pole zero in the system

[shanmei]

For example, the transfer function H(s)=1/(1+S).
the definition of pole is 1+S=0, then H(s) is infinite, but the pole's role is to make the system signal decrease, for example, the gain of an amplifier would go down when the signal frequency pass though a pole, but now H(s) is infinite, it seems that it would increase the gain of the signal.

so what is the problem ? thansk..

 

[varunkant2k]

S = -1 = sigma+j*w. Do you think, frequency can be negative?
Invert Laplase ( of the function) is e^-t. the signal decreases exponentially. At t = infinite, it will be zero but at w = 0 Hz, H(t) will be one.
We assign characteristic equation to zero to get the pole location in S-plane.Representaion of Pole/Zero in s domain helps us to find the stability.
Hope your doubt clears here.

 

[LvW]

Hi shanmei, here comes a more general explanation:

First, you have to notice that the meaning and the usage of the symbol"s" is twofold:
* "s" ist nothing else than an abbreviation for jw to be applied for small signal analysis (calculation by hand like in 1/sC=1/jwC.)
* s=sigma+jw is a so called "complexe frequency" (which do NOT exist in reality). However, its implementation opens the way to a very useful interpretation of transfer functions and their properties (e.g. stability analyses). More than that - in conjunction with the Laplace transform - it allows an easy transition to the time domain behaviour.
Thus, transient waveforms can be described very easily.

Regarding your example: Replace s by jw and you have the frequency dependent function that describes a first order lowpass. Of course, the magnitude of this function shows absolutely no increase for rising frequencies.
Replacing s by (sigma+jw) and setting the denominator=0 leads to (as mentioned by varunkant) the expression sigma+jw=-1. Of course, such a "frequency" does not exist.
If you could excite your circuit with such a "frequency" it would indeed produce a very large (infinite) output - but you can't !
However, the system theory tells us that the location of this "pole" at sigma=-1 (on the negative-real axis of the complex s-plane) is a good and very instructive description of frequency-dependent system properties. The real benefits of this principle can only be seen for higher order functions.