#### 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

$\ldots+\ m\ddot{x}+d\left({\dot{x}-\dot{x}}_{in}\right)+c\left(x-x_{in}\right)+\ldots=F_e$

Step 2: Laplace transform (using $x=x$, $\dot{x}=sx$, $\ddot{x}=s^2x$, …)

$\ldots+\ ms^2x+ds\left(x-x_{in}\right)+c\left(x-x_{in}\right)+\ldots=F_e$

Step 3: Transfer function (choose in and out) $H\left(s\right)=\frac{out\left(s\right)}{in\left(s\right)}$

$H(s)=\frac{x(s)}{x_{in}(s)}=\frac{ds+c+\ldots}{\ldots+ms^2+ds+c+\ldots}$

Step 4: Magnitude & Phase response $s=j\omega$, $j^2=-1$

$\left|H\right|=\sqrt{\left[Re\left(out\right)\right]^2+\left[Im\left(out\right)\right]^2}/\sqrt{\left[Re\left(in\right)\right]^2+\left[Im\left(in\right)\right]^2}$

$arg\left(H\right)=atan\left(\frac{Im\left(out\right)}{Re\left(out\right)}\right)-atan\left(\frac{Im\left(in\right)}{Re\left(in\right)}\right)$

Step 5: Bode plot, $f\left[Hz\right]=\frac{\omega}{2\pi}$

$A=\left|H\right|$ , $A[dB]=20log(A)$ , $A={10}^{A\left[dB\right]/20}$

$\varphi\left[deg\right]=\arg{\left(H\right)}\cdot\frac{180}{\pi}$

Magnitude plot: loglog($f,A$), phase plot: semilogx($f,\varphi$)

#### Magnitudes conversion

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 |

#### Slope & (Bode’s) gain/phase relation

Decrease in magnitude is related to phase lag: $\pm20dB/dec\sim\pm{90}^\circ$ phase shift for stable non-minimum phase systems -> effect of 1 pole or 1 zero.

#### Zeros ($z_i$) and poles ($p_i$):

$H=\frac{N\left(s\right)}{D(s)}=K\frac{\left(s-z_1\right)\left(s-z_2\right)\ldots\ (s-z_n)}{\left(s-p_1\right)\left(s-p_2\right)\ldots\ (s-p_m)}$

$z_1,\ldots.,\ z_n,p_1,\ldots,p_m$ 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:

$H\left(s\right)=\frac{A}{s^2}=\frac{1}{ms^2}$

#### Complex poles/zeros (resonance/anti-resonance)

For complex (conjugate) poles and zeros, the gain/phase relation is doubled: $\pm40dB/dec\sim\pm{180}^\circ$ phase. The behavior around the complex poles/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.

$H(s)=\frac{s+2\zeta_z\omega_n+\omega_n^2}{s+2\zeta_p\omega_n+\omega_n^2}$

#### Bandwidth (open-loop definition)

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