An Op-Amp Multiplier Consider the op-amp circuit shown in Figure 6.33, in which one resistor R2 is in parallel connection with an op-amp, and the resulting parallel circuit is in series connection with another resistor R1. Determine the relation between the input voltage vi and the output voltage

Question

An Op-Amp Multiplier

Consider the op-amp circuit shown in Figure 6.33, in which one resistor [latex]R_{2}[/latex] is in parallel connection with an op-amp, and the resulting parallel circuit is in series connection with another resistor [latex]R_{1}[/latex]. Determine the relation between the input voltage [latex]v_{\mathrm{i}}[/latex] and the output voltage [latex]v_{\mathrm{o}}[/latex]. Assume that the current drawn by the op-amp is very small.

Accepted Answer

Label the currents at the nodes with unknown voltages. The system has only one significant node: node 1. Applying Kirchhoff's current law to node 1 gives

[latex]i_{1}-i_{2}-i_{3}=0.[/latex]

Because the current drawn by the op-amp is very small, that is, [latex]i_{3} \approx 0[/latex], we have

[latex]i_{1} \approx i_{2} ext {. }[/latex]

Using the voltage-current relation for each resistor yields

[latex]\frac{v_{\mathrm{i}}-v_{1}}{R_{1}}=\frac{v_{1}-v_{0}}{R_{2}} .[/latex]

Note that the input terminal marked with the plus sign is connected to the ground. From the op-amp equation [latex]v_{+} \approx v_{-}[/latex], we have

[latex]v_{1}=v_{-} \approx v_{+}=0.[/latex]

Thus, the relation between the input voltage [latex]v_{\mathrm{i}}[/latex] and the output voltage [latex]v_{\mathrm{o}}[/latex] is

[latex]\frac{v_{\mathrm{i}}}{R_{1}}=-\frac{v_{\mathrm{o}}}{R_{2}}[/latex]

or

[latex]v_{\mathrm{o}}=-\frac{R_{2}}{R_{1}} v_{\mathrm{i}}.[/latex]

This circuit is known as an op-amp multiplier and is widely used in control systems. Op-amps can also be used for integrating and differentiating signals.

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