Электрлік тізбектер теориясы: Фостер теоремасы

Electrical network 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 strictly monotonically increases with frequency. It is easily seen that the reactances of inductors and capacitors individually increase with frequency and from that basis a proof for passive lossless networks generally can be constructed. The proof of the theorem was presented by Ronald Martin Foster in 1924, although the principle had been published earlier by Foster's colleagues at American Telephone & Telegraph. 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.

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

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. 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. 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. The theorem was first published by Campbell in 1922 but without a proof. 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. Foster published his paper the following year which included his canonical realisation forms. 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 the extension of Foster's work to all 2 element kind networks after discovering an isomorphism between them. Cauer was interested in finding the necessary and sufficient condition for realisability of a rational one port network from its polynomial function, a condition now known to be a 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. Foster networks are only a subset of realisable networks,