Network Synthesis Part I - Cairo...

Network Synthesis

Part I

Dr. Mohamed Refky Amin

Electronics and Electrical Communications Engineering Department (EECE)

Cairo University

elc.n112.eng@gmail.com

http://scholar.cu.edu.eg/refky/

OUTLINE

• References

• Definition

• Network Functions

• Realizability Conditions

2Dr. Mohamed Refky

Gabor C. Temes & Jack W. Lapatra, “Introduction to Circuit

Synthesis and Design”, McGraw-Hill Book Company.

M.E. Van Valkenburg, “Introduction to Modern Network

Synthesis”, John Wiley Inc.

References

3Dr. Mohamed Refky

What we have used to do so far is the calculation of the response

of a known circuit to a given excitation.

This is called analysis of circuits.

Network Synthesis

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Definition

Dr. Mohamed Refky

In network synthesis we try to find a new circuit that provides a

required response to a given input excitation

Network Synthesis

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Definition

Dr. Mohamed Refky

Synthesis solutions are not unique

In network synthesis, complex frequency

𝑠 = 𝛿 + 𝑗𝜔

is used to analyze the circuits because it simplifies algebraic work

by including the imaginary part in 𝑠.

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Definition

Dr. Mohamed Refky

𝐼 =𝑉

𝑅 + 𝑗𝜔𝐿 +1𝑗𝜔𝐶

𝐼 =𝑉

𝑅 + 𝑠𝐿 +1𝑠𝐶

For a single port network, synthesis may be operated on the

following functions:

Network Synthesis

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One-Port Networks

Dr. Mohamed Refky

𝑉 𝑠

𝐼 𝑠

𝐼 𝑠

𝑉 𝑠

Driving point impedance

𝑍 𝑠 =

Driving point admittance

𝑌 𝑠 =

For a two port network, synthesis may be operated on the

following functions:

Network Synthesis

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Two-Port Networks

Dr. Mohamed Refky

𝑉1 𝑠

𝐼1 𝑠

𝑉2 𝑠

𝐼2 𝑠

Driving point impedance

𝑍11 𝑠 =

Driving point admittance

𝑍22 𝑠 =

For a two port network, synthesis may be operated on the

following functions:

Network Synthesis

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Two-Port Networks

Dr. Mohamed Refky

𝑉1 𝑠

−𝐼2 𝑠

−𝐼2 𝑠

𝑉1 𝑠

Driving point impedance

𝑍12 𝑠 =

Driving point admittance

𝑌21 𝑠 =

For a two port network, synthesis may be operated on the

following functions:

Network Synthesis

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Two-Port Networks

Dr. Mohamed Refky

𝑉2 𝑠

𝑉1 𝑠

−𝐼2 𝑠

𝐼1 𝑠

Driving point impedance

𝐺21 𝑠 =

Driving point admittance

𝛼21 𝑠 =

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

𝑍 𝑠 or 𝑌 𝑠

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One-Port Networks

Dr. Mohamed Refky

=𝜙 𝑠

𝜓 𝑠=𝛼𝑚𝑠

𝑚 + 𝛼𝑚−1𝑠𝑚−1 +⋯+ 𝛼0

𝛽𝑛𝑠𝑛 + 𝛽𝑛−1𝑠

𝑛−1 +⋯+ 𝛽0

=𝛾𝑚𝛾𝑛

𝑠 − 𝑧1 𝑠 − 𝑧2 … 𝑠 − 𝑧𝑚𝑠 − 𝑝1 𝑠 − 𝑝2 … 𝑠 − 𝑝𝑛

𝑍 𝑠 =1

𝑠𝐶// 𝑅 + 𝑠𝐿

=

1𝑠𝐶

𝑅 + 𝑠𝐿

𝑅 + 𝑠𝐿 +1𝑠𝐶

=𝑅 + 𝑠𝐿

𝑠𝐶𝑅 + 𝑠2𝐿𝐶 + 1

Network Synthesis

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Example (1)

Dr. Mohamed Refky

Find the impedance of the shown circuit

𝑍 𝑠 or 𝑌 𝑠

𝛼’s and 𝛽’s are positive constants

𝑚 is the orders of 𝜙 𝑠 .

𝑛 are the orders of 𝜓 𝑠 .

Network Synthesis

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One-Port Networks

Dr. Mohamed Refky

=𝜙 𝑠

𝜓 𝑠=𝛼𝑚𝑠

𝑚 + 𝛼𝑚−1𝑠𝑚−1 +⋯+ 𝛼0

𝛽𝑛𝑠𝑛 + 𝛽𝑛−1𝑠

𝑛−1 +⋯+ 𝛽0

𝑍 𝑠 or 𝑌 𝑠

𝑧1, 𝑧2, …, 𝑧𝑚 are the zeros of 𝑍 𝑠 or 𝑌 𝑠

𝑝1, 𝑝2, …, 𝑝𝑛 are the poles of 𝑍 𝑠 or 𝑌 𝑠

Network Synthesis

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One-Port Networks

Dr. Mohamed Refky

=𝜙 𝑠

𝜓 𝑠=𝛼𝑚𝑠

𝑚 + 𝛼𝑚−1𝑠𝑚−1 +⋯+ 𝛼0

𝛼𝑛𝑠𝑛 + 𝛽𝑛−1𝑠

𝑛−1 +⋯+ 𝛽0

𝛾𝑚𝛾𝑛

𝑠 − 𝑧1 𝑠 − 𝑧2 … 𝑠 − 𝑧𝑚𝑠 − 𝑝1 𝑠 − 𝑝2 … 𝑠 − 𝑝𝑛

𝛾𝑚𝛾𝑛

is the scale factor

For series impedances

𝑍 𝑠 = 𝑍1 𝑠 + 𝑍2 𝑠

For parallel impedances

𝑍 𝑠 =1

𝑌1 𝑠 + 𝑌2 𝑠

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

=1

1𝑍1 𝑠

+1

𝑍2 𝑠

For series impedances

𝑌 𝑠 =1

𝑍1 𝑠 + 𝑍2 𝑠

For parallel impedances

𝑌 𝑠 = 𝑌1 𝑠 + 𝑌2 𝑠

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

=1

𝑍1 𝑠+

1

𝑍1 𝑠

For a combination of series and

parallel impedances

𝑍 𝑠 = 𝑍1 𝑠 + 𝑍𝑝 𝑠

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Realization of a Function

Dr. Mohamed Refky

= 𝑍1 𝑠 +1

1𝑍2 𝑠

+1

𝑍3 𝑠

= 𝑍1 𝑠 +1

𝑌2 𝑠 + 𝑌3 𝑠

For a combination of series and

parallel impedances

𝑍 𝑠 =1

𝑌1 𝑠 + 𝑌2 𝑠

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

=1

1𝑍1 𝑠

+1

𝑍2 𝑠 + 𝑍3 𝑠

For a combination of series and

parallel impedances

𝑌 𝑠 =1

𝑍1 𝑠 + 𝑍𝑝 𝑠

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

=1

𝑍1 𝑠 +1

𝑌2 𝑠 + 𝑌3 𝑠

=1

𝑍1 𝑠 +1

1𝑍2 𝑠

+1

𝑍3 𝑠

For a combination of series and

parallel impedances

𝑌 𝑠 = 𝑌1 𝑠 + 𝑌𝑠 𝑠

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

=1

𝑍1 𝑠+

1

𝑍2 𝑠 + 𝑍3 𝑠

𝑍 𝑠 = 𝑍1 𝑠 +1

𝑌2 𝑠 +1

𝑍3 𝑠 +1

𝑌4 𝑠 +1

𝑍5 𝑠 + ⋯

𝑌 𝑠 = 𝑌1 𝑠 +1

𝑍2 𝑠 +1

𝑌3 𝑠 +1

𝑍4 𝑠 +1

𝑌5 𝑠 + ⋯

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

𝑍 𝑠 =1

𝑠𝐶 +1

𝑅 + 𝑠𝐿

𝑍 𝑠 =𝑠𝐿 + 𝑅

𝑠2𝐿𝐶 + 𝑠𝑅𝐶 + 1

Network Synthesis

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Realization of a Function

Dr. Mohamed Refky

𝑍 𝑠 =1

𝑠𝐶 +1

𝑅 + 𝑠𝐿

𝑍 𝑠 =𝑠𝐿 + 𝑅

𝑠2𝐿𝐶 + 𝑠𝑅𝐶 + 1

In network synthesis, we try to find a way to convert the function

(𝑍 𝑠 or 𝑌 𝑠 ) into a form that is easier to be realized into a

circuit.

1) The function must be a Positive Real (PR)

𝑅𝑒𝑎𝑙 𝑍 𝑠 or 𝑌 𝑠 ≥ 0 for 𝑅𝑒𝑎𝑙 𝑠 ≥ 0

This condition means that the power flows from the source to the

circuit

Network Synthesis

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Realizability Conditions

Dr. Mohamed Refky

𝑍1 𝑠 = 𝑅 + 𝑠𝑋 𝑍2 𝑠 = −𝑅 + 𝑠𝑋

𝑌1 𝑠 = 𝐺 +1

𝑠𝑋𝑌2 𝑠 = −𝐺 +

1

𝑠𝑋

1) The function must be a Positive Real (PR)

𝑅𝑒𝑎𝑙 𝑍 𝑠 or 𝑌 𝑠 ≥ 0 for 𝑅𝑒𝑎𝑙 𝑠 ≥ 0

This condition means that the power flows from the source to the

circuit

The poles of the function are negative or, if complex, they have

a negative real part. This condition makes the circuit stable.

The poles on the 𝑗𝜔 axis must be simple poles.

𝑍 𝑠 or 𝑌 𝑠 must not have multiple zeros or poles at the

origin.

Network Synthesis

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Realizability Conditions

Dr. Mohamed Refky

2) For the function

𝑍 𝑠 or 𝑌 𝑠

The power of the numerator and denominator in 𝑠 must differ at

most by ±1.

This is because the function must be reduced to one of the

elements

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Realizability Conditions

Dr. Mohamed Refky

=𝜙 𝑠

𝜓 𝑠=𝛼𝑚𝑠

𝑚 + 𝛼𝑚−1𝑠𝑚−1 +⋯+ 𝛼0

𝛽𝑛𝑠𝑛 + 𝛽𝑛−1𝑠

𝑛−1 +⋯+ 𝛽0

𝑅, 𝑠𝐿,1

𝑠𝐶𝐺,

1

𝑠𝐿, 𝑠𝐶or

In first foster form, partial fraction is used to factorized 𝑍 𝑠

Network Synthesis

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First Foster Form

Dr. Mohamed Refky

𝑍 𝑠 =𝑘1

𝑎1𝑠 + 𝑏1+

𝑘2𝑎2𝑠 + 𝑏2

𝑍 𝑠 =𝛼1𝑠 + 𝛼0

𝛽2𝑠2 + 𝛽2𝑠 + 𝛽0

Use the first foster form to synthesize the function

𝑍 𝑠 =𝑠2 + 4𝑠 + 3

𝑠2 + 2𝑠

Network Synthesis

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Example (1)

Dr. Mohamed Refky

In second foster form, partial fraction is used to factorized 𝑌 𝑠

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Second Foster Form

Dr. Mohamed Refky

𝑌 𝑠 =𝑘1

𝑎1𝑠 + 𝑏1+

𝑘2𝑎2𝑠 + 𝑏2

𝑌 𝑠 =𝛼1𝑠 + 𝛼0

𝛽2𝑠2 + 𝛽2𝑠 + 𝛽0

Use the second foster form to synthesize the function

𝑌 𝑠 =4𝑠4 + 7𝑠2 + 1

𝑠 2𝑠2 + 3

Network Synthesis

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Example (2)

Dr. Mohamed Refky

𝑌 𝑠 =4𝑠4 + 7𝑠2 + 1

2𝑠3 + 3𝑠

= 2𝑠 +𝑠2 + 1

𝑠 2𝑠2 + 3

First Cauer Form of 𝑍 𝑠 starts with

𝑍 𝑠 =𝛼𝑚𝑠

𝑚 + 𝛼𝑚−1𝑠𝑚−1 +⋯+ 𝛼0

𝛽𝑛𝑠𝑛 + 𝛽𝑛−1𝑠

𝑛−1 +⋯+ 𝛽0

then Continued Fraction Expansion (CFE) is applied to 𝑍 𝑠 to

put it in the form

Network Synthesis

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Cauer Form (Continued Fraction

Expansion)

Dr. Mohamed Refky

𝑍 𝑠 = 𝑍1 𝑠 +1

𝑌2 𝑠 +1

𝑍3 𝑠 +1

𝑌4 𝑠 +1

𝑍5 𝑠 + ⋯

Realize the following function in the first Cauer form

𝑍 𝑠 =𝑠4 + 4𝑠2 + 3

𝑠3 + 2𝑠

Network Synthesis

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Example (3)

Dr. Mohamed Refky

First Cauer Form of 𝑌 𝑠 starts with

𝑌 𝑠 =𝛽𝑛𝑠

𝑛 + 𝛽𝑛−1𝑠𝑛−1 +⋯+ 𝛽0

𝛼𝑚𝑠𝑚 + 𝛼𝑚−1𝑠

𝑚−1 +⋯+ 𝛼0

then Continued Fraction Expansion (CFE) is applied to 𝑌 𝑠 to

put it in the form

Network Synthesis

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Cauer Form (Continued Fraction

Expansion)

Dr. Mohamed Refky

𝑌 𝑠 = 𝑌1 𝑠 +1

𝑍2 𝑠 +1

𝑌3 𝑠 +1

𝑍4 𝑠 +1

𝑌5 𝑠 + ⋯

Realize the following admittance function in the first Cauer Form

𝑌 𝑠 =𝑠2 + 4𝑠 + 3

𝑠2 + 2𝑠

Network Synthesis

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Example (4)

Dr. Mohamed Refky

Secound Cauer Form of 𝑍 𝑠 starts with

𝑍 𝑠 =𝛼0 +⋯+ 𝛼𝑚−1𝑠

𝑚−1 + 𝛼𝑚𝑠𝑚

𝛽0 +⋯+ 𝛽𝑛−1𝑠𝑛−1 + 𝛽𝑛𝑠

𝑛

then Continued Fraction Expansion (CFE) is applied to 𝑍 𝑠 to

put it in the form

Network Synthesis

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Cauer Form (Continued Fraction

Expansion)

Dr. Mohamed Refky

𝑍 𝑠 = 𝑍1 𝑠 +1

𝑌2 𝑠 +1

𝑍3 𝑠 +1

𝑌4 𝑠 +1

𝑍5 𝑠 + ⋯

Realize the following admittance function in the second Cauer

Form

𝑍 𝑠 =𝑠4 + 4𝑠2 + 3

𝑠3 + 2𝑠

Network Synthesis

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Example (5)

Dr. Mohamed Refky

Secound Cauer Form of 𝑌 𝑠 starts with

𝑌 𝑠 =𝛽0 +⋯+ 𝛽𝑛−1𝑠

𝑛−1 + 𝛽𝑛𝑠𝑛

𝛼0 +⋯+ 𝛼𝑚−1𝑠𝑚−1 + 𝛼𝑚𝑠

𝑚

then Continued Fraction Expansion (CFE) is applied to 𝑌 𝑠 to

put it in the form

Network Synthesis

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Cauer Form (Continued Fraction

Expansion)

Dr. Mohamed Refky

𝑌 𝑠 = 𝑌1 𝑠 +1

𝑍2 𝑠 +1

𝑌3 𝑠 +1

𝑍4 𝑠 +1

𝑌5 𝑠 + ⋯

Realize the following admittance function in the second Cauer

Form

𝑌 𝑠 =𝑠2 + 4𝑠 + 3

𝑠2 + 2𝑠

Network Synthesis

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Example (6)

Dr. Mohamed Refky