Bode/Nyquist Control Lab
move poles with sliders and see robustness margins change instantly
Complex Frequency Domain 🖖
Bode and Nyquist plots map the frequency response of LTI systems...
Predicting oscillation without solving equations 🖖
Instead of solving the differential equation, you can shake the system with sine waves across many frequencies and watch how it answers. The Bode plot records how much each frequency is amplified and how far it is delayed in phase. Trouble appears where the gain passes 0 dB while the phase nears −180°: feedback then reinforces itself and the loop rings. Phase margin measures how far you sit from that edge — more margin means calmer, steadier control.
Phase margin sets a hidden delay budget 🖖
A healthy phase margin quietly tells you how much pure time delay the loop can survive. A delay adds phase lag that grows with frequency (−ωτ), so it drains margin fastest where the loop is busiest. The tolerable delay is roughly τ = PM / ω_c, with PM in radians and ω_c the gain-crossover frequency. So a fast, wide-bandwidth loop can be wrecked by a few milliseconds of sensor lag that a slower loop with the same margin shrugs off.
Example problems
- stable baseline - Well-damped second-order loop with comfortable phase margin.
- near oscillation - Low damping and higher gain move phase/gain margins toward instability.
- slow but robust - Conservative low-bandwidth tuning with strong robustness.
- high-gain risk - High gain can create fast response but poor robustness margins.