Input/output characteristics of opamp circuits involving diodes (fast peak detector and sine-to-block wave converter)

Question

I have these relatively simple circuits of which I need to find the transfer functions. Both circuits involve the use of an OpAmp and a diode after the output of the OpAmp. As an input I have a regular sine wave.

Finding the input/output characteristics of a simple inverting amplifier or passive filter is easy, but I get stuck when trying to find the transfer functions for these circuits:

^{simulate this circuit – Schematic created using CircuitLab}

• The first circuit (with output \$V_{out1}\$) is a peak detector circuit. The capacitor holds the output voltage, while the resistor is there for slowly discharging the capacitor.
• The second circuit (with output \$V_{out2}\$) is a 0 Volt comparator for turning the sine wave into a square wave. I put a diode after the output to only allow positive voltages. The resistor to ground is to reduce the output voltage.

My question is: How can I find the input/output characteristics of these circuits (\$V_{out1} = ... * V_{in}\$ and \$V_{out2} = ... * V_{in}\$)?

Answer 1

This is homework, so I'll help you find answers, but not directly give such – that'd rob you of anything to learn.

How can I find the transfer functions of these circuits

Vout2/Vin

This is an opamp without feedback; i.e. an ideal opamp would act like a comparator. A TL081 will likely latch up under these conditions. It's generally the most terrible Opamp you're likely to find anywhere. Don't use it.

For the ideal opamp case, I'm sure that you can just plug in a few hypothetical voltages and see how this behaves. It's easy, don't be afraid to try.

Vout1/Vin

Not at all – the diodes make this a nonlinear system (made worse by the unfortunate of opamp model, but I assume you're assuming an ideal opamp). That doesn't mean this doesn't have a transfer function – but paired with C1, the system becomes time-variant and thus doesn't have a transfer function, as the diode's I/V curve will be biased to the difference between voltage across C1 and OA1's output voltage.

So, you've got both a frequency-dependent behaviour and an amplitude nonlinearity. Vout1/Vin hence depends on Vin's "history" and can't be reduced to a single curve.