## Saturday, March 13, 2021

### Audio Amplifier Feedback - Lag Compensation

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

Lag compensation involves an RC network connected between the inputs of an opamp:

It may be a bit difficult to understand, as feedback reduces the voltage across the compensation network, and thus reduces the effect of that network on the closed loop response.

For the loop gain, however, lead-lag compensation does make a difference. Consider the following schematic:
From the point of view of the feedback signal, the compensation network $R_N C_N$ is connected in parallel to $R_I$ and modifies the transfer function of the feedback network as follows:$$B={Z_I \over {Z_I + Z_F}}={{R_I \over {R_I + R_F}} \times {{s R_N C_N +1} \over {s (R_N + R_I||R_F)C_N +1}}}$$
The compensation network adds to the loop gain a pole at $\omega_p = {1 \over {(R_N + R_I||R_F) C_N}}$ and a zero at $\omega_z = {1 \over {R_N C_N}}$; note that $\omega_p < \omega_z$ (green is the forward gain of the amplifier, blue is $1/B$):

The loop gain is reduced by 3dB at $\omega_p$ and keeps falling until the zero cancels the effect of the pole, including the extra phase lag. The crossover shifts to a lower frequency, where the phase lag is smaller, increasing the phase margin (here, green is the loop gain without compensation, blue is the loop gain with lead-lag compensation):
The net effect of lag compensation is a reduction of the loop gain at higher frequencies without the phase lag that would normally be associated with such a reduction.

Compare lag compensation to dominant pole (e.g. Miller) compensation with the same bandwidth (green is the loop gain without compensation, blue - with lead-lag compensation, red - with dominant pole compensation):

Lag compensation allows more loop gain at lower frequencies while providing a very similar phase margin. Due to the finite gain of the amplifier, the closed loop frequency response with lead-lag compensation (blue) is slightly different from that with a dominant pole (red):

Sometimes in the lead-lag compensation network, either R of C is omitted. If R is omitted and C is left alone:
the feedback network transfer function becomes $$B={Z_I \over {Z_I + Z_F}}={{R_I \over {R_I + R_F}} \times {1 \over {s (R_I||R_F)C_C +1}}}$$ There is no zero anymore to compensate for the additional pole, and for stability, a zero would usually need to be introduced separately:

If C is omitted and R is left alone:
then the feedback network transfer function becomes $$B={Z_I \over {Z_I + Z_F}}={{R_G \over {R_G + R_F}} \times {R_1 \over {R_1 + R_G||R_F}}}$$ Since ${R_1 \over {R_1 + R_G||R_F}}<1$, the loop gain is decreased across all frequencies.