Network Analysis and Synthesis (ECEG-3122)
Chapter five
Synthesis of Driving-point Networks
5.1 Introduction
In this chapter we will study methods for synthesizing one-port networks with
two/three kinds of elements. Since we have three elements to choose from, the networks
to be synthesized are either . First we will discuss the
properties of a particular type of one-port network, and then we will synthesize it.
Elementary Synthesis Procedure
The basic philosophy behind the synthesis of driving point functions is to break up a
p.r. function into a sum of simpler p.r. functions and then to
synthesize these individual as elements of the overall network whose driving-
point impedance is .
First, consider the “breaking-up” process of the function into the sum of
functions . One important restriction is that all must be positive real.
Certainly, if all were given to us, we could synthesized a network whose driving-
point impedance is by simply connecting all the in series. How if we were to
start with to give us the individual ? Suppose is given in general as
Consider the case where has a pole at (that is, ). Let us divide and
to give a positive real quotient and remainder , which we can be denoted
as and .
Where both are positive real
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Network Analysis and Synthesis (ECEG-3122)
From the foregoing discussion, it is seen that if has a pole at , a partial fraction
expansion can be made such that one of the terms is of the form and the other terms
combined still remain p.r. A similar argument shows that if has a pole at
(that is, ), we can divide the numerator by the denominator to give a quotient
and a remainder term , again denoted as and . Then,
. Here is also p.r.
If has a pair of conjugate imaginary poles on the imaginary axis, for example, poles
at , then can be expanded into partial fractions so that
( ) ( )
Finally, if is minimum at some point , and as shown in
Fig. 4.1, we can remove a constant from so that the remainder is still
p.r. This is because will still be greater than or equal to zero for all values.
Fig. 4.1:
Properties and Synthesis of One-port Two-element Immittances
There are a number of methods of synthesizing (or realizing) a one port network. But in
this chapter we will only consider the following four basic forms as follows:
1. Foster I or Foster series form
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Network Analysis and Synthesis (ECEG-3122)
2. Foster II or Foster parallel form
3. Cauer I form and
4. Cauer II form as shown in the following Fig.4.1 below.
Fig.4.2: General representation of Foster I, Foster II, Cauer I and Cauer II forms.
The Foster I and II forms are obtained by partial fraction expansion of and
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.
Impedance or Admittance Function
Consider the impedance of a passive one port network which is represented as,
Where, are even and are odd parts of the numerator and
denominator. The average power dissipated by one-port passive network is
| | where, is the input current.
For a pure (reactive) network, it is known that the power dissipated is zero.
Therefore, real part of is zero, i.e.,
From condition of positive realness, we know that
In order for i.e., and for the existence
of , either of the following cases must hold.
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Network Analysis and Synthesis (ECEG-3122)
(a)
(b)
Consider the example of an immittance function given by
We see from this development the following properties of function:
1. and are the ratio of even to odd or odd to even polynomials. (this
property is called as “Fosters Reactance Theorem”)
2. Since both and are Hurwitz, they have imaginary roots, and it follow
that the poles and zeros of or are on the imaginary axis (including
origin).
3. The poles and zeros interlace (alternate) on the axis.
4. The highest and lowest powers of numerator and denominator must differ by
unity.
5. There must be either a zero or a pole at the origin and infinity.
Exercise: Check whether the following functions are or not. If not, give reason.
1. 3.
2. 4.
Note: Case I: when and are in series;
and
Case II: when and are in parallel;
and
Synthesis of L-C Immittance Functions
The partial fraction expansion of an L-C function is expressed in general terms as
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Network Analysis and Synthesis (ECEG-3122)
The synthesis is accomplished directly from the partial fraction expansion by
associating the individual terms in the expansion with network elements. If is an
impedance then the term represents a capacitor of farads: the term is an
inductor of henrys, and the term is a parallel tank circuit that consists of a
capacitor of farads in parallel with an inductor of . Thus a partial fraction
expansion of general L-C impedance would yield the network shown in Fig. 4.3. This
method of synthesis that is based on partial fraction expansion is called Foster
synthesis.
Fig. 4.3. Foster Synthesis of an L-C impedance function
Another option partial fraction expansion of Y(s) which is given by the equation below
gives us a circuit consisting of parallel branches as shown in Fig.4.4. This is Foster II
synthesis method of an network.
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Network Analysis and Synthesis (ECEG-3122)
Fig. 4.4. Foster II synthesis
Exercise: An impedance function has the pole zero pattern shown in Fig. 4.2. If
, synthesize the impedance in
a) Foster –I and II forms.
b) Cauer –I and II forms.
Fig.4.5:
Impedance or Admittance Function
impedance or admittance function has the following properties.
1. Poles and zeros lie on the negative real axis (including origin) of the plane,
and they interlace (alternate).
2. The residues of the poles of are real and positive.
3. The residues of the poles are real and negative; however, the
residues of the poles of must be real and positive.
4. The singularity nearest to (or at) the origin must be a pole, i.e.,
function .
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Network Analysis and Synthesis (ECEG-3122)
5. The singularity nearest to (or at) the negative infinity ( ) must be a zero, i.e.,
function .
Exercise: Check if the following impedance functions are R-C impedance functions.
1. 3.
2. 4.
Note: An impedance, also can be realized as an admittance, .
All the properties of admittance are the same as the properties of
impedances. It is therefore important to specify whether a function is to be realized as
an impedance or admittance.
Case I: when and are in series;
and
Case II: when and are in parallel;
and
Synthesis of R-C Impedances or R-L Admittances
Referring to the series foster form for an L-C impedance given in the Fig. 4.6, we can
obtain a foster realization of an R-C impedance by simply replacing all the inductances
by resistance so that a general RC impedance could be represented as shown in Fig.
4.x4.
Fig. 4.6: Foster realization of an R-C Impedance function
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Network Analysis and Synthesis (ECEG-3122)
Where ⁄ ⁄
Instead of letting represent an impedance consider the case where is an
admittance . If we associate the individual terms in the expansion to network
elements, we then obtain the network shown in Fig. 4.7.
Fig. 4.7. R-C impedance function can be realized as R-L admittance
Exercise: an immittance function is given by
Find the and R-L representation of in: (a) Foster-I and II forms, (b) Cauer- I
and II forms.
Impedance or Admittance Function
impedance or admittance function has the following properties.
1. Poles and zeros lie on the negative real axis (including origin) of the plane, and
they interlace (alternate).
2. The residues of the poles are real and negative; however, the
residues of the poles of must be real and positive.
3. The singularity nearest to (or at) the origin must be a zero, i.e.,
function .
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Network Analysis and Synthesis (ECEG-3122)
4. The singularity nearest to (or at) the negative infinity ( ) must be a pole, i.e.,
function .
Exercise: Check if the following impedance functions are R-L impedance or R-C
admittance functions.
1. 3.
2. 4.
Note: An impedance, also can be realized as an admittance, .
All the properties os admittance are the same as the properties of
impedances. It is therefore important to specify whether a function is to be realized as
an impedance or admittance.
Case I: when and are in series;
and
Case II: when and are in parallel;
and
The immittance that represent series Foster impedance or a parallel Foster
admittance is given by
The significant difference between an impedance and impedance is that the
partial fraction expansion term for the tank circuit is ⁄ whereas for the
impedance the corresponding term must be multiplied by an in order to give an
tank circuit consisting of a resistor of value ( ) in parallel with an inductor of
value ( ⁄ ).
Exercise: an impedance function is given by
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Network Analysis and Synthesis (ECEG-3122)
Find the and representation of in: (a) Foster-I and II forms, (b) Cauer- I
and II forms.
Synthesis of Certain Functions
Under certain conditions, R-L-C driving point functions may be synthesized with the
use of either partial fractions or continued fractions. For example, the function
is neither L-C, R-C, nor R-L, nevertheless, the function can be synthesized by continued
fractions as shown below
The network derived from this expansion is given in Fig. 4.8.
Fig. 4. 8
Consider the following admittance function. The poles and zeros of the admittance
function are all on the negative real axis but they do not alternate.
The partial fraction expansion of is
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Network Analysis and Synthesis (ECEG-3122)
Since one of residues is negative we cannot use this expansion for synthesis. An
alternate method would be to expand and then multiply the whole expansion by .
Multiply by s we obtain,
Note that also has a negative term. If we divide the numerator of this negative
term to the denominator we can rid ourselves of any terms with negative signs.
[ ]
The network that is realized from the expanded function is given in Fig. 4.9 below.
Fig. 4.9:
To synthesize the same example in cauer form, expanding by continued fraction
expansion results in negative quotients. However we can expand by
continued fraction. Although the expansion is not as simple or straight forward as in
the case of an functions, because we sometimes have to reverse the order of
division to make the quotients all positive. The continued fraction expansion of is
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Network Analysis and Synthesis (ECEG-3122)
As we see the division process giving the quotient of involves a reversal of the order of
the polynomials involved. The resulting ladder network is given on Fig. 4.10.
Fig. 4.10:
In the beginning of this section, it was stated that only under special conditions can an
R-L-C driving point function be synthesized with the use of a ladder form or the Foster
forms. These conditions are not given here because they are rather involved instead.
When a positive real function is given, and it is found that the function is not
synthesizable by using two kinds of elements only, it is suggested that a continued
fraction expansion or a partial fraction expansion be tried first.
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Network Analysis and Synthesis (ECEG-3122)
Tutorial Problems
1. Which of the following functions are L-C driving point impedances? Why? Also
synthesize the realizable impedances in a foster-I&II and Cauer-I&II forms.
( )( ) ( )( )( )
A) C)
B) D)
2. Indicate which of the following function is either , of
impedance functions.
A) C) E)
B) D)
3. An impedance function has the pole zero pattern shown in the figure below. If
synthesize the impedance in Foster-I & II and Cauer-I & II forms.
4. From the following functions pick out the ones which are R-C admittances and
synthesize in Foster-I & II and Cauer-I & II forms.
A) C)
B) D)
5. Find the networks for the following function. Both foster and ladder forms are
required.
A) B)
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6. For the network shown, find when
Synthesize as an admittance.
7. Synthesize by continued fractions the function
8. Synthesize the following functions in Foster-I&II and Cauer-I&II forms.
A) C) E)
B) D) F)
9. Of the three pole-zero diagrams shown below, pick the diagram that represents
an impedance function and synthesize in series Foster form.
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