Why −180° matters: Nyquist and margins
Negative feedback can become reinforcing feedback when the returning signal is inverted and strong enough.
Calculate a phase margin while keeping the full Nyquist stability conditions in view.
Bode plots locate the frequency where loop magnitude reaches one. At that frequency, phase determines how close negative feedback is to reinforcing itself.
Review the previous lesson: Bode plots: gain and phaseFirst, picture it
The summing junction subtracts the returning signal. If the loop adds another −180° phase shift, subtraction of an inverted signal becomes addition. If its magnitude is also 1, the return can sustain itself. This is why the critical Nyquist point is −1, not simply any point at −180°.
Build the idea step by step
A Nyquist plot places each complex loop value on a plane. The critical point is minus one because one plus the loop value then becomes zero in the closed-loop denominator. Reaching minus 180 degrees alone is not enough: the magnitude must also reach one. Phase margin measures how much extra lag would reach minus 180 degrees at a gain crossover. For complicated loops, a single margin is not a stability test. The full Nyquist test counts how the complete curve goes around minus one and accounts for unstable open-loop poles.
Read the equations
Under the usual single-crossover convention, phase margin is the extra lag from gain-crossover phase to minus 180 degrees.
What the symbols mean
- open-loop transfer function
- gain-crossover frequency, in radians per second
- phase margin, in degrees
The 45-degree example assumes a suitable single crossover and the appropriate phase branch. Multiple crossings or open-loop unstable poles require the full test. The interactive point below illustrates geometry at one frequency, not the complete curve.
A 45 degree phase margin
Suppose an appropriate single-crossover loop has magnitude one at a chosen frequency, with phase minus 135 degrees.
Subtract the phase lag from 180 degrees. The margin is 45 degrees.
Another 45 degrees of lag at that same crossover would point to minus one. A margin alone is not a universal stability proof when there are several crossings or unstable open-loop poles.
More detail and model assumptions
Why −1 is the critical point
The closed-loop denominator of unity negative feedback is 1+L. It vanishes when L=−1: magnitude one and phase an odd multiple of 180° at the same frequency. Thus −180° phase by itself is not forbidden. If |L|=.1 there, the point is −.1, not −1; the remaining distance is relevant to gain margin.
At a single 0 dB crossing, phase margin is the additional phase lag needed to reach −180°. Gain margin is the additional loop gain needed to reach magnitude one at a −180° crossing. These are local readings under stated loop assumptions, not probabilities of safety and not guarantees against saturation.
The Nyquist curve carries more information
Nyquist plots L as a path in the complex plane. Closed-loop stability depends on how that path encircles −1 together with the number of open-loop unstable poles and the contour convention. Just seeing the curve pass near −1, or seeing positive phase margin at one crossing, is not the full Nyquist stability test.
The Lab shows the positive-frequency branch and its mirror for the nominal sampled motor. Its margin labels are suppressed when there is no single relevant crossing. Compare the time run separately when the actuator saturates or the actual motor differs from the nominal one.
Distance to −1: 0.77. This is a geometric illustration at one frequency, not a stability test for an entire loop.
Connect the example to the lab
At 0 dB crossover, a phase of −135° gives a 45° phase margin under the standard single-crossover convention. An additional 45° of lag would reach the critical direction.
Inspect margins and saturationMake a prediction before moving a control.
- PREDICTIs −180° automatically unstable?
- CHANGE ONE THINGPlace points L=−.1 and L=−1 on a complex plane and compute 1+L for each.
- OBSERVEIn the motor workbench, read phase crossing, gain magnitude and the Nyquist −1 marker together.
- EXPLAINExplain what a local margin says, and which open-loop pole information a full stability claim still needs.
'−180° means unstable' is false. The gain must also be considered, and Nyquist encirclements account for unstable open-loop poles.
Check your understanding+
At phase −180°, is a loop with |L|=0.1 necessarily at the critical point?
No. It is at −0.1, not −1. The standard gain margin there is 10×, or 20 dB, if the usual assumptions apply.Why the next lesson follows
We now turn to a controller used in many real loops: PID. Its three actions change low-frequency error and transient behavior.