# BODE PLOT – COMPOSITION & INTERPRETATION

## Dynamics & Control

### INTRODUCTION

This sheet provides the steps to compose a bode plot of an arbitrary ordinary differential equation. In the resulting bode plot some insights and interpretations are presented, which are also valid for frequency response functions.

###### Composition of bode

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, )

dB Gain
-401/100
-200.1
21.26 (∼ +25 %)
31.41 (∼ +40 %)
62
2010
3030
40100
601000
###### Slope & (Bode’s) gain/phase relation

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.

###### Zeros () and poles ():

,…, ,,…, 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.

Suspension mode
Slope 0 with damped resonance followed by -2 slope in the low frequency range. Typical for suspension behavior in a motion system.

Mass line
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.

Notch filter
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.

###### Bandwidth (open-loop definition)

Zero-gain / 0 dB crossing in the open-loop Bode plot.