Ulaby Problem 1.28
Problem Statement
A series RLC circuit is connected to a generator with a voltage \(v_s = V_0 \cos{(\omega t + \frac{\pi}{3})}\) (V).
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(a) Write the voltage loop equation in terms of current \(i(t)\), R, L, C, and \(v_s(t)\).
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(b) Obtain the corresponding phasor-domain equation.
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(c) Solve the equation to obtain an expression for the phasor current.
Solutions
Part (a) Loop Equation in Time Domain
\[v_s(t) = \boxed{i(t)R + L \frac{di(t)}{dt} + \frac{1}{C} \int i(t) dt}\]
Part (b) Loop Equation in Phasor Domain
\[\tilde{V_s} = \boxed{\tilde{I}R + j \omega L \tilde{I} + \frac{1}{j \omega C} \tilde{I}}\]
Part (c) Solve for Phasor Current
\[\begin{aligned} \tilde{I} &= \frac{\tilde{V_s}}{R + j \omega L + \frac{1}{j \omega C}} \\ \tilde{I} &= \frac{V_0 e^{j \frac{\pi}{3}}}{R + j \omega L + \frac{1}{j \omega C}} \\ \tilde{I} &= \frac{ j \omega CV_0 e^{j \frac{\pi}{3}}}{j \omega CR + j \omega Cj \omega L + 1} \\ \tilde{I} &= \frac{ j \omega CV_0 e^{j \frac{\pi}{3}}}{1 + j^2 \omega^2 Cj \omega L + j \omega CR} \\ \tilde{I} &= \boxed{\frac{ j \omega CV_0 e^{j \frac{\pi}{3}}}{1 - \omega^2 Cj \omega L + j \omega CR}} \\\end{aligned}\]