Impedance and Equivalent Voltages and Currents for Non-TEM Lines

1
Lecture 3

• Impedance and Equivalent Voltages and Currents
  for Non-TEM Lines
•  Impedance Properties of One-Port Networks
• Impedance, Admittance and Scattering Matrices
•  Signal Flow Graphs

2
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• show how circuit and network concepts can be
  extended to handle many microwave analysis and
  design problems of practical interest

3
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• equivalent voltage and current can be defined
  uniquely for TEM-type lines (require two
  conductors) but not so for non-TEM lines

4
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• for non-TEM lines, voltage and current are only
  defined for a particular waveguide mode, V is
  related to Et and I to Ht where t denotes the
  transverse component

5
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• the product of the equivalent V and I should
  yield the power flow of the mode
• V/I for a single traveling wave should be equal
  to the characteristic impedance of the line or
  can be normalized to 1

6
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• for an arbitrary waveguide mode with a ve and
  -ve (in z) traveling waves

7
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• we can write the voltage and current of an
  equivalent transmission line as

8
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• i.e., we are only interested in certain
  quantities and these quantities can be derived
  using circuit and network theory

9
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• the incident power is given by
• Where

10
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• the characteristic impedance of the equivalent
  transmission line is

11
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• the wave impedance is given by

12
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• if we choose the characteristic impedance of the
  line equal to that of the wave impedance, i.e.,
• which can be either TE or TM modes

13
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• therefore, if one can measure the voltage and
  current for each mode, the field in the waveguide
  can be determined as sum of the field for each
  mode

14
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines
15
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• we reiterate that there are various types of
  impedance
• intrinsic impedance
• depends on material parameters but is equal to
  the wave impedance of a plane wave in a
  homogeneous medium

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Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• wave impedance of a particular type of wave,
  namely, TEM, TE and TM
• depends on frequency, materials and boundary
  conditions

17
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• characteristic impedance
• unique for TEM waves, non-unique for TE and TM

18
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• we will calculate the wave impedance of the TE10
  mode waveguide
• now consider the field equations for the TE10
  rectangular waveguide mode, the field components
  are

19
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines
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Impedance and Equivalent Voltages and Currents
for Non-TEM Lines
21
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• compare with the transmission line equations and
  the incident power

22
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines
23
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• therefore, we can relate the field and circuit
  parameters for the TE10 waveguide mode
• we can also the transverse resonance technique to
  look at the wavenumber of the TE10 mode in the y
  direction (height of the waveguide)

24
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• the waveguide can be regarded as a transmission
  line with certain characteristic impedance and is
  shorted at both ends

25
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• using the transmission line equation to transfer
  the short circuit to y y'
• according to the transverse resonance technique,
  we have
• impedance looking downward impedance looking
  upward 0

26
Impedance and Equivalent Voltages and Currents
for Non-TEM Lines

• for the above equation to be true for any value
  of y', ky must be zero
• this is correct as ky np/b where n 0 for TE10
  mode

27
Impedance Properties of One-Port Networks

• consider the arbitrary one-port network shown here

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Impedance Properties of One-Port Networks

• assume that
• if then

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Impedance Properties of One-Port Networks

• the input impedance
• for a lossless network, R 0, therefore

30
Impedance Properties of One-Port Networks

• the input impedance is purely imaginary
• the reactance is positive for an inductive load
  (Wm gt We) and is negative for a capacitive load

31
Impedance Properties of One-Port Networks

• Foster ?s reactance theorem the rate of change
  of the reactance for the lossless one-port
  network with frequency is

32
Impedance Properties of One-Port Networks

• from Maxwell?s equations
• and the vector identity

33
Impedance Properties of One-Port Networks
34
Impedance Properties of One-Port Networks
35
Impedance Properties of One-Port Networks

• Note that
• and therefore,
• 
  0

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Impedance Properties of One-Port Networks

• V j XI for lossless line

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Impedance Properties of One-Port Networks

• poles and zeros must alternate in position as the
  slope is always positive

38
Even and Odd Properties of Z(w) and G(w)

• define the Fourier transform as
• note that v(t) must be a real quantity, i.e.,
  v(t) v(t), therefore,
• or V(-w) V(w)

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Even and Odd Properties of Z(w) and G(w)

• note that we can only measure V(w), we need its
  complex conjugate to obtain v(t), similar
  arguments hold for I(w)
• V(-w) V(w)Z(w)I(w)Z(- w)I(- w)Z(-w)I(w)
• Z(w) Z(- w)

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Even and Odd Properties of Z(w) and G(w)

• therefore, the real part of Z, i.e, R is an even
  function of w while the imaginary part X is an
  odd function of w
• the reflection coefficient G also has an even
  real part and an odd imaginary part

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Even and Odd Properties of Z(w) and G(w)
42
Impedance, Admittance and Scattering Matrices
43
Impedance, Admittance and Scattering Matrices

• N-port microwave network, each port has a
  reference plane tn
•  
• at the reference plane of port N, we have

44
Impedance, Admittance and Scattering Matrices

• if we are only interested in knowing the
  relationship among the voltages and currents at
  the ports, we can define a impedance matrix Z so
  that

45
Impedance, Admittance and Scattering Matrices

• the element Zij of the impedance matrix is given
  by
• similar equations can be written for the
  admittance matrix 

46
Impedance, Admittance and Scattering Matrices

• for reciprocal network, the impedance
  (admittance) matrix is symmetric
• for lossless network, all matrix elements are
  purely imaginary

47
Impedance, Admittance and Scattering Matrices

• the scattering matrix relate the voltage waves
  incident on the ports to those reflected from the
  ports
• the scattering parameter is written as

48
Impedance, Admittance and Scattering Matrices

• each element is given by
• Sij is found by driving port j with an incident
  wave of voltage Vj, and measuring the reflected
  amplitude coming Vi-, out of port i

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Impedance, Admittance and Scattering Matrices

• the incident waves on all ports except the jth
  port are set to zero which implies that all these
  ports are terminated with match loads
• for a reciprocal network
• S St, i.e., the matrix is symmetric

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Impedance, Admittance and Scattering Matrices

• for a lossless network
• for all I,j
• the scattering parameters can be readily measured
  by a Network Analyzer

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A Shift in Reference Planes

• the S parameters relate the amplitude of
  traveling wave incident on and reflected from a
  microwave network, phase reference planes must be
  specified for each port of the network
• we need to know how the S parameters change when
  the reference planes are moved

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A Shift in Reference Planes
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A Shift in Reference Planes

• let the original reference at zl0, the incident
  and reflected port voltages are related by
• at the new reference planes at znln

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A Shift in Reference Planes

• for a lossless transmission line

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A Shift in Reference Planes

• In matrix form, we have

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A Shift in Reference Planes

• note that each diagonal term is shifted by twice
  the electrical length of the shift in the
  reference plane, i.e., the shift is a round trip
  shift

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Generalized Scattering Parameters

• note that not all the ports are of the same
  characteristic impedance, let the nth port has a
  characteristic impedance of Zon

58
Generalized Scattering Parameters

• we define a new set of wave amplitude as
• an represents an incident wave at the nth port
  and bn represents a reflected wave from that port

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Generalized Scattering Parameters

• at the reference plane, we have

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Generalized Scattering Parameters

• the average power delivered to the nth port is
• the average power delivered through port n is the
  incident power minus the reflected power

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Generalized Scattering Parameters

• a generalized scattering matrix can be defined
  with the matrix element given by
• or

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Generalized Scattering Parameters

• note that it only depends on the ratio of the
  characteristic impedances, not the characteristic
  impedance themselves

63
Signal Flow Graph

• the primary components of a signal flow graph are
  nodes and branches

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Signal Flow Graph

• for a two-port network, we have

65
Signal Flow Graph

• each port has two nodes, node a is identify with
  a wave entering the port while node b is identify
  with a wave reflected from the port
• nodes a and b are connected by a branch and each
  branch is associated with a scattering parameter

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Simplification of Signal Flow Graphs

• series rule two branches, whose common node has
  only one incoming and one outgoing wave many be
  combined to form a single branch

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Simplification of Signal Flow Graphs

• parallel rule two branches that are parallel may
  be combined as

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Simplification of Signal Flow Graphs

• parallel rule two branches that are parallel may
  be combined as

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Simplification of Signal Flow Graphs

• self-loop rule when a loop has a self-loop, it
  can be eliminated

70
Simplification of Signal Flow Graphs

• splitting rulea node may be split into two
  separate nodes

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Masons rule

• independent variable node is the node of an
  incident wave
• dependent variable node is the node of a
  reflected wave

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Masons rule

• path is a series of codirectional branches from
  an independent node to a dependent node, along
  which no node is crossed more than once, the
  value of a path is the product of all the branch
  coefficients along the path

73
Masons rule

• first-order loop is the product of branch
  coefficient encountered in a round trip from a
  node back to that same node, without crossing the
  node twice
• second-order loop is the product of any two
  nontouching first-order loop
• third-order loop is the product of three
  nontouching first-order loop

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Masons rule

• the Masons rule for the ratio T of the wave
  amplitude of a dependent variable to the wave
  amplitude of an independent variable is given as
• ,are the coefficients of the
  possible paths connecting the independent and
  dependent variables

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Masons rule

• are the sums of all the first-order,
  second-order, loops
• are the sums of all first-order, second-order,
  loops that do not touch the first path between
  the variables

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Masons rule

• are the sums of all first-order, second-order,
  loops that do not touch the second path between
  the variables and so on, for all the path between
  the independent and dependent variables

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flow graph simplification

• find Gin

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flow graph simplification

• Splittling rule

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flow graph simplification

• self-loop rule

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flow graph simplification

• series rule follows by the parallel rule yields

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flow graph simplification

• Masons rule
• Two paths

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flow graph simplification

• first-order loop

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flow graph simplification

• there is no second-order loop
• the sum of all the first-order loop not touching
  P1
• the sum of all the first-order loop not touching
  P2 is zero

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flow graph simplification

• Therefore,
• this is the same result obtained by the other
  method