Stability of Differentiator Circuit

Question

let's consider a generic differentiator circuit, i.e. a circuit whose transfer function is \$\text{constant}\cdot j\omega\$. For instance, you may consider this circuit with Op-Amp:

I was reading this paragraph about it ("Pulse And Digital Circuits", Prakash Rao):

Which is the precise reason for its unstability? In general, I know that a linear system is stable if and only if its transfer function has only poles with negative real part. In this case I do not see any poles, and so?

Moreover, I have seen that usually negative feedback (as that present in this circuit) improves the stability: is, despite this fact, this circuit unstable?

Answer 1

With the ideal differentiator, there is a phase lag in the negative feedback network. Combined with the phase shift in a real opamp this can result in making the overall phase shift around the loop reach 360 deg and cause oscillation. Even if it doesn't cause oscillation it can reduce the phase margin at high frequencies.

To understand this imagine that Vin is grounded. The network R and C will give a phase shift that is asymptotic to 90 deg at high frequency.

A typical opamp is designed with dominant pole compensation to provide stability - this adds another 90 deg of phase shift at high frequencies. In addition, there are usually one or two additional poles internal to the opamp that can give more phase shift.

Note from the diagram that although the phase shift is ~90 deg over much of the operating range it goes beyond that above ~100kHz while the gain is still >1. This can cause instability when combined with the phase-shift of the negative feedback network. (note: the curve that is horizontal for much of the graph is the phase shift, the sloping line is the open-loop gain).

OPA227 Datasheet

Answer 2

At high frequencies, it becomes more prone to noise. When noised is introduced to an amplifier, well... you can imagine what an amplifier will do with noise. If you get into control theory, you'll find that a differentiator has an infinite gain margin with 90 degrees of phase margin. The gain margin won't make the differentiator unstable; the phase margin will. The oscillation goes rampant at high frequencies.

You can also prove instability comparing this system to a Linear Time-Invarient (LTI) system. There's a concept called: Bounded-Input, Bounded Output (BIBO) stability and when you do the math on that, you'll find that the system is also unstable, which is a more accurate way to determine if the system is stable.

The physics behind this unstable behavior may be another topic but think about the nonlinearity of the capacitor.

There are ways to combat this and STMicrosystems has a good application note to one of their amplifiers on that. It is a good read if you are curious about how to compensate for this unstable system.