Kombolcha Institute of Technology

School of Electrical and Computer Engineering

Chapter Five:-Synthesis of Driving Point Function

Jemal H. ( Msc )
jjemalassen@gmail.com
December, 2024

Network Analysis and Synthesis 1 Lecture # 5 1

 Introduction

• The main synthesis of networks elements are L-C, R-C or R-L elements.
• Evaluation of elements of a network from its driving point impedance or
admittance function is known as driving point synthesis.
• For a given positive real function F(s), the network can be synthesized by any
one of the following two methods:
• Foster forms
• Cauer forms.
• The Foster I and II forms are obtained by partial fraction expansion of
Z(s)and Y(s), respectively.
• While the Cauer I and Cauer II forms are obtained by continued fraction
expansion of immittance function by arranging both the numerator and
denominator polynomial in descending and ascending orders respectively.

Network Analysis and Synthesis 2 Lecture # 5

 Cont…

• Foster Forms:
• The networks synthesized by partial fraction expansion methods are called
Foster type of networks.
• Foster form I: Provides series impedance network realization Z(s),
• Foster form II: Provides parallel admittance network realization Y(s).
• Cauer Forms
• The networks are synthesized by Continued Fraction Expansion Methods are
called Cauer type of networks.
• By this technique, the synthesized network is of a ladder type.
– Cauer I Form: Since the numerator and denominator polynomials of an
LC function always differ in degrees by unity, there is always a zero or a
pole at s = ∞.
– The Cauer I Form is obtained by successive removal of a pole or a zero at
infinity from the function.
– Power of s are put in increasing order.
Network Analysis and Synthesis 3 Lecture # 5
 Synthesis of L–C Networks

• These networks have only inductors and capacitors.

• The average power consumed in these kind of networks is zero. (Because an
inductor and a capacitor don’t dissipate energy.
• The immittance function (that is, Z(s) or Y(s)) is a ratio of odd to even
polynomial or even to odd polynomial
• The highest order terms in the numerator and denominator differ by one.
• There is a pole or zero at origin Or at infinity and Interlace (alternate) on the axis
• The poles and zeros occur in complex conjugate pairs
Foster Form-I of L–C Network
• The partial fractions of (S) are given by

Network Analysis and Synthesis 4 Lecture # 5

 Cont…

• The Foster form-I of the L–C network is shown in Figure below.

Figure Foster Form-I of L–C Network

Network Analysis and Synthesis 5 Lecture # 5

 Foster Form-II of L–C Network

• The equation is given as follows

• The Foster form-II for the L–C network is shown in Figure

Network Analysis and Synthesis 6 Lecture # 5

 Cauer Form-I and II of L–C Network

• The Cauer form-I of L–C network is shown in Figure below.

• Inductors are in serious and

Capacitors are in parallel

Cauer Form-II of L–C Network

• Cauer form-II of the L–C network is shown in Figure below.

• Inductors are in parallel and

Capacitors are in serious

Network Analysis and Synthesis 7 Lecture # 5

 Synthesis of R–C Network

• The poles and zeros of the or lie on the negative real axis of
the complex s-plane.
• The poles and zeros are interlacing, that is, they alternate along the negative
real axis.
• The poles and zeros are simple. There are no multiple poles and zeros.
• The singularity nearest to (or at) the origin must be a pole, i.e., function .

• It is obtained by using the partial fraction expansion of (s).

Foster Form-I of R–C Network

Network Analysis and Synthesis 8 Lecture # 5

 Cont….

• Accordingly, the Foster form-I of R–C network is shown in Figure below.

Figure Foster Form-I of R–C Network

Network Analysis and Synthesis 9 Lecture # 5

 Foster Form-II of R–C Network

• It is obtained from the following:

Network Analysis and Synthesis 10 Lecture # 5

 Cauer Forms of R–C Network

• The R–C network for Cauer form-I and Cauer form-II is shown in Figure (a)
and (b) below respectively.
• In Cauer form-I resistors are in series and capacitors are in parallel.
• In Cauer form-II capacitors are in series while the resistors are in parallel

(a) Cauer Form-I of R–C Network

(b) Cauer Form-II of R–C Network

Network Analysis and Synthesis 11 Lecture # 5

 Synthesis of R–L Network

• The poles and zeros are simple. There are no multiple poles or zeros.
• The poles and zero interlace (that is, alternate) each other along the negative
real axis.
• The poles and zeros lie on the negative real axis of the complex s-plane.
• The singularity nearest to (or at) the origin must be a Zero, i.e., function .

Foster Form-I of R–L Network

• The partial fraction expansion of ZRL(s) is given as follows:

Network Analysis and Synthesis 12 Lecture # 5

 Foster Form-II of R–L Network

• Consider the foster form-II of R–L network shown in Figure below.

• The network is designed by using the partial fraction method.
• Here, we consider the partial fraction expansion of Y(s).

Network Analysis and Synthesis 13 Lecture # 5

 Cauer Form-I and form II of R–L Network

• The figure below shows the representation of cauer form-I for R–L network

• The figure below shows the representation of cauer form-II for R–L
network.

Network Analysis and Synthesis 14 Lecture # 5

 Example -1

• Obtain the Cauer forms of the RC impedance function

• The Cauer I form is obtained by continued fraction expansion

• By continued fraction expansion

Network Analysis and Synthesis 15 Lecture # 5

 Cont…

• The impedances are connected in the series branches whereas the

admittances are connected in the parallel branches.
• The circuit is shown in the figure above.

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 Cont…

• The Cauer II form is obtained by continued fraction expansion.

• Arranging the polynomials in ascending order of s

• Since negative term results, continued fraction expansion of Y(s) is carried

out.

• By continued fraction
expansion,

Network Analysis and Synthesis 17 Lecture # 5

 Cont…

Network Analysis and Synthesis 18 Lecture # 5

 Example -2

• Find the Foster forms of the RL impedance function

• The Foster I form is obtained by partial-fraction expansion of impedance
function

• By partial-faction expansion,

Network Analysis and Synthesis 19 Lecture # 5

 Cont…

• The first term represents the impedance of the resistor of Ω.

• The other two terms represent the impedance of a parallel RL circuit

Network Analysis and Synthesis 20 Lecture # 5

 Cont…

• The Foster II form is obtained by partial fraction expansion of Y (s).

• Since the degree of the numerator is equal to the degree of the denominator,
division is first carried out.

• By partial-fraction expansion,

Network Analysis and Synthesis 21 Lecture # 5

 Cont…

• The first term represents the admittance of a resistor of 1 Ω.

• The other two terms represent the admittance of a series RL circuit

Network Analysis and Synthesis 22 Lecture # 5

Network Analysis and Synthesis 23 Lecture # 5