Question
Show that the amplitude of the steady-state current in the $L R C$ -series circuit in Example 10 is given by $E_{0} / Z$, where $Z$ is the impedance of the circuit.
Step 1
We have a $LRC$ series circuit and we are asked to find the amplitude of the steady-state current. The amplitude of the current is given by $E_{0}/Z$, where $E_{0}$ is the peak voltage and $Z$ is the impedance of the circuit. Show more…
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Show that the impedance of a series $R L C$ circuit can be written $$Z=\sqrt{R^{2}+\omega^{2} L^{2}\left(1-\omega_{0}^{2} / \omega^{2}\right)^{2}}$$
$\bullet$ (a) Use the phasor diagram for a parallel $R-L-C$ circuit (see Figure 22.21 ) to show that the current amplitude $I$ for the current i through the source is given by $I=\sqrt{I_{R}^{2}+\left(I_{C}-I_{L}\right)^{2}}$ (b) Show that the result of part (a) can be written as $I=V / Z,$ with $1 / Z=\sqrt{1 / R^{2}+(\omega C-1 / \omega L)^{2}}$
Determine the current in the general RLC circuit with $R^{2} < 4 L / C,$ if $E(t)=E_{0} e^{-a t},$ where $E_{0}, a$ are constants.
Linear Differential Equations of Order $n$
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