# Foster’s Form of Network Synthesis

## Foster’s Form of Synthesis

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

• Series combination of parallel LC networks
• Parallel combination of series LC networks

Foster 1st 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) represent 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 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 exists one at ω =0 and another at ω=∞, therefore there will be the presence of first element as capacitor and the last element as 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 parallel network is calculated as

C1    = 1/(2A1)

= 1/(2*19.45)

and

L1    = 2A12

= 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  synthesis of admittance function, and is: As observed:

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: Updated: April 22, 2020 — 12:43 pm