# Root Locus question

1. May 14, 2008

### bill nye scienceguy!

If have a system with 4 open loop poles, all at say s=-2, how would the locus approach each of the four asymptotes?

thanks

2. May 14, 2008

### rbj

are you really the bill nye on the radio? just curious. after looking at your history, i somehow doubt it.

this thing sounds like what happens with a Moog 4-pole Low Pass Filter. once you surround it with a loop and gain, you just solve the new transfer function for the poles and you can see that they move outward from the original place like 4 corners of a square centered at the original location. are you gonna make me do the math to show you?

3. May 15, 2008

### bill nye scienceguy!

unfortunately not, sorry!

anyway, do you mean that the open loop transfer function G(s)=1/(s+2)^4 would become something like Gc(s)=K/[(s+2)^4+K]?
I still can't really visualise what the locus would look like though. Does it just expand outwards from s=-2, following the four asymptotes?

4. May 15, 2008

### rbj

yes it does. but it's not really asymptotes. the loci of the roots really are the four corners of a square centered at s=-2. as K gets bigger, the square gets bigger (not proportionately). (i am presuming that K is the signed, linear gain in series with G(s) and that the feedback is negative feedback with unity gain.)

$$G_c(s) = \frac{K G(s)}{1 \ + \ K G(s)} = \frac{K}{1/G(s) \ + \ K}$$

now, to find the poles of Gc(s), you set the denominator to zero and solve for s. if you are not already familiar with the nth roots of unity (involving complex numbers), you should get that down first.

Last edited: May 15, 2008