Foster has given two types of network synthesis for one-port reactive network

Series combination of parallel LC networks

The parallel combination of series LC networks

Foster 1^{st} Form

When numerator polynomial is of higher degree than the denominator polynomial

The first element capacitor is represented by A0/s

The last element Inductor is represented by Hs corresponds t a pole at infinity

2Ai/(s^{2} + ω^{ 2}) represents the conjugate poles results in LC resonance.

When no terms in the denominator of Z(s), there will not be any A_{0}/s term indicating the absence of capacitor.

Example: Obtain the first form of the Foster Network for the driving point impedance of LC network given as:

Z(s) = 10(s^{2}+4)(s^{2}+16) / s(s^{2}+9)

Solution:

It is observed from the function Z(s) that the numerator has one higher degree of s than the denominator, hence two poles exist one at ω =0 and another at ω=∞, therefore there will be the presence of the first element as the capacitor and the last element as the inductor

The partial fraction expansion of the network function is: