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 τp = L/(R1 + R2) and zero time constant of τz = L/(R2) . 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: ωp > ωz, 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 fβ , where β (s) starts rolling off, and fT, where β (s) = 1 is the high-frequency (hf) region of the transistor, where base impedance is gyrated by +90o at the emitter. Base resistance RB 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 fT and used as τT in the formula shown for L, where τT = 1/ ωT = 1/2 x π x fT. 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 RB.
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 R1 >> R2. (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. RB 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 fβ , it is merely RB/( β0 + 1).
A variation on this theme is shown below, with parallel instead of series L, R2.
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 ωz = –1/(L/R1) and ωp = –L/[R1 || R2] 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 ωp < ωz to reduce loop gain at a low frequency, then flare out the magnitude response with a zero as fT is approached for a smooth transition of –1 log-log slope across fT. The same active-device inductor synthesis can be used to synthesize active lag-lead circuits.