Q: How does an integrator K/s appear on a Bode plot?

An integrator G(s) = K/s has a gain plot that is a straight line with slope -20 dB/decade, passing through 0 dB at ω = K rad/s. The phase is constant at -90° across all frequencies. This means an integrator alone contributes exactly -90° of phase lag. When combined with other elements, the phase margin is measured at the frequency where the total gain is 0 dB.

Q: How do I enter custom numerator and denominator coefficients?

Enter polynomial coefficients from highest to lowest power, separated by commas. For example, a numerator of '2s+1' is entered as '2,1', and a denominator of 's²+3s+2' is entered as '1,3,2'. So G(s) = (2s+1)/(s²+3s+2) is specified as numerator '2,1' and denominator '1,3,2'. Coefficients can be any real numbers.

Question 1

What are gain margin and phase margin in a Bode plot?

Accepted Answer

Gain margin (GM) is the additional gain (in dB) that can be applied to the open-loop system before it becomes unstable. It equals -20·log₁₀|G(jω_pc)| dB where ω_pc is the phase crossover frequency (where phase = -180°). Phase margin (PM) is the additional phase lag before instability, equal to 180° + ∠G(jω_gc) where ω_gc is the gain crossover frequency (where |G| = 0 dB). A system is stable when both GM > 0 and PM > 0.

Question 2

When does a 2nd-order system show a resonance peak in its Bode plot?

Accepted Answer

A 2nd-order system G(s) = ωn²/(s²+2ζωns+ωn²) shows a gain peak (resonance peak) when the damping ratio ζ < 1/√2 ≈ 0.707. The peak frequency is ωr = ωn√(1-2ζ²) and the peak magnitude is Mr = 1/(2ζ√(1-ζ²)). Lower damping gives a sharper, higher peak. The phase decreases monotonically from 0° to -180°.

Question 3

How does an integrator K/s appear on a Bode plot?

Accepted Answer

An integrator G(s) = K/s has a gain plot that is a straight line with slope -20 dB/decade, passing through 0 dB at ω = K rad/s. The phase is constant at -90° across all frequencies. This means an integrator alone contributes exactly -90° of phase lag. When combined with other elements, the phase margin is measured at the frequency where the total gain is 0 dB.

Question 4

How do I enter custom numerator and denominator coefficients?

Accepted Answer

Enter polynomial coefficients from highest to lowest power, separated by commas. For example, a numerator of '2s+1' is entered as '2,1', and a denominator of 's²+3s+2' is entered as '1,3,2'. So G(s) = (2s+1)/(s²+3s+2) is specified as numerator '2,1' and denominator '1,3,2'. Coefficients can be any real numbers.

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