28-12-2012, 06:19 PM
Circuit Analysis Problems On Hurwitz polynomials
Circuit Analysis Problems.doc (Size: 772 KB / Downloads: 27)
Abstract:
A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this. However, for the most part, they assume that the components of the network are all linear. The methods described in this article are only applicable to linear network analysis except where explicitly stated.
A useful procedure in network analysis is to simplify the network by reducing the number of components. This can be done by replacing the actual components with other notional components that have the same effect. A particular technique might directly reduce the number of components, for instance by combining impedances in series. On the other hand it might merely change the form in to one in which the components can be reduced in a later operation. For instance, one might transform a voltage generator into a current generator using Norton's theorem in order to be able to later combine the internal resistance of the generator with a parallel impedance load.
KEYWORDS:
Hurwitz polynomial, real, complex, imaginary axis, right half-plane, even & odd polynomials & continued fractions, etc.
INTRODUCTION:
A: Hurwitz Polynomials:-
In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative. One sometimes uses the term Hurwitz polynomial simply as a (real or complex) polynomial with all zeros in the left-half plane (i.e., a Hurwitz stable polynomial).
A polynomial is said to be Hurwitz if the following conditions are satisfied:
Let P(s) =
Where ……., are the coefficients
For the polynomial P(s) to be Hurwitz,
1. P(s) is real when s is real
2. The roots of P(s) have real parts which are zero or negative.
A simple example of a Hurwitz polynomial is the following:
The only real solution is −1, as it factors to:
B: Properties
For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. For all of a polynomial's roots to lie in the left half-plane, it is necessary and sufficient that the polynomial in question pass the Routh-Hurwitz stability criterion. A given polynomial can be tested to be Hurwitz or not by using the continued fraction expansion technique.
1. All the poles and zeros of a function are in the left half plane or on its boundary the imaginary axis.
2. Any poles and zeroes on the imaginary axis are simple (have a multiplicity of one).
3. Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative.
4. Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle).
5. There have no any missing term of’s’
CIRCUIT ANALYSIS PROBLEMS ON HURWITZ POLYNOMIALS:-
In the analysis and design of linear electronic circuits, it is often necessary and desirable to restrict attention to first-order effects only. Simplifications of the circuit may take the form of a dominant natural frequency (or time-constant) description. That is, of the total natural frequencies of the circuit, one may be found to be dominant over a frequency or time interval of interest. For the low-pass situation, necessary conditions are presented in this paper to establish the dominant zero of a polynomial in terms of the coefficients of the polynomial. Simple expressions are also given to estimate the second-order effect of excess phase (or dead time). A simple transistor amplifier is used as an example.
A: Methods to solve Hurwitz Polynomials:-
For the stability of system function H(s), it must satisfy following conditions,
1. H(s) should not have a pole in right half of s-plane.
2. The poles of H(s) on the imaginary axis must be non- repeated. There should not be multiple poles of H(s) on imaginary axis.