Questions: Compensator Realization: Active and Passive Networks

Question 1

A control engineer designs a lead compensator with transfer function C(s) = 10(s + 2)/(s + 5). They attempt to realize it using only resistors and capacitors (passive RC network). What fundamental limitation prevents a valid passive realization?

• A. Passive RC networks cannot realize transfer functions with a zero to the left of a pole
• B. Passive RC networks cannot produce voltage gain greater than unity, but this compensator requires gain = 10
• C. The pole and zero are too close together for a passive network to separate them
• D. RC networks are only valid for transfer functions with a zero at the origin

Answer: (B)

The DC gain of this compensator is 10 × (0 + 2)/(0 + 5) = 4, and the high-frequency gain is 10. Both exceed unity. Passive RC voltage dividers can only attenuate — their transfer function magnitude is always ≤ 1 at all frequencies. Achieving gain > 1 requires an active element (op-amp) to supply energy to the signal. This is the fundamental physical reason: passive networks consume or store energy; they cannot create it, so output can never exceed input.

Question 2

An engineer inserts a passive lead compensator between a sensor stage and an amplifier stage. After connection, the measured frequency response differs significantly from the designed transfer function. What is the most likely cause?

• A. The RC components have tolerances that shift the pole and zero frequencies
• B. The compensator's finite output impedance loads the downstream amplifier, altering the effective transfer function
• C. Lead compensators cannot be realized passively — only lag compensators can
• D. The amplifier stage's bandwidth is too narrow to pass the compensated signal

Answer: (B)

Passive networks have non-zero output impedance that interacts with the input impedance of the downstream stage, forming an unintended voltage divider. The resulting transfer function includes loading effects not captured in the isolated compensator design. This is the fundamental practical disadvantage of passive realizations — every passive compensator must be analyzed with its load in place. Active op-amp realizations avoid this because the op-amp's low output impedance and high input impedance decouple adjacent stages, making the transfer function nearly independent of loading.

Question 3

An active op-amp compensator achieves near-ideal isolation between stages because its low output impedance and high input impedance prevent the compensator from loading adjacent circuit stages.

• T. True
• F. False

Answer: True

This is the key practical advantage of active realizations. An ideal op-amp has infinite input impedance (draws no current from the preceding stage) and zero output impedance (maintains the designed output voltage regardless of the load). Real op-amps approximate this well within their bandwidth, effectively buffering the compensator from both the upstream signal source and the downstream load. This means the transfer function measured in isolation is what you get in the full circuit — a guarantee passive networks cannot provide.

Question 4

A transfer function with more zeros than poles (improper transfer function) can generally be realized as a physical circuit by choosing appropriate resistor and capacitor values.

• T. True
• F. False

Answer: False

A physically realizable circuit cannot have more zeros than poles. An improper transfer function would require pure differentiation of arbitrarily high order — for example, a transfer function with two more zeros than poles would require computing the second derivative of the input signal. In practice, this means amplifying high-frequency noise without bound, which is physically impossible and unstable. Before building any compensator, engineers must verify properness (degree of numerator ≤ degree of denominator) and, if the designed function is improper, add high-frequency poles to regularize it — accepting a tradeoff between ideal performance and physical realizability.

Question 5

Explain why a transfer function with more numerator zeros than denominator poles cannot be physically realized as a circuit, and what a designer must do to make such a function realizable.

Model answer: Each zero in excess of the poles corresponds to a derivative operation on the input. A transfer function H(s) = s²/(s + 1) would require computing the second derivative of the input — which means amplifying high-frequency components without limit. No real circuit can do this because all physical signals contain noise at high frequencies, which would be infinitely amplified. To make an improper function realizable, the designer adds high-frequency poles (e.g., multiplying the numerator and denominator by 1/(τs+1)^n to restore properness), accepting that the compensator deviates from ideal at high frequencies in exchange for physical stability.

This constraint is sometimes called the 'properness' or 'realizability' condition: a causal, stable, physically implementable transfer function must have the number of poles ≥ number of zeros. It reflects the fundamental reality that physical systems have inertia — they cannot respond instantaneously to arbitrarily rapid input changes. Adding roll-off poles at high frequencies is the standard engineering fix, and the frequency at which they are placed is itself a design variable: high enough not to affect the control bandwidth, low enough to keep the circuit stable and physically constructable.