Related questions with answers

The average power dissipated in a series $R L C$ circuit is $\bar{P}=\frac{1}{2} I_{\max }^2 R$ , where $I_{\max }$ is given by Eq. ( $I_{\max }=\frac{\mathcal{E}_{\max }}{Z}=\frac{\mathcal{E}_{\max }}{\sqrt{R^2+[\omega L-(1 / \omega C)]^2}}$ ). We can define the width of resonance in terms of the half-power points $\bar{P}_{1 / 2}$ of the resonance curve; these have half the peak value, or $\bar{P}_{1 / 2}=\frac{1}{2}\left(\frac{1}{2} \mathcal{E}_{\max }^2 / R\right)$ . By setting $\bar{P}=\bar{P}_{1 / 2}$ , obtain an equation for the values of $\omega$ at the half-power points. Solve the resulting quadratic equation and show that the two roots are separated by

\Delta \omega=\frac{R}{L}

Thus we can verify the equivalence of Eqs. ( $\mathrm{Q}=\frac{\omega_0 L}{R}$ ) and ( $\mathrm{Q}=\frac{\omega_0}{\Delta \omega}$ ), $\mathrm{Q}=\omega_0 L / R=\omega_0 / \Delta \omega$ .

A series $R L$ circuit is often used as a low-pass filter. Consider a resistor $R$ and inductor $L$ connected in series to an oscillating source of emf with amplitude $\mathcal{E}_{\text {max }}$ . In terms of $\mathcal{E}_{\text {max }}$ , what is the amplitude of the voltage across the resistor for a low frequency, when $\omega L / R=1 / 10$ ? For a high frequency, when $\omega L / R=10$ ? At what frequency is the amplitude of the voltage across the resistor equal to $\mathcal{E}_{\max } / \sqrt{2}$ ?

A series $R C$ circuit is often used as a high-pass filter. Consider a resistor $R$ and capacitor $C$ connected in series to an oscillating source of emf with amplitude $\mathcal{E}_{\max }$ . In terms of $\mathcal{E}_{\max }$ , what is the amplitude of the voltage across the resistor for a low frequency, when $\omega R C=1 / 10$ ? For a high frequency, when $\omega R C=10$ ? At what frequency is the amplitude of the voltage across the resistor equal to $\mathcal{E}_{\max } / \sqrt{2}$ ?

Question

Consider that an emf is connected to a parallel $R L C$ circuit. In this case, the voltage $V_{\max }$ is the same across the source and each element, but the net current is the sum of the individual currents, so a phasor diagram for the currents can be drawn instead. Draw this diagram. By dividing the magnitude of each current phasor by the common value of $V_{\max }$ , obtain a phasor diagram that relates impedance, reactances, and resistance. Use the geometry of the diagram to show that
$Z=\left[\frac{1}{R^2}+\left(\frac{1}{X_L}-\frac{1}{X_C}\right)^2\right]^{-1 / 2}$

Solution

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In this problem, we have a parallel RLC circuit connected to a source with an amplitude of $V_\text{max}$ as shown in the figure below. Our objective is to draw the net phasor diagram for the currents in this circuit. Also, we want to determine an expression for the net impedance in the circuit using the phasor diagram geometry.