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Design >> Analog Design >> Poles and Zeros
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Message started by ARJ on Nov 14^th, 2007, 1:25pm

Title: Poles and Zeros
Post by ARJ on Nov 14^th, 2007, 1:25pm

Hi all,

I guess i have a very foolish question.

We have all learned that poles and zeros are contributed by capacitor either parasitic or external. Poles tend to increase the gain to infinity and zero to 0.
I have learned that in most cases the output load capacitance , or in the case of Op-amp the capacitance at the output of first stage, is the dominant pole.

The capacitance impedance decreases as the frequency increases. Suppose F0 be the frequency of the dominant pole. when f > F0, the dominant capacitance tend to decrease its impedance, hence the quantum of voltage dropping across it reduces.

Hence according to the above explanation the gain should decrease and that's what happens when u see the Bode plot.

Then why is that poles are defined as those which increase the gain to infinity.

Please correct me.

Thanks

ARJ

Title: Re: Poles and Zeros
Post by RFICDUDE on Nov 14^th, 2007, 7:18pm

THe reason for this is that the pole location is out in the left half plane of the S-plane s=sigma + j*omega, and the pole can either be on the negative real axis or exist as a complex conjugate pair.

So if s actually took on complex values when evaluating H(s) then the function would blow up when s equals the pole value. But transfer functions are only evaluated at s=j*omega (sigma=0). Therefore we only see the cross sectional view of the function H(s) as it cuts through the j*omega plane. Negative real poles look like low pass functions and complex conjugate pairs can look like peaked low pass, band pass, or high pass functions. Really high Q poles have very small real parts (sigma), so these really do have high peaks near the j*omega axis.

Hope this helps.

Title: Re: Poles and Zeros
Post by buddypoor on Nov 16^th, 2007, 1:16am

Hello ARJ !

The answer of RFICDUDE is totally correct. However, perhaps it is not quite understood by you as it is pure theory. Therefore I recommend a very illustrative application note which shows the effect of a transfer function pole.

Here is the reference: www.maxim-ic.com/appnotes.cfm/appnote_number/733/ - 65k

(look at Fig. 4)

Perhaps this can help as well.

Lutz

Title: Re: Poles and Zeros
Post by ARJ on Nov 28^th, 2007, 12:20pm

Thank you guys,

That was really helpful.. it cleared many of my doubts.....

By the way sorry for replying so late.

ARJ

Title: Re: Poles and Zeros
Post by joel on Dec 28^th, 2007, 2:15pm

I appreciate this discussion on the basics. To revive this thread and embarrass myself by exposing my limited understanding...

I've been trying to make sense of the Laplace Transform method. I can follow the math, but not the meaning. In particular,
when transforming from t to s spaces, one is going from one-dimensional (time) to two-dimensional (sigma, omega) spaces.
So what is the physical meaning of sigma? What constrains its value when transforming from t to s? What (if any) information is lost
when transforming from s to t? Does my question even make sense? I don't know...

Title: Re: Poles and Zeros
Post by thechopper on Dec 28^th, 2007, 8:18pm

Hi joel,

The actual interpretation for sigma is not difficult:
The reason for the two dimensional space involved in the Laplace transform is that it represents any type of signal by using not only periodic functions (like in the case of the Fourier transform, i.e. sigma =0), but also their "non-periodic" components.
The two dimensions in the Laplace transform then can be interpreted as the "periodic" (imaginary axes) and "non-periodic" or transient components.
Therefore, sigma represents the transient component for the signal that was Laplace transformed. For stability (in feedback systems) you know it is required left plane poles. Why? because the transient component needs to vanish for t -> ∞. Otherwise it will let the periodic part of the signal to grow with time rather to disappear.

Hope this helps
Tosei

Title: Re: Poles and Zeros
Post by buddypoor on Dec 29^th, 2007, 5:24am

Hello joel,

here is a rather clear and evident interpretation of "sigma" and "omega" in the s-plane:

Take a simple RLC-circuit and stimulate it with a step function. Than calculate the step reponse in the time domain. The solution of the corresponding differential equation is a time function exp(s*t). By introducing the presently unknown factor "s" (which has the dimension (1/sec) into the characteristic equation the solution shows that "s" consists of two parts: one is real (called "sigma", with a negative sign) and the other one is pure imaginary (jw). The latter one - if substituted back into the e-function - has to be interpreted as a sine function : exp(jwt).
Therefore, the real part exp(sigma*t) determines the amplitude of the sinus, which decreases continuously with increasing time. So we have a frequnecy with a decaying amplitude.
Thus, the solution of the differential equation of a system leads directly to a parameter which is called the complex frequency variable "s=sigma+j*omega".

Perhaps this can help to understand the physical meaning of "s".
Regards
Lutz

Title: Re: Poles and Zeros
Post by joel on Dec 29^th, 2007, 6:48pm

Thank you! A few words of explanation and the light begins to shine!
I've been studying from Schaum's Signals & Systems, which does not
indulge in a great deal of discussion. My intro electronics text doesn't
address Laplace or the pole/zero techniques, and my 'analog vlsi' books
assume I've already learned it. Can you recommend a good book
about this stuff? Thanks again, /jd

Title: Re: Poles and Zeros
Post by buddypoor on Dec 30^th, 2007, 2:41am

Hi joel.

I cannot recommend any specific textbook in english since I am familiar with german textbooks only.
However, any good textbook on analog active filter circuits should explain the meaning of the complex frequency variable since the LAPLACE transform is used extensively in filter theory.
For example, the application note

http://www.maxim-ic.com/appnotes.cfm/appnote_number/733/

explains in a nice manner the difference between "s" and "j*omega". Of course, in the real world only "omega" exists, however, the use of "s" opens a very elegant way to describe filter responses with poles in the complex s-plane. Here, the pole location can be described by two specific parameters (pole-Q and pole frequency), which appears also in the filter transfer function - if "s" instead of "omega" is used.
Regards
Lutz