoff frequency obLalned:
! ! !"!!"#$ !"#!!
! ! !!!""# !"
!
!"
! !!!"#!!"
!
!
! ! !!!"#$!!"
!!
!

1he followlng uC operaLlng polnLs were obLalned (ln kllo-volLs):
node volLage volLage ulfference
v(4) 3.03146 0.03032
v(3) 3.10198
1he lollowlng 8ode ÞloL Was obLalned:


+)$(')# -
1hls clrculL conLalns a 8eslsLor ln serles wlLh Lhe 33 Mega-Chm reslsLor. 1he volLage drop across Lhe
dlode wlll be approxlmaLely 32 volLs. 1he value of 8eq changes ln Lhe Lransfer funcLlon.
llgure 4

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!

! ! !!!"#$ !"
!
!"
! !!!"!!!"
!
!
! ! !!!"#!!"
!!
!
1he followlng uC operaLlng polnLs were obLalned:
node volLage volLage ulfference
v(3) 3.09961 0.03216
v(3) 3.04743

+)$(')# +
1hls clrculL conLalns anoLher 976 kllo-Chm reslsLor ln serles wlLh Lhe 33 Mega-Chm 8eslsLor. 1he value
of 82 was also reduced Lo 40 Mega-Chm:
llgure 3

1he cuLoff frequency obLalned:
! ! !"!!"#$ !"#!!
! ! !"!!"#$ !"
!
!"
! !!!"!!!"
!
!
! ! !!!!"#!!"
!!
!
1he followlng uC operaLlng polnLs were obLalned:

node volLage volLage ulfference
v(3) 3.13097 0.03431
v(3) 3.09646

+)$(')# .
1hls clrculL ups Lhe value of 83 Lo 40 Mega-Chms. 1he value of 82 ls kepL aL 82 Mega-Chms.
llgure 6

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!"#
!

! ! !!!!"# !"
!"# ! !!!"" !!"
!

! ! !!!"!#!!"
!!

1he followlng uC operaLlng polnLs were obLalned :
node volLage volLage ulfference
v(3) 3.10436 0.06108
v(3) 3.04328

+)$(')# /
1hls clrculL reduces Lhe value of 83 Lo 30 Mega-Chm ln order Lo lower Lhe volLage drop.
llgure 7

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!
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! !!!"#!!"
!
!
! ! !!!"#$!!"
!!
!

1he followlng uC operaLlng polnLs were obLalned:
node volLage volLage ulfference
v(3) 3.10096 0.04399
v(3) 3.03497

+)$(')# 0
1hls clrculL reduces Lhe value of 83 Lo 27.4 Mega-Chm:
llgure 8

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! ! !!!"## !"
!
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!
!
! ! !!!"#$!!"
!!
!
node volLage volLage ulfference
v(3) 3.10007 0.04203
v(3) 3.03802

ln LoLal clrculLs A Lhrough l can be descrlbed by Lhe followlng Lable:

ClrculL A ClrculL 8 ClrculL C
lrequency (Pz) 8.3113 8.4331 12.2087
LqulvalenL lmpedance (Clga-Chms) 2.613 2.616 2.373
CurrenL (mlcro-Amperes) 1.3296 1.329 1.3334
volLage ulfference (kllo-volLs) 0.03032 0.03216 0.03431
ClrculL u ClrculL L ClrculL l
lrequency (Pz) 8.1143 8.6967 8.8633
LqulvalenL lmpedance (Clga-Chms) 2.622 2.612 2.6084
CurrenL (mlcro-Amperes) 1.3236 1.3314 1.3329
volLage ulfference (kllo-volLs) 0.06108 0.04399 0.04203

,&&)"1 2 .+ .3(%'45)"1 +242()#%$
A decoupllng CapaclLor ls Lo be added beLween Lhe ouLpuL of Lhe fllLer clrculL and Lhe lnpuL Lo Lhe
preamp. 1he purpose of Lhls capaclLor ls Lo cuL ouL Lhe hlgh volLage uC and send Lhrough Lhe slgnal. As
prevlously sLaLed Lhe capaclLor behaves as shorL for hlgh frequencles and Lhus wlll allow Lhe slgnal Lo
pass. 1he addlLlonal capaclLance wlll noL cause a change ln Lhe uC operaLlng polnLs, buL wlll creaLe a
new Lransfer funcLlon. 1he followlng clrculL demonsLraLes Lhls. 1he 2 Clga-Chm reslsLor ls added as a
ºdraln" for Lhe decoupllng capaclLor.
llgure 9

lor compleLeness a new Lransfer funcLlon wlLh Lhe lncluslon Lhe decoupllng capaclLor and Lhe draln
reslsLor. Agaln, Lhe dlode was LreaLed as an open clrculL (refer Lo flgure 10).
llgure 10

ln Lhls case Lhe volLage aL node 3 dlvldes beLween Lhe lmpedance from C2 and Lhe 83. lf one conslders
Lhe Lwo elemenLs ln serles and Lhen comblnes Lhem wlLh 84 ln parallel.
1he equlvalenL lmpedance looklng Lhrough Lermlnal 3 ls:
!
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!
!
!
!!
!
!" ! !!
!
!
!
!
! ! ! !
!
!
!

lf Lhe equlvalenL lmpedance ls subsLlLuLed lnLo Lhe orlglnal Lransfer funcLlon, a new Lransfer funcLlon ls
obLalned for node 3.
! ! !
!
!
!
!
!
!
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!
!"
!!
!"
!
!"
! !
!
!
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!!!
!
! !
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! !
!
!!
!
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1o obLaln Lhe ouLpuL volLage, a volLage dlvlder musL be applled beLween Lhe capaclLor and Lhe reslsLor.
1hls glves Lhe followlng relaLlonshlp beLween v3 and vo:
!
!
!
!
!
!
!
! ! !
!
!
!
!
!
!

MulLlplylng Lhe Lwo equaLlons, one obLalns Lhe followlng Lransfer funcLlon:
! ! !
!
!
!
!
!
!
!
!
!
!
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!
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!!
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!
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! !
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!!!
!
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! !
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!!
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!!! ! !
!
!
!
!!

1hough Lhe Lransfer funcLlon ls more complex, Lhe addlLlonal capaclLor does noL do much Lo affecL Lhe
frequency response of Lhe sysLem. An lnpuL slgnal ls fllLered by Lhe flrsL capaclLor (C1). So even Lhough
Lhe decoupllng capaclLor behaves as a shorL ln AC, Lhere ls no AC slgnal presenL from Lhe lnlLlal volLage
source Lo go Lhrough anyway. LssenLlally, Lhe decoupllng capaclLor allows for [usL Lhe slgnal from Lhe
dlode Lo pass (l.e. cuLs ouL Lhe uC slgnal, hence Lhe name decoupllng capaclLor).
1o demonsLraLe Lhe lmporLance of Lhe decoupllng capaclLor, a swlLch aLLached Lo a volLage source was
placed ln poslLlon of Lhe dlode. 1he volLage source was glven a volLage of 600 mv, whlch ls generally Lhe
volLage assoclaLed wlLh a dlode. 1he followlng was creaLed uslng lClrculL, an Apple clrculL slmulaLlng
sofLware.
llgure 11

When Lhe volLage source ls flrsL closed, Lhe clrculL musL reach Lhe sLeady sLaLe and Lhus Lhe clrculL goes
Lhrough a LranslenL response. 8oLh Lhe fllLer capaclLor and Lhe decoupllng capaclLor are assumed Lo
have no lnlLlal volLage. 1he volLage source Lurn on can be represenLed as a negaLlve lmpulse sLep
funcLlon. 1he volLage Lhen decays from Lhe capaclLor exponenLlally. 1he followlng dlagram represenLs
Lhe lnlLlal lmpulse and decay:

When Lhe swlLch ls closed, Lhe clrculL has (assumedly) reached Lhe sLeady sLaLe and Lhus Lhe capaclLor
musL undergo Lranslence agaln. 1he clrculL on Lhe lefL represenLs Lhe capaclLor's volLage rlghL afLer
swlLch has been closed, Lhe clrculL on Lhe rlghL represenLs Lhe volLage afLer Lhe swlLch ls opened agaln.

.)6('66)%"6
1hls pre-experlmenLal analysls shows LhaL Lhe volLage dlvldes, low pass fllLer, and Lhe decoupllng
capaclLor oughL Lo work effecLlvely. Several dlfferenL analyLlcal Lools were used ln order Lo ensure LhaL
Lhe clrculL wlll be able Lo funcLlon properly. 1hese Lools lnclude hand calculaLlons, SÞlCL, MulLlSlm,
MA1LA8, and lClrculL. MA1LA8 was qulLe useful ln checklng hand calculaLlons for arlLhmeLlc errors as
well as slmulaLlng 8ode ploLs. 1he analysls could noL have been compleLed as effecLlvely wlLhouL Lhe
clrculL slmulaLlon sofLware. lL wlll be lnLeresLlng Lo see how Lhe experlmenLally lmplemenLed resulLs wlll
maLch up wlLh Lhese slmulaLlons.
1he MulLlSlm analysls was conducLed wlLh a volLage source of 4 volLs, whereas Lhe experlmenL wlll be
conducLed wlLh a volLage source of 4 kv. 1he slmulaLlon could noL be done wlLh 4 kv, as some of Lhe
reslsLors would burn ouL as soon as Lhe slmulaLlon began. 1hls, of course, ls because Lhe deslgn crlLerla
assumed LhaL Lhe reslsLors were noL hlgh volLage. MulLlSlm ls an offshooL of SÞlCL and has Lhe same
deslgn parameLers as SÞlCL, buL for compleLeness, slmulaLlons were run ln boLh. lL was noL Loo Llme
consumlng Lo do boLh, as Lhe programs are generally easy Lo work wlLh.
1he LranslenL analysls was slmulaLed uslng lClrculL. 1hls clrculL program ls new sofLware LhaL was
developed a few years ago. lL musL be noLed LhaL lL ls generally beLLer Lo use for quallLaLlve analysls.
MulLlSlm and SÞlCL are much more preclse Lools. 1hus, Lhls analysls only used lL for LesLlng Lhe response
of Lhe uC decoupllng capaclLor. lClrculL ls easy Lo use, and conLalns a 'swlLch' elemenL, whlch ls generally
how a dlode ls descrlbed ln clrculL analysls. 1here are Lwo baslc models for dlode analysls, Lhe consLanL
drop model and one ln whlch Lhe dlode can be a shorL or an open. ln Lhe consLanL drop model, Lhe dlode
ls assumed Lo have a drop of 600 mv when on and Lo be an open when off. ln Lhe oLher model, Lhe
dlode ls a shorL when on and an open when off. 1hls slmulaLlon used Lhe consLanL drop model, buL lL
should be noLed LhaL lL doesn'L make Loo blg a dlfference ln whlch model ls used because Lhe 4 kv
source ls so much blgger Lhan Lhe dlode volLage. 1hus, Lhe lmpulse senL Lo Lhe decoupllng capaclLor wlll
noL depend on Lhe volLage drop of Lhe dlode, buL raLher Lhe volLage across Lhe reslsLor ln parallel wlLh
Lhe dlode. 1he clrculL slmulaLlon lllusLraLes Lhls clearly.
lL should be lnLeresLlng Lo see how Lhe clrculL holds up Lo Lhls pre-experlmenLal analysls. lL mlghL be
worLhwhlle Lo use a pulse generaLor Lo LesL Lhe capablllLles of Lhe decoupllng capaclLor. lL also mlghL be
equally lmporLanL Lo use an AC source Lo demonsLraLe Lhe effecLlveness of Lhe fllLer. 1he cuLoff
frequency glven ln Lhls reporL represenLs Lhe momenL aL whlch Lhe value of Lhe ouLpuL beglns Lo
decrease. 1he 8ode ÞloL shows Lhe magnlLude of Lhe ouLpuL decreaslng raLher sharply.







,443"&)78 0'"(#)%"6 963& )" ,"25:6)6

MA1LA8 code used Lo LesL hand-calculaLlons and 8ode ÞloL done by hand:

R1 = input('value of R1: ');
R2 = input('value of R2: ');
R3 = input('value of R3: ');
R4 = input('value of R4: ');
C = 200 * (10^-12);
Rx = R2 + R3;
% H(s) = (R4*Rx)/((R4+Req-R1*R4)*(R1+Rx+C*s*R1*Rx))
a = R4*Rx;
b = R4+Rx - (R1*R4);
d = R1+Rx;
e = C*R1*Rx;

% H(s) = (a/be)/((bd/be)+ s);

x = (b*d)/(b*e);
y = a/(b*e);

X = [1 x]
Y = [0 y]

bode(Y,X)
grid on

MA1LA8 Code Lo deLermlne reslsLor values for speclflc volLage drops
V = input('voltage drop across diode (in Volts): ');
R4 = input('value of R4: ')
R3 = ((45/3000)*(R4))/(1-(45/3000))
Rxx = R3 + R4;

Req = ((1/4)*(Rxx))/(1-(1/4))
Rx = Req;

R1 = Req/2;

C = 750 * (10^-12);
a = R4*Rx;
b = R4+Rx - (R1*R4);
d = R1+Rx;
e = C*R1*Rx;
% H(s) = (a/be)/((bd/be)+ s);
x = (b*d)/(b*e);
y = a/(b*e);

X = [1 x]
Y = [0 y]

bode(Y,X)
grid on

MA1LA8 Code Lo deLermlne cuLoff frequencles, currenL, and equlvalenL lmpedance
R1 = input('value of R1: ')
R2 = input('value of R2: ')
R3 = input('value of R3: ')
C = 200 * (10^-12);
Req = R2+R3;
s = (R1+Req)/(R1*Req*C)
f = s/(2*pi)
R4 = 2*10^9;
Rx = R1+R2+R3+R4


Vin = 4*10^3;

I = Vin/Rx