S4-3.4 Frequency Response Analysis

What this is and why it exists

Frequency response asks the system a different question: not "how do you step?" but "how do you treat a sinusoid of every frequency?". The answer can be MEASURED on real hardware with a signal generator — no model required — which is why practising engineers reach for Bode plots first. The unit delivers the two safety margins that quantify how close a loop is to oscillation, and the two standard fixes, lead and lag compensation, that buy speed or accuracy when the margins allow.

The vocabulary

• Frequency response — the steady-state gain and phase a system applies to sinusoids, as functions of frequency.
• Bode plot — gain in decibels and phase in degrees, each against log frequency; straight-line segments make it sketchable by hand.
• Nyquist plot — the same information drawn as one curve in the complex plane, gain and phase together.
• Nyquist criterion — stability read from how that curve encircles the minus-one point, grounded in the principle of argument.
• Gain margin — how much extra gain the loop tolerates before oscillation; read where phase reaches minus 180 degrees.
• Phase margin — how much extra phase lag it tolerates; read where gain crosses unity (zero decibels).
• Lead compensator — adds phase in a chosen band: buys stability margin and speed.
• Lag compensator — raises low-frequency gain: buys steady-state accuracy while leaving the crossover region nearly alone.

The mental model

A loop oscillates when its own signal returns in phase with itself at unity strength — the minus-one point is where that self-reinforcement condition lives, and both plots are ways of watching how close the loop comes to it. The Bode plot watches from two windows: at the frequency where gain crosses unity, how far is phase from minus 180 (phase margin)? At the frequency where phase reaches minus 180, how far below unity is gain (gain margin)? Healthy loops carry a phase margin around 45 to 60 degrees; less rings, none oscillates. The Nyquist plot watches the same approach as geometry — the curve's distance from minus one — and its encirclement count, via the principle of argument, keeps scoring stability even when the open loop is itself unstable, where Bode reasoning alone misleads.

Bode sketching stays quick because the pieces are straight lines: each pole bends the gain slope down another 20 decibels per decade and contributes 90 degrees of lag spread over a decade each side; each zero does the opposite. Sum the segments and the sketch is done — then software refines it.

Compensation is margin arithmetic. A lead compensator is a phase bump you slide over the gain-crossover frequency: phase margin rises, crossover moves higher, the loop speeds up — the derivative idea in frequency clothing. A lag compensator boosts gain far below crossover: steady-state error shrinks as if gain were raised, while at crossover the device is nearly transparent, so margins survive — the integral idea. Diagnose which deficit you have, speed-and-stability or accuracy; that names the compensator.

What you should now be able to explain or do

Sketch a Bode plot from a transfer function by segments. Read both margins off a Bode or Nyquist diagram. State the Nyquist criterion and why it works from encirclements. Choose lead or lag for a stated deficit and say where its action concentrates.

Check yourself