Post-Graduate
ECE-601
Active Circuits
Lecture #2
Scattering Matrices
Instructor:
Dr. Ahmad El-Banna
© Ahmad El-Banna
November 2014
Benha University
Faculty of Engineering at Shoubra
Microwave Network Analysis
Scattering Matrix Representations
Signal Flow Graphs
© Ahmad El-Banna
Introduction
ECE-601 , Lec#2 , Nov 2014
Agenda
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INTRODUCTION
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ECE-601 , Lec#2 , Nov 2014
© Ahmad El-Banna
© Ahmad El-Banna
• Circuits operating at low frequencies can be treated as an
interconnection of lumped passive or active components with
unique voltages and currents defined at any point in the circuit.
• In this situation the circuit dimensions are small enough such that
there is negligible phase delay from one point in the circuit to
another.
• In addition, the fields can be considered as TEM fields supported by
two or more conductors.
• This leads to a quasi-static type of solution to Maxwell’s equations
and to the well-known Kirchhoff voltage and current laws and
impedance concepts of circuit theory .
• In general, the network analyzing techniques cannot be directly
applied to microwave circuits, but the basic circuit and network
concepts can be extended to handle many microwave analysis and
design problems of practical interest.
ECE-601 , Lec#2 , Nov 2014
Introduction
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MICROWAVE NETWORK ANALYSIS
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ECE-601 , Lec#2 , Nov 2014
© Ahmad El-Banna
• Hybrid (h)
• Inverse hybrid (g)
• Scattering Transfer (T)
© Ahmad El-Banna
• Impedance (Z) Matrix
• Admittance (Y) Matrix
• Scattering (S) Matrix (focus of the lecture)
• Transmission (ABCD) Matrix
ECE-601 , Lec#2 , Nov 2014
Famous Matrices for Network Analysis
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General Properties:
Reciprocal networks: A network is said to be reciprocal if the voltage
appearing at port 2 due to a current applied at port 1 is the same as the
voltage appearing at port 1 when the same current is applied to port 2.
Symmetrical networks: A network is symmetrical if its input impedance is
equal to its output impedance.
Lossless network: A lossless network is one which contains no resistors or
other dissipative elements.
© Ahmad El-Banna
A two-port network (a kind of four-terminal network or quadripole):
is an electrical network (circuit) or device with two pairs of terminals to
connect to external circuits.
ECE-601 , Lec#2 , Nov 2014
Matrices for 2-Port Network
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• Y-Matrix
• ABCD-Matrix
• S-Matrix
© Ahmad El-Banna
• Z-Matrix
ECE-601 , Lec#2 , Nov 2014
Matrices Overview
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Port 1
N-Port
Network
Port N
• Z-Matrix
• Y-Matrix
Port N
© Ahmad El-Banna
Port 1
ECE-601 , Lec#2 , Nov 2014
Z&Y Matrices General view
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SCATTERING MATRIX
REPRESENTATIONS
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ECE-601 , Lec#2 , Nov 2014
© Ahmad El-Banna
© Ahmad El-Banna
• At high frequencies, Z, Y, h & ABCD parameters are difficult
(if not impossible) to measure.
• V and I are not uniquely defined
• Even if defined, V and I are extremely difficult to measure (particularly I).
• Required open and short-circuit conditions are often difficult to achieve.
• In other words, direct measurements can’t be done since all are EM waves at
high frequencies
• Scattering (S) parameters are often the best representation for multi-port
networks at high frequency.
• We can directly measure reflected, transmitted and incident wave with a
network analyzer.
• Once the parameters are known they can be converted to any other matrix
parameters.
ECE-601 , Lec#2 , Nov 2014
Motivation
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a1
b1
Z 01 ,  1
1
Z 02 ,  2
2
z1
z2
Local coordinates
On each transmission line:
Vi  zi   Vi 0 e  i zi  Vi 0e  i zi  Vi   zi   Vi   zi 
Vi   zi  Vi   zi 
I i  zi  

Z 0i
Z 0i
Incoming wave function  ai  zi   Vi
i  1, 2
 zi 

Outgoing wave function  bi  zi   Vi  zi 

Z 0i
Z 0i
a2
b2
© Ahmad El-Banna
S-parameters are defined
assuming transmission lines are connected to each port.
ECE-601 , Lec#2 , Nov 2014
Scattering Parameters
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L 
V1  0 
V1  0 


b1  0 
a1  0 
 S11
Z 01
Z 01
L
a1
Z 01
b1
l1
 b1  0   La1  0
 S11 a1  0 
For a one-port network,
S11 is defined to be the
same as L.
© Ahmad El-Banna
• Reflection Coefficient
ECE-601 , Lec#2 , Nov 2014
One Port N/w
• Return loss
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Z 01 ,  1
1
Z 02 ,  2
2
b1
z1
z2
b1  0   S11a1  0   S12a2  0 
b2  0   S21a1  0   S22a2  0 
 b1  0    S11


b2  0    S21
Scattering
matrix
S12   a1  0  

  b   S  a 

S22   a2  0  
a2
b2
© Ahmad El-Banna
a1
ECE-601 , Lec#2 , Nov 2014
For a Two-Port Network
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Output is
matched
b1  0 
S12 
a2  0  a  0
Input is
matched
b2  0 
S21 
a1  0  a 0
Output is
matched
b2  0 
S22 
a2  0  a  0
Input is
matched
2
1
2
1
input reflection coef.
w/ output matched
reverse transmission coef.
w/ input matched
forward transmission coef.
w/ output matched
output reflection coef.
w/ input matched
For a general multiport network:
bi  0 
Sij 
a j  0
All ports except j are semi-infinite (or matched)
ak 0 k  j
© Ahmad El-Banna
b1  0 
S11 
a1  0  a 0
ECE-601 , Lec#2 , Nov 2014
Scattering Parameters
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© Ahmad El-Banna
Example 1
Find the S parameters for a series impedance Z.
Z0
b1
V1
Z



V

a2
Z0
2
b2
z2
z1
Semi-infinite
S11 Calculation:
a1
V1
Z0
b1


Z
2
Z in
z1
b1  0
V1  0 Zin  Z 0
S11 
 

a1  0 a 0 V1  0 Zin  Z 0
2
S11 
Z
Z  2Z0

V

z2
Z  Z0   Z0


 Z  Z0   Z0
a 0
2
By symmetry:
Z0
b2
ECE-601 , Lec#2 , Nov 2014
a1
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S22  S11
Semi-infinite
a1
Z0
b1
V1


Z
b2  0 
a1  0  a 0
2
V2  0 
 
V1  0  a 0
2

1
V
 0  a1  0
2
Z in
z1
S21 

V

Z0
b2
z2
a2  0 V2  0   V2  0 
 Z0 
V2  0   V1  0  

Z

Z
0 

V1  0   a1 Z 0 1  S11 
Z0
 Z0 
V2  0   V2  0   a1 Z 0 1  S11  

Z

Z
0 

© Ahmad El-Banna
S21 Calculation:
ECE-601 , Lec#2 , Nov 2014
Example 1..
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a1
Z0
V1
b1


Z

V

2
Z in
z1
Z0
b2
z2
 Z0 
a1  0  Z 0 1  S11  

Z

Z
0 

S21 
a1  0  Z 0
 Z0  
Z   Z0   2Z  2Z0   Z0 
 1  S11  

1

 




Z

Z
Z

2
Z
Z

Z
Z

2
Z
Z

Z
0 
0 
0 
0 
0 



© Ahmad El-Banna
Semi-infinite
ECE-601 , Lec#2 , Nov 2014
Example 1…
18
Hence
2 Z0
S21 
Z  2 Z0
S12  S21
Your turn !
© Ahmad El-Banna
Find the S parameters for a Parallel impedance Zp.
ECE-601 , Lec#2 , Nov 2014
Example 2
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© Ahmad El-Banna
For reciprocal networks, the S-matrix is symmetric.
 S   S 
T
Note :
If
For lossless networks, the S-matrix is unitary.
  S T  S *   S *  S T  U 
 A B   U 
then
 B  A  U 
Identity matrix
Notation:  S    S    S 
†
Equivalently,
S   S 
T*
1
so
T*
H
S   S 
†
1
ECE-601 , Lec#2 , Nov 2014
Properties of the S Matrix
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total voltage and current on a transmission
line in terms of the incident and reflected
voltage wave amplitudes
•
the incident and reflected voltage wave
amplitudes in terms of the total voltage
and current
ECE-601 , Lec#2 , Nov 2014
•
© Ahmad El-Banna
Power Waves
• The average power delivered to a load
• The reflection coefficient for the reflected power wave
ZR : reference impedance
ZL : load impedance
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SIGNAL FLOW GRAPH
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ECE-601 , Lec#2 , Nov 2014
© Ahmad El-Banna
Nodes: Each port i of a microwave
network has two nodes, ai and bi.
Node ai is identified with a wave
entering port i, while node bi is
identified with a wave reflected
from port i. The voltage at a node is
equal to the sum of all signals
entering that node.
Branches: A branch is a directed
path between two nodes
representing signal flow from one
node to another. Every branch has
an associated scattering parameter
or reflection coefficient.
© Ahmad El-Banna
Components of a signal flow graph
are nodes and branches:
ECE-601 , Lec#2 , Nov 2014
2-Port Representation
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© Ahmad El-Banna
Special Networks
ECE-601 , Lec#2 , Nov 2014
(a) One Port Network
(b) Source Representation
(c) Load Representation
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(c)
© Ahmad El-Banna
(a) Series rule.
(b) Parallel rule.
(c) Self-loop rule.
(d) Splitting rule.
ECE-601 , Lec#2 , Nov 2014
Decomposition Rules
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© Ahmad El-Banna
Use signal flow graphs to derive expressions for Γin and Γout for the shown
microwave network
ECE-601 , Lec#2 , Nov 2014
Application
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• The lecture is available online at:
• http://bu.edu.eg/staff/ahmad.elbanna-courses/11983
• For inquires, send to:
• ahmad.elbanna@fes.bu.edu.eg
© Ahmad El-Banna
• Chapters 4, Microwave Engineering, David Pozar_4ed.
ECE-601 , Lec#2 , Nov 2014
• For more details, refer to:
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