Foster's reactance theorem

Foster's reactance theorem is an important theorem in the fields of electrical

The theorem can be extended to admittances and the encompassing concept of immittances. A consequence of Foster's theorem is that zeros and poles of the reactance must alternate with frequency. Foster used this property to develop two canonical forms for realising these networks. Foster's work was an important starting point for the development of network synthesis.

It is possible to construct non-Foster networks using active components such as amplifiers. These can generate an impedance equivalent to a negative inductance or capacitance. The negative impedance converter is an example of such a circuit.

Explanation

Reactance is the

${\displaystyle Z=iX\,}$

where ${\displaystyle \scriptstyle X}$ is reactance and ${\displaystyle \scriptstyle i}$ is the imaginary unit, then the admittance is given by

${\displaystyle Y={\frac {1}{iX}}=-i{\frac {1}{X}}=iB}$

where ${\displaystyle \scriptstyle B}$ is susceptance.

If X is monotonically increasing with frequency then 1/X must be monotonically decreasing. −1/X must consequently be monotonically increasing and hence it is proved that B is increasing also.

It is often the case in network theory that a principle or procedure applies equally well to impedance or admittance—reflecting the principle of duality for electric networks. It is convenient in these circumstances to use the concept of immittance, which can mean either impedance or admittance. The mathematics is carried out without specifying units until it is desired to calculate a specific example. Foster's theorem can thus be stated in a more general form as,

Foster's theorem (immittance form)

The imaginary immittance of a passive, lossless one-port

strictly

monotonically increases with frequency.

Foster's theorem is quite general. In particular, it applies to distributed-element networks, although Foster formulated it in terms of discrete inductors and capacitors. It is therefore applicable at microwave frequencies just as much as it is at lower frequencies.^[1][2]

Examples

The following examples illustrate this theorem in a number of simple circuits.

Inductor

The impedance of an inductor is given by,

${\displaystyle Z=i\omega L\,}$

${\displaystyle \scriptstyle L}$ is inductance

${\displaystyle \scriptstyle \omega }$ is angular frequency

so the reactance is,

${\displaystyle X=\omega L\,}$

which by inspection can be seen to be monotonically (and linearly) increasing with frequency.^[3]

Capacitor

The impedance of a capacitor is given by,

${\displaystyle Z={\frac {1}{i\omega C}}}$

${\displaystyle \scriptstyle C}$ is capacitance

so the reactance is,

${\displaystyle X=-{\frac {1}{\omega C}}}$

which again is monotonically increasing with frequency. The impedance function of the capacitor is identical to the admittance function of the inductor and vice versa. It is a general result that the dual of any immittance function that obeys Foster's theorem will also follow Foster's theorem.^[3]

Series resonant circuit

A series LC circuit has an impedance that is the sum of the impedances of an inductor and capacitor,

${\displaystyle Z=i\omega L+{\frac {1}{i\omega C}}=i\left(\omega L-{\frac {1}{\omega C}}\right)}$

At low frequencies the reactance is dominated by the capacitor and so is large and negative. This monotonically increases towards zero (the magnitude of the capacitor reactance is becoming smaller). The reactance passes through zero at the point where the magnitudes of the capacitor and inductor reactances are equal (the

A consequence of Foster's theorem is that the zeros and poles of any passive immittance function must alternate as frequency increases. After passing through a pole the function will be negative and is obliged to pass through zero before reaching the next pole if it is to be monotonically increasing.^[1]

The poles and zeroes of an immittance function completely determine the frequency characteristics of a Foster network. Two Foster networks that have identical poles and zeroes will be equivalent circuits in the sense that their immittance functions will be identical. There can be a scaling factor difference between them (all elements of the immittance multiplied by the same scaling factor) but the shape of the two immittance functions will be identical.^[5]

Another consequence of Foster's theorem is that the phase of an immittance must monotonically increase with frequency. Consequently, the plot of a Foster immittance function on a Smith chart must always travel around the chart in a clockwise direction with increasing frequency.^[2]

Realisation

A one-port passive immittance consisting of discrete elements (that is, not distributed elements) can be represented as a rational function of s,

${\displaystyle Z(s)={\frac {P(s)}{Q(s)}}}$

where,

${\displaystyle \scriptstyle Z(s)}$ is immittance

${\displaystyle \scriptstyle P(s),\ Q(s)}$ are polynomials with real, positive coefficients

${\displaystyle \scriptstyle s}$ is the Laplace transform variable, which can be replaced with ${\displaystyle \scriptstyle i\omega }$ when dealing with

steady-state AC

signals.

A Foster network must be passive, so an active network, containing a power source, may not obey Foster's theorem. These are called non-Foster networks.[9] In particular, circuits containing an amplifier with positive feedback can have reactance which declines with frequency. For example, it is possible to create negative capacitance and inductance with negative impedance converter circuits. These circuits will have an immittance function with a phase of ±π/2 like a positive reactance but a reactance amplitude with a negative slope against frequency.^[6]

These are of interest because they can accomplish tasks a Foster network cannot. For example, the usual passive Foster

History

The theorem was developed at

Bibliography

• Foster, R. M., "A reactance theorem", Bell System Technical Journal, vol.3, no. 2, pp. 259–267, November 1924.
• Campbell, G. A., "Physical theory of the electric wave filter", Bell System Technical Journal, vol.1, no. 2, pp. 1–32, November 1922.
• Zobel, O. J.,"Theory and Design of Uniform and Composite Electric Wave Filters", Bell System Technical Journal, vol.2, no. 1, pp. 1–46, January 1923.
• Matthew M. Radmanesh, RF & Microwave Design Essentials, AuthorHouse, 2007
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• James T. Aberle, Robert Loepsinger-Romak, Antennas with non-Foster matching networks, Morgan & Claypool Publishers, 2007
  ISBN 1-59829-102-5
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• Colin Cherry, Pulses and Transients in Communication Circuits, Taylor & Francis, 1950.
• K. C. A. Smith, R. E. Alley, Electrical circuits: an introduction, Cambridge University Press, 1992
  ISBN 0-521-37769-2
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• Carol Gray Montgomery, Robert Henry Dicke, Edward M. Purcell, Principles of microwave circuits, IET, 1987
  ISBN 0-86341-100-2
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• E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900–1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000. Retrieved 19 September 2008.
• Bray, J, Innovation and the Communications Revolution, Institute of Electrical Engineers, 2002
  ISBN 0-85296-218-5
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