Foster's reactance theorem

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 or decrease with frequency respectively and from that basis a proof for passive lossless networks generally can be constructed.

Foster's work was an important starting point for the development of network synthesis.

The network consequently must consist entirely of inductors and capacitors and the impedance will be purely an imaginary number with zero real part.

In particular, it applies to distributed-element networks, although Foster formulated it in terms of discrete inductors and capacitors.

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.

[4] A consequence of Foster's theorem is that the zeros and poles of any passive immittance function must alternate as frequency increases.

[1] The poles and zeroes of an immittance function completely determine the frequency characteristics of a Foster network.

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.

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.

[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 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.

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
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.
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 inductor. If the polynomial function has a zero at ω =∞ one of the LC sections will reduce to a single capacitor. If both zeroes are present then two sections reduce to a parallel LC circuit.