Question

Section 3: Laplace transform for RLC circuit analysis (10 marks) A second-order RLC circuit with a dependent source is shown in Fig. 3. 22 + VO - 132 1F + 15e-2 u(t) V ) 9[1-u(t)] V Y 0.50 » Fig. 3 A second-order RLC circuit with a dependent source Take the Laplace transform of the circuit and hence find the response io(t) for t > 0. Specify whether it is an underdamped, critically damped or overdamped case. Sketch the response.

Answer 1

connected solution step assume anitially cirwit is for tco. corront trough srost carvite are anductor soo t o is 10 = 9 – 3A. 2.12 a 0.SK Affry kit so he given direction of Loop. 1. Lo + VGA 65% av for tto. Circuit whese Von – x9, 10) = x3 = -37 1*3 +VO) 40.5% (-39=9v 3+1D) -1.5=9=> VE) = 7.5V i. Initial conditions are 10) =118 - 12) = 3A V16) =V=Y (7)=75V Stepe: now assume that the ciswit is connecter for to, Apply krz is the Loop don 1.10+ 1.diosts - 0.5% = 1 Surladt ? dt =0 5e vol 500 listy + diot -ashots & Stadt dt t circuit for tso. Apply Laplace transtorn to the above equation.

LoL) + 3 Toks - Soy-0.599) - E-S-Te Voy - Se + 50.53903) - 3-50 Iat 15²0.55-1) Do's) = 10.55 → Afflz Kr. en Loop2. sia & Siastoly Jat + 0.31 = 15e 21a+ Slanity Idt & asi = 15e2t Apply Laplace Transform to the above eah. z lat Ia-Iols + rol+ 0.5 Tds = 15 SAL 25 Pet Ba-Iols) + 517.5+0.55Pols) = 155 St2 (25+18 + (0-55-1990's) = 155 - 7.55 = -7.55 S+ (25+1) Ia = -7.58² (1-055) Bols) Ia = -71582 + 1-0.55 Tolso. (Sta) (2541) 25+ substitute Ta aman Ios) + [33+0.55-1] Bols) = 10.55 -7. 52 E & 1-0.55 ( 24) 1st2) 25+1 =10.55. 2 st 125t) (SAS) + [1-0.55 +(2541) 2570.55-1]] Tols) [1-0.55 + 25 + 25 +liss-Bols) = 10.55 + 7.552 1854) (+2)

3 [23+25+ 1] Sois) = s [9.5 (2541) 6523+ Tussy (25H) 1S+2) (25+25+1) Iols) — 215² + + 21 +7.55 2st - 52.55 Pols) = 2157 395+21 = 3 75TLBS+7) 123735+)(542) (23+25+1) sans Pors) = 2,8 bost21 (25 +251) 1ST) partial fraction gethod to fols). Apply 215 +60s +21 = ASTB a u (25+25+1) (A2 25 72st sta 2153 +6Ost2 = (ASTB) (ST2) + c(23372541) (25725+1) [S+2) (23725+) (ST2) 345+ cost21 = (A+20)3 +(24448426) 5+(2842) compose s power coefficier to on both sides of above avation. Atac =21 2+B+ 20=60 2B+c=21 solve above equations. A-27, B 12, C--3.

a Pols) = 27St12 - 3 25725t = 13.55+6 stas - 3 sta Tols) = 13.55th sts (S+0.5)+(6.52 Jos) - 1358 (sto.5) – 1.5x os (sto.5) 710.52 5+0.5) 70.53% - Apply Doverge Laplare Transform to solo . -0.55 -0.5t costat ze 2- 13.5e cos o.5t - 1.5e sin ast-e .. - 2 -0.5t l (13.5 coso.5t-1.5sinost-se Josy forto 3) The gives Cerwit is a onden - dampede

Response of Tolt) : -0.5XT got to 10o = 113. 5coso - 1.5sino -30 = 30.5 * nt tay Tony =ē 12510581559)-3237 Ni At tot, Tolt)= -0.317 01435.. ...... r) At =3T, Tolli -8032 At t=2T, 12ty=-0.583 3514 Foto 312 2 --0-54 NH Response of lolly