IEEE Circuits and Systems Magazine - Q1 2020 - 38

Cp are always positive, which means that this narrow
band equivalent circuit for a lossless network is always
realizable! We finally note that in a similar manner a
corresponding conclusion can be made for an equivalent circuit featuring an LC series circuit. Example 3
shows the characteristics of the different approximations for our worked example in a susceptance versus
frequency plot.

Example 3
The graph below shows the quality of the narrowband
approximation for the susceptance associated with the
example network (see Example 1) by means of an LC
parallel tank in comparison to the approximation by a
single inductance. The approximation has been performed for ~ 0 = 2, where the susceptance values of the
exact characteristics and the approximations coincide.
However, the LC parallel tank also correctly reproduces
the slope of the susceptance spectrum at ~ 0 = 2, which
means that it represents the network truthfully for narrowband signals. Accordingly, when applying a narrowband signal leading to the stationary state, the energy
stored in the network can be calculated from this approximately equivalent LC tank. Also, when approaching
a parallel resonance frequency, in this example at ~ = 1,
the values L p and C p obtained for the approximation at
the considered frequency asymptotically match the LC
parallel tank corresponding to this resonance frequency
in the" Foster 1" realization. As it was proven that L p
and C p are always greater or equal to zero (and thus represent a realizable network), it is therefore also proven
that the components in the Foster 1 realization are always greater or equal to zero.

2
Original Circuit
LC Parallel
Tank Approx.
L Approx.

Susceptance B (ω)

1.5
1
0.5
0
-0.5
-1
-1.5
-2

0.5

1.5
2
Frequency ω

IEEE CIRCUITS AND SYSTEMS MAGAZINE

2.5

IV. Further Relations to Foster's
Original Conditions
In the above sections we showed, how the consideration
of the energy that is stored in the lossless network after
building up the stationary state can lead to Foster's conditions (1) and (2). In fact, Foster's paper [1] provides
much more than this condition. It provides the wellknown equivalent circuits, i.e. Foster's first and second
form of a canonical realization as shown in Fig. 1, which
can be related to a partial fraction decomposition of the
immittance functions as discussed above and illustrated in the examples.
Foster's equivalent circuits feature resonant LC circuits corresponding to natural modes of oscillations.
As every lossless LC circuit can, in principle, indefinitely sustain oscillations, the total energy contained
in the network depends on the potential excitation of
these natural modes and thus on the way the stationary state was approached. Which particular natural
modes of oscillation are actually excited crucially depends on the driving circuit. For instance, including
an internal resistance in the chosen source introduces losses in the entire circuit, i.e. the LC network and
source network, and thus potentially excited natural
oscillations can be dampened such that they vanish
as time progresses. Alternatively, the excitation of
these natural modes can be avoided if the spectrum
of the excitation signal features no significant contributions at the frequencies associated with these natural modes of oscillation, which is the consideration
that led us to the application of a slowly increasing
sinusoid above.
As mentioned in the Introduction, while the partial
fraction expansion of a rational function, i.e. the immittance function, and the interpretation of individual
terms as resonant LC circuits in an equivalent circuit
is straightforward, it is not obvious that the resulting
LC component values are real-valued and positive and
that the equivalent circuit thus corresponds to an actually realizable one - this fact was explicitly proven
by Foster.
This proof can also be provided by virtue of narrow band approximation introduced in the previous
section. The considered approximate parallel LC circuit also features a resonance, which, in general, is of
no physical significance, though. The circuit merely
approximates the admittance (and thus also impedance) of the original circuit in a narrow band around
the particularly selected driving frequency ~ 0 . However, if ~ 0 is chosen to approach one of the parallel
resonance frequencies of the circuit (where the susceptance B (~) approaches zero), the equivalent approximate circuit will asymptotically represent one of the
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