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### 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.

*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 | |
---|---|---|

-40 | ∼ | 1/100 |

-20 | ∼ | 0.1 |

2 | ∼ | 1.26 (∼ +25 %) |

3 | ∼ | 1.41 (∼ +40 %) |

6 | ∼ | 2 |

20 | ∼ | 10 |

30 | ∼ | 30 |

40 | ∼ | 100 |

60 | ∼ | 1000 |

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.

**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.

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

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