Question

a) Consider the electrical circuit in Figure 1. Determine the state space representation of the circuit where the output is voltage across the capacitor. (6 marks) b) From result obtained in (a), predict the transfer function. (4 marks) [10 MARKS] 1 Ω 192 112 W vi(t) 1 H elle 1 H Illl 1 F volt) Figure 1

Answer 1

solution O M dz , IH lu Vo(4) IF le We assume 20,- én dla for dt x2= e, icz dlzz S #3 Vc at x 3 = dvc at Now VL, = Vir did, Vi-Vc leit ( e, -ée) (e-ex, ez) ber l3 ei = 2/3 es, + 3 in - 33Vc + 1/3 vi + 00 value of I put in ③ VL - Zee, - 13 lis + 2 Ve + 2 vi CS

2 put ve = L, deu der dt dt dee, 2 dt ei, - 1/5 es, + 1/ s Uc+? Vi similar to *, = -2x, -1/ X₂ + 1/3 x3 + 2/3 u Now: V;-V2 = 2 V; - VL - é; (b) + ez (2) put lz= ei-ee, vi u ₂ = zei-ée value of e, taken from (8,3 bei ď v tý v;) ent 2 Now Vez: -24,+ le, tu; Jan put here V2 2 = -1/3 ei, - 2/3 ecz +2 33 Vc + }}"; dina -) 2 = Lui La 2 (2 + 2 / 3 Vc + 1 at ICS

im it is similar to x = -1/32, 0 - 2 / 3 x₂ + 2/3 X3 + 1 2 3 1 B Now le = ls= ez-ein put ezt ei-e le=e,-ec, - ei (put value of e, from de = -1/3 le. - 2 / 2 - 3 Vc to vi i Gisesinttanto lc duc dt le = 2 it is similar to is **-**-* CS Scanned with CamScanner

from @ ► and . 를 C + 13 13 같지 름자 구름성을 2 넣절름자 공적 같이 퉈 Xi 3 [] 22 늘 U X3 3 where X, éi élr vc 3 dle, 131 at diez at CS

6 y = Vo = vc=X3 y = ox, + 0x2 + 1Xz+ou 1 y = [ool ] (o=o) we need to function calculate Transfer from @ "-2/3 -113 / AS -1/ 3-2/3 2/3 ,-/ -2% -13 B= 2/3 13 13 c= [oo] b=0 Transfer function: c (SI-A)*B+D - © CS

23 1 개 73 13 23 ㅔ 23 개 R R3-R2 3 ,R>-12, 2 - 1 A- [] - C, → CG-2 ㅔ3 시13 13 A= C 3 옄 1, AC O 0 - ( 2C24 CS ned with canner

© -1/3 Va Ao O 0 - 1 g» (₂+(3 1/3 -2/3 A= 6 O Now this is like ann -anoz ao A = O 1 and h. 2 bras bo + Tof.- st any shit cs Scanned with CamScanner

where y: [bure bnrz' 6.] com this with value of A and o pore get we 92 = '3 a = 1,96=2/ boot bz0, I T. F. = 83 3 8² 18+} ANS 8 Both parts are solved here. ICS Scanned with CamScanner