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respSin - RC Circuit response to Sinusoidal Input ut A Sin...

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RCCircuitresponse toSinusoidalInput u H t L = ASin @ 2 Π ft + Φ D v H t L = e - kt v H 0 L + A 0 t e - k H t - s L Sin @ 2 Π fs + Φ D s 4Sin @ 20 Π t + Π 4 D , 8 t, 0, 5 < 1 2 3 4 5 - 4 - 2 2 4 A 0 t e - k H t - s L Sin @ 2 Π fs + Φ D s = Ae - kt 0 t e ks H Cos Φ Sin2 Π fs + Sin Φ Cos2 Π fs L s = Ae - kt B Cos Φ 0 t e ks Sin2 Π fsds + Sin Φ 0 t e ks Cos2 Π fsds F e ks Sin2 Π fs = e ks Ime i2 Π fs = Ime k + 2 Π is H kisreal ! L
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0 t e ks Sin2 Π fsds = 0 t e ks Ime i2 Π fs ds = Im 0 t e ks + 2 Π ifs ds 0 t e ks Cos2 Π fs s = Re 0 t e ks + 2 Π ifs s = Im 0 t e H k + 2 Π if L s s = Im e H k + 2 Π if L t - 1 k + 2 Π if e H k + 2 Π if L t - 1 k + 2 Π if = e H k + 2 Π if L t - 1 k + 2 Π if k - 2 Π if k - 2 Π if = I e H k + 2 Π if L t - 1 M k 2 + 4 Π 2 f 2 H k - 2 Π if L since H k - 2 Π if L H k + 2 Π if L = k 2 - H 2 Π if L 2 = k 2 + 4 Π 2 f 2 = A k I e H k + 2 Π if L t - 1 M - 2 Π if I e H k + 2 Π if L t - 1 ME I k 2 + 4 Π 2 f 2 M 1 k 2 + 4 Π 2 f 2 I ke kt H Cos2 Π ft + iSin2 Π ft L - k - 2 Π ife kt H Cos2 Π ft + iSin2 Π ft L + 2 Π if M = 1 k 2 + 4 Π 2 f 2 AI ke kt Cos2 Π ft - k + 2 Π fe kt Sin2 Π ft M + i I ke kt Sin2
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