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| '''Foster's reactance theorem''' is an important theorem in the fields of electrical [[Network analysis (electrical circuits)|network analysis]] and [[Network synthesis filters|synthesis]]. The theorem states that the [[electrical reactance|reactance]] of a passive, lossless two-terminal (one-port) network always [[monotonic]]ally increases with frequency. The proof of the theorem was first presented by [[R. M. Foster|Ronald Martin Foster]] in 1924.<ref name="Foster, 1924">Foster, 1924.</ref>
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| ==Explanation==
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| Reactance is the [[imaginary number|imaginary]] part of the complex [[electrical impedance]]. 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. Other than that, the theorem is quite general, in particular, it applies to [[distributed element model|distributed element]] circuits although Foster formulated it in terms of discrete inductors and capacitors. Foster's theorem applies equally to the [[admittance]] of a network, that is the [[susceptance]] (imaginary part of admittance) of a passive, lossless one-port monotonically increases with frequency. This result may seem counterintuitive since admittance is the reciprocal of impedance, but is easily proved. If an impedance,
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| :<math> Z = iX \,</math>
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| :where,
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| :<math>\scriptstyle Z </math> is impedance
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| :<math>\scriptstyle X </math> is reactance
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| :<math>\scriptstyle i </math> is the [[imaginary unit]]
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| then the admittance is given by
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| :<math> Y = \frac{1}{iX} = - i\frac{1}{X} =iB </math>
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| :where,
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| :<math>\scriptstyle Y </math> is admittance
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| :<math>\scriptstyle B </math> is susceptance | |
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| 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 apply equally to impedance or admittance as they do here. It is convenient in these circumstances to use the concept of [[immittance]] which can mean either impedance or admittance. The mathematics are carried out without stating which it is or specifying units until it is desired to calculate a specific example. Foster's theorem can thus be stated in a more general form as,
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| :;Foster's theorem (immittance form)
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| :''The imaginary immittance of a passive, lossless one-port monotonically increases with frequency.''<ref name=Aberle8-9>Aberle and Loepsinger-Romak, pp.8-9.</ref><ref name=Radmanesh459>Radmanesh, p.459.</ref>
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| ==Examples==
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| {|style="float:right;"
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| |-
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| |[[File:Reactance L.svg|thumb|150px|Plot of the reactance of an inductor against frequency]]
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| |[[File:Reactance C.svg|thumb|150px|Plot of the reactance of a capacitor against frequency]]
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| |[[File:Reactance LC.svg|thumb|150px|Plot of the reactance of a series ''LC'' circuit against frequency]]
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| |[[File:Reactance anti-LC.svg|thumb|150px|Plot of the reactance of a parallel ''LC'' circuit against frequency]]
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| |}
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| The following examples illustrate this theorem in a number of simple circuits.
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| ===Inductor===
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| The impedance of an [[inductor]] is given by,
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| :<math> Z = i \omega L \,</math>
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| :<math> \scriptstyle L </math> is [[inductance]]
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| :<math> \scriptstyle \omega </math> is [[angular frequency]]
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| so the reactance is,
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| :<math> X = \omega L \,</math> | |
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| which by inspection can be seen to be monotonically (and linearly) increasing with frequency.<ref name=Cherry100>Cherry, pp.100-101.</ref>
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| ===Capacitor===
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| The impedance of a [[capacitor]] is given by,
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| :<math> Z = \frac {1}{i \omega C} </math>
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| :<math> \scriptstyle C </math> is [[capacitance]]
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| so the reactance is,
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| :<math> X = - \frac {1}{\omega C} </math>
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| 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 impedance|dual]] of any immittance function that obeys Foster's theorem will also follow Foster's theorem.<ref name=Cherry100/>
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| ===Series resonant circuit===
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| A series [[LC circuit|''LC'' circuit]] has an impedance that is the sum of the impedances of an inductor and capacitor,
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| :<math> Z = i \omega L + \frac {1}{i \omega C} = i \left ( \omega L - \frac {1}{\omega C} \right )</math>
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| 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 [[resonant frequency]]) and then continues to monotonically increase as the inductor reactance becomes progressively dominant.<ref name=Cherry102>Cherry, pp.100-102.</ref>
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| ===Parallel resonant circuit===
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| A parallel ''LC'' 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,
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| :<math> Y = i \omega C + \frac {1}{i \omega L}</math>
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| The impedance function is,
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| :<math> Z = i \left ( \frac{\omega L}{1 - \omega^2 LC} \right ) </math>
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| At low frequencies the reactance is dominated by the inductor and is small and positive. This monotonically increases towards a [[pole (complex analysis)|pole]] at the [[antiresonance|anti-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.<ref name=Cherry102/>
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| ==Poles and zeroes==
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| [[File:Reactance first form.svg|thumb|200px|Plot of the reactance of Foster's first form of canonical driving point impedance showing the pattern of alternating poles and zeroes. Three anti-resonators are required to realise this impedance function.]]
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| A consequence of Foster's theorem is that the [[pole (complex analysis)|poles]] and [[zero (complex analysis)|zeroes]] of any passive immittance function must alternate with increasing frequency. 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.<ref name=Aberle8-9/>
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| With the addition of a scaling factor, 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.<ref>Smith and Alley, p.173.</ref>
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| Another consequence of Foster's theorem is that the plot of a Foster immittance function on a [[Smith chart]] must always travel around the chart in a clockwise direction with increasing frequency.<ref name=Radmanesh459/>
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| ==Realization==
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| [[File:Foster driving point 1st kind.svg|thumb|'''Foster's first form of canonical driving point impedance realisation.''' If the polynomial function has a pole at ''ω''=0 one of the ''LC'' sections will reduce to a single capacitor. If the polynomial function has a pole at ''ω''=∞ one of the ''LC'' sections will reduce to a single inductor. If both poles are present then two sections reduce to a series ''LC'' circuit.]]
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| [[File:Foster driving point 2nd kind.svg|thumb|'''Foster's second form of canonical driving point impedance realisation.''' If the polynomial function has a zero at ''ω''=0 one of the ''LC'' sections will reduce to a single capacitor. If the polynomial function has a zero at ''ω''=∞ one of the ''LC'' sections will reduce to a single inductor. If both zeroes are present then two sections reduce to a parallel ''LC'' circuit.]]
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| A one-port passive immittance consisting of discrete elements (that is, not a distributed element circuit) is described as rational in that in can be represented as a [[rational function]] of ''s'',
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| :<math>Z(s) = \frac {P(s)}{Q(s)}</math>
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| :where,
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| :<math>\scriptstyle Z(s) </math> is immittance
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| :<math>\scriptstyle P(s), \ Q(s) </math> are [[polynomial]]s with real, positive coefficiencts
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| :<math>\scriptstyle s </math> is the [[Laplace transform|Laplace]] operator, which can be replaced with <math>\scriptstyle i\omega </math> when dealing with [[steady-state]] [[alternating current|AC]] signals.
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| This is sometimes referred to as the [[driving 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. Foster in his paper describes how such a lossless rational function may 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 is [[Wilhelm Cauer|Cauer's]] ladder realisation from filter design.<ref name=Aberle9/><ref>Cherry, pp.106-108.</ref><ref>Montgomery ''et al.'', pp.157-158.</ref>
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| ==Non-Foster networks==
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| 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.<ref name=Aberle9>Aberle and Loepsinger-Romak, p.8.</ref> 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.<ref name=Aberle9>Aberle and Loepsinger-Romak, p.9.</ref>
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| These are of interest because they can accomplish tasks a Foster network cannot. For example, the usual passive Foster [[impedance matching]] networks can only match the impedance of an [[antenna (radio)|antenna]] with a [[transmission 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.<ref name=Aberle9>Aberle and Loepsinger-Romak, p.8.</ref> This would allow the creation of compact antennas that have wide bandwidth, violating the [[Chu-Harrington limit]]. Practical non-Foster networks are an active area of research.
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| ==History==
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| The theorem was developed at [[American Telephone & Telegraph]] as part of ongoing investigations into improved filters for telephone [[multiplexing]] 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.<ref>Bray, p.62.</ref> The theorem was first published by [[George Ashley Campbell|Campbell]] in 1922 but without a proof.<ref>Cherry, p.62.</ref> Great use was immediately made of the theorem in filter design, it appears prominently, along with a proof, in [[Otto Julius Zobel|Zobel]]'s landmark paper of 1923 which summarised the state of the art of filter design at that time.<ref>Zobel, pp.5,35-37.</ref> Foster published his paper the following year which included his canonical realisation forms.<ref name="Foster, 1924"/>
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| [[Wilhelm Cauer|Cauer]] in Germany grasped the importance of Foster's work and used it as the foundation of [[network synthesis]]. Amongst Cauer's many innovations was to extend Foster's work to all 2-element-kind networks after discovering an [[isomorphism]] between them. Cauer was interested in finding the conditions for realisability of a rational one-port network from its polynomial function (the condition of being a Foster network is not a [[necessary and sufficient condition]], for that, see [[positive-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.<ref>E. Cauer ''et al.'', p.5.</ref>
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| ==References==
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| {{reflist|2}}
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| ==Bibliography==
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| *Foster, R. M., "[http://www3.alcatel-lucent.com/bstj/vol03-1924/articles/bstj3-2-259.pdf A reactance theorem]", ''Bell Systems Technical Journal'', '''vol.3''', no. 2, pp. 259–267, November 1924.
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| *Campbell, G. A., "[http://www3.alcatel-lucent.com/bstj/vol01-1922/articles/bstj1-2-1.pdf Physical theory of the electric wave filter]", ''Bell Systems Technical Journal'', '''vol.1''', no. 2, pp. 1–32, November 1922.
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| *Zobel, O. J.,"[http://www3.alcatel-lucent.com/bstj/vol02-1923/articles/bstj2-1-1.pdf Theory and Design of Uniform and Composite Electric Wave Filters]", ''Bell Systems Technical Journal'', '''vol.2''', no. 1, pp. 1–46, January 1923.
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| *Matthew M. Radmanesh, ''[http://books.google.com/books?id=bJrYx827oWsC&pg=PA459 RF & Microwave Design Essentials]'', AuthorHouse, 2007 ISBN 1-4259-7242-X.
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| *James T. Aberle, Robert Loepsinger-Romak, ''[http://books.google.com/books?id=4jt4gBgiDbIC&pg=PA5 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.
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| *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, ''[http://books.google.com/books?id=Sex_282iULMC&pg=PA157 Principles of microwave circuits]'', IET, 1987 ISBN 0-86341-100-2.
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| *E. Cauer, W. Mathis, and R. Pauli, "[http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf 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.
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| *Bray, J, ''Innovation and the Communications Revolution'', Institute of Electrical Engineers, 2002 ISBN 0-85296-218-5.
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| [[Category:Circuit theorems]]
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