Fe exam prep: rlc circuit phasor notation

  • electromagnetics
  • phasors

posted on 02 Sep 2016

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).

  • (a) Write the voltage loop equation in terms of current \(i(t)\), R, L, C, and \(v_s(t)\).

  • (b) Obtain the corresponding phasor-domain equation.

  • (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

\[\begin{aligned} \tilde{I} &= \frac{\tilde{V_s}}{R + j \omega L + \frac{1}{j \omega C}} \\ \tilde{I} &= \frac{V_0 e^{j \frac{\pi}{3}}}{R + j \omega L + \frac{1}{j \omega C}} \\ \tilde{I} &= \frac{ j \omega CV_0 e^{j \frac{\pi}{3}}}{j \omega CR + j \omega Cj \omega L + 1} \\ \tilde{I} &= \frac{ j \omega CV_0 e^{j \frac{\pi}{3}}}{1 + j^2 \omega^2 Cj \omega L + j \omega CR} \\ \tilde{I} &= \boxed{\frac{ j \omega CV_0 e^{j \frac{\pi}{3}}}{1 - \omega^2 Cj \omega L + j \omega CR}} \\\end{aligned}\]