One transfer function, four views. Drag a pole across the s-plane and watch all of them move together — the Bode curves, the Nyquist locus and the step response are not separate calculations, they are the same numbers read differently.
A transfer function is a product of simple factors. In dB (20·log₁₀|G|) and degrees, multiplying them becomes adding their magnitude and phase contributions.
The break point of a pole or zero. Above it a pole rolls off at −20 dB/decade and a zero rises at +20, each asymptotically.
The number of pure integrators in the loop: Type 0 has steady-state error to a step; Type 1 tracks a step but lags a ramp.
Phase margin at gain crossover (target roughly 45–60°); gain margin at phase crossover (at least 6–10 dB).
Each real pole contributes exactly one factor of frequency to the rolloff well past its corner. Stack two and you get −40 dB/decade, three −60: additively. Each pole also adds, eventually, another −90° of phase lag, which is why stacking poles steadily erodes phase margin toward the −180° instability threshold.
A system built only from left-half-plane poles and zeros, like everything this tool can build, has the smallest possible phase lag for a given magnitude shape. Right-half-plane zeros add extra lag without changing the magnitude at all.
Two poles arriving together reinforce each other in phase at their natural frequency when damping is low: the lower the damping ratio ζ, the taller and narrower the peak, until at ζ ≥ 1/√2 (about 0.707) it disappears. An integrator, by contrast, is a pure 90° at every frequency, with no corner.
At the corner the real gain is −3.01 dB, not 0, and the phase exactly −45°. The gain error is largest at the corner and 0.04 dB a decade away; the phase asymptote is exact at the corner and worst at the decade marks, 5.7° out. The two errors never coincide, so a margin read off straight lines is taken from where the phase construction is least reliable.
How much extra phase lag the loop can tolerate at the gain-crossover frequency (where the loop gain is exactly 1, 0 dB) before the phase reaches −180° and the feedback that was stabilising the system starts reinforcing it instead. Textbook target: roughly 45–60° for a well-damped response.
How much extra gain the loop can tolerate at the phase-crossover frequency (where phase is exactly −180°) before the magnitude reaches 0 dB and the same instability sets in. A healthy design usually wants at least 6–10 dB of room here.
Two poles arriving together (a complex-conjugate pair) briefly reinforce each other in phase right at their natural frequency when damping is low — the lower the damping ratio ζ, the taller and narrower that peak, until at ζ ≥ 1/√2 (≈0.707) the peak disappears entirely.
1/(jω) is a pure 90° phase shift at every frequency — there's no corner to it, which is exactly why integrators (and the "type" of a control system) contribute a fixed phase lag rather than a frequency- dependent one.
The straight-line construction is exact far from the corner and wrong precisely at it. At the corner frequency the real gain is −3.01 dB, not 0, and the phase is exactly −45°.
| Frequency | Asymptote | True gain | Asymptote | True phase |
|---|---|---|---|---|
| ÷10 | 0.00 dB | -0.04 dB | 0.0° | -5.7° |
| ÷5 | 0.00 dB | -0.17 dB | -13.5° | -11.3° |
| ÷2 | 0.00 dB | -0.97 dB | -31.5° | -26.6° |
| ×1 | 0.00 dB | -3.01 dB | -45.0° | -45.0° |
| ×2 | -6.02 dB | -6.99 dB | -58.5° | -63.4° |
| ×5 | -13.98 dB | -14.15 dB | -76.5° | -78.7° |
| ×10 | -20.00 dB | -20.04 dB | -90.0° | -84.3° |
The gain error is largest at the corner and negligible a decade away — 0.04 dB at ten times out, which no instrument would resolve. The phase behaves the opposite way: the asymptote is exact at the corner and worst at the decade marks, where it is 5.7° out. So the two errors never coincide, and a stability margin read off the straight lines is being taken from the one place the phase construction is least reliable.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.