IEEE Circuits and Systems Magazine - Q1 2020 - 34

The slope of the frequency response (of reactance or susceptance) is always
larger or equal than that of an inductor or capacitor yielding the same
reactance or susceptance at the particularly considered frequency.
2X $ X
2~
~

and 2B $ B .
2~
~

(2)

These conditions can be readily verified for the simple
special cases of a single inductor (here the slope 2X/2~
is simply given by the inductance L) and a single capacitance C, where X = -1/ (~C ) and thus 2X/2~ = 1/ (~ 2 C ).
We note that when passing frequencies where the susceptance or reactance functions show poles, the sign
of these functions change from positive to negative but
mathematically the derivative is non-existent at those
"singular" frequencies such that the above conditions
are not violated, i.e. they hold at all frequencies where
the derivative exists.
In the years after its introduction, Foster's theorem
has been generalized to the case of electromagnetic fields
(see, e.g. [6]) and considered in the context of many particular cases such as, e.g., resonant circuits [7] and the
impedance of resonant antennas [8].
Foster's proof given in [1] is based "upon the solution of the analogous dynamical problem of the small
oscillations of a system about a position of equilibrium with no frictional forces acting"; thorough discussions of this approach can be found, e.g., in [4]
and [9]. In [10] Papoulis provides an alternative proof
based on the determinants associated with lossless
networks of lumped elements. In [11] Nedlin 2 showed
that the Foster condition (1) can alternatively be obtained by considering the average energy stored in a
lossless network, when it is sinusoidally driven at a
particular frequency.
In the following we will first show an approach similar to that in [11] for deriving the strict Forster condition
(2) by considering the time-depending energy taken up
by a lossless network. Then we will discuss an approximate equivalent circuit valid in a narrow band around
a chosen, particular frequency and use this equivalent
circuit to again prove (2) in an even simpler manner. We
also show that this equivalent circuit is always realizable (i.e. contains real-valued and positive component
values). By considering Foster's full equivalent circuit
close to resonances, we finally show that from the previous considerations, it can also be proven that Foster's
equivalent circuit is always realizable, which yields an
alternative path to Foster's statements in [1].
2

See also the comments on the paper in [12].

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While the theorem has been proven before, the author feels that the presented approach is somewhat
simpler to grasp and in particular does not require advanced complex analysis. Furthermore, the presented
approach does not assume to the network as being
composed of lumped components which means that it
also holds for systems featuring distributed components
such as microwave resonators or transmission lines.
Also, it illustrates the implications of the spurious excitation of natural modes of oscillations, which, in contrast
to lossy networks, are undampened and thus infinitely
sustained in purely reactive networks. As it turns out,
only by carefully avoiding the excitation of these spurious oscillations, the reactance theorem can be obtained
from this approach. And finally, an even simpler version
of the proof can be obtained by considering equivalent
circuits, which reproduce the reactance and its slope
(with respect to frequency) accurately around a particularly chosen frequency, which are moreover, as mentioned above, always realizable and thus could also be
used to represent the network in case of other narrowband considerations.
II. Derivation of Foster's Condition from
Energy Considerations
As mentioned earlier, the following approach is similar
to that presented in [11] but extends the consideration.
It is also related to an approach presented by Paschke in
his lecture notes [13], which, however, has never been
published before to the best of our knowledge [14]. Note
that the following approach does not consider that the
network is composed of lumped circuit elements; the
only assumption which is made is that no losses whatsoever occur in the one-port device.
When applying a sinusoidal signal to a lossless circuit
(one-port), energy is provided to the circuit and transients occur, which depend on the manner the sinusoid
is applied to the circuit, e.g., abruptly with a particular
initial phase or with gradually increasing amplitude as it
will be considered below. Once the stationary state has
developed, we are faced with the well-known phenomena
of reactive power, i.e. that at certain instants in time the
network takes up power from the source while in others
it provides power back to the source. In this stationary
state, the average power taken up by the network vanishes, though. The way the sinusoidal signal is applied to
the network in general makes a difference in the amount
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