Foster’s Form of Synthesis

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

1. The first element capacitor is represented by A0/s
2. The last element Inductor is represented by Hs corresponds t a pole at infinity
3. 2Ai/(s² + ω ²) represents the conjugate poles results in LC resonance.
4. When no terms in the denominator of Z(s), there will not be any A₀/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(s²+4)(s²+16) / s(s²+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:

Z(s) = A₀/s + A₁/(s+j3)  + A₁*/(s-j₃)   + Hs

The values of residues is found to be as:

A₀ at s=0 is  = 640/9 = 71.11;

so we find C₀ = 1/A₀

= 1 / 71.11    =   0.0141 Farad

A₁ at s= -j₃ is= 350/18  =  19.45

Now we know Li and Ci of the parallel network is calculated as

C₁    = 1/(2A₁)

= 1/(2*19.45)

= 0.0257 Farads

and

L₁    = 2A₁/ω²

= 2*19.45 / 9

= 4.322 Henry

The end element inductor is ‘H’     = 10 henry

Therefore the realization of the impedance function is:

Foster’s 2nd Form

2nd for is for the synthesis of admittance function, and is:

As observed:

Given an admittance function:

i. The inductor is represented by the term B₀/s and this corresponds to a pole at origin

ii. The capacitor  C_{∞   is represented by Hs and corresponds to a pole at infinity.}

iii. The series combination of L and C is determined from 2*B₁ / (s² + ω ²)

iv. In case there is no pole at  ω=0 or at ω=∞ or at both, signifies absence of end elements.

Example: