ABSTRACT
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
(75.6)
where
ω
≥
0. This equation can be expanded in partial fraction as
(75.7)
where
ω
=
ω
, and the residues
H
,
K
and
K
are all real and positive. Substituting
s
=
j
ω
and writing
Z
(
j
ω
) = Re
Z
(
j
ω
) +
j
Im
Z
(
j
ω
) results in an odd function known as the
reactance function
X
(
ω
):
(75.8)
Taking the derivatives on both sides yields
(75.9)
Since every factor in this equation is positive for all positive and negative values of
ω
, we conclude that
(75.10)
It states that the slope of the reactance function versus frequency curve is always positive, as depicted in Fig. 75.1. Consequently, the poles and zeros of
Z
(
s
) alternate along the
j
ω
-axis. This is known as the
separation property
for reactance function credited to Foster [9]. Because of this, the pole and zero frequencies of (75.6) are related by
(75.11)
We now consider the realization of
Z
(
s
). If each term on the right-hand side of (75.7) 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 corresponds to a capacitor of capacitance 1/
K
. Each of the remaining term can be realized as a parallel combination of an inductor of inductance 2
K
/
ω
and a capacitor of capacitance 1/2
K
. The resulting realization is shown in Fig. 75.2 known as the
first Foster canonical form
. Likewise, if we consider the admittance function
Y
(
s
) = 1/
Z
(
s
) and expanded it in partial fraction, we obtain
(75.12)
Z s m n
m n
n
m ( ) + ==
Z s H s s s
s s s
( ) = +( ) +( ) +( ) +( ) +( )
ω ω ω
ω ω
...
