The Routh Hurwitz criterion
Routh Hurwitz criterion
The Routh Hurwitz criterion provides a simple procedure that is known to confirm whether all the roots of a polynomial A(s) have negative real parts without having to compute the roots of A(s). Having real parts mean that it lies in the left half of the s-plane.
If the order n of the polynomials A(s) is even, and therefore coefficient a0, belongs to the row n, then a zero is placed under a0 in row n-1. The next step will be to construct row n-2 by using the entries of rows n and n-1 in accordance with the following mentioned formula:
Row n-2 = (an-1 an-2 - an an-3/ an-1) (an-1 an-4 - an an-3/ an-1)
It is to be noted that the entries in this row have determinants like quantities as their numerators. Therefore, the Routh Hurwitz criterion can be stated as follows:
The roots of the given polynomial lie in the left half of the s-plane if all the entries in the leftmost column of the Routh array are nonzero and have the same sign. If any changes in the signs are encountered during the scanning of the left-most column, the number of such changes is the number of roots of A(s) in the right half.
The Routh Hurwitz criterion can also be used to determine the critical value of the loop gain K for which the polynomial A(s) has a pair of roots on the jw-axis of the s-plane. This is a special criterion, if A(s) has a pair of roots on the jw-axis, the Routh-Hurwitz test terminates prematurely in that an entire row of zeros is encountered in constructing the Routh array. When this happens, the feedback system is said to be on verge of instability. The critical value of K is deduced from the entries of the particular row as specified. The corresponding pair of roots on the jw-axis is found in the auxiliary polynomial formed from the entries of the preceding row.
Sinusoidal oscillators
While designing sinusoidal oscillators, feedback is applied to an amplifier with the specific objective of making the system unstable. In this application, the oscillator contains an amplifier and a frequency-determining network. They form a closed-loop feedback system. The amplifiers set the necessary condition for oscillation. To avoid distortion of the output signal, the degree of non-linearity in the amplifier is maintained at a very low level.
Root Locus method
A root locus method is an analytical tool for the design of a linear feedback system, with emphasis on the locations of the poles of the system’s closed-loop transfer function. As we already know, the poles of a system’s transfer function determine its transient response, therefore, by already knowing the locations of a closed-loop pole.
The name root locus comes from the locus traced out by the roots of the system’s characteristics equation in the s-plane as a parameter.
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