Lecture #2 Scattering Matrices 2014
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Lecture #2 Scattering Matrices Instructor: Dr. Ahmad El-Banna Benha University Faculty of Engineering at Shoubra November 2014 Post-Graduate ECE-601 Active Circuits © Ahmad El-Banna
Transcript of Lecture #2 Scattering Matrices 2014
Lecture #2 Scattering Matrices Instructor: Dr. Ahmad El-Banna
Benha University Faculty of Engineering at Shoubra
No
vem
ber
2014
Post-Graduate ECE-601 Active Circuits
© A
hmad
El-B
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Agenda
Introduction
Microwave Network Analysis
Scattering Matrix Representations
Signal Flow Graphs 2
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INTRODUCTION 3
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Introduction
• 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. 4
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MICROWAVE NETWORK ANALYSIS 5
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Famous Matrices for Network Analysis
• Impedance (Z) Matrix
• Admittance (Y) Matrix
• Scattering (S) Matrix (focus of the lecture)
• Transmission (ABCD) Matrix
• Hybrid (h)
• Inverse hybrid (g)
• Scattering Transfer (T)
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Matrices for 2-Port Network
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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.
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.
Matrices Overview
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• Y-Matrix • Z-Matrix
• S-Matrix
• ABCD-Matrix
Z&Y Matrices General view
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• Z-Matrix
• Y-Matrix
N-Port Network
Port 1 Port 1
Port N Port N
SCATTERING MATRIX REPRESENTATIONS
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Motivation
• 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.
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Scattering Parameters
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S-parameters are defined assuming transmission lines are connected to each port.
On each transmission line:
0 0
0 0
i i i iz z
i i i i
i i i i
i
i
i i i
i
i
iV zV z V e V e
V z V zI z
Z Z
V z
1, 2i
1 2
1z2z
1a
1b 2b
2a
01 1,Z 02 2,Z
Local coordinates
0
0
i i i i i
i i i i i
a z V z Z
b z V z Z
Incoming wave function
Outgoing wave function
One Port N/w
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01
01
1
1
1
1
11
0
0
0
0
L
Z
Z
V
V
S
b
a
1l
1a
1b01Z
L
1 1
11 1
0 0
0
Lb a
S a
For a one-port network, S11 is defined to be the same as L.
• Return loss
• Reflection Coefficient
For a Two-Port Network
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Scattering matrix
1 11 1 12 2
2 21 1 22
1 11 12 1
2 21 22 2
2
0 0 0
0 0
0
0
0
0
0
b S a S a
b S a S
b S S ab S a
b S S
a
a
1 2
1z2z
1a
1b 2b
2a
01 1,Z 02 2,Z
Scattering Parameters
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Output is matched
Input is matched
input reflection coef. w/ output matched reverse transmission coef. w/ input matched forward transmission coef. w/ output matched output reflection coef. w/ input matched
2
1
2
1
1
11
1 0
1
12
2 0
2
21
1 0
2
22
2 0
0
0
0
0
0
0
0
0
a
a
a
a
bS
a
bS
a
bS
a
bS
a
Output is matched
Input is matched
For a general multiport network:
All ports except j are semi-infinite (or matched)
0
0
0k
i
ij
j a k j
bS
a
Example 1
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1a
1b
2a
2b0Z 0Z
Z
2V1V
1z 2z
Find the S parameters for a series impedance Z.
02 2
1 1 011
1 10
0
00
0
0
0 0
0 0
in
ina a
b V Z ZS
a V Z Z
Z Z Z
Z Z Z
11
02
ZS
Z Z
By symmetry:
S11 Calculation: 1a
1b 2b0Z 0Z
Z
2V1V
inZ
1z 2z
Semi-infinite
22 11S S
Example 1..
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2
2
2
21
1 0
2
1 0
0
0
0
0
a
a
bS
a
V
V
2 2 2
02 1
0
1 1 0 11
02 2 1 0 11
0
0 0 0
0 0
0 1
0 0 1
a V V
ZV V
Z Z
V a Z S
ZV V a Z S
Z Z
S21 Calculation:
1a
1b 2b0Z 0Z
Z
2V1V
inZ
1z 2z
Semi-infinite
1 1 00 0V a Z
Example 1…
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01 0 11
0
21
1 0
0 0 0 011
0 0 0 0 0
0 1
0
2 21 1
2 2
Za Z S
Z ZS
a Z
Z Z Z Z Z ZS
Z Z Z Z Z Z Z Z Z Z
021
0
2
2
ZS
Z Z
Hence
1a
1b 2b0Z 0Z
Z
2V1V
inZ
1z 2z
Semi-infinite
12 21S S
Example 2
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Find the S parameters for a Parallel impedance Zp.
Your turn !
Properties of the S Matrix
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For reciprocal networks, the S-matrix is symmetric.
For lossless networks, the S-matrix is unitary.
Identity matrix
T
S S
* *T T
S S S S U
Equivalently,
* 1T
S S
Notation: † *H T
S S S
† 1
S S
so
A B U
B A U
Note :
If
then
Power Waves
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ZR : reference impedance
ZL : load impedance
• The reflection coefficient for the reflected power wave
• The average power delivered to a load
• 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
SIGNAL FLOW GRAPH 22
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2-Port Representation
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Components of a signal flow graph are nodes and branches: 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.
Special Networks
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(a) One Port Network (b) Source Representation (c) Load Representation
(c)
Decomposition Rules
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(a) Series rule. (b) Parallel rule. (c) Self-loop rule. (d) Splitting rule.
Application
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Use signal flow graphs to derive expressions for Γin and Γout for the shown microwave network
• For more details, refer to:
• Chapters 4, Microwave Engineering, David Pozar_4ed.
• The lecture is available online at: • http://bu.edu.eg/staff/ahmad.elbanna-courses/11983
• For inquires, send to:
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