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I have a question regarding the stability of an ideal op amp differentiator. Say I have the following circuit:

schematic

simulate this circuit – Schematic created using CircuitLab

Now calculating its transfer function I get: $$ \frac{V_\text{out}}{V_\text{in}}=-sRC $$

Please correct me if I am mistaken here but as far as I know the condition for oscillating behaviour is that my gain is bigger than 1 and I have a phase shift of a multiple of \$2 \pi\$, so that's the condition I want to avoid. Now why is this ideal op amp considered to be unstable? The inverting behaviour gives me a phase shift of 180°. I get another -90° from the ideal differentiator. This still leaves 90° phase margin. I assume that this is problematic as the op amp itself behaves like a first order low pass filter and adds another 90°?

So far my knowledge, please feel free to correct me here!

Now some half-knowledge back from my electronic circuit design courses tells me that I can use a transimpedance amplifier to build a stable ideal differentiator. Unfortunately I cannot find anything about that on the web and was wondering if there is any truth to that?

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Please correct me if I am mistaken here but as far as I know the condition for oscillating behaviour is that my gain is bigger than 1.

Damped oscillations can appear as a loop gain of 1 is approached. If the gain equals 1, then their will be constant amplitude oscillations.

The inverting behaviour gives me a phase shift of 180°. I get another -90° from the ideal differentiator. This still leaves 90° phase margin.

This is correct for an ideal op-amp.

Now why is this ideal op amp considered to be unstable?

I expect that you mean "ideal op-amp differentiator". It isn't unstable.

This still leaves 90° phase margin. I assume that this is problematic as the op amp itself behaves like a first order low pass filter and adds another 90°?

This is correct. Real op-amps (most are internally compensated) have an open-loop first order low-pass gain response. This does indeed introduce a 90 degree phase shift above the open loop corner, so at least it will ring and at worst oscillate continuously.

I have included the loop-gain response for two values of the compensating resistor R3: 0 ohms (blue) and 400 ohms (red). The sharp peak on the blue line can promote instability.

An ideal amplifier's response will be flat.

enter image description here

And the following is the differentiator output The blue line without compensation rings. The red line with R2=400 ohms shows nice pulses. The input is a 10mV, 10ms pulse repeating every 100ms. It is too small to see on the diagram.

enter image description here

I can't advise on transimpedance differentiators.

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The ideal opamp differentiator is not inherently unstabe - it is susceptible to noise. These concepts are not interchangeable. In other words, you won't be able to create an unwanted oscillations with this circuit alone. The reson it is susceptible to noise is that the gain increases (to infinity) with increasing frequency. The way to mitigate that effect is to add series resistance with the capacitor to provide an upper limit on gain no matter how high the frequency is. A more agressive appoach would be to add a capacitor across the feedback resistor to roll off and attenuate the gain at increasing frequencies.

Now some half-knowledge back from my electronic circuit design courses tells me that I can use a transimpedance amplifier to build a stable ideal differentiator.

You can't find any information on this because a differentiator and a transimpedance amplifier are fundamentally different circuits nor are they related. There is an instability phenomenon that is well-known with transimpedance amplifiers when the source is capacitive which will be conducive of oscillations and instability, but that's beyond the scope of your question.

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