Impedance Method: An RLC Circuit For the electric circuit in Example 6.4, use the impedance method to derive the differential equation relating the output voltage vo(t) to the input voltage va(t). Assume zero initial conditions.

Question

Impedance Method: An RLC Circuit

For the electric circuit in Example 6.4, use the impedance method to derive the differential equation relating the output voltage [latex]v_{o}(t)[/latex] to the input voltage [latex]v_{\mathrm{a}}(t)[/latex]. Assume zero initial conditions.

Accepted Answer

The original electric circuit is shown in Figure 6.55a. We can replace the passive elements with their impedance representations and redraw the circuit in the [latex]s[/latex] domain as shown in Figure 6.55b. Note that the resistor [latex]R[/latex] is in parallel connection with the capacitor [latex]C[/latex]. The corresponding equivalent impedances are

[latex] \begin{aligned} & Z_{1}(s)=L s \ & \frac{1}{Z_{2}(s)}=\frac{1}{R}+\frac{1}{1 / C s}, \end{aligned} [/latex]

or

[latex]Z_{2}(s)=\frac{R}{R C s+1} .[/latex]

For the equivalent impedance circuit in Figure 6.55c, we apply Kirchhoff's voltage law,

[latex]Z_{1}(s) /(s)+Z_{2}(s) I(s)-V_{\mathrm{a}}(s)=0,[/latex]

where the current is related to the output voltage by

[latex]V_{\mathrm{o}}(s)=Z_{2}(s) I(s).[/latex]

Thus, we have

[latex]Z_{1}(s) \frac{V_{\mathrm{o}}(s)}{Z_{2}(s)}+V_{\mathrm{o}}(s)=V_{\mathrm{a}}(s),[/latex]

which gives the transfer function relating the input voltage [latex]v_{a}[/latex] and the output voltage [latex]v_{o}[/latex],

[latex]\frac{V_{\mathrm{o}}(s)}{V_{\mathrm{a}}(s)}=\frac{Z_{2}(s)}{Z_{1}(s)+Z_{2}(s)}=\frac{R}{R L C s^{2}+L s+R} .[/latex]

By transforming [latex]V_{\mathrm{o}}(s) / V_{\mathrm{a}}(s)[/latex] from the [latex]s[/latex] domain to the time-domain with the assumption of zero initial conditions, we obtain the differential equation of the system

[latex]R L C \ddot{v}_{\mathrm{o}}+L \dot{v}_{\mathrm{o}}+R v_{\mathrm{o}}=R v_{\mathrm{a}^{\prime}},[/latex]

which is the same as the one obtained in Example 6.4.

Question 6.12: Impedance Method: An RLC Circuit For the electric circuit in...

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