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Subject: Re: Poles of Amplifiers

Date: 01/26/01 at 6:08 AM
Posted by: John Dunn - Consultant
E-mail: ambertec@ieee.org
Message Posted:

In Reply to: Poles of Amplifiers posted by Cm on 01/12/01 at 1:45 AM:

Hello, Cm.

Please let me try to explain matters this way:

The input-to-output transfer function of a linear system is expressed as a fraction of two polynomials, one polynomial as the numerator and the other polynomial as the denominator. I like to call the complete transfer function H(S), the numerator N(S) and the denominator D(S). Thus, H(S) = N(S)/D(S).

The term "S" referes to a complex number of the form "sigma + j * omega" where "j" is the square root of negative one and the terms "sigma" and "omega" are the "real" and "imaginary" parts of a complex frequency. The "omega" by itself is in units of radian frequency in units of radians per second where omega = 2 * pi * f, where "f" is frequency expressed in hertz.

There are roots in S of both of the polynomials N(S) and D(S). The roots of the numerator polynomial are the zeros of the transfer function. The roots of the denominator polynomial are the poles of the transfer function. The poles act to reduce the magnitude of H(S) at higher frequencies and the zeros tend to increase that magnitude.

A linear system is considered to be stable if it non-oscillatory. That mns, if it is lone, it eventually settles to some quiescent state. For example, a weight suspended from a spring may bob up and down for quite some time if disturbed, but it will eventually come to a stop as enough time transpires for all of the weight's kinetic energy to be dissipated in friction of the spring's bending, air resistance, and so forth. The spring and weight may seem "unstable" because it is easily stirred into motion, but it qualifies as "stable" because it will eventually come to a stop.

A linear system's stability can be analyzed by studying the roots of the denominator. If the denominator's value of "sigma" is positive, the system will be unstable. In an x-y coordinate plane diagram where sigma is drawn along the x-axis and omega is drawn along the y-axis, for sigma being positive, the poles of an unstable system are said to lie in the "right half plane". Conversely, if the sigma is negative and the poles lie in the "left half plane", the system is stable.

The magnitude of H(S) versus frequency determines stability. Where feedback is involved, stability is achieved when the transfer function's magnitude has no frequency at which the magnitude is greater than one, or unity, with an in-phase feedback. Since it is assumed that feedback is designed to be negative with a magnitude of greater than unity, we call that negative feedback our "zero degrees". In-phase feedback then means a phase angle of 180 degrees. At that particular frequency where the magnitude of H(S) has descended to unity, the difference between the actual phase angle versus the 180 degrees is called the "phase margin". For example, if we had a stable system with unity gain at a phase shift of 150 degrees, our phase margin would be 30 degrees.

This is just a start. I hope I've shed some light, but I recommend finding a good textbook because this subject constitutes the bulk of a college level course.

Good luck.

John Dunn - President
Ambertec, Inc.
ambertec@ieee.org

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