Network Synthesis
Part II
Dr. Mohamed Refky Amin
Electronics and Electrical Communications Engineering Department (EECE)
Cairo University
2nd.year.circuits@gmail.com
http://scholar.cu.edu.eg/refky/
OUTLINE
• Previously on ELC 202B
• Synthesis with LC Elements
• Synthesis with RC Elements
• Synthesis with RL Elements
Dr. Mohamed Refky 2
Previously on ELC 202B
Definition
In network synthesis we try to find a new circuit that provides a
required response to a given input excitation
Synthesis solutions are not unique
Dr. Mohamed Refky 3
Previously on ELC 202B
One-Port Networks
We will focus on the synthesis of driving point functions for one-
port networks.
The functions used are generally in the form of ratios of
polynomials
𝜙 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 or 𝑌 𝑠 = =
𝜓 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝛾𝑚 𝑠 − 𝑧1 𝑠 − 𝑧2 … 𝑠 − 𝑧𝑚
=
𝛾𝑛 𝑠 − 𝑝1 𝑠 − 𝑝2 … 𝑠 − 𝑝𝑛
Dr. Mohamed Refky 4
Previously on ELC 202B
Realization of a Function
𝑠𝐿 + 𝑅 1
𝑍 𝑠 = 2 𝑍 𝑠 =
𝑠 𝐿𝐶 + 𝑠𝑅𝐶 + 1 1
𝑠𝐶 + 𝑅 + 𝑠𝐿
Dr. Mohamed Refky 5
Previously on ELC 202B
First Foster Form
In first foster form, partial fraction is used to factorized 𝑍 𝑠
𝛼1 𝑠 + 𝛼0
𝑍 𝑠 =
𝛽2 𝑠 2 + 𝛽2 𝑠 + 𝛽0
𝑘1 𝑘2
𝑍 𝑠 = +
𝑎1 𝑠 + 𝑏1 𝑎2 𝑠 + 𝑏2
1 1
= +
𝑎1 𝑏 𝑎2 𝑏
𝑠+ 1 𝑠+ 2
𝑘1 𝑘1 𝑘2 𝑘2
Dr. Mohamed Refky 6
Previously on ELC 202B
Second Foster Form
In second foster form, partial fraction is used to factorized 𝑌 𝑠
𝛼1 𝑠 + 𝛼0
𝑌 𝑠 =
𝛽2 𝑠 2 + 𝛽2 𝑠 + 𝛽0
𝑘1 𝑘2
𝑌 𝑠 = +
𝑎1 𝑠 + 𝑏1 𝑎2 𝑠 + 𝑏2
1 1
= +
𝑎1 𝑏 𝑎2 𝑏
𝑠+ 1 𝑠+ 2
𝑘1 𝑘1 𝑘2 𝑘2
Dr. Mohamed Refky 7
Previously on ELC 202B
Cauer Form (Continued Fraction
Expansion)
First Cauer Form of 𝑍 𝑠 starts with
𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 =
𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
then Continued Fraction Expansion (CFE) is applied to 𝑍 𝑠 to
put it in the form
1
𝑍 𝑠 = 𝑍1 𝑠 +
1
𝑌2 𝑠 + 1
𝑍3 𝑠 + 1
𝑌4 𝑠 +
Dr. Mohamed Refky 𝑍5 𝑠 + ⋯ 8
Previously on ELC 202B
Cauer Form (Continued Fraction
Expansion)
Secound Cauer Form of 𝑍 𝑠 starts with
𝛼0 + ⋯ + 𝛼𝑚−1 𝑠 𝑚−1 + 𝛼𝑚 𝑠 𝑚
𝑍 𝑠 =
𝛽0 + ⋯ + 𝛽𝑛−1 𝑠 𝑛−1 + 𝛽𝑛 𝑠 𝑛
then Continued Fraction Expansion (CFE) is applied to 𝑍 𝑠 to
put it in the form
1
𝑍 𝑠 = 𝑍1 𝑠 +
1
𝑌2 𝑠 + 1
𝑍3 𝑠 + 1
𝑌4 𝑠 +
Dr. Mohamed Refky 𝑍5 𝑠 + ⋯ 9
Previously on ELC 202B
Cauer Form (Continued Fraction
Expansion)
First Cauer Form of 𝑌 𝑠 starts with
𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 =
𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
then Continued Fraction Expansion (CFE) is applied to 𝑌 𝑠 to
put it in the form
1
𝑌 𝑠 = 𝑌1 𝑠 +
1
𝑍2 𝑠 + 1
𝑌3 𝑠 + 1
𝑍4 𝑠 +
Dr. Mohamed Refky 𝑌5 𝑠 + ⋯ 10
Previously on ELC 202B
Cauer Form (Continued Fraction
Expansion)
Secound Cauer Form of 𝑌 𝑠 starts with
𝛽0 + ⋯ + 𝛽𝑛−1 𝑠 𝑛−1 + 𝛽𝑛 𝑠 𝑛
𝑌 𝑠 =
𝛼0 + ⋯ + 𝛼𝑚−1 𝑠 𝑚−1 + 𝛼𝑚 𝑠 𝑚
then Continued Fraction Expansion (CFE) is applied to 𝑌 𝑠 to
put it in the form
1
𝑌 𝑠 = 𝑌1 𝑠 +
1
𝑍2 𝑠 + 1
𝑌3 𝑠 + 1
𝑍4 𝑠 +
Dr. Mohamed Refky 𝑌5 𝑠 + ⋯ 11
Network Synthesis
Synthesis with LC Elements
A network (given in terms of 𝑍 𝑠 or 𝑌 𝑠 ) is synthesized into 𝐿𝐶
elements (coil and capacitor) if
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 or 𝑌 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑍 𝑠 or 𝑌 𝑠 is the ratio of an even 𝑁 𝑠 and an odd 𝐷 𝑠 or
vice versa.
The degrees of 𝑁 𝑠 and 𝐷 𝑠 differ by exactly 1.
𝛼𝑚 𝑠 𝑛+1 + 𝛼𝑚−1 𝑠 𝑛 + ⋯ + 𝛼0 𝛼𝑚 𝑠 𝑛 + 𝛼𝑚−1 𝑠 𝑛−1 + ⋯ + 𝛼0
=
𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0 𝛽𝑛 𝑠 𝑛+1 + 𝛽𝑛−1 𝑠 𝑛 + ⋯ + 𝛽0
Dr. Mohamed Refky 12
Network Synthesis
Synthesis with LC Elements
A network (given in terms of 𝑍 𝑠 or 𝑌 𝑠 ) is synthesized into 𝐿𝐶
elements (coil and capacitor) if
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 or 𝑌 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑍 𝑠 or 𝑌 𝑠 is the ratio of an even 𝑁 𝑠 and an odd 𝐷 𝑠 or
vice versa.
The degrees of 𝑁 𝑠 and 𝐷 𝑠 differ by exactly 1.
There is either a zero or a pole at 𝑠 = 0.
There is either a zero or a pole at 𝑠 → ∞.
Dr. Mohamed Refky 13
Network Synthesis
Synthesis with LC Elements
A network (given in terms of 𝑍 𝑠 or 𝑌 𝑠 ) is synthesized into 𝐿𝐶
elements (coil and capacitor) if
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 or 𝑌 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑍 𝑠 or 𝑌 𝑠 has only simple poles and zeros located
alternatively on the 𝑗𝜔 axis .
The residues at all poles are real and positive.
Dr. Mohamed Refky 14
Network Synthesis
Synthesis with LC Elements
Applying partial fractions to an LC impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 or 𝑌 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘0 𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + + 2 + ⋯+ 2
𝑠 𝑠 + 𝜔1 𝑠 + 𝜔𝑛
The order of the numerator
is greater that that of the The function has a
denominator pole at zero
Dr. Mohamed Refky 15
Network Synthesis
Example (1)
Find the two Foster Form realizations of the function :
𝑠2 + 1 𝑠2 + 9
𝑍 𝑠 =
𝑠 𝑠2 + 4
Dr. Mohamed Refky 16
Network Synthesis
Example (2)
Find the first and second Cauer realizations for the impedance
function given by
𝑠2 + 1 𝑠2 + 9
𝑍 𝑠 =
𝑠 𝑠2 + 4
First Cauer Form
𝑠 4 + 10𝑠 2 + 9
𝑍 𝑠 =
𝑠 3 + 4𝑠
Dr. Mohamed Refky 17
Network Synthesis
Synthesis with RC Elements
Applying partial fractions to an RC impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
Dr. Mohamed Refky
𝑠 + 𝛿 1 𝑠 + 𝛿𝑛
Network Synthesis
Synthesis with RC Elements
Applying partial fractions to an RC impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
All poles and zeros are simple and lie on the – 𝛿 (negative real)
axis of the 𝑠-plane.
Poles and zeros alternate on the – 𝛿 axis.
Dr. Mohamed Refky
Network Synthesis
Synthesis with RC Elements
Applying partial fractions to an RC impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The first critical frequency (the smallest in absolute value) on
the – 𝛿 axis is a pole (may be at 𝑠 = 0).
The last critical frequency is a zero (may be at 𝑠 = ∞).
Dr. Mohamed Refky
Network Synthesis
Synthesis with RC Elements
Applying partial fractions to an RC impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The residues at the poles of 𝑍 𝑠 are real and positive.
Dr. Mohamed Refky
Network Synthesis
Synthesis with RC Elements
Applying partial fractions to an RC impedance or admittance
function will produce the following possible parts
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
𝑠 + 𝛿1 𝑠 + 𝛿𝑛
All poles and zeros are simple and lie on the – 𝛿 (negative real)
axis of the 𝑠-plane.
Poles and zeros alternate on the – 𝛿 axis.
Dr. Mohamed Refky
Network Synthesis
Synthesis with RC Elements
Applying partial fractions to an RC impedance or admittance
function will produce the following possible parts
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The first critical frequency (the smallest in absolute value) on
the – 𝛿 axis is a zero (may be at 𝑠 = 0).
The last critical frequency is a pole (may be at 𝑠 = ∞).
Dr. Mohamed Refky
Network Synthesis
Synthesis with RC Elements
Applying partial fractions to an RC impedance or admittance
function will produce the following possible parts
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The residues at the poles of 𝑌 𝑠 are real and negative (No
circuit can be synthesized). The residues of 𝑌 𝑠 /𝑠 are real and
positive .
Dr. Mohamed Refky
Network Synthesis
Example (3)
Find the Foster realizations of the impedance function
2 𝑠+1 𝑠+3
𝑍 𝑠 =
𝑠 𝑠+2
Dr. Mohamed Refky 25
Network Synthesis
Example (4)
Find two Cauer realizations for the function given by
2 𝑠+1 𝑠+3
𝑍 𝑠 =
𝑠 𝑠+2
First Cauer Form
2𝑠 2 + 8𝑠 + 6
𝑍 𝑠 =
𝑠 2 + 2𝑠
Dr. Mohamed Refky 26
Network Synthesis
Synthesis with RL Elements
Applying partial fractions to an RL impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
𝑠 + 𝛿1 𝑠 + 𝛿𝑛
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
Dr. Mohamed Refky
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
Network Synthesis
Synthesis with RL Elements
Applying partial fractions to an RL impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
𝑠 + 𝛿1 𝑠 + 𝛿𝑛
All poles and zeros are simple and lie on the – 𝛿 (negative
real) axis of the 𝑠-plane.
Poles and zeros alternate on the – 𝛿 axis.
Dr. Mohamed Refky
Network Synthesis
Synthesis with RL Elements
Applying partial fractions to an RL impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The first critical frequency (the smallest in absolute value) on
the – 𝛿 axis is a zero (may be at 𝑠 = 0).
The last critical frequency is a pole (may be at 𝑠 = ∞).
Dr. Mohamed Refky
Network Synthesis
Synthesis with RL Elements
Applying partial fractions to an RL impedance or admittance
function will produce the following possible parts
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑍 𝑠 = =
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑘1 𝑠 𝑘𝑛 𝑠
= 𝑘∞ 𝑠 + 𝑘0 + + ⋯+
𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The residues at the poles of 𝑍 𝑠 are real and negative (No
circuit can be synthesized). The residues of 𝑍 𝑠 /𝑠 are real
and positive.
Dr. Mohamed Refky
Network Synthesis
Synthesis with RL Elements
Applying partial fractions to an RL impedance or admittance
function will produce the following possible parts
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
All poles and zeros are simple and lie on the – 𝛿 (negative
real) axis of the 𝑠-plane.
Poles and zeros alternate on the – 𝛿 axis.
Dr. Mohamed Refky
Network Synthesis
Synthesis with RL Elements
Applying partial fractions to an RL impedance or admittance
function will produce the following possible parts
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The first critical frequency (the smallest in absolute value) on
the – 𝛿 axis is a pole (may be at 𝑠 = 0).
The last critical frequency is a zero (may be at 𝑠 = ∞).
Dr. Mohamed Refky
Network Synthesis
Synthesis with RL Elements
Applying partial fractions to an RL impedance or admittance
function will produce the following possible parts
𝐷 𝑠 𝛽𝑛 𝑠 𝑛 + 𝛽𝑛−1 𝑠 𝑛−1 + ⋯ + 𝛽0
𝑌 𝑠 = =
𝑁 𝑠 𝛼𝑚 𝑠 𝑚 + 𝛼𝑚−1 𝑠 𝑚−1 + ⋯ + 𝛼0
𝑘0 𝑘1 𝑘𝑛
= 𝑘∞ + + + ⋯+
𝑠 𝑠 + 𝛿1 𝑠 + 𝛿𝑛
The residues of the poles of 𝑌(𝑠) are real and positive.
Dr. Mohamed Refky
Network Synthesis
Example (5)
Find the Foster realizations of the impedance function
𝑠 𝑠+1
𝑍 𝑠 =3
1
𝑠+2 𝑠+2
Dr. Mohamed Refky 34
Network Synthesis
Example (6)
Find the second Cauer realization of the impedance function given
by
𝑠+1 𝑠+3
𝑍 𝑠 =
𝑠+2 𝑠+6
Poles and zeros alternate on the – 𝛿 axis
and the first critical frequency is a zero
The network is RL
Dr. Mohamed Refky 35
