Question

Question 1 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] 192 192 192 m V:(t) 1 H 1F volt) Figure 1

Answer 1

Solution: 122 122 122 Ila =Vc 1 H 1 H 1F volt) As we can see, here three storage elements are present let current inductor is į, , ¿z Vc and voltage across capacitor If we apples nodal analysis at node and VL2 we get- At node vc, -V t ILI + V., -VL2=0 nade vi ) Vi - 2v, -V2 - Vį tIs=0 sence, V, = h, d Izi Ili dt . L=1 L, là H e V2= 월 2 Ini I22 + ILI -vio

iL2 = 212, + Ili - Vi At node V2 Vz-V + Ilz + Vez-Vc=0 > 2 V2 - V1 + I22 -Ve=0 I Li +2 122+ Iz - V=0 from ear (2) - Ili + 2(2 İut Il-vi) + Ilz-Vc = 0 - Iut 4 ILI +2 I - zvi tt I22-vc=0 3 ILI - 2 I-ILz tv tzvi =

» Thi 2.IL - -|Izt I vete vi 3 (ii) 3 3 3 So, I 2 = 21-23 I L1 - 11 IL 2 + 1/ 3 ret 2 vi) + I., - vi ( 3 + 1) = 1 - 2 fiz r 2 v1 + (1 - 1) vi 12 * 1.- EL2+vc + javi - 3 – (ii) At noke Vc, Vc-\z + cidve =0 It Vc - Id2 + vc co {:C=1f}

3 ů= Izz - Vc C " 3 Iul 2. 3 Ilz. 2tVit르 vc 3 V 느 - 그리 1 C 2. 3 IL2 3 tv) ret! vi 3. 3 from eq (1), (ii) & (iv) 놈 ILI - 2 3. 3 ILI Ilzl 2. 3 + vi - 3. 들 들 은 들 2. 3 - ". 3.

In State Space representation X = A xit) + BULE) Y = C Oc(t) + D ult) Hore - Ć - 2 3 Ill H = -ایر) B = ult)= vi Ila wilt with win ابرا / ما داده 2 3 ما يم) - ادم -دم 3 - 2 3 3 3 - Vc so, Y is output and here Ocelfut Y = [o o ] Thus, c=[ool C: [ 0 ], 4 = 0 Ini + Izz Vc

b w/- wla win 3 A= 1-2/2 - 1 1 B = C = Cool], d= [o] 1/3 - 2/3 - 1 3 - 2 1 / We know transfer function is given as, Y(s) a(s) C(sI-a) 's B t 015) where, I = Unit matrix But it can be easily solved MATLAB using. 552&f fraction S 35²+25²+45ti - en 2 2 we got, la(s) = I S 3 s3+² st s+I 3 3

- Х X +7X Workspace +1 O 7 X L Select data to plot AMATLAB 7.12.0 (R2011a) File Edit Debug Parallel Desktop Window Help % ... Ė Current Folder: C:\MATLAB Current Folder It O Command Window U.SSSS -0.000T 0.0007 C: MATLAB E- -0.3333 -0.6667 -0.3333 Name wave_generated_again.m >> B=[2/3,1/3,1/3] I trademarks.txt B = patents.txt license.txt = uninstall 0.6667 = toolbox 0.3333 0.3333 - stateflow >> C=[0 0 1] Esimulink - runtime artw - resources 0 0 1 - notebook - My Program >> D=0; E licenses >> [NUM, DEN]=ss2tf(A,B,C,D) . lib Name Value Min Max HA (-0.6667,-0.333... -0.6... 0.6... HB [0.6667;0.3333;... 0.3... 0.6... HC [0,0,1] 0 HD 0 0 0 DEN (1,1.6667,1.333... 0.3... 1.6... EN... [9,0.3333,1.110... 0 0.3... 1 - sys > java NUM = Details 0 Command History +1 O 7 X dsolve (odel, initials) clc clear all "odel=('Dx=1000-10*x'); initials-('x (0)=0); x=dsolve (odel, initials) 12-07-2009 00:00 --% "A=[-2/3 -1/3 1/3; -1/3 "B=[2/3,1/3,1/3] C=[0 0 1] D=0; [NUM, DEN]=ss2tf(A,B,C,D) 0.3333 0.0000 0.0000 Select a file to view details DEN = 1.0000 1.6667 1.3333 0.3333 fx >> Start OVR