This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.
In the previous post, I explained how current dumping works on an intuitive, qualitative level. Let's go into the next level of detail.
A current dumping amplifier is a feedback amplifier consisting of a low distortion, low power amplifier A and a high-power, distorting buffer ("current dumper", here Tr1 and Tr2) inside a common negative feedback loop:
Negative feedback reduces, but not completely eliminates, the distortion that the current dumper adds to the signal. From my earlier post, the share of that distortion that remains at the output (point D at the schematic above) after the negative feedback is applied is given by the Error Transfer Function $${ETF={1 \over {1- A \times B}}}$$ where $A$ is the transfer function (basically, the frequency-dependent gain) of the integrator composed of A,R1,C1, and $B$ is the transfer function of the feedback network. In this case, the feedback network has unity gain and is connected to the inverting input of the integrator, so $B=-1$, and $${ETF={1 \over {1+ A}}}$$ Therefore of the total open loop distortion $\epsilon$ of the current dumper, at point D we observe $$\epsilon _D={\epsilon \times ETF}=\epsilon {1 \over {1+ A}}$$ Since the current dumper adds distortion $\epsilon$, for this to happen, the input of the current dumper (point A at the schematic above) should see "pre-distortion" $$ \epsilon _A = {\epsilon_D - \epsilon}={{\epsilon {1\over {1+ A}}}-\epsilon}=-\epsilon{A \over {1+ A}}$$ At the load, $\epsilon_A$ and $\epsilon_D$ combine in reverse proportion to the impedances of R2 and L1: $$\epsilon_{LOAD}\propto {\epsilon_A Z_{L_1} + \epsilon_D Z_{R_2}}$$ Note that the load impedance affects the absolute level of the combined signal at the load but not the proportion of $\epsilon_A$ to $\epsilon_D$.
Combining the last three equations and dropping the common denominator $1+A$: $$\epsilon_{LOAD}\propto {\epsilon (Z_{R_2} - A Z_{L_1})}$$
Clearly, the perfect cancellation of the current dumper's distortion occurs when $$Z_{R_2} = A Z_{L_1}$$
For the implementation above, under ideal conditions, $Z_{R_2}= {R_2}$, $Z_{L_1}= s {L_1}$, $A=1/{(s R_1 C_1)}$, and the perfect cancellation means
$$R_2={L_1 \over {R_1 C_1}}$$
which is the result from [1]: "For the linearity of Tr1 and Tr2 to be immaterial then L must equal RRC".
However, by going through the algebra above, we obtained a more general and quite remarkable result: a perfect distortion cancellation requires the ratio of $Z_{R_2} / Z_{L_1}$ to mimic the amplifier's loop gain $A$. This gives us the freedom to make current dumping work under less than ideal conditions and in different implementations than above. Stay tuned.
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