Re: Poles an Zeros

Rob wrote:
Hello,

Could someone explain to me Poles and Zeros.
What does a Pole do and what does a zero do and how they interact
together.
How does phase, gain, and delays relate to them?

First, you have to wrap your mind around a plane that represents all possible frequencies in two dimensions, called the S plane.

The two dimensions of frequency are real and imaginary or time constant and sinusoidal. Time constants (decaying or growing) can be described as e (2.7 something, the base of natural logarithms) raised to a real power. Positive powers represent exponentially growing signals and negative exponents represent exponentially decreasing signals. The amount that any signal is growing or shrinking can be represented as a line across the S plane that is positioned along the real axis. If you make the exponent imaginary (has a square root of -1 factor in it) the result is a sine wave. Positive and negative imaginary exponentials are just sine waves going forward or backward in time (and they look exactly the same, since a sine wave is the same for all time). Imaginary frequencies are represented as a line across the S plane at right angles to the real exponential signals. Low frequencies are represented as being near the origin and higher frequencies are represented as being further from the origin. A zero frequency, representing DC, is the origin. So any exponentially growing or decaying sine wave of any frequency and decay or growth rate can be represented as a point on the S plane. Whew!

The amplitude of the signal of any frequency is represented as a distance at right angles to the S plane, like a rod sticking up out of it, for big values, or holes down into it for very small values. Actual linear systems frequency response to all frequencies can be represented as a surface or rubber *** that is lifted up of held down at frequency points that represent important (resonant or characteristic) frequencies for the system. If the system produces some combination of sine wave and decay or growth rate if energized and left otherwise un-driven, (zero continuous input energy resulting in a finite result energy), that natural response can be said to be an infinite response (because it happens with zero input energy). Such points on the S plane are represented as having the rubber response surface held up at that point by an infinitely tall "pole". There may also be particular combinations of sinusoid and decay rate or growth that cannot be produced, regardless of how hard you drive the system. Those are "zero response points on th S plane rubber surface, and they can be thought of points tacked down to infinitely deep holes at those points on the rubber response surface. If the "rubber" surface is given the right kind of elasticity, mathematically, once a system is described at its "poles" and "zeros", the rubber surface predicts its response to all combinations of wave and decay/growth frequency by the height of its surface.

Is that enough abstraction for you to work on visualizing for a while?

If a book or a link could be recommended along with a what math needs
to be used. (maybe a course outline to work from)

The S plane frequency representation is based completely on representing the signals and responses as complex exponentials (e raised to an exponent with real and imaginary parts, representing the two dimensions of the S plane. If you Google "S plane" and "exponential frequency" you should have more hits than you need.

I am right now working on systems which I have an good idea on how
they work but I don't seem to have the underlying understanding or how
to calculate.

Some fully worked out examples I can look at would also be helpful. I
just seem to be missing parts. I have found information. But I am
having some issues putting it all together. A number of fields all
converge and I am not able to find a path and seem to be jumping
around and not getting anywhere.
The area I am interested in are digitally controlled power converters
(DC/AC and DC/DC power supplies).

The S plane is an extremely useful visualization tool for capturing every possible response of a linear system to excitation by almost any signal.
--
Regards,

John Popelish
.