BODE PLOT – COMPOSITION & INTERPRETATION

Dynamics & Control

Bode plot - Composition and interpretation

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.

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Composition of bode

Step 1: Derive the (Ordinary) Differential Equation
+m{\" x}+d(\dot x-\dot x_{in})+c(x-x_{in})+...=F_e

Step 2: Laplace transform (using x=x, \dot x={sx}, \" x=s^2x, …
...+ms^2x+ds(x-x_{in})+c(x-x_{in})+{\dots}=F_e

Step 3: Transfer function (choose in and out) H(s)=\frac {out(s)}{in(s)}
H(s)=\frac{x(s)}{x_{in}(s)}=\frac{ds}+c+...}{...+ms^2+ds+c+...}

Step 4: Magnitude & Phase response s={j\omega }, j^2=-1
|H|=\sqrt{[Re(out)]^2+[Im({out})]^2}/\sqrt{[Re(in)]^2+[Im(in)]^2}
arg(H)=atan\left(\frac{Im(out)}{Re(out)}\right)-atan\left(\frac{Im(in)}{Re(in)}\right)

Step 5: Bode plot, f[Hz]=\frac{\omega }{2\pi }
A=|H|, A[dB]=20\log (A), A=10^{A[dB]/20}
\varphi [deg]=arg(H)\ast \frac{180}{\pi }
Magnitude plot: loglog(f,A), phase plot: semilogx (f, \varphi)

Magnitudes conversion
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 (z_i) and poles (p_i): H=\frac{N(s)}{D(s)}=K\frac{(s-z_1)(s-z_2){\dots}(s-z_n)}{(s-p_1)(s-p_2){\dots}(s-p_m)}

z_1,…, z_n,p_1,…,p_m obtained by factorization

Poles and zeros determine the asymptotic values in the bode plot.

Bode plot - Composition and interpretation color

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(s)=\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: ±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.

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.

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