Stability analysis for Bandpass Filter (Multiple Feedback op-amp circuit)

Question: How to analyze the stability of a 2nd order bandpass filter which is implemented using a multiple feedback topology...

In a more general scenario , a circuit transfer function involving an op-amp and feedback components around the op-amp can be calculated with an ideal op-amp model. Does this transfer function tell us anything about stability of the circuit? Or, under ideal op-amp model, there will be no problem for stability? If so, then the analysis of stability problem will be based on the feedback theory, which we'll need to determine the op-amp open loop gain (A(s)) and feedback factor beta(s)? Is it so?

Thanks

Calculating the poles of the transfer function will tell you how damped the circuit is in the time domain, as well as general stability(s-plane analysis, etc.). If you have the bode plot of your circuit, you could also analyze the gain and phase margins.

dewasiuk said:

Calculating the poles of the transfer function will tell you how damped the circuit is in the time domain, as well as general stability(s-plane analysis, etc.). If you have the bode plot of your circuit, you could also analyze the gain and phase margins.

Click to expand...

thanks for replying..but maybe I didn't make myself clear...
For example, If you check T.I.'s famous "op-amp for everyone", section 16.5.1.2 Multiple Feedback Topology, you'll see there's a transfer function calculated for the multiple feedback topology implementation of the bandpass filter circuit. (Actually, there shouldn't be the terms omiga_m in the transfer function, but never mind it)...my question is if you look at this transfer function, the dominator polynomial's coefficients are all positive (with physical resistors and capacitors implementation), which means the root of this second order polynomial is to the left of the y-axis on the s-plane. So does this mean that the circuit is unconditionally stable? Notice that the derivation of this transfer function is based on the assumption of ideal op-amps. How is the transfer function derived this way different from the way if we look at the problem from a feedback system's point of view? i.e. consider the open loop gain of op-amp to be A and try to calculate the feedback factor beta...

You should definitely consider the real OP characteristic in stability analysis. A second order filter has two poles, so a circuit with ideal OP is at least asymptotically stable, phase margin may be however low. Adding OP poles to the loop gain can bring up instability.

Determining the loop gain will tell you about filter stability in any case.

zhangyi17 said:

:::::::: if you look at this transfer function, the dominator polynomial's coefficients are all positive (with physical resistors and capacitors implementation), which means the root of this second order polynomial is to the left of the y-axis on the s-plane. So does this mean that the circuit is unconditionally stable? Notice that the derivation of this transfer function is based on the assumption of ideal op-amps. How is the transfer function derived this way different from the way if we look at the problem from a feedback system's point of view? i.e. consider the open loop gain of op-amp to be A and try to calculate the feedback factor beta...

Click to expand...

Perhaps it's helpful to recall some fundamental aspects of filter design:
*The selectivity of a bandpass (quality factor) is directly connected to the pole location in the s-plane
*When the pole-Q is finite the poles, of course, are within the left-hand half of this plane (LHP)
* This means: the system is stable (instability/oscillations occur for Q approaching infinity with poles on the imag. axis), but the stability margin, of course, will be relatively small for high Q-values. That's something like a "natural law".
* However, it is common practice to design the filter stage for an ideal opamp. Thus, real opamp properties will lead to an Q-enhancement and a degradation of stability margins. The situation becomes even worse due to capacitive loading effects. These effects can cause stability problems.
* Therefore, to limit these Q-enhancement effects the opamp bandwidth (gain-bandwidth-product) must be large enough if compared with the filter pole frequency (factor 50...100, depending on the accuracy requirements).
* It should be mentioned that it is possible to include the dominant opamp pole in the filter design (parts values). This method is called "pre-distortion" and, of course, requires the exact knowledge of the 1st opamp pole. In this case, the unwanted Q-enhancement effect is reduced.
* Another simulation-based method modifies the parts values of a certain feedback path to compensate the parasitic opamp influence. In this case, a realistic opamp model is required that reflects the real frequency dependence of the opamp gain.
____________
I hope, this answers your question.

A Spice simulation of the circuit with real op amp models will help in determining the real world filter response, including doing a Bode plot to examine stability. You can also do some Monte Carlo runs to see the effect of component tolerances.