Foster's reactance theorem is an important theorem in the fields of electricalnetwork analysis andsynthesis. The theorem states that thereactance of a passive, lossless two-terminal (one-port) network always strictlymonotonically increases with frequency. It is easily seen that the reactances ofinductors andcapacitors individually increase or decrease with frequency respectively and from that basis a proof for passive lossless networks generally can be constructed. The proof of the theorem was presented byRonald Martin Foster in 1924, although the principle had been published earlier by Foster's colleagues atAmerican Telephone & Telegraph.

The theorem can be extended toadmittances and the encompassing concept ofimmittances. A consequence of Foster's theorem is thatzeros and poles of the reactance must alternate with frequency. Foster used this property to develop twocanonical forms for realising these networks. Foster's work was an important starting point for the development ofnetwork synthesis.

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

Reactance is theimaginary part of the complexelectrical impedance. Bothcapacitors andinductors possess reactance (but of opposite sign) and are frequency dependent. The specification that the network must be passive and lossless implies that there are no resistors (lossless), or amplifiers or energy sources (passive) in the network. The network consequently must consist entirely of inductors and capacitors and the impedance will be purely an imaginary number with zero real part. Foster's theorem applies equally to theadmittance of a network, that is thesusceptance (imaginary part of admittance) of a passive, losslessone-port monotonically increases with frequency. This result may seem counterintuitive since admittance is the reciprocal of impedance, but is easily proved. If the impedance is

${\displaystyle Z=iX}$

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

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

where${\displaystyle B}$ is susceptance.

IfX is monotonically increasing with frequency then 1/X must be monotonically decreasing. −1/X must consequently be monotonically increasing and hence it is proved thatB 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 ofduality for electric networks. It is convenient in these circumstances to use the concept ofimmittance, 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-portstrictly monotonically increases with frequency.

Foster's theorem is quite general. In particular, it applies todistributed-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]

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

The impedance of aninductor is given by,

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

${\displaystyle L}$ isinductance

${\displaystyle \omega }$ isangular frequency

so the reactance is,

${\displaystyle X=\omega L}$

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

The impedance of acapacitor is given by,

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

${\displaystyle C}$ iscapacitance

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 thedual of any immittance function that obeys Foster's theorem will also follow Foster's theorem.^[3]

A seriesLC 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 (theresonant frequency) and then continues to monotonically increase as the inductor reactance becomes progressively dominant.^[4]

A parallelLC circuit is the dual of the series circuit and hence its admittance function is the same form as the impedance function of the series circuit,

${\displaystyle Y=i\omega C+\frac{1}{i\omega L}}$

The impedance function is,

${\displaystyle Z=i\left(\frac{\omega L}{1-{\omega }^{2}LC}\right)}$

At low frequencies the reactance is dominated by the inductor and is small and positive. This monotonically increases towards apole at theanti-resonant frequency where the susceptance of the inductor and capacitor are equal and opposite and cancel. Past the pole the reactance is large and negative and increasing towards zero where it is dominated by the capacitance.^[4]

A consequence of Foster's theorem is that thezeros 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 thefrequency characteristics of a Foster network. Two Foster networks that have identical poles and zeroes will beequivalent 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 theshape of the two immittance functions will be identical.^[5]

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

A one-port passive immittance consisting of discrete elements (that is, notdistributed elements) can be represented as arational function ofs,

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

where,

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

${\displaystyle P(s),Q(s)}$ arepolynomials with real, positive coefficients

${\displaystyle s}$ is theLaplace transform variable, which can be replaced with${\displaystyle i\omega }$ when dealing withsteady-stateAC signals.

This follows from the fact the impedance ofL andC elements are themselves simple rational functions and any algebraic combination of rational functions results in another rational function.

This is sometimes referred to as thedriving point impedance because it is the impedance at the place in the network at which the external circuit is connected and "drives" it with a signal. In his paper, Foster describes how such a lossless rational function may be realised (if it can be realised) in two ways. Foster's first form consists of a number of series connected parallel LC circuits. Foster's second form of driving point impedance consists of a number of parallel connected series LC circuits. The realisation of the driving point impedance is by no means unique. Foster's realisation has the advantage that the poles and/or zeroes are directly associated with a particular resonant circuit, but there are many other realisations. Perhaps the most well known isWilhelm Cauer'sladder realisation from filter design.^[6][7][8]

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 anamplifier withpositive feedback can have reactance which declines with frequency. For example, it is possible to create negative capacitance and inductance withnegative 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 Fosterimpedance matching networks can only match the impedance of anantenna with atransmission line at discrete frequencies, which limits the bandwidth of the antenna. A non-Foster network could match an antenna over a continuous band of frequencies.^[9] This would allow the creation of compact antennas that have wide bandwidth, violating theChu-Harrington limit. Practical non-Foster networks are an active area of research.

The theorem was developed atAmerican Telephone & Telegraph as part of ongoing investigations into improved filters for telephonemultiplexing applications. This work was commercially important; large sums of money could be saved by increasing the number of telephone conversations that could be carried on one line.^[10] The theorem was first published byCampbell in 1922 but without a proof.^[11] Great use was immediately made of the theorem in filter design, it appears prominently, along with a proof, inZobel's landmark paper of 1923 which summarised the state of the art of filter design at that time.^[12] Foster published his paper the following year which included his canonical realisation forms.^[13]

Cauer in Germany grasped the importance of Foster's work and used it as the foundation ofnetwork synthesis. Amongst Cauer's many innovations was the extension of Foster's work to all 2-element-kind networks after discovering anisomorphism between them. Cauer was interested in finding thenecessary and sufficient condition for realisability of a rational one-port network from its polynomial function, a condition now known to be apositive-real function, and the reverse problem of which networks were equivalent, that is, had the same polynomial function. Both of these were important problems in network theory and filter design. Foster networks are only a subset of realisable networks,^[14]

• 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, 2007ISBN 1-4259-7242-X.
• James T. Aberle, Robert Loepsinger-Romak,Antennas with non-Foster matching networks, Morgan & Claypool Publishers, 2007ISBN 1-59829-102-5.
• 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, 1992ISBN 0-521-37769-2.
• Carol Gray Montgomery, Robert Henry Dicke, Edward M. Purcell,Principles of microwave circuits, IET, 1987ISBN 0-86341-100-2.
• 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, 2002ISBN 0-85296-218-5.

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