Open and Closed Loop Frequency Response

The open-loop frequency response is the frequency response of the amplifier with no feedback — before feedback, or with the feedback network deliberately broken.

The closed-loop frequency response is the frequency response of the amplifier with feedback.

These two are closely related. The theoretical closed-loop frequency response is equal to the open-loop frequency response times the amount of feedback. If you have 40dB (100 times) of feedback, and an open-loop response of 1kHz, the closed-loop frequency response is 100kHz.

The formula relating these two is:

where $D$ is the feedback factor and $A_{ol}$ is the open-loop gain. The total denominator expression is what we call feedback.

The open-loop frequency response is determined by the internal compensation (intended or not) of the amplifier. Many amplifiers are designed with one stage having a very high output impedance, so the stray capacitance of that stage’s output determines the open-loop frequency response. For integrated circuits the open-loop frequency response is either specified, or you can see it graphically as a function of gain — in the latter case, look at the maximum gain, which means zero feedback.

Just to remind you: the closed-loop gain $A_{cl}$ is related to the open-loop gain $A_{ol}$ in exactly the same way as the frequency response, although inversely.

At Electrocompaniet the thinking favored a large open-loop bandwidth. This is also my opinion, but I feel it shouldn’t be larger than necessary. There is always a tradeoff, and if you go for too high an open-loop bandwidth, you reduce the possible amount of feedback you can have. My thinking is that as long as the open-loop bandwidth is high enough, you should use the rest of your gain for feedback. This will give you a more optimal design, because the overall distortion will be reduced.

What determines the open-loop bandwidth

Mostly it is determined by the last voltage-amplification stage. The collectors of this stage (assuming transistor amplifiers, which these articles are all about :-)) are connected to the bases of the drivers of the output stage. The input impedance of these drivers is normally very nonlinear, and strongly frequency-dependent. This means you can very well get a major pole here which varies strongly with signal level and the load (loudspeaker and cables) of the output stage. The solution to this is to voltage-drive the output stage, thus loading down the amplification stage. This will also have the effect of pushing up the cutoff frequency at this point. The EC amplifiers have this pole around 500kHz. The benefits of a voltage-driven output stage are described elsewhere on this site, including in an AES paper (not yet ported to this site).