Form Of Characteristic Equation Research Articles

Stability Analysis of Multi-Dimensional Linear Time Invariant Discrete Systems within the Unity Shifted Unit Circle

This technical brief proposes a new approach to multi-dimensional linear time invariant discrete systems within the unity shifted unit circle which is denoted in the form of characteristic equation. The characteristic equation of multi–dimensional linear system is modified into an equivalent one- dimensional characteristic equation. Further formation of stability in the left of the z-plane, the roots of the characteristic equation f(z) =0 should lie within the shifted unit circle. Using the coefficients of the unity shifted one dimensional equivalent characteristic equation by applying minimal shifting of coefficients either left or right and elimination of coefficient method to two triangular matrixes are formed. A single square matrix is formed by adding the two triangular matrices. This matrix is used for testing the sufficient condition by proposed Jury’s inner determinant concept. Further one more indispensable condition is suggested to show the applicability of the proposed scheme. The proposed method of construction of square matrix consumes less arithmetic operation like shifting and eliminating of coefficients when compare to the construction of square matrix by Jury’s and Hurwitz matrix method.

A Modified Stability Analysis of Two-Dimensional Linear Time Invariant Discrete Systems within the Unity-Shifted Unit Circle

This paper proposes a method to ascertain the stability of two dimensional linear time invariant discrete system within the shifted unit circle which is represented by the form of characteristic equation. Further an equivalent single dimensional characteristic equation is formed from the two dimensional characteristic equation then the stability formulation in the left half of Z-plane, where the roots of characteristic equation f(Z) = 0 should lie within the shifted unit circle. The coefficient of the unit shifted characteristic equation is suitably arranged in the form of matrix and the inner determinants are evaluated using proposed Jury’s concept. The proposed stability technique is simple and direct. It reduces the computational cost. An illustrative example shows the applicability of the proposed scheme.

Application of tricomi functions in the eigenvalue analysis of the circular gyrotropic waveguide

The complex Tricomi confluent hypergeometric functions Φ (a, c; x) are introduced as exact functions for propagation of normal TEom modes in circular guiding structures, containing coaxially positioned azimuthally magnetized ferrite layer of double-connected cross-section. An original form of characteristic equation of the ferrite-fiiled coaxial waveguide is derived in terms of the arguments of Tricomi functions. Using the specially tabulated values of the latter, a numerical solution of this equation is accomplished, giving the eigenvalue spectrum and nonreciprocal phase characteristics of the structure. An analysis of its phase shifting capabilities is presented for a variety of geometry and ferrite parameters. The potentialities of the gyrotropic waveguide as digital nonreciprocal phaser, operating in the TE01 mode, are discussed.