Complex number methods for op. amp. circuits

If one rederives the standard rules governing the behaviour of op. amp. circuits in terms of the complex number representation of voltage, current & impedance, one finds that the rules are the same as before or translate easily in to say the complex expressions for impedance. One then has the usual convenience of using complex numbers to represent amplitude and phase in the one entity. (It is not that op. amp. circuits are a special case. This easy translation occurs for any circuit provided only that the time-domain differential equations governing the circuit are linear.)

The circuit diagram on the left is an application of the previous discussion. The complex number representation of the voltage gain is:

We recall from lectures that the time-domain analysis of the circuit in which the impedances & are purely resistive ( & ), yields a voltage gain . But then the circuit on the right is a different case. We would have to think again in time-domain analysis. But with the complex number method we can go straight ahead.

Show that:

Hence write down expressions for the phase shift through the circuit and the magnitude of the voltage gain.

If the actual voltage input is , show that the actual output voltage is:

In case you wonder after this example why we ever do direct time-domain analysis, remember that the complex number method gives easy answers only for sinusoidal inputs. It wouldn't easily tell us the output voltage if the input was a step function. For that we would go back to direct time-domain solution. The Laplace transform method we study later in this course is good for that sort of problem.

SOLUTION:

The input voltage is and the complex number form is .
In complex number notation then the gain is:

The phase shift through the amplifier is .

The magnitude of the gain is:

The complex number form of the output voltage is:

Then the actual output voltage is:

since and .
Hence the required result.