week102 - R C Circuit Response Kirchoff Law ut G RQ t 1 R 0...

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R C Circuit Response Kirchoff Law u H t L = R Q L H t L + G Q H t L , t > 0 B G = 1 C F R > 0; C > 0 R ' Resistance' is measured in ' Ohms'. C ~ ' Condensor'. ' Capacitance' C is measured in ' Farads'. G ' Elastance' v H t L = G Q H t L = ' Output' voltage u H t L source H ' Input' L voltage t 0 ** ** ** ** ** ** ** ** ** ** ** ** ** k = G R = 1 RC Q H t L = e - k t Q H 0 L + 1 R 0 t e - k H t - s L u H s L s, t 0 v H t L = e - kt GQ H 0 L + k 0 t e - k H t - s L u H s L s t > 0 . GQ H 0 L = v H 0 L = e - k H t L GQ H Τ L + k Τ t e - k H t - s L u H s L s, t > Τ . GQ H Τ L = v H Τ L ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** Printed by Mathematica for Students
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Rewrite as Q L H t L = - G R Q H t L + 1 R u H t L Differential Equation : x L H t L = - k x H t L + 1 R u H t L , k = G R = 1 RC SOLVE Technique of Solution Details Multiply both sides by e kt H a nonzero number ! called ' Integrating Factor' L : and get : e kt x L H t L + k e kt x H t L = 1 R e kt u H t L Left - side = d dt I e kt x H t LM Hence d ds I e ks x H s LM = 1 R e ks u H s L s > 0 Take Τ Τ £ t Then integrate from Τ to t on both sides Τ t d ds I e ks x H s LM s = e kt x H t L - e k Τ x H Τ L = Τ t 1 R e ks u H s L s Or, x H t L = e - k H t L x H Τ L + 1 R e - kt Τ t e ks u H s L s = e - k H t L x H Τ L + 1 R Τ t e - k H t - s L u H s L s ** ** ** ** ** ** ** ** ** ** ** 2 review102.nb Printed by Mathematica for Students
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Now back to our problem.
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