This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.
Since my previous post was about shaping the open loop gain of a long-tail pair (LTP) with an RC network across the load, it seems to be a convenient moment to talk about LR compensation of an LTP:
The compensation network consists of an inductor $L$ in series with an optional resistor $R$, connected between emitters of the differential amplifier. Intuitively, at low frequencies where the impedance of the $RL$ network $Z_{RL}=R+sL$ is low compared to $R_e$, the $RL$ network reduces the degeneration added by $R_e$ and increases the transconductance of the LTP. At high frequencies, where $Z_{RL}$ is high compared to $R_e$, the $RL$ has little effect, and the LTP is degenerated by $R_e$:
To find out exactly how the transconductance changes in between these two extremes, I need to consider the small signal equivalent emitter circuit of the LTP:
Here, $r_e = {V_T \over I_E}$ is the emitter intrinsic resistance. The transconductance of the LTP with $RL$ compensation is $$g_m={1 \over {2 r_e+(R+sL)||(2 R_e)}}={{sL+(R+2R_e)} \over {2(R_e+r_e)(sL+R+2(R_e||r_e))}}$$ and has a pole at $s=-{{R+2(R_e||r_e)} \over L}$ and a zero at $s=-{{R+2R_e} \over L}$.
For example, the Bode plot above was generated with $R_e=220 ohm$, $R=0ohm$, $L=100 \mu H$ and $r_e={V_T \over I_E} = {{25mV} \over {1mA}}=25ohm$. With these values, the pole is at $s=-{{R+2(R_e||r_e)} \over L} = -{{0+2(220||25)} \over 10^{-4}} \approx -450k$, or $71.5kHz$, and the zero is at $s=-{{R+2R_e} \over L} = -{{0+2 \times 220} \over 10^{-4}} = -4.4M$ or $700kHz$.
One word of caution is that low $R$ in the $LR$ network undoes the degeneration added by $R_e$ and thus makes the LTP more susceptible to overload and nonlinearity, as discussed in the post of input stage linearity. The intended outcome is, of course, that the extra transconductance available with low $R$ increases loop gain, which reduces the differential input voltage seen by the LTP and improves its linearity. However, depending on the specific implementation, the loss of degeneration with low $R$ may become problematic.
Another potential problem is that the inductor is an effective antenna and will increase inductive coupling of noise from e.g. power supply rails into the LTP. The standard remedy is to use a shielded inductor and/or use two inductors in series connected in anti-phase, so that noise coupled into one is cancelled by the other.
