ABSTRACT
To make Re Z(jω) = 0, there are three nontrivial ways: (i) m1 = 0 and n2 = 0, (ii) m2 = 0 and n1 = 0, (iii) m1m2 – n1n2 = 0. The first possibility leads Z(s) to n1/m2, the second to m1/n2. For the third possibility, we require that m1m2 = n1n2 or
(7.4)
Z s m n
m n ( ) = +
r s m m n n
m n ( ) = −
−
r j Z jω ω ω( ) = ( ) =Re 0 for all
m n m m n n1 1+( ) = +( )2 2 2 1
In Chapter 6, we showed that any positive-real function can be realized as the input immittance of a
(7.5)
Therefore, the driving-point immittance of a lossless network is always the quotient of even to odd or odd to even polynomials. Its zeros and poles must occur in quadrantal symmetry, being symmetric with respect to both axes. As a result, they are simple and purely imaginary from stability considerations, or Z(s) can be explicitly written as
(7.6)
where ωz1 ≥ 0. This equation can be expanded in partial fraction as
(7.7)
where ωpi = ωi, and the residues H, K0 and Ki are all real and positive. Substituting s = jω and writing Z(jω) = Re Z(jω) + j Im Z(jω) results in an odd function known as
the reactance function X(ω):
(7.8)
Taking the derivatives on both sides yields
(7.9)
Since every factor in this equation is positive for all positive and negative values of ω, we conclude that
(7.10)
It states that the slope of the reactance function versus frequency curve is always positive, as depicted in
separation property for reactance function credited to Foster [9]. Because of this, the pole and zero frequencies of (7.6) are related by
(7.11)
We now consider the realization of Z(s). If each term on the right-hand side of (7.7) can be identified as the input impedance of the LC one-port, the series connection of these one-ports would yield the desired realization. The first term is the impedance of an inductor of inductance H, and the second term
combination of an inductor of inductance 2Ki/ω 2 i and a capacitor of capacitance 1/2Ki. The resulting
admittance function Y(s) = 1/Z(s) and expanded it in partial fraction, we obtain
(7.12)
Z s m n
m n
n
m ( ) + ==
Z s H s s s
s s s
( ) = +( ) +( ) +( ) +( ) +( )
ω ω ω
ω ω
...
