:::::::: 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...
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.
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I hope, this answers your question.