## 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

- 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:

Z(s) = A_{0}/s + A_{1}/(s+j3) + A_{1}*/(s-j_{3}) + Hs

The values of residues is found to be as:

A_{0} at s=0 is = 640/9 = 71.11;

so we find C_{0} = 1/A_{0}

= 1 / 71.11 = 0.0141 Farad

A_{1} at s= -j_{3} is= 350/18 = 19.45

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

C_{1} = 1/(2A_{1})

= 1/(2*19.45)

= 0.0257 Farads

and

L_{1} = 2A_{1}/ω^{2}

= 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 B_{0}/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*B_{1} / (s^{2} + ω^{ 2})

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

**Example:**