Bode plots: gain and phase
Two aligned plots show how a loop responds across slow and fast inputs.
Convert between amplitude ratio and decibels, and recognize the zero-decibel crossover.
A single frequency gave one magnitude and phase. Bode plots put those results on aligned axes over a range of frequencies.
Review the previous lesson: Why analyze frequency and phase?First, picture it
A Bode magnitude plot answers 'how much does this frequency get through?' The phase plot answers 'how late does it arrive?' The logarithmic axis compresses a huge frequency range into a readable map.
Build the idea step by step
A Bode plot puts two measurements on the same frequency axis: amplitude ratio and phase. Its horizontal spacing follows multiplication, such as 1, 10, and 100, rather than equal additions. The magnitude uses decibels to display large and small ratios together. Zero decibels means a ratio of one, so the signal passes at the same amplitude. First check whether the plot shows the plant, the open loop, or the closed loop. The meaning changes even when the axes look the same.
Read the equations
Decibels express an amplitude ratio on a logarithmic scale. Zero decibels means a ratio of one.
What the symbols mean
- open-loop transfer function, usually controller times plant
- angular frequency, in radians per second
- amplitude ratio in decibels
The factor 20 converts an amplitude ratio to decibels. A gain of 10 is 20 dB; a gain of 0.1 is minus 20 dB. At a loop gain crossover, magnitude is one. We will use its phase to discuss margin next.
For a first-order low-pass plant, high frequencies are attenuated and increasingly delayed. This is a shape sketch, not numerical data from the lab.
What plus six decibels means
At one frequency, suppose the open-loop output amplitude is twice its input amplitude.
Convert this ratio to decibels: about plus 6.02 dB.
At a frequency where the ratio falls to 1, the magnitude plot reaches zero decibels. Phase at that frequency will matter for the next lesson.
More detail and model assumptions
Read the two panels together
Magnitude shows the ratio of output to input amplitude. In decibels, 20log₁₀|L|=0 means |L|=1; +20 dB means ten times the amplitude and −20 dB means one tenth. Phase shows how far the sinusoidal output leads or lags. Neither panel alone tells the whole feedback story at crossover.
The horizontal axis is logarithmic frequency: equal distances represent ratios, not equal rad/s differences. At the gain-crossover frequency, read phase on the phase panel to estimate phase margin. At a −180° phase crossing, read magnitude to estimate gain margin. If there are several crossings, one headline number may hide the important one.
A design plot, not a verdict
A Bode plot of open-loop L helps compare low-frequency tracking, crossover and high-frequency noise exposure. It is not itself the closed-loop step response. To connect the two, compute the closed-loop relation and then test the time response under the same nominal, unsaturated assumptions.
When P, I or D changes, the controller changes L and the gain crossover may move. Read the new margin at the new crossover. The Lab's plotted loop is discrete and excludes load, noise, parameter mismatch and voltage saturation; its label describes exactly this scope.
Gain crossover 7.86 rad/s, phase margin 52°. Raising K moves the magnitude curve up, so crossover moves right where the lag has taken more phase. This idealized loop never reaches −180°; a real motor adds sampling delay, saturation and unmodeled dynamics, so a margin here is not a guarantee for hardware.
Connect the example to the lab
If a loop is +6 dB at some frequency, its sinusoidal output amplitude is roughly twice its input amplitude at that frequency.
Open a live motor Bode plotMake a prediction before moving a control.
- PREDICTDoes moving crossover also move the margin?
- CHANGE ONE THINGPin one motor run and change only P; open Bode in the frequency workbench.
- OBSERVERead both gain crossover and phase margin before and after, then compare the step response.
- EXPLAINExplain why a faster rise does not automatically mean greater robustness.
Do not confuse plant G, controller C, open loop L=CG and closed loop T=L/(1+L). Their Bode plots answer different questions.
Check your understanding+
What does 0 dB mean on a loop-gain Bode plot?
The magnitude of the open-loop transfer function is one at that frequency.Why the next lesson follows
At gain crossover, the amount of phase lag tells us how near the loop is to the critical minus-one point.