Foundations — Gain margin, phase margin — stability margins

This page assumes nothing. Before you can read the parent note on stability margins, you must own every symbol it throws at you. We build each one from a picture, then show why the topic needs it.

0. What is a "signal", and what is a "loop"?

Look at Figure s01 below: the blue sent wave enters on the left, journeys around the loop, and returns as the red wave. Two things may have happened to it on the trip — it may have grown or shrunk (its height changed, yellow arrow), and it may have slid left or right in time (its timing changed, green arrow).

Those two changes — height and timing — are the only two things the entire topic cares about. Every symbol below is a tool for measuring one of them.

1. Frequency $\omega$ — how fast the wave wiggles

Why the topic needs it: a control loop behaves differently at every wiggle-speed. It might be perfectly calm for slow signals and go berserk at one specific fast one. So we must test every $\omega$ — that is why later formulas are functions of $\omega$.

2. The complex plane — a home for "height AND timing" at once

Height and timing are two numbers. The clever trick of this whole field is to store both in a single point on a flat map called the complex plane.

• Length 1, angle $-180^{\circ }$ = the arrow pointing straight left with length one = the point $-1$. That is the disaster point.
• Angle $-180^{\circ }$ means "flipped upside down" = the wave comes back inverted in timing.
• Length $1$ means it comes back exactly as strong.

Why the topic needs it: the single sentence "signal returns inverted and equally strong ⟹ instability" is "the arrow lands on $-1$." Margins measure the distance from your arrow to $-1$.

3. Transfer function $L(j\omega )$ — the machine that changes the arrow

The parent note calls $L(s)=G(s)H(s)$ the open-loop transfer function: controller $G$ times plant/sensor $H$, i.e. everything the wave passes through on one trip before we close the loop.

Figure s04 makes this concrete: the imaginary axis is where $e^{st}$ is a non-fading spin.

• $|L(j\omega )|$ = the length of the returned arrow = the height-change (the gain).
• $\angle L(j\omega )$ = the angle of the returned arrow = the timing-shift (the phase).

Why the topic needs it: margins are just special readings of $|L|$ and $\angle L$ at special frequencies.

4. Reading gain in decibels ($\text{dB}$) and the $\log$

The parent note keeps writing gains in dB. Here is that symbol, from zero.

Why the topic needs it: "gain margin in dB" is just how many dB the length sits below $0$ dB at the danger frequency.

5. Angles, sign convention, degrees, radians, and $-180^{\circ }$

Converting for the delay formula: the parent's ${\tau }_{max}=\text{PM(rad)}/{\omega }_{gc}$ needs PM in radians, so $\text{PM(rad)}={\text{PM}}^{\circ }\times \frac{\pi }{180}.$

6. The tangent, arctangent, and why phase uses them

The parent note computes phase as sums of $\arctan \omega$. Here is why that tool, and no other, shows up.

Why the topic needs it: the total phase $\angle L$ is a sum of small angles, one per factor of $L$, and each factor's angle is an $\arctan$ of "imaginary-over-real."

7. Putting it together: the two crossover questions

Now every symbol in the parent's key definitions is earned. The whole topic is two readings, drawn in Figure s06:

1. Phase crossover ${\omega }_{pc}$ — the frequency where the arrow points to $-180^{\circ }$ (straight left). Ask: how much shorter than length 1 is it? → that spare length factor is the gain margin $\text{GM}=1/|L(j{\omega }_{pc})|$.
2. Gain crossover ${\omega }_{gc}$ — the frequency where the arrow has length $1$ (0 dB). Ask: how far is its angle from $-180^{\circ }$? → that spare angle is the phase margin $\text{PM}=180^{\circ }+\angle L(j{\omega }_{gc})$.

Each margin is measured where the other quantity already sits at its critical value — remember the parent's "GaP" mnemonic.

How the foundations feed the topic

Figure s07 is the annotated version of the dependency map — read it if the code block below feels abstract: each box is one thing you learned above, arrows point from a prerequisite to what it enables, ending at the stability-margins topic.

Equipment checklist

Self-test — cover the right side and see if each answer is automatic. Everything here is built on this page (the section is noted in brackets); you do not need the parent note to answer.