Impedance Method: An Op-Amp Circuit For the op-amp circuit in Example 6.10, use the impedance method to derive the differential equation relating the output voltage vo(t) to the input voltage vi(t). Assume zero initial conditions.

Question

Impedance Method: An Op-Amp Circuit

For the op-amp circuit in Example 6.10, use the impedance method to derive the differential equation relating the output voltage [latex]v_{\mathrm{o}}(t)[/latex] to the input voltage [latex]v_{\mathrm{i}}(t)[/latex]. Assume zero initial conditions.

Accepted Answer

The original op-amp circuit is shown in Figure 6.56a. Replacing the passive elements with their impedance representations gives the equivalent op-amp circuit as shown in Figure 6.56b, where

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

or

[latex] \begin{aligned} & Z_{1}(s)=\frac{R_{1}}{R_{1} C_{1} s+1}, \ & Z_{2}(s)=\frac{R_{2}}{R_{2} C_{2} s+1} . \end{aligned} [/latex]

Because the current drawn by the op-amp is very small, applying Kirchhoff's current law to node 1 yields,

[latex] \begin{aligned} I_{1}(s) & =I_{2}(s), \ \frac{V_{\mathrm{i}}(s)-V_{1}(s)}{Z_{1}(s)} & =\frac{V_{1}(s)-V_{\mathrm{o}}(s)}{Z_{2}(s)}, \end{aligned} [/latex]

where the voltage at node 1 obeys

[latex]V_{1}(s)=V_{-}(s)=V_{+}(s)=0.[/latex]

Consequently, we have

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

which gives the transfer function relating the input voltage [latex]v_{\mathrm{i}}[/latex] and the output voltage [latex]v_{\mathrm{o}^{\prime}}[/latex]

[latex]\frac{V_{\mathrm{o}}(s)}{V_{\mathrm{i}}(s)}=-\frac{Z_{2}(s)}{Z_{1}(s)}=-\frac{R_{1} R_{2} C_{1} s+R_{2}}{R_{1} R_{2} C_{2} s+R_{1}}.[/latex]

By transforming [latex]V_{\mathrm{o}}(s) / V_{\mathrm{i}}(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_{1} R_{2} C_{2} \dot{v}_{\mathrm{o}}+R_{1} v_{\mathrm{o}}=-R_{1} R_{2} C_{1} \dot{v}_{\mathrm{i}}-R_{2} v_{\mathrm{i}},[/latex]

which is the same as the one obtained in Example 6.10.

Question 6.13: Impedance Method: An Op-Amp Circuit For the op-amp circuit i...

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