Step 1: Derive the (Ordinary) Differential Equation
Step 2: Laplace transform (using , , , …
Step 3: Transfer function (choose in and out)
Step 4: Magnitude & Phase response ,
Step 5: Bode plot,
Magnitude plot: loglog(f,A), phase plot: semilogx (f, )
|2||∼||1.26 (∼ +25 %)|
|3||∼||1.41 (∼ +40 %)|
Decrease in magnitude corresponds is related to phase lag: ±20dB/dec ∼ ±90° phase shift for stable non-minimum phase systems -> effect of 1 pole or 1 zero.
,…, ,,…, obtained by factorization
Poles and zeros determine the asymptotic values in the bode plot.
Slope -1 at low frequencies
If this is the case in the Bode plot of the controller, an integral action is present.
Slope 0 with damped resonance followed by -2 slope in the low frequency range. Typical for suspension behavior in a motion system.
The -2 slope part of the system response. The gain of the response equals the inverse mass of the system:
Complex poles/zeros (resonance/anti-resonance)
For complex (conjugate) poles and zeros, the gain/phase relation is doubled: ±40dB/dec ∼ ±180° phase. The behavior around the comples polse/zero frequency is determined by the damping factor.
A special case of a complex pole zero pair is the notch filter, with more damping in the poles than in the zeros. The result is a dip (notch) in the response.
Zero-gain / 0 dB crossing in the open-loop Bode plot.