# Understanding poles and zeros

1. Feb 4, 2008

### edmondng

Hi,

I'm trying to gain a better understanding on poles and zeros. Seems everywhere i read it says the poles and zeros can tell you the system response, characteristic and you can do lots with it from filter design to controls. Its a broad area but was hoping someone can provide further information to enhance my understanding like.

Some questions:-
-Why do you always want the poles to be on the left hand side of the complex plane? Do you always want poles to be on the left hand side of the plane?
-How can you design filters from the poles/zeros. Is there a direct relationship between resistors/caps/inductors to poles zeros? Lets say i have an equation for my transfer function , can you pick values for the resistors/inductors/caps to make the n-order filter?
-How do you tell the stability/poles zeros of a control system? I don't see how you can come up with an equation to look at the poles/zeros of a system.
-The closer to a pole means higher gain/amplitude while closer to a zero means lower gain/amplitude.
-A reactive component like caps/inductors produces 1 pole. So if you want a 5 pole system you can get away with 'T' like figure (combine with inductor/resistor) with 2 extra caps on each side of the 'T', for a total of 5 caps. But how do you find the values based on your desired cutoff frequency?

Thanks

2. Feb 4, 2008

### chroot

Staff Emeritus
Poles in the right half plane correspond to growing amplitudes; for example, a sine wave that keeps getting louder and louder without bound. That's unstable. Poles in the left half plane correspond to exponentially decreasing amplitudes.

Essentially, every reactive element (capacitor or inductor) adds a pole or zero to the transfer function, depending upon how you wire it up. You can indeed go directly from a transfer function to an implementation with discrete RC or RL networks. Normally, you need op-amps or other circuitry to keep one stage from loading the others.

If you have a transfer function, you have everything you can possibly know about your system. You can easily find the frequencies for which the response is zero -- these are the zeroes, of course. Same for the poles.

Essentially, yes.

Again, in practice you would generally isolate each stage from the others with an op-amp or buffer of some kind. Look at the Sallen-Key active filter configuration; it is popular in industry.

- Warren

3. Jul 8, 2008