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To find frequency response, substitute jo for s and findthe magnitude and phase shift of the transfer functionfor different values of f.A r,( co ) - - 1 Take magnitude of this functionY \ R' C' j 'cD and Phase anglez = comprex varue l?,i,.; ;eat part of z im(z) = imaginaryPhase ShiftMagnitudeScale _-/im(z) \| - | i---,-,2 =.-- t-,2 0 = ?fCtani- ilzl =;re(z) - lm(z) \ re(z) jdb(rrl) = 20- log( | z(r) l)Gain and Phase Plot of Practical OP AMP Integrator-RyA v(c,r) =Ri(1 *ftrcj ')et438a-8.MCD 8;.t.:..Practical integrator onlyiexhibits integral action forfrequencies much higher (10 times) 1/RfC.A v(ar)RyR1GainPhase Shift0(r,t) = 180' 180R1C'crr );atan (= tan-1atanUsing Matlab script to generate Bode plots andtransfer function.ri=input('Enter value of input resistance: ');c=input('Enter value of capacitance: ');rf=input('Enter value of feedback resistance: ');% compute transfer function model parmeters forparactical% integratorta u = rf*c;ki=-rf./ri;et438a-8.MCD I= arctan% compute parameter for ideal integrator'tau1=.11*".':.]'.',,']..'...:".''j:o/o build transfer function% denominator form a1*s^2+a2s+a3Av=tf([ki], [tau 1])Av1 =tf([-1 J, [tau 1 0])%plot both on the same graphsbode(Av,Avl );Bode Diagrams\.20cn910(Dfn.E(t,(g -1n6 2oo(DIo 100U'ctEnLv-100-200-3001ot1otFrequency (radlsec)1ooet438a-8.MCD 10R f= 100kC = O.O1 pFIntegtral action on time varying error signalsEt 4 Al-l\ i = lul\ItimeIntegral of constant is line. Integrator produceslinearly increasing output for constant error input.Negative error causes output to decrease. Zeroerror maintains value.et438a-8.MCD 11Exampfe: an ide?lltgsla!o1h3-._e sain of Kr = 0.1V/s''lts ihital outlut at tro-iJ u=l.'s v. oetermine theoutputs if the error nas sLf in."ases of le(t) - o o<=t<-1 se(t) = 2.5 1<t<-2 se(t)=e 2<t<-3se(t)=Q 3<t<-4se(t) = -1.5 4<t<=5 s\lL,\ t) 1tvi[t/: o.J* e(+)ox +vo {0" ist se.i,h{,r* t,=o* t| -l 'V,(t): J.,{'o Jx + (,s : c + r.s ; V,it) ;3i c (t < I\, \ ,tv2(t\ -'c i f ;.sJx+ t,s: o.zsxlt .:.r\= o,tst-oZr+t.sl/ '! "v'v.tr). o r.r'i ;;;; r<t <L {,^d r,.,Jo-l vgruc( he.t-f sfepV.tz) -- o zr(z) + /'Zs = / ?s1 6',{ r a i c"^ d ,{ron {o " frext S+e Po--[ t=s V-(s)'-o,ts(s) uz.?s= Z5r-/ 'vV =r..Ct*t PqT bico?e,4sESv c"){/L,4x /ItrU-,a' , n t€ 3 <t<.1 hr Cl^...,,.,E lr\ €f,^t{^\odx+2.rS = l"''\ c:.-\\ ' t-Iui" - L ,ta- r\-r.Sdx +J.rS' -o,/Sx/ +2,/sJ^{u " ner* slai€t /?s .- c.1t -c,yr r"?s'< at [=3 Vr(:)=c.{(:\tolrVr(:) ; ?, /sVocrl = o ')VsG)= o,)Vt(+)'et438a-8.MCDL2Pfot o[ integr"to-t:*-o#Jp'_t1jr. :, ;: 1*,,,,'Mgtle'.h,,,PJ0", : .,,,.t1-11nspiceto.il3J..,l....''.,.-. x-ones (Lr length (t1),) 'xl=x. *1.5tz=Iinspace (1.,2,51 ;x2=0. 25 .t(t2+L.25;t3=Iinspace(2,3r 5);x3=0.4.*t3+0.95;t =Iinspace (3, 4,51 ;x4=ones ( L, length (t4) ) ;x4=x. *2 .L5 ;' t5=Iinspace ( 4 ,5,5) ;x5=-0.15.*t5+2.75;t5=Iinspace ( 5, 6, 5) ;x6=><. * 2 ;plot (t1, x1 ,E2,x2, t3, x3, t4,x4, t5, x5 , t6, x6l ;axis(t0 61 2.51);tj"tIe ( 'Integrator Error' ) ;xlabel ('Time (sec) ');ylabel ('Output Voltager ) ,'lntegrator Enor2.50123456Time (sec)oo)(U6o1.5et438a-8.MCD 13Proportional-l nteg ral ControlTime Functionrtv(t)=Kp'e(t) " Kp rcl I e(t) dtr vo' J0Laplace FunctionK-'K'V(s) = K p'E(s) . -:---t'E(s)Transfer FunctionffiKp(T)Adds one pole and one zero to the transfer functionof the system.Pl control used on process with large load changesProportional action only can not reduce error toreasonable limits.Intergral mode provides reset action that drivessteady-state error to zero. Eliminates offset erroret438a-8.MCD 14IntegralIt4odet\:\ProportionalIr/odeProportional modeerror while integralIntegralIr/odegives instantaneous response tomode decreases error over timeet438a-8.MCD 15Transfer FunctionSimplifyAu(s) =_Proportional GainA v(s)R 1C's * IR;'c'sKp=t/*t 1 \=-\*. r RI.c"/K, = -1-' Ri C-21(s)=_.-=-Z;(s)Rs * 1I C.sR;Select Values of R;,Ry and C and cornpute Bode plotRi=10k Rf=100k C=0.01 pFAdds a pole at s=0 and zero at s = -flRfCet438a-8.MCD 167660a!a o230or(urnAcDuoo -20tnC'f -40-ou-80c-input'( rEnteri value' of, capacltance: t);rf,=input(rEnter value of feedback resistance: t);t compute transfer function model parmgters for* PI controllerI compute numerator parametert,au=rf *c;t compute parameter for denominatortaul = ri-*c;I buil-d transfer functionI denoruinator form a1*s^2*a2s*a3Av=tf ( [tau 1] , Itaul 0] )*n l nf nranhbode (Av) ,'Bode Diagrams-1001011o2 103Frequency (rad/sec)104Integral action below 1000 rad/sec1/RfC sets break pointet438a-8.MCD 17.:Examp|e:DesignaPloPAMF-control|erwith......-,; "*""Eit:'i6o Ena lffint+i5r' oi#n'rrdfitieiic/ of . : '"':j100 rad/sec. Rin = 10kCIR1 v {nn R;^ = 10OOO=Kp so KP::100 Rin =RinR1:= Rin'Kp Rf =1.106 Ohmso = 1000 rad/secr=-L so c:==1- c=1.10-9 FRt'C Rf.t vDerivative Control ModeThis mode is never used alone. Only produces anoutput when error is changingUse with proportional or proportional and integralGives and output that is proportional to the rate ofchange in the error signalAnticipates the error by observing rate of changeet438a-8.MCD 18.:, : Ideal DeriygtlVe Modg.,E-quatigns, . ,,r.iii+*r,:r:: pr; i+ii:-:'., l. ', .., ,;:$fi- Fl*r' -'.' ,.,, ,,'i *l' t tr.1 ,;liti;f{f": 'Time Function v(t) = K D.d "(t)v dtLaplaceTransferFu nctionV(s)=Kp's'E(s)V(s) ,,E(s) "OP AMP Realizations of Differentiatorsldeal OP AMP differentiatorTransfer FunctionVCI^\"---Av(s) =-Ry'C'sIntroduces 1 zeroat s=0Differentiators add a zero to the controller that usesthem.Differentiators are highpass filters to sine signals.Increase sensitivity to rapid changes.et438a-8,MCD 19A / a\^V\D/ =L^\_---_lIntroducesPole s=- 1R i'cZero s=0Pole restricts the high frequency response of thed ifferentiator ci rcu itGain and Phase Plot of Practical OP AMP Differentiator-RfCj'cDA v(ro) = -1 + R;'C's(r* R;'C'j t j0(ar)et438a-8.MCD 20R 1'C'rrl- atan(*,.C.or) TA v(c,l)=90| ,. .Used in processes that have sudden load changesthat proportional only can not handle.EquationsTime Functionv(t) = K p'e(t)+ Kp'Ko qe(t) - o.KD +v(t) * Vo'J dt u dtNote: ,^lo'KD dtv(t)ffi=Kpis a rate limit to highfrequency changesLaplaceV(s) = K p'E(s) + K p'K p's'E(s) o'K p's'V(s)Transfer Function1 + K D'set438a-8.MCD 211 * cr'K p'SOP AMP Realization'of PD Controller, 1.rj1,:;t:;:,.e:'.,-r446i.1.Y"1,fi:,.,.:;.''ll:':,,g$.!fel.&1r',t , t-, r. , ..,,59=-KoE(s) P:1 * J( p'sVRd1R1+ Qf,'K D'sKp=R6'C rr,=Rg. R6R1R1+ R6KP=Ro=RfNote: 1/Ko = the derivative action break point f1/crlQ = limiter action break point f0<cr< 1et438a-8.MCD 22Kp= lo t= /oo roJ/se( *.=


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SIU ET 438A - LECTURE NOTES

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