Derived Positive Real Functions and Matrices and Consequent Coefficient Relations

It was shown previously [1],[2] by one of the authors that the positive real property of rational positive real functions representing two-element kind networks remains invariant under simultaneous differentiation of the numerator and denominator. Some time back, it was shown by Talbot[3] that the result holds for any positive real function. (In fact, Talbot showed that the result holds for all positive but not necessarily real functions.) In this correspondence, by using an artifice due to Brune, it will be shown that the invariance of the positive real property under the above-mentioned operation can be extended to rational positive real matrices (open circuit impedance and short-circuit admittance), whose elements have only common poles. These results are used to arrive at a set of ordered inequalities which serve as a set of necessary conditions for two-element kind realizability. A set of necessary conditions to determine the positive real nature of the type of matrices mentioned is derived.