Question

i(t) 5 k Ohm R2 10 k Ohm V1 C1 2 UF 2i(t) W1 12V t-0 CSW Figure 3 a) Find the capacitor voltage v(1) for > 0, for the circuit in Figure 3 above. b) Using appropriate equations show that the system in Figure 3 is unstable. c) Using appropriate equations and diagrams show what must be done to make the system stable. [4 marks] [5 marks) [6 marks]

Answer 1

Que som Given ckt as > ilt) All indicated Value on ckt is already given question. 5kn + Re lokn + 2 144) (+ Vect) 12V 2MG الا sw tro (9) Calculation Velt) for too to In ckt The switch (wis is closed at one dependent source is also given circuit becomes. by closing the switch, the Step mm Skn it) + R lokn ✓ aict) G lay t quf Velt) The Value of Velt) can be coloulated by the lablace transform method which is quite easy eſprachable In Laplace the capacitor c is gepresented as and est- CS and all the Resister Remans Same The and Currents can be written in their aplace Voltag foon.

Step 2 → as given below The laplace equivalent ckt becomes RI Vc(s) P Р 5 kr 175) IP R2 lokohm Vi (s) sa 27%) (V 12 V G=2uF (P is assummed Point) Taking o as reference Point. Kel at Point P. By Applying TIS) + 2 Ils) + Vels) Vels) + R2 1/5G Vels) I(5) + VelS) R2 + a 1154 b here I(S) = V. (5) Vels) RI foom equation ; put the Value of I(s) So equation to becomes. in en @ Vils) - Vels) + Ve(s) R2 + Vels) /sa RI Now we have Vi (5) 12 R = 5kn, R2 = 10kn, 4=2uF > By putting all the valen in egn 1215 - Ve(s) Vels ) Vels) + + 5K TOK 's(ax106)

By rearranging the tears $x 10-6x2 12 Vels) + 5x12 t 70x1072 5(5K) 12 + 2x156s f 315x703) Vers) [ **] rets 10****) Vels) (-10%*+221045 + 2x1033) (-01-2300] + 12 X 10 X10-3 55 Vcls) ( to tax10 9.4 10 + o y Vels) -2-4 ş . => VIS) -2.4 *** -O.17 @xluss x1095) V (5) 2.4 2116-354-50+5) Vis) = 1200 C S (5-50 d By partial fractional Rule V(S) - А B + S-50 A is given A 1200 lim So 11 1200 S-50 = -24 0-50

Bis given By 1200 12 00 B= lim s S-50 50 B = 24 By Putting A = 124, B= 24 in equation (d Vls) 24 -24 S + 50- كه By taking Inverse both sides Velt) 50t -24 Ult) + 24 e ult) Velt) (-24+24 e sotJv; v; +>0 in figure So the capacitor Volterge velt) fro, tso, for the circuit Justo. som is esot ...]vstro 24 Parot (b) wwith the help of Paot (a). we have a equatim (@ in part (a). By referring in. Part (a) So Vels) -24 24 t S oo 5-50 $ = 50, which lies sight The roots of Vels) is s plane Side of If any

the If any lies 00t side of s plane 5 signe becomes unstable. system Velt) Value. 07 We com understand by we have sto Velt) = is The exponential form in Velt) anfest 17 st increasing --]" So at t=0 sel 50X 00 Vc (t=00) 94 re Vel=a) = 0o v That is unstable system. Conclusion so the system If any ooot lies right side ofis-plane ®, becomes unstable fen of increasing If any exponential tem is in order system The m. becomes instable

Past) from To sto make System Unilable to stable : Step o = - In order to make system Stable ) we have to do a mechanism in which the odot of V(s) lier in left side s Plane. so Now In order to do that we are the Current Source which is dependent of removing Step@: The circuit becomes 1 R = 5kn 17 • R2 = loke G + Vill) 12V quf Noting that switch is already closed at tro To solve the Velt) Now We Can lise ) any approach. Gon across Capacitor In Conventional Approach Voltage across Com be given as below -t/Rc Velt) - Vcloo) + (veco) – velco) e Wher Ve 100) Voltage acrooss Capacitor at t=0 see Velo) Voltage acouss Capaitos at tao Circut time Constant. equivalent Resistance acouss rests capautoo. Capacitor lralue Rc R C

Step © > Calculatim Perot, Now R Calculatim R= R = 5kn R2 = 10kr Soyoce R is calculated and resistanco deactivating all voltage is seen from Capacitor by 5K R TOKS 10 K x 5K R = (10k) || 5K = - 50 15 TO K+ 5K 10 R = 3 Velo) = V(10) ov (because before t-o, capacitor cuas totally dischared ) at t=0, Vc(oo) 5k 10 X 12 L) 12V lok 10+5 Vc100) Velco 10 X12 15 120 Vc100) 8 Volt 15 Now - 8 volt Vclou) 3 Al Values is R = lºkn, C=2MF, Velo) = OV, putting in en -- +/(x10x28100) 8 + (0-8)e so com Vel)

Velt) Bli-e-tris) Isot Velt) = 8(1-e :) volt it>o so decreasing by seeing the Vel), the Velt), we have exponential team which is e system em becomer stahle 150t so Now Conclusion: The Stable System is R,=5kn lilt) Vi 12V R2 lok + G 206 CSW and this stahle system The value of capauter voltage is 8(1-e-150t) v.