EE-2100 Circuits and Fields
Synthesis of Passive Networks
By
Manuja Gunawardana
Department of Electrical Engineering, University of Moratuwa
Sep, 2022
Overview
• Synthesis of passive networks:
• LC Networks
• RC & RL Networks
• RLC Networks
Analysis Vs Synthesis
Circuit Synthesis Methods
Synthesis of LC Networks: Impedance Function
• Z or Y is the ratio of odd to even or even to odd polynomials.
• All poles and zeros lie on the jω axis.
• Poles and zeros are simple and imaginary.
• Poles and zeros alternate (on the imaginary axis).
• Highest and lowest powers of the numerator and
denominator should differ by 1.
• There must be a pole or a zero at the origin and infinity. 0, zero at ∞
lim 𝑍 𝑠 = '
!→# ∞, pole at ∞
First Foster Form – LC Networks
Example 1
Second Foster Form – LC Networks
Example 1
First Cauer Form
The first Cauer form is generated by the continued fractions expansion of a reactance function about the point at infinity
Example 1
Second Cauer Form
The second Cauer form is generated by the continued fraction expansion of a reactance function about the origin.
This too results in a realisation where the series arms are impedances while
the shunt arms are admittances. We will consider the same examples as
before. However, in this realisation the series arms contain capacitors while
the shunt arms have inductors.
Summary
Submission
Take <your index number>%3. Pick the impedance function that matches the answer (0/1/2). Synthesis the impedance
function using 1st Foster, 2nd Foster, 1st Cauer and 2nd Cauer Methods.
0)
1)
2)
Scan and submit your answers to moodle under “Submission: Synthesis of LC Networks” in .pdf format
Synthesis of RC Networks: Impedance Function
• The poles and zeros of an RC driving point function lie
on the nonpositive real axis.
• They are simple.
• Poles and zeros alternate.
• The critical frequency of the smallest magnitude is a
pole.
Possible Forms of Expansion of Z(s)
Example
Example 1
Synthesis of RL Networks
RC Networks RL Networks
Recap
Submission Answers: Q0
Submission Answers: Q1
Submission Answers: Q2
Synthesis of RL Networks
RC Networks RL Networks
RC: Example 2
RC: Example 3
RC: Example 4
Synthesis of RLC Networks
Many procedures available in classical design for the synthesis of RLC circuits, we will consider the Brune Synthesis,
which starts off with what is called the Foster Preamble.
Foster Preamble
Step 1: Reactance Reduction
Foster Preamble
Step 2: Susceptance Reduction
Foster Preamble
Step 3: Rreactance Reduction
Note: Steps 1 and 2 may be
interchanged. We should
choose the sequence that
gives rise to a simpler
realization.
Example
Submission: Example2
Dual Properties of RC and RL Networks
Z Y
Series Parallel
Parallel Series
Capacitor Inductor
Inductor Capacitor
Submission: Example2
RC: Example 2
Submission: Example3
RC: Example 3
Submission: Example4
RC: Example 2
Synthesis of RLC: Brune Synthesis
Brune Synthesis: Step 1
Brune synthesis starts with the Foster preamble, to remove reactive and
susceptive components to yield a minimum reactive and minimum susceptive
function. This would be of the form:
Example:
Brune Synthesis: Step 2
Remove a “minimum resistance” from the remainder to obtain a minimum
resistive function
𝑍 𝑠 = 𝑅! + 𝑍! (𝑠)
Example:
Let 𝜔$ be such that:
Brune Synthesis: Step 3
At the frequency at which the resistance is a minimum, we now have a pure
reactive network
𝑍! (𝑗𝜔" ) = 𝑗𝑋!
Remove this inductance from the network
𝑍! (𝑠) = 𝑠𝐿! + 𝑍% (𝑠)
* Since 𝑍(𝑠) had no poles or zeros on the 𝑗𝜔 axis, the removal of 𝐿!𝑠 from 𝑍!(𝑠) to yield
𝑍"(𝑠) has introduced a zero at 𝑠! = ± 𝑗𝜔! 𝑡𝑜 𝑍"(𝑠).
Example:
There are two possible cases to be considered in going through this cycle. One is the possibility that 𝐿! is negative, leaving 𝑍"(𝑠)positive real, and
the other is when 𝐿! is positive, leaving 𝑍"(𝑠) non-positive real.
Brune Synthesis: Step 4
Remove the added zero to the impedance by considering as a pole of the
admittance
𝑠
𝐿
𝑍# 𝑠 ⟹ 𝑌# 𝑠 = # # # + 𝑌& 𝑠
𝑠 + 𝜔"
∗ 𝑍# (𝑠) now has a pole at infinity
Example:
Brune Synthesis: Step 5
Remove the added pole as an inductor.
𝑍$ 𝑠 = 𝑠𝐿$ + 𝑍' 𝑠
Example:
Brune Synthesis: Step 6
The combination of 𝐿( , 𝐿% and 𝐿& may be realised with a coupled transformer as
Example:
We have just completed one cycle of the Brune synthesis!
Brune Synthesis: Back to Step 2
The same procedure is now applied to 𝒁𝟒 𝒔 . 𝑍' (𝑠) is of the same form as 𝑍(𝑠),
except that it is of order (𝑛 − 2) instead of 𝑛
Example:
Brune Synthesis: When 𝑋! > 0: Step 2,3
Example:
Brune Synthesis: When 𝑋! > 0: Step 4
Example:
Brune Synthesis: When 𝑋! > 0 : Step 5
Example:
Brune Synthesis: When 𝑋! > 0 : Step 6
Example:
Brune Synthesis: When 𝑋! > 0 : Step 6
Example:
We have just completed one cycle of the Brune synthesis!
Brune Synthesis: When 𝑋! > 0 : Step 2
The same procedure is now applied to 𝒁𝟒 𝒔 . 𝑍' (𝑠) is of the same form as 𝑍(𝑠),
except that it is of order (𝑛 − 2) instead of 𝑛
Example:
