Question

Question 4: Consider the circuit below. The voltage sources are cos(2t) and cos(t). From left to right, the component values are XQ2, 2X Henries, and 2/X Farads. The voltage Vo(t) is the voltage on the inductor. You are required to use Fourier Transformation to analyze this circuit. a) Find Vo(6). (10 pts) b) Find vo(t) by applying inverse Fourier transformation. (15 pts) XR F dx Volt) __ cas at cost $ 2xt

Let X (which is the last digit of your id) be 8. If the question involves a component value of 4X, that value is 4.8=32

x=8

Answer 1

~ 4) X=8 8w AM 114F HE 16H +1 It Cosat Cost a) Taking Focerier Transform. 8 W Volw) Aliw HE x ti Y = + 7 [SC0-1)+8(WH)] le 16 jw = = T[86-2) + scato)] Apply Ксе at node Vocw), Vo-X Vo + + 16 W vo-Y (415w) =O L Y 4 Viesu + 59 ] = Spy Vo C ajw+'+ 48w"] = + je Vo Cws - hebreo + 13 Y 16jw

Vo = ဒ0 46w+ajw+1 [x + @wy] Volw) 2 ) [s-w4]tajw 11 [80W-2) +(wta)] 461 •TJ [8(W-1)+ s(w+) [1-wa] taiw '; X(6). 86w-wo) = X(W) SCO-wo) QjQ2) Vo (W) = 11-16)+ j4 TT. [ 8(W-] 25 (-2) + TJ 8(6+2) 11-(-2)"4] +aj(-2) 4(1) TT 8(0-1) [1-14)]+aji 4 (-)" T 8(6+1) [1-6-04]+ajco) Vocw) - TT. 86-2) 45 -15+j4 75-14 T 8(wta) 4 IT 86-1) - - taj 4 IT 8(6+1) -3-21

Vocw) 4j -15-14 -151475-14 X T 8(6-2) 4j -15 tj4 15+j4 T scwra) --15-14 4. X -3+aj -3-aj -3-aj T SCw-l) -3taj 1 - 2 x 3 +23 IT scw+i) -3-2) 16 - 360 TT 8(6-2) - -16-36o. TT 8(wa) 241 241 241 - 12 h 175001). -14,418 11 8w) Yow) = 46. [1 (566-2) + SCw+a)] (S06-2)- 8(6+2))] + ten (t C6604) +SC+1)]] + [T (8Cw4) - $(w+i))] b) Taking Inverse -j6o 2417 T Fourier transform

16 Cosat Vo(t) = 360 (j sinat) 241 241 12 cost + ј8 13 Cisinat ) 13 Vo(t) = 16 241 60 Cosatt sinat 241 + he cost + sint