What are poles and zeros ?

[norman king]

Please can anybody provide me, or tell me where to look, for a simple explanation of "poles" and "zeros" as encountered in closed loop systems.
I want to know how to add them to make closed systems stable.

 

[Roff]

Don't get me wrong here. I'm not putting you down. This is just my observation based on my personal experience.
I'm betting that, without a class in control system (feedback, servo, ...) theory, and maybe a Laplace transforms class, you will never understand how to stabilize a closed loop. I studied both in college, and it was and still is a struggle to understand and apply the theory.

 

[_3iMaJ]

Mathematically speaking, poles are the points at which the denominator of a rational polynomal are zero. Zeros are the points at which the numerator is zero.

#poles = #of zeros, always.

 

[Roff]

#poles = #of zeros, always.

Not true. Where did you get that idea? What about a multi-pole lowpass filter? It has any number of poles, but no zeroes.

 

[i_build_stuff]

Assuming you know how to analyze the circuit in the s domain, what you do is take your equations and solve for Vout(s) / Vin(s). This gives the transfer function, which usually works out like this:

Vout(s) = Vin(s) * H(s)

Which is usually the ratio of two polynomials in s (if it's NOT a ratio of polynomials, or if you don't get an H(s) that multiplies by Vin(s), then your circuit is nonlinear and the normal tricks don't apply). Here is an example of what you might get when they're factored:

H(s) = (s+1)(s-2) / ((s-3)(s+5)(s-10))

Now, the poles can be read off as 3, -5, and 10, and the zeros are -1 and 2. Note that this is a simplified example, and the zeros / poles are usually complex.

 

[Russlk]

here is a rule of thumb: If the open loop transfer function passes thru 0 dB at 6 dB per octave, it will be stable.

 

[norman king]

THANK YOU ALL FOR YOUR REPLIES

TO RON H.. YOU WON YOUR BET ... LIKE YOU YEARS AGO I HAD A CLASS IN CONTROL THEORY AND LAPLACE THEORY AND STILL CANT UNDERSTAND HOW TO APPLY WHAT I WAS SUPPOSED TO LEARN. MAYBE A HANDS-ON LAB SESSION WAS NEEDED TO GET TO GRIPS ON THE SUBJECT. TOO LATE NOW ?

TO RUSSLK... IS WHAT YOU SAY THE SAME AS SAYING THE GAIN MUST BE LESS THAN ONE WHEN THE PHASE SHIFT THROUGH THE LOOP IS 180° ?

 

[Russlk]

Yes, Norman

 

[norman king]

Thank you.

 

[rakarlin]

Cut your loop at some convenient point. You may need to add components so the cut ends still see the same source and terminating impedance. Apply a frequency sweep to the input side of the cut. Measure the response at the output side of the cut. The "transfer function" is the Output divided by the Input, usually volts, although other units such as current and power are possible. Plot the transfer function against the input variable, usually voltage at changing frequency. Any place the transfer function blows up to infinity (or very large) is a pole. Any place it falls to zero, (usually at zero frequency and infinite frequency), we have a zero. By stability, most engineers mean unconditional stability: stable hot, cold, new, old, high line, low line, parts anywhere in their specified range.... You can tinker one or two units into stability by playing pole-zero games. It may not be unconditional, and forget this for production. Take the lowest rolloff at the high frequency end and lower it with more C until the loop gain drops below 1.000 before the second lowest rolloff is significant. Some margin may be necessary. I go for the first stage which I always design for the highest gain. I add a capacitor from stage output to stage inverting input. (E.G. Plate to grid, Collector to base, or drain to gate.) This is a "Miller Effect" capacitor. The foregoing applies to linear amplifiers. Servo systems are much more complex, but the same definition applies to poles and zeroes.

 

[rakarlin]

Gain less than 1.000 when phase shift hits 180 is the stability rule, but keep in mind all the things that can change. Some margin is essential.

 

[Roff]

Rakarlin, you are responding to a 4 year old post.

 

[Mikebits]

I to have fell victim to the RTOP (responding to old post) syndrome, would be nice if the monitor could bonk ya on the head when doing so A software warning would be cool, is that possible?

 

[dknguyen]

But then we would miss out on getting to bonk you on the head instead.

 

[rakarlin]

Yes. That is fairly obvious from the dates. But the entry is still up, and the physics is still the same. Incidentally, Wikipedia has a pretty good article on poles, zeros, and such. I do not know of much that is good on stability. Most of it is totally impractical. You could make it work for one unit on a good day at a fixed temperature and supply voltage, but that's it. Not quite what we mean by unconditional stability. Thanks for the response.

 

[Nigel Goodwin]

Yes. That is fairly obvious from the dates. But the entry is still up, and the physics is still the same.

But the OP hasn't been on these forums since 2006, it's pointless replying to such old posts - check the dates before you post.

 

[rakarlin]

If the site managers wish to do so, they can lock out further posts. When this is done, it is customary to append a statement that the thread is now closed. If you feel that they should close the thread, by all means tell them so!! The thread was open, so I felt free to add to it. I understand that this could create problems for very slow readers, and for those who have trouble with dates, and I apologize for any such trouble, but I am only a member and decisions such as which threads to lock and when to lock them are well above my pay grade.

 

[MrAl]

Hi,

Couldnt help but wonder if you got a message in your email about this thread or something. I got a real old one one time, where it was several months later after the post had been submitted here. Thought that was very strange too.
If that's not the case, i would have to wonder how you even found this old thread

 

[Roff]

Hi,

Couldnt help but wonder if you got a message in your email about this thread or something. I got a real old one one time, where it was several months later after the post had been submitted here. Thought that was very strange too.
If that's not the case, i would have to wonder how you even found this old thread

Rakarlin has made 4 posts, and they are all in this thread. His first was a couple of days ago.

 

[aljamri]

Yes. That is fairly obvious from the dates. But the entry is still up, and the physics is still the same. Incidentally, Wikipedia has a pretty good article on poles, zeros, and such. I do not know of much that is good on stability. Most of it is totally impractical. You could make it work for one unit on a good day at a fixed temperature and supply voltage, but that's it. Not quite what we mean by unconditional stability. Thanks for the response.

During my study, I've used to get written problems and solve them using MATLAB by applying Root Lucus and another two theorms.

Some where during my course, I've been told that this can be applied on electronics. Since then I'm looking for time to study this.

In case this is not violating any rules of the Forum, I'm interested on how to make it step by step, from real circuit values up to adding / omitting poles and zeros; If any one interested.