A system has adequate phase margin but Kv = 2, and you need Kv = 20. You place a lag compensator with zero at z_c = 1 rad/s and pole at p_c = 0.1 rad/s, with ωgc = 10 rad/s. What primarily determines that the phase margin is preserved?
AThe gain K_c is chosen to be exactly 1, contributing no gain at crossover
BThe zero and pole are placed one decade below ωgc, so the phase dip occurs far below the crossover frequency
CThe compensator adds a pole at the origin, increasing low-frequency gain without affecting phase
DThe lag compensator adds positive phase near crossover, offsetting any reduction from other factors
The negative phase contribution of a lag compensator (up to −90°) occurs between its pole p_c and zero z_c frequencies. By placing both well below ωgc (the rule of thumb is z_c ≤ ωgc/10), the phase dip peaks at the geometric mean of p_c and z_c — far from the crossover region. At ωgc itself, the compensator's phase contribution is small (roughly −6° or less), leaving the phase margin nearly intact. If z_c were placed near ωgc, the negative phase dip would directly erode the phase margin you designed.
Question 2 Multiple Choice
An engineer places a lag compensator with z_c = ωgc/2 instead of the recommended ωgc/10. Compared to correct placement, what is the primary consequence?
AThe low-frequency gain boost β is reduced by a factor of 5
BThe system type increases by one, eliminating steady-state error to ramp inputs
CThe negative phase contribution at ωgc is significantly larger, eroding the designed phase margin
DThe compensator has no effect because the zero is still below the crossover frequency
Phase margin erosion is the primary danger of incorrect lag compensator placement. With z_c = ωgc/2, the geometric mean of p_c and z_c is √(p_c · z_c) = √(z_c/β · z_c) = z_c/√β — now potentially close to ωgc. The lag network's phase dip, which can reach −90°, occurs in this range. Even if the dip peak is still somewhat below ωgc, the phase at the crossover frequency is now substantially negative, potentially reducing phase margin by 20–40°. This can destabilize a system designed with tight margins.
Question 3 True / False
A lag compensator adds a pole at the origin to the open-loop transfer function, changing the system type and enabling zero steady-state error to step inputs.
TTrue
FFalse
Answer: False
A lag compensator C(s) = K_c(s + z_c)/(s + p_c) has a pole at s = −p_c, not at the origin (s = 0). The system type — determined by the number of open-loop poles at the origin — is unchanged. A lag compensator improves steady-state accuracy by multiplying the error constant (Kp, Kv, or Ka) by the finite factor β = z_c/p_c, not by adding an integrator. A Type 0 system with a lag compensator still has finite steady-state error to a step input — the error is simply smaller by factor β. Adding a pure integrator (pole at origin) would change system type and is an entirely different design choice.
Question 4 True / False
A lag compensator increases the velocity error constant Kv by the ratio z_c/p_c = β, which reduces steady-state ramp tracking error by the same factor.
TTrue
FFalse
Answer: True
For a unity-feedback system, Kv = lim(s→0) s·G(s)C(s). The lag compensator at DC (s→0) contributes z_c/p_c = β to this limit, multiplying Kv by β. Since steady-state ramp error = 1/Kv (for a Type 1 system), a β-fold increase in Kv produces a β-fold reduction in steady-state error. This is the compensator's purpose: if the uncompensated system gives Kv = 2 and you need Kv = 20, set β = 10 and design z_c and p_c accordingly. The gain improvement at DC is exactly the ratio of the zero and pole distances from the origin.
Question 5 Short Answer
Explain why a lag compensator's zero and pole must be placed well below the gain crossover frequency, and what goes wrong if they are placed too close to it.
Think about your answer, then reveal below.
Model answer: Between its pole p_c and zero z_c, a lag compensator contributes negative phase — up to −90° at the geometric mean. If z_c is near ωgc, this phase dip occurs in the crossover region and erodes the phase margin that was carefully designed to give the desired transient response. The rule of placing z_c at least a decade below ωgc (z_c ≤ ωgc/10) ensures the phase dip peaks far below crossover, contributing only a few degrees of negative phase at ωgc. The low-frequency gain boost β is unaffected by placement — it is determined by z_c/p_c regardless of where in the frequency spectrum those frequencies fall.
This placement rule is the central practical discipline of lag compensator design. A common error is treating the lag compensator as a pure gain boost and ignoring its phase contribution. But every real compensator has both magnitude and phase responses. The 'free lunch' of increased low-frequency gain without reduced phase margin is only available when placement is correct. The slow pole-zero pair near the origin also introduces a long-duration tail in the step response — acceptable in most applications but worth verifying.