Microwave Network Analysis

1
Chapter 4

• Microwave Network Analysis

2
Equivalent Voltage and Current

• For non-TEM lines, the quantities of voltage,
  current, and impedance are nor unique, and are
  difficult to measured. Following considerations
  can provide useful result
• 1) Voltage and current are defined only for a
  particular mode, and are defined so that the
  voltage is proportional to the transverse
  electric field, and the current is proportional
  to transverse magnetic field.
• 2) The product of equivalent voltage and current
  equals to the power flow of the mode.
• 3) 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 is usually selected as equal to the
  wave impedance of the line.

3
The Concept of Impedance

• Various types of impedance
• 1) ?(?/?)1/2 intrinsic impedance of medium.
  This impedance is dependent only on the material
  parameters of medium, and is equal to the wave
  impedance of plane wave.
• 2) ZwEt /Ht1/Yw wave impedance, e.g. ZTEM,
  ZTM, ZTE. It may depend on the type of line or
  guide, the material, and the operating frequency.
• 3) Z0 (L /C)1/2 1/Y0 characteristic
  impedance. It is the ratio of voltage to current.
  The characteristic impedance is unique definition
  for TEM mode but not for TM or TE modes.
• The real and imaginary parts of impedance and
  reflection coefficient are even and odd in ?0
  respectively.

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Impedance and Admittance Matrices

• The terminal plane (e.g. tN) is important in
  providing a phase reference for the voltage and
  current phasors.
• At the nth terminal (reference) plane, the
  relations are given as

• Reciprocal Networks

• If the arbitrary network is reciprocal ( no
  active devices, ferrites, or plasmas), Y and
  Z are symmetric matrices .

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• Lossless Networks

• If the arbitrary network is lossless, then the
  net real power delivered to the network must be
  zero. Besides

Example4.1 Find the Z parameters of the two-port
network?
Solution
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The Scattering Matrices

• The scattering parameter Sij is the transmission
  coefficient from j port to port i when all other
  ports are terminated in matched loads.

• Z or Y ? S

• S ? Z

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Example4.2 Find the S parameters of the 3 dB
attenuator circuit?
Solution
A matched 3B attenuator with a 50 O
Characteristic impedance
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• Reciprocal Networks

• Lossless Networks

• No real power delivers to network.Besides

• S is symmetric matrix

Example4.3 Determine if the network is
reciprocal, and lossless? If port 2 terminated
with a matched load, what is the return loss at
port 1? If port 2terminated with a short circuit,
what is the return loss seen at port 1?
Solution

• Since S is not symmetric, the network is not
  reciprocal.

• So the network is not lossless.

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When port 2 terminated with a matched load,
?S110.15.
When port 2 terminated with a short circuit,
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Summary

• Reciprocal Networks (symmetric)
• No active elements, no anisotropic material

• Lossless Networks
• No resistive material, no radiation

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Example4.4 Determine if the network is
reciprocal, and lossless ?
Solution
From the result of example 4.2
A matched 3B attenuator with a 50 O
Characteristic impedance
Since the network is reciprocal but not lossless,
S should be symmetric but not unitary.
Since the network is reciprocal but not lossless,
Z should be symmetric but not imaginary.
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Example4.5 Determine if the network is
reciprocal, and lossless ?
Solution
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• Problem 1Determine if the inductance networks
  are reciprocal, and lossless ?

• Problem 2Determine if the capacitance networks
  are reciprocal, and lossless ?

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Three Port Network

• Reciprocal Network

• Matching at all ports

• Lossless Network

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Applications

• A counter-clockwise circulator

• S is unitary and matched at all ports, but not
  symmetric. Therefore, circulator is lossless and
  matched, but not reciprocal.

• Power splitters

• S is symmetric and matched at all ports, but
  not unitary. Therefore, circulator is reciprocal
  and matched, but not lossless.

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• A Shift in Reference Planes

• Twice the electric length represents that the
  wave travels twice over this length upon
  incidence and reflection.

• Generalized Scattering
• Parameters

• If the characteristic impedances of a multi-port
  network are different,

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Generalized Scattering Matrices

• The scattering parameter Sij defined earlier was
  based on the assumption that all ports have the
  same characteristic impedances ( usually Z050?).
  However, there are many cases where this may not
  apply and each port has a non-identical
  characteristic impedance.
• A generalized scattering matrix can be applied
  for network with non-identical characteristic
  impedances, and is defined as following

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The Transmission (ABCD) Matrix

• ABCD matrix has the advantage of cascade
  connection of multiple two-port networks.

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• Table 4-2 Conversions between two-port network
  parameters

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• For reciprocal network, Z is is symmetric.
  Hence, Z12Z21

Example4.6 Find the S parameters of network?
Solution

• From Table 4-1

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• From Table 4-2

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The Transmission T Matrix

• At low frequencies, ABCD matrix is defined in
  terms of net voltages and currents. When at high
  frequencies, T matrix defined in terms of
  incident and reflected waves will become very
  useful to evaluate cascade networks.

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Equivalent Circuit for Two-Port Networks
A coax-to-microstrip transition and equivalent
circuit representations. (a) Geometry of the
transition. (b) Representation of the transition
by a black box. (c) A possible equivalent
circuit for the transition.
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• Equivalent circuits for some common microstrip
  discontinuities. (a) Open-ended. (b) Gap. (c)
  Change in width. (d) T-junction.

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• Equivalent circuits for a reciprocal two-port
  network.
• (a) T equivalent (b) ? equivalent

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Example4.7 Find the network as equivalent T and
? model at 1GHz?
Solution

• From Table 4-1

• From Table 4-2

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Equivalent T model

• From Table 4-2

Equivalent ? model
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Example4.8 Find the equivalent ? model of
microstrip-line inductor?
Solution

• From Table 4-1

• From Table 4-2

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Equivalent ? model
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Example4.9 Find the equivalent T model of
microstrip-line capacitor?
Solution

• From Table 4-1

• From Table 4-2

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Equivalent T model
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• Problem3 Design a 6GHz attenuator ?
• (Hint -20logS216 ? S210.501
  )

• Problem4 Design a 6nH microstrip-line inductor
  on a 1.6mm thick FR4 substrate. The width of line
  is 0.25mm. Find the length (l ) and parasitic
  capacitance?

• Problem5 Design a 2pF microstrip-line capacitor
  on a 1.6mm thick FR4 substrate. The width of line
  is 5mm. Find the length (l ) and parasitic
  inductance?