IEEE Circuits and Systems Magazine - Q1 2020 - 39

LC tanks shown in the "Foster 2" realization Fig. 1 (left).
As we have proven that the component values for this
equivalent circuit are always real and positive, this
also means that the component values of the "Foster
1" circuit have to be positive. A similar consideration
can be made using an approximate LC series (rather
than parallel) circuit yielding the realizability of the
components for the "Foster 2" circuit when considering frequencies featuring series resonance (i.e. vanishing reactance X (~)). Hence, besides the reproduction
of the conditions (2), this approach also provides a
simple alternative proof for the realizability of Foster's
equivalent circuits.
V Conclusion
It was shown that the property that the reactance
and susceptance of a lossless network always increases with frequency, which is frequently referred
to as Foster's theorem, can be derived by considering
the amount of energy that is delivered to the network
when building up the stationary state. This method
readily proves the more stringent version of the theorem stating that 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. Also, this consideration
is not restricted to lossless networks composed of
lumped elements.
This particular approach has also led to the consideration of an equivalent LC circuit that yields a narrow band approximation of the frequency responses
by not only yielding the correct value of the reactance
or susceptance, but also the correct slope. Foster's
condition can be readily deduced by considering the
energy in these equivalent components. It was also
proven that these equivalent components are always
positive and thus the narrow-band equivalent circuit
is always realizable. As the equivalent LC circuit (series or parallel) converges to one of the resonant LC
circuits in Foster's equivalent circuits when approaching a particular resonance frequency, this can also be
used to prove the realizability of Foster's circuits (type
1 and type 2).
In summary, the presented framework provides a
simple alternative route to Foster's theorem.
Acknowledgment
The author is deeply indebted to Fritz Paschke, a dedicated scientist, inventor and teacher, who inspired the
considerations in this paper a long time ago and who
presented a special version of this approach in his lectures at Vienna University of Technology [13].
FIRST QUARTER 2020

Bernhard Jakoby (SM'98) became a
Member of IEEE in 1990, and a Senior Member (SM) in 1998. He was born in Neuss,
Germany in 1966. He obtained his Dipl.Ing. (M.Sc.) in Communication Engineering and his doctoral (Ph.D.) degree in
electrical engineering from the Vienna University of Technology (VUT), Austria, in 1991 and 1994, respectively. In
2001 he obtained avenia legendi for Theoretical Electrical
Engineering from the VUT. From 1991 to 1994 he worked
as a Research Assistant at the Institute of General Electrical Engineering and Electronics of the VUT. Subsequently
he stayed as an Erwin Schrödinger Fellow at the University of Ghent, Belgium, performing research on the electrodynamics of complex media. From 1996 to 1999 he held
the position of a Research Associate and later Assistant
Professor at the Delft University of Technology, The Netherlands, working in the field of microacoustic sensors.
From 1999 to 2001 he was with the Automotive Electronics
Division of the Robert Bosch GmbH, Germany, where he
conducted development projects in the field of automotive liquid sensors. In 2001 he joined the newly formed Industrial Sensor Systems group of the VUT as an Associate
Professor. In 2005 he was appointed Full Professor of Microelectronics at the Johannes Kepler University Linz,
Austria. He is currently working in the field of fluidic sensors and monitoring systems.
References
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[14] F. Paschke, private communication, 2019.
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