Part 2 continues with pole-zero compensators of feedback loops and presents some methods not usually found in textbooks (See Part 1 here: Lead-Lag Pole-Zero Frequency Compensators, Part 1: Passive Compensators). The dual of the RC lead-lag compensator is the RL lead-lag compensator, shown below, with a transfer function of

Instead of a parallel-R pole, the dual has a series-R pole of time constant *τ*_{p} = *L*/(*R*_{1} + *R*_{2}) and zero time constant of *τ*_{z} = *L*/(*R*_{2}) . The series-R of the pole is greater than that of the zero, and being in the denominator of the time constant, the pole frequency is larger: *ω*_{p} > *ω*_{z}, as for the RC lead-lag circuit.

It is usually more desirable to implement circuits with capacitors than inductors, but there is one interesting exception: where the inductor is synthesized by a transistor and resistor, as shown below.

Between *f*_{β} , where *β* (*s*) starts rolling off, and *f*_{T}, where *β* (*s*) = 1 is the *high-frequency* (hf) *region* of the transistor, where base impedance is *gyrated* by +90^{o} at the emitter. Base resistance *R*_{B} in this hf region takes on an equivalent circuit as shown on the right, that of a parallel RL, where the inductance value depends on the speed of the BJT as expressed by *f*_{T} and used as *τ*_{T} in the formula shown for *L*, where *τ*_{T} = 1/ *ω*_{T} = 1/2 x π x *f*_{T}. This equivalent circuit conveniently applies itself to the lead-lag compensator for compensation in the hf region. It differs from the passive version in that *L* is shunted by *R*_{B}.

When the hf equivalent circuit is solved for the transfer function, it is

The circuit has one zero and one pole, and the ratio of the pole/zero frequencies is

Pole and zero are separated by making *R*_{1} >> *R*_{2}. (A pole and zero that are close to or at the same frequency are a *doublet*.) This has the same effect on quasistatic gain as for the passive circuit. *R*_{B} affects placement of pole and zero and shifts both in frequency. The pole and zero are constrained to be in the hf region or else *L* “disappears” when gyration stops outside this region. Below *f*_{β }, it is merely *R*_{B}/(* β*_{0} + 1).

A variation on this theme is shown below, with parallel instead of series *L*, *R*_{2}.

The transfer function for the RL lead-lag compensator is

It is similar in form to the RC compensator in having a parallel-R pole but differs in having a zero at the origin. Zero and pole frequencies are ω_{z} = –1/(*L*/*R*_{1}) and *ω*_{p} = –*L*/[*R*_{1} || *R*_{2}] where the zero frequency is that at which the gain of the zero crosses a gain of one. The same general considerations apply.

Lag-lead compensators place *ω*_{p} < *ω*_{z} to reduce loop gain at a low frequency, then flare out the magnitude response with a zero as *f*_{T} is approached for a smooth transition of –1 log-log slope across *f*_{T}. The same active-device inductor synthesis can be used to synthesize active lag-lead circuits.