IEEE Circuits and Systems Magazine - Q1 2020 - 33
particular observation that was made in [1], i.e. that the
reactance and susceptance functions always increase
with frequency, which implies the poles and zeros of
these immittance functions all lie on the real ~ axis and
alternate. This is frequently stated as1
2X 2 0 and 2B 2 0.
2~
2~
Foster's realization implies an even stricter requirement
(see, e.g., [4], [5]), i.e.
(1)
As pointed out by Foster himself [1], this property
was actually proven earlier by Zobel [2] based on previous work of Campbell [3]. In fact, it can be shown that
1
We provide the condition in the way it is usually cited. Strictly the `2_
sign should rather be a ` $ _ sign as the degenerate case of a short or
open circuit in principle also represents a lossless network.
(a) Foster 1
(b) Foster 2
Figure 1. First (left) and second (right) canonical realization
of a lossless network as described by Foster. The component
values can be obtained by means of a partial fraction expansion of the impedance or admittance function, respectively.
See also Example 1.
Example 1
Consider the following lossless LC network (in all examples
The partial fraction decompositions of this rational func-
we consider suitably scaled, dimensionless quantities [9]):
tion yield for the impedance
2
1
s
Z (s ) = s + 1 +
2s 2(s 2 + 1)
1
and the admittance
Y (s ) = 1 2 s 2 + 1 2 s 2 ,
2 s - s1 2 s - s2
1
where
s 21 = -1 + 1
2
The impedance of this network is given by
and s 22 = -1 - 1
2
represent the two complex conjugate pairs of poles of Y (s )
4
(corresponding, in turn, to resonance frequencies). The
2
Z (s ) = 2s +3 4s + 1
2s + 2s
poles of the impedance (at ~ = 0, ~ " 3 and ~ = !1)
as can be readily verified. The susceptance versus frequency characteristics is given here
appear as zeros of the admittance and vice versa.
Each term of these partial fraction decompositions can
be represented by a resonant circuit in Foster's equivalent
circuit. In case of the decomposition for Z (s ) for this ex-
10
ample (i.e., the" Foster 1" realization), the first two terms
Susceptance B (ω)
correspond to poles at infinity and zero yielding a single
component rather than an LC tank.
5
0
1
2
-5
-10
1/2
2
2
0
0.5
1
1.5
2
Frequency ω
2.5
3
Foster 1
2
1
1
2 - √2
2 + √2
Foster 2
B. Jakoby is with the Institute for Microelectronics and Microsensors, Johannes Kepler University Linz, 4040 Linz Austria e-mail: bernhard.jakoby@jku.at.
FIRST QUARTER 2020
IEEE CIRCUITS AND SYSTEMS MAGAZINE
33
IEEE Circuits and Systems Magazine - Q1 2020
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