Chapter 3 Microwave Network Analysis

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Lecture 6
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Chapter 3 Microwave Network Analysis

• 3.1 Impedance and Equivalent Voltages and
  Currents
• 3.2 Impedance and Admittance Matrices
• 3.3 The Scattering Matrix
• 3.4 The Transmission (ABCD) Matrix

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Microwave network analysis

• Advantages
• Extend circuit and network concepts in the
  low-frequency circuits to handle microwave
  analysis and design problems of interest.

Basic procedure (1) Field analysis and Maxwell
equations ? (2) obtain quantities (ß, z0, etc.) ?
(3) treat a TL or WG as a distributed component ?
(4) interconnect various components and use
network and/or TL theory to analyze the whole
system (R, T, Loss, etc.)
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3.1 Impedance and Equivalent Voltages and
Currents
Equivalent voltages and Currents
(a) Two-conductor TEM line
(b) Non-TEM line
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Define equivalent voltages and currents in
non-TEM line

• C1 Voltage and current are defined only for a
  particular waveguide mode, and are defined so
  that the voltage is proportional to the
  transverse electric field, and the current is
  proportional to the transverse magnetic field.

C2 In order to be used in a manner similar to
voltages and currents of circuit theory the
equivalent voltages and currents should be
defined so that their product gives the power
flow of the mode.
C3 The ratio of the voltage to the current for
a single traveling wave should be equal to the
characteristic impedance of the line. This
impedance may be chosen arbitrarily, but is
usually selected as equal to the wave impedance
of the line, or else normalized to unity.
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• Transverse fields for an arbitrary WG mode

½VI
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Apply C3
Propagation constant ß and the field intensity
A, A- are calculated from the field analysis.
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Equivalent voltages and currents for higher WG
modes

• Higher order modes can be treated in the same
  way, so that a general field in a waveguide can
  be expressed in the following form

where V and I, are the equivalent voltages and
currents for the n-th mode, and C1n, and C2n arc
the proportionality constants for each mode.
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Example-Equivalent voltage and current for TE10
mode of a rectangular WG
Waveguide fields Transmission line model
If choose Z0 ZTE, then
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Impedance

• Intrinsic impedance of the medium
• (dependent only on the material parameters of
  the medium, and is equal to the wave impedance
  for plane waves.)
• (2) Wave impedance
• (a characteristic of the particular type of
  wave. TEM, TM, and TE waves each have different
  wave impedances, which may depend on the type of
  line or guide, the material, and the operating
  frequency.)
• (3) Characteristic impedance
• (The ratio of voltage to current for a traveling
  wave on a transmission line uniquely defined for
  TEM waves defined in various ways for TE and TM
  waves.

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Reflection of a rectangular WG discontinuity for
TE10
er 2.54
a 2.286cm b 1.016 cm
Characteristic impedance
Reflection coefficient
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3.2 Impedance and Admittance Matrices
Port any type of transmission line or
transmission line equivalent of a single
propagating waveguide mode (one mode, add one
electric port).
tN terminal plane providing the phase reference
for V, I
The total voltage and current on the n-th port
An arbitrary N-port microwave network
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• The impedance (admittance) matrix Z (Y) of
  the microwave network relating these voltages and
  currents

Z or Y is complex number and totally 2N2
elements but many networks are either reciprocal
(symmetric matrix) or lossless (purely imaginary
matrix), or both and the total number will be
substantially reduced.
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Transfer impedance, Zij (between ports i and j)
Calculated by driving port j with the current Ij,
open-circuiting all other ports (so Ik 0 for k
? j), and measure the open-circuit voltage at
port i.
Find the impedance matrix elements

• Input impedance, Zii
• Seen looking into port i when all other ports
  are open-circuited.

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Transfer admittance, Yij (between ports i and
j) Calculated by driving port j with the
voltage Vj, short-circuiting all other ports (so
Vk 0 for k ? j), and measure the short-circuit
current at port i.
Find the admittance matrix elements

• Input admittance, Yii
• Seen looking into port i when all other ports
  are short-circuited.

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Example Evaluation of impedance parameters
Find the Z parameters of the two-port T-network
shown below.
Solutions
It can be verified that Z21 Z12, indicating the
circuit is reciprocal.
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3.3 The scattering Matrix

• The scattering matrix S relates the voltage waves
  incident on the ports to those reflected from the
  ports.
• For some components and circuits, the scattering
  parameters can be calculated using network
  analysis techniques.
• Otherwise, the scattering parameters can be
  measured directly with a vector network analyzer

S is symmetric for reciprocal network and
unitary for lossless network.
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Vector network analyzer
Find the admittance matrix elements
(directly measure S-parameters)
Sij is found by driving port j with an incident
wave of voltage V , and measuring the reflected
wave amplitude V-, coming out of port i. The
incident waves on all ports except the jth port
are set to zero, which means that all ports
should be terminated in matched loads to avoid
reflections.
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Conversion between Scattering and impedance
parameters
For a N-ports network, assuming every port has
the same impedance and set Z0n 1, we have
Identity matrix
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3.4 The transmission (ABCD) matrix

• Define a 2 x 2 transmission, or ABCD matrix, for
  each two-port network according to the total
  voltages and currents
• The ABCD matrix of the cascade connection of two
  or more two-port networks can be found by
  multiplying the ABCD matrices of the individual
  two-ports.

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ABCD matrix for a cascaded connection of two-port
network
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The ABCD parameters of some Useful two-Port
circuits
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Conversion between two-port network parameters