ABSTRACT
In this chapter, we demonstrate that any rational positive-real function can be realized as the input immittance of a passive one-port network terminated in a resistor, thereby also proving the sufficiency
Consider the even part
(6.1)
of a given rational positive-real impedance Z(s). As in (5.26), we first separate the numerator and denominator polynomials of Z(s) into even and odd parts, and write
(6.2)
Then, we have
(6.3)
showing that if s0 is a zero or pole of r(s), so is –s0. Thus, the zeros and poles of r(s) possess quadrantal symmetry with respect to both the real and imaginary axes. They may appear in pairs on the real axis, in pairs on the jω-axis, or in the form of sets of quadruplets in the complex-frequency plane. Furthermore, for a positive-real Z(s), the jω-axis zeros of r( jω) are required to be of even multiplicity in order that Re Z(jω) = r( jω) never be negative.
