# Bode Phase plot of a second order system

1. Jun 11, 2008

### bumclouds

Hey guys.

I need to know how to draw a Phase Bode plot of a Second order system.

I understand and can draw the Gain(Magnitude) Bode plot, but I can't seem to get the grip of the Phase one.

As far as I know there is an asymptote at 0$$^{o}$$ at low frequencies and an asymptote at 180$$^{o}$$ at high frequencies. But the transition between the two changes with damping factor.

Do any of you know the relationship between damping factor and the asymptotes so that I am able to draw an accurate Phase Plot of 2nd Order Systems?

Thanks guys

2. Jun 11, 2008

### Staff: Mentor

Dunno about "damping factor". A second order system might have two poles, or a pole and a zero. Each pole or zero will introduce a phase change in the frequency region near it:

http://en.wikipedia.org/wiki/Bode_plot

3. Jun 11, 2008

### asynchronous13

Let's say you've got a transfer function

$$\frac{X(s)}{U(s)}=\frac{G\cdot\omega^2}{s^2+2\zeta\omega s+\omega^2}$$

Then $$\zeta$$ is your damping ratio. If I remember correctly, the damping ratio determines the shape of the step response and impulse response of the system.

If $$\zeta > 1$$ then the system is over-damped and the system response is slow.
If $$\zeta < 1$$ then the system is under-damped and the output has overshoot or ringing before reaching steady-state.
If $$\zeta = 1$$ then it is critically damped, which is the fastest response without oscillations.
If $$\zeta = 0$$ then it oscillates.

The location of the poles on the s-plane can be determined from damping ratio (or the damping ratio can be calculated if you know the location of the poles).

Set the denominator in the above equation to zero and solve for s to get the locations of the poles.

(sorry, running out of time, hope this helps you get started....)

4. Jun 11, 2008

### rbj

that "zeta" coefficient that is related to Q. i think that's what it is.

i wouldn't call a system with one pole and one zero a "second-order system". it's a first-order system.

i thought that Bode plots (using these asymptotes) were good when the poles and zeros were real (so the system can be modeled as a cascaded sections of smaller first-order systems). if the poles/zeros are complex and not so awful close to the real axis, then you need to plot out the frequency response legitimately (using a plotting function in software) instead of the asymptotic approximation approach using Bode plots. you can determine where the asymptotes go, but the behavior around the corners (where asymptotes meet) will be a lot different for complex pole pairs compared to real poles.

5. Jun 11, 2008

### Staff: Mentor

Thanks for the clarifications, rbj. Good stuff.

I like this article about the Relationship between the Root Locus plot and 3-D Bode Diagrams -- the 3-D relationship is a neat way to think about it:

http://www.ae.gatech.edu/people/ptsiotra/Papers/3DBodePlot.pdf

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