Question

QUESTION 5 (25 marks) Using Laplace transform techniques, determine the current i, (t) through the inductor in the circuit shown in Figure 5. 202 + Vo w 5 exp(-2t)u(t) V 10 IF + 3u(-t) V + 0.5V 1Η io M Figure 5

Answer 1

T Vo 2-2 + w wm If It c5ē uit) LC LE CCCC _2t ti 341-t) volt O.Svo IH io we have two case here and to to = 0 i fos u(-t) - 1 t<o 4rt) and ult) - 1 t>O: 41-t) 20 Case ir t <o: 2t ult) lie short ciruit = 0 5e-24 341-t) = 3 volt IR tvo £) 3V IF 20.5 V. IH You Now Apply Kaplane Laplane transform nog abone ciriif

222 Iofs) 12 + Vuls- S + s 0.5V(s) 렇 I, som I g(s) - I, TO) we have two loops 4, and is Apply KVL at a al Io, (s) – 11) - $ I, to.5 Vols) = 0 and. I Vols) Vols) = 1 Io, (s) : Toils) 2150, 1s) - Iv) - D 0.5 To, (s) = 0 :) i =0 : 0. Vols) + 3 tita To, () [2 +0.5] + I(-2 - $). 10. 0.5 Tyls) - (2+5) I, Ij = 2.5 Toils)/(2+$) opply kvk at at tz Ž +SIo, (s) 0. Svols) To, (s) + š + S To, (s) - 0.5 tolls) IL, 30 To, is)[1 + 0.5) 7 II + 2.5 foils) Toil) ( 5+0.536 * * * * + + 3 O. 1 +5 - mia f 11 0 of Stl. 8 » >>

islo 2 .): - Io, (s) 115) [ S+0.5 2.5 (45+1) To, (s) 1 (+1) (5+0.5) 2.5 #5+1) I, () = -3 (25+1) s(as? + 5 + 5 +0.562.5) 31354) Toils): - 3125+1) s(232 +25470) S(252 +25-2) Case (1) to. uit) =1 ul-t) - 0 le short circuit 2 tro M ma 12 IF + 5e-20 .O.Svo IH Tols) Connest abone ckt into sdomain 22 112 1 S vo 5 st2 iti 0.5VO 7 Iz sport for(s) Igls)-12

loop sy Apply kvk at loop - + ? (For Lo) – Iz) - Iz +0.8V20 and Vo = Toz (S) St3 - Stz + 2 Iog(s) - 21₂ - Iz + 0.5 for(s) =0 5 - Tal»[2 +0.5) (5+2) + Iz (2+$) (-1) Iz (osts :) Lis Iools) - s 5+2) KVL at loop ty - hody V4 + s Tea Is) - 0 5 % - Ta telepekto C C Coltea Foo (s) + sfor (s) - 0.5 for (3) Too (1+5-05) - Toa (O.S+s) ss Ia ) S > C o > 2.5 Togls) - Toasts+0.5) (25+1) 5 sta کر

Toz (s) (s+0.5) (25+1) – 2.5 5 S +2 Foz (s) - C 5. S+2) [0432 +45 +0.5 -2.5 5. (5+2) (&s? +63-2) Tocals): C Tos)= Io, (s) to fools) on combining r Case (1) and Case cii) 5 (s+2) (252 +23-2) Tois) 3(25+1) s1232+2549 c C C 31045+1) (5+2) -51s) sis+2)(252 +25-2) Tols) = -3(~52 +55 +2) - 55 s(s+2) (as? +25 -2) -652-155-5-6 s15+2) (252 +25-29 C C ☺

Io lb) = -( 632 +205+6) s(s+2) (2,32+25-2) Dols) = -669+2) (s+1) sis+2) (+23-2) To (s) = - 6( 5+3) (s+ s(s+2) (232 +23-2) innerse laplace of Ils) we will get dolt). by taking Molt)=- at blos cethal cosh (2 5+) 3 sº s8 (3 **)}- ] Abone result is diff cult to solne on ney by write the sof), (MATLAR Code written MATLAB Code me easing uning as

Syms S F = (6*5^2+20*5+6) / ((3) * (3+2) * (2* (3)^2+2*3-2)) ilaplace (F)

>> lapades F = (6*3^2 + 20*5 + 6)/(s* (s + 2) * (2*9^2 + 2*3 - 2)) ans (5*exp(-2*t))/2 - exp(-t/2)* (cosh ( (5^(1/2)*t)/2) 3*5* (1/2) *sinh ((5^(1/2) *t)/2)) - 3/2