## Thursday, October 28, 2021

### Hawksford's Error Correction (H.ec)

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

In 1983, Malcom Hawksford presented at the 74th AES convention a paper "Power Amplifier Output Stage Design Incorporating Error Feedback Correction With Current Dumping Enhancement". The idea was simple - compare the signal after the output stage with that at its input and add the difference back to the input:

On the schematic, H is the output stage with near-unity gain; $\epsilon$ is the error (distortion) introduced by the output stage; and B is the feedback network. Working from right to left: $$v_o = \epsilon + v_e H$$ $$v_e = v_i - v_c = v_i - B (v_o - v_e)$$ From the second equation we find $$v_e={{v_i - v_o B} \over {1-B}}$$ Substituting back into the first equation: $$v_o = \epsilon + v_e H = \epsilon +{{H(v_i - v_o B)} \over {1-B}} = \epsilon {{1-B} \over {1-B+HB}} + v_i {H \over {1-B+HB}}$$ That is, we have the error transfer function $$ETF = {{1-B} \over {1-B+HB}}$$ and the signal transfer function $$STF={H \over {1-B+HB}}$$

How the magic works: when $B=1$, $ETF=0$ and $STF=1/B=1$. Neither the transfer function of the output stage $H$ nor its distortion $\epsilon$ affect the output signal.

H.ec can also be analyzed as a positive feedback loop with the transfer function $1/(1-B)$ nested inside a negative feedback loop with forward gain $H$ and feedback gain $B$ - which gives the loop gain $H B / (1-B)$. It is equivalent to replacing $H$ with $H^\prime = {H/(1-B)}$.

One example of practical implementation is Thule Audio's Spirit IA-100 integrated amplifier. The simplified schematic is as follows:

Here, Q7 and R4 convert the difference between the output and input of the emitter follower Q2 Q5 into the collector current of Q7. R4 sets the transimpedance, that is, the rate of voltage-to-current conversion. Q7's collector current flows into the current mirror Q1 Q3 Q4 Q6. At the output of the current mirror, this current adds to the the current from the voltage amplification stage, represented by Ivas. The sum of the two currents is converted to voltage by R1 and becomes the input signal for the emitter follower. The value of R1 defines the proportion of the two currents at the summing point and hence $B$, so R4 needs to be adjusted for the lowest distortion. (The real IA-100 uses a push-pull emitter follower at the output and no copuling capacitor, but that is not essential for the discussion of H.ec.)

Another implementation, the paX amplifier by Jan Didden, published in Elektor Magazine in 2008, employs the AD844 instead of Q7 and the current mirror in the schematic above, but otherwise works the same.