Question

10-76 The circuit in Figure P10-76 is shown in the t domain with initial values for the energy storage devices. (a) Transform the circuit into the s domain and write a set of node-voltage equations. (b) Transform the circuit into the s domain and write a set of mesh-current equations. (c) With the circuit in the zero state, use symbolic operations in MATLAB to solve for the node voltages. UR ) R W Vol a ich Llo_c() W VAT) yp) i), VoC vs(t)=VA) V ist)=1/ (t) A FIGURE P10-76

Answer 1

Consider the given circuit diagram: VO Voi L 10_10() Wir VA VDO in(o, VO2C2 IBCO vs(t)=VA ) V is(tele () A Figure 1 (a) The voltages of capacitor C, and C2 are Voi and Vo2 and initial current of the inductor is Io. Laplace transform equivalent for vs (t)=VĂu(t)is, Laplace transform equivalent for Is(t)= Ige-fu(t)is, Is(s) = 11 S + a Draw the Laplace transform equivalent for the above circuit. Ls LlO V.(s) : V.(s) Figure 2

From Figure 2, ...... (1) Vą (s) + V1_VA(s) Vg (s)-V) (s)_. R; VA(s) = Apply Kirchhoff's current law at node. Vß (s). ¿ D OP=0 (680); -(s)s+()-10-6 ska(9)+(cs+&%(%)+Voc (@)Po(*)=0 GA()(cist 2007-236() =oC..2 Apply Kirchhoff's curent law at node. Vc (s). re(a)=40-%56, elo, volb)=156) -6 Vo(9–46=@+(r()-Kon}<<($976) -0 Esra (9)+(Es+Cystem rccs)-(79)210-Voc:-(, ) o (5)=0 (5),(6)+(Esteist ær(97-6,7005-(] 1o+yo.C....(3)

Apply Kirchhoff's current law at node, Vo(s). V>(s)-V}(s), V. (s)-Vc(s) = 0 R . R2 sta Thus, node voltage equations are (680)+(as the 600)-(B)<co), 366- 6)• , eo (0)- -.- VA (s)=" -G572(6)+(cis tp 604)36(6)=6oC (1) kno(+6;$this yr67-C )'06)=()40 +Vo.C. (360) Cayrolo)-( * )* (*) - (b) From Figure 2, apply Kirchhoff s voltage law to bottom left side mesh. ņa + L[46-1c0)-40+ d.) 40-16)+"px -0 (1st en 2007-C 156)=Late(o)="_PO4 + 110 (15+ y)246-(0)216)-Lite(6)="s_Von * Los ...,

Apply Kirchhoff's voltage law to bottom right side mesh. la(s) =- we ...... (6) Apply Kirchhoff's voltage law to top mesh. Llo +[$[1c(9)-4a()]+(c ) 10 (5)+ You + R/C($)+R[(5)–19()]=0 +(Ls)ZA(s)–R:!3 (5)+(Is + this+ R, + R 16 ()–LI- YOU LIOS +V01 -(Ls)IA (S)-R21B (s)+ Ls+= +R1 + R2 Ic(s)=- Cs The mesh current equations are 1 _VA-V02 +LIOS IB (S)-LsIc(s)=- CS Ls + (s) 1₂5) 100) -

(C) It is given that the circuit is at zero-state. The s-domain equivalent circuit for the circuit shown in figure 1 is, cs Vos Figure 3 From the above circuit we have VA(n)="A ...... (8 Apply Kirchhoff's current law at nodeVB(s). V8 (s) –VA (s) VB (s)-Vp(s), R - +- R1 (V6 (5)-VA(;s+ $(9)=* (*) -6 -6,%(+(Gs+)68053-6)()=0 ...... (9)

Apply Kirchhoff's current law at nodeVc(s). Vc(s)-VA(s) Vc(s) Vc(s) – V) (s) - 0 IST R + +- R Vc ()-AO)+(66)C>${"c()=600)=0 C)*6+ E +c=5+ -)ko (0) = o... (10) Apply Kirchhoff's current law at node, Vo(s). 1 (05)-1(0), 65 (O)=VC(OLEO Como(0)=(@)pc(*)+(te ($)-64.. (11) ) + — +- ...... (11) S+a Equations (8) to (11) are in standard form (117A ($)+()Vo(s)+(OV.(*)+%(5)=” G465)+(+)po 0+ovc(-)-(366)-0 )()+(0)73(9)+(Ls ++mPC($)-C 36(5)=0 (ow()-&)(0)-2676(5)+(8 + 16 (6) + — + C2S+ :)=0

Above four equation in matrix form as below [ 1 0 -48 Gs+ h (97 +Cos + Ls Vo(s)] IA LS + a Write the MATLAB code. >> clear all >> syms s R1 R2 C1 C2 L VA IA alpha >> admitance=[1 0 0 0;-Cl*s Cl*s+ (1/R1) 0- 1/R1;-1/(L*s) 0 (1/(L*s)) +C2*5+ (1/R2) - (1/R2);0 - (1/R1) - (1/R2) ((1/R1) + (1/R2))]; >> current=[VA/s;0;0;IA/ (s+alpha)]; >> voltage=inv (admitance) *current; >> simplify (voltage) ans = VALIA* (C2*h = C1*R**2* VA/S (IA* (C2*L*R2*s^2 + L+S + R2)) / ((alpha + s) * (C1*R1*s + C1*R2*s + C1*L*s^2 + C2*L*s^2 + C1*C2*L*R1*3^3 + C1*C2*L*R2*s^3 + 1)) + (VA* (C1*R1*s + C1*R2*s + C1*L*s^2 + C1*C2*L*R1*s^3 + C1*C2*L*R2*s^3 + 1))/(s* (C1*R1*s + C1+R2*s + C1*L* s^2 + C2*L*s^2 + C1*22*1*R1*s^3 + C1*C2*L*R2*3^3 + 1)) (VA*alpha + VA*S + IA*L*s^2 + C1*L*VA*S^3 + C1*R1*VA*S^2 + C1*R2*VA*S^2 + C1*R1*VA*alpha*s + C1*R2*VA*alpha*s + C1*IA*L*R1*s^3 + C1*L*VA*alpha+s^2)/(s* (alpha + s) * (C1*R1*s + C1+R2*s + C1*L*5^2 + C2*L*s^2 + C1*C2*L*R1*s^3 + C1*C2*L*R2+5^3 + 1)) (VA* (C1*R1*s + C1*R2*s + C1*L*$^2 + C1*C2*L*R2*3^3 + 1))/(s* (C1*R1*s + C1+R2*s + C1*L*s^2 + C2*1*9^2 + C1*C2*L*R1*s^3 + C1*C2*L*R2*5^3 + 1)) + (IA* (C1*R1*s + 1)* (C2*L*R2*^2 + L*S + R2))/((alpha + s) * (C1*R1*s + C1*R2*s + C1*L*3^2 + C2*L*s^2 + C1*C2*L*R1*s^3 + C1*C2*L*R2*^3 + 1))