# Foster's reactance theorem

Foster's reactance theorem is an important theorem in the fields of electrical network analysis and synthesis. The theorem states that the reactance of a passive, lossless two-terminal (one-port) network always monotonically increases with frequency. The proof of the theorem was first presented by Ronald Martin Foster in 1924.[1]

## Explanation

Reactance is the 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 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 the impedance is

$Z = iX \,$

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

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

where $\scriptstyle B$ 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 monotonically increases with frequency.[2][3]

## Examples

 Plot of the reactance of an inductor against frequency Plot of the reactance of a capacitor against frequency Plot of the reactance of a series LC circuit against frequency Plot of the reactance of a parallel LC circuit against frequency

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

### Inductor

The impedance of an inductor is given by,

$Z = i \omega L \,$
$\scriptstyle L$ is inductance
$\scriptstyle \omega$ is angular frequency

so the reactance is,

$X = \omega L \,$

which by inspection can be seen to be monotonically (and linearly) increasing with frequency.[4]

### Capacitor

The impedance of a capacitor is given by,

$Z = \frac {1}{i \omega C}$
$\scriptstyle C$ is capacitance

so the reactance is,

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

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.[4]

### Series resonant circuit

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

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

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 resonant frequency) and then continues to monotonically increase as the inductor reactance becomes progressively dominant.[5]

### Parallel resonant circuit

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,

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

The impedance function is,

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

At low frequencies the reactance is dominated by the inductor and is small and positive. This monotonically increases towards a pole at the 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.[5]

## Poles and zeroes

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.

A consequence of Foster's theorem is that the poles and 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.[2]

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.[6]

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.[3]

## Realization

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.
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.

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

$Z(s) = \frac {P(s)}{Q(s)}$
where,
$\scriptstyle Z(s)$ is immittance
$\scriptstyle P(s), \ Q(s)$ are polynomials with real, positive coefficiencts
$\scriptstyle s$ is the Laplace tranform variable, which can be replaced with $\scriptstyle i\omega$ when dealing with steady-state AC signals.

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 Cauer's ladder realisation from filter design.[7][8][9]

## Non-Foster networks

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.[7] 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.[7]

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 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.[7] 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.

## History

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.[10] The theorem was first published by Campbell 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, in Zobel'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.[1]

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.[13]

## References

1. ^ a b Foster, 1924.
2. ^ a b Aberle and Loepsinger-Romak, pp.8-9.
3. ^ a b Radmanesh, p.459.
4. ^ a b Cherry, pp.100-101.
5. ^ a b Cherry, pp.100-102.
6. ^ Smith and Alley, p.173.
7. ^ a b c d Aberle and Loepsinger-Romak, p.8.
8. ^ Cherry, pp.106-108.
9. ^ Montgomery et al., pp.157-158.
10. ^ Bray, p.62.
11. ^ Cherry, p.62.
12. ^ Zobel, pp.5,35-37.
13. ^ E. Cauer et al., p.5.