Example of Signal Flow Graphs

1
Lecture 4

• Example of Signal Flow Graphs
• Microstrip Line Design and Matching
• Multisection Transformer
•  
• Binomial Multisection Matching Transformers
• Chebyshev Multisection Matching Transformers

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Example of Signal Flow Graphs

• use signal flow graphs to find the power ratios
  for the mismatched three-port network shown below
  (Problem 5.32, Pozar)
  

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Example of Signal Flow Graphs

• the signal flow graph is as follows

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Example of Signal Flow Graphs

• Alternatively, we have

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Example of Signal Flow Graphs

• to relate b2 and a1, we have the signal flow
  graph is as follows
  

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Example of Signal Flow Graphs

• To relate b3 and b2,

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Example of Signal Flow Graphs

• the power ratio must be

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Example of Signal Flow Graphs
9
Microstrip Line Design and Matching

• to design and fabricate a 50 W microstrip line
• to design and fabricate a quarter-wave
  transformer and open-stub matching circuits for
  matching a 25 W load to a 50 W transmission line
  at 4 GHz
• to use design curves (or computer code) for
  circuit design and simulations

10
Design of a Microstrip Line

• using the closed-form formulas discussed earlier,
  calculate the width of a 50 W microstrip line
  
• the printed-circuit board has a dielectric
  constant of 2.6 and thickness of 1.59 mm
  
• assuming the conductor thickness is small, obtain
  the effective dielectric constant

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Design of a Microstrip Line

• from design curves, we found that W4.3980 and
  ee 2.1462
  
• fabricate a microstrip transmission line using a
  conducting tape
  
• the width should be close to the size W
• press the conducting tape to eliminate any air
  gap between the substrate and the conductor

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Design of a Microstrip Line

• for more accurate fabrication, one can use
  etching techniques
  
• attach one SMA connector as shown below

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Design of a Microstrip Line

• do a one-port calibration of the vector network
  analyzer (VNWA) from 0.5 to 10GHz at the end of
  the flexible cable, assume a fixed load (50W) is
  a broadband load in the one-port calibration

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Design of a Microstrip Line

• APC-7 is a sexless precision connector which can
  be used up to 20 GHz
  
• to obtain an accurate amplitude and phase of
  DUT(device under test), the VNWA must be
  calibrated at a reference point
  
• the most commonly used OSL method utilizes three
  standards, Open, Short and Load (50W)
  

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Design of a Microstrip Line

• the front panel of VNWA has two ports which are
  designated as Port 1 and Port 2
  
• some devices have only one port and they are
  called the one-port devices
  
• TV has only one input, if we want to measure the
  input impedance of a TV antenna, connect it to
  either Port 1 or Port 2 and measure the
  reflection from the antenna

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Design of a Microstrip Line

• because it is common to use Port 1 for the
  one-port device measurement, we will discuss the
  S11 (Port 1) calibration
  
• first we need to choose the point at which the
  calibration is performed
  
• for example, if we want to perform S11
  calibration at the end of a long cable,
  calibration standards Open, Short and Load must
  be connected at this point

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Design of a Microstrip Line

• after the correct S11 one-port calibration, Short
  connected at the calibration point should show
  the reflection coefficient of -1 (0dB and 180o
  phase)
• the calibration point also corresponds to zero
  second in the time domain

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Design of a Microstrip Line

• note that the APC-7 and the SMA connectors are of
  different size and therefore, we need an
  APC-7-SMA adapter
  

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Design of a Microstrip Line

• as a result, we need to shift the reference plane
  to the end of the adapter
  
• two options, i.e., port extension and electrical
  delay can be used
  
• port extension requires the time delay from the
  original plane to the new calibration plane while
  electrical delay requires the round-trip time

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Design of a Microstrip Line

• after one-port S11 calibration has been done,
  attach a short to the end of the microstrip line,
  obtain the electrical delay to the short from the
  reference position

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Design of a Microstrip Line

• the short can be achieved by using conducting
  tape
  

22
Design of a Microstrip Line

• remove the short and attach a SMA connector

23
Design of a Microstrip Line

• connect the 25 W load to the SMA connector
• measure the input impedance at the load, note
  that due to imperfect connections, the measured
  load may have a small imaginary part
  
• we can use a Smith Chart to find out the location
  on the microstrip line where the input impedance
  becomes 25 W, here let us assume it is exactly 25
  W

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Design of a Microstrip Line

• make a quarter-wave transformer using a
  conducting tape
  
• quarter-wave transformer can be explained by the
  following equation
  

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Design of a Microstrip Line

• there will be no reflection is
• Therefore,
• and

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Design of a Microstrip Line

• because of the presence of the SMA connector at
  the end of the microstrip line, it is not
  convenient to put the quarter-wave transformer
  there we can move the 25 W point to l/2 from
  the load toward the APC-7-SMA adapter

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Design of a Microstrip Line

• the effective dielectric constant is 2.1462,
  therefore,
  

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Design of a Microstrip Line

• the length of the quarter-wave transformer is
  le/4, however, le is different from the one for
  the 50 W line
  
• the characteristic impedance of the transformer
  is 35.25 W and from the previous equations, we
  found W7.2261 and ee 2.2193

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Design of a Microstrip Line

• the length of the quarter-wave transformer is

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Design of a Microstrip Line

• the microstrip line and the quarter-wave
  transformer are depicted below
  
• assuming that the total length of the
  transmission line is 100 mm

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Design of a Microstrip Line

• the magnitude of S11 measured at the left SMA
  connector looks like
  

32
Design of a Microstrip Line

• note that this result may be different from
  measurement, one of the reasons is that we assume
  the characteristic impedance and effective
  dielectric constant are independent of frequency
• we can make a rough estimation of the bandwidth
  of this quarter-wave transformer

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Design of a Microstrip Line

• at the designed frequency fo , the reflection
  coefficient is
  

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Design of a Microstrip Line

• Assuming a TEM line

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Design of a Microstrip Line
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Design of a Microstrip Line

• nearby the design frequency, and therefore

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Design of a Microstrip Line

• this function is symmetric about the design
  frequency, we can define a bandwidth for a
  maximum value of the reflection coefficient that
  can be tolerated

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Design of a Microstrip Line

• , the lower value is
  while the upper value is
  

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Design of a Microstrip Line

• For a TEM line,
• the fractional bandwidth is given by

40
Design of a Microstrip Line

• make an open stub using a conducting tape
• to derive the formulas for location d and length
  l of the stub, consider the following equations
  with ttan(bd)
  

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Design of a Microstrip Line

• to match the line, we need G Yo 1/Zo

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Design of a Microstrip Line

• solving for t gives

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Design of a Microstrip Line

• the two principal solutions for d are

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Design of a Microstrip Line

• here XL 0,
• d 51.2(35.26/360)5.0148mm
• this is too close to the SMA connector, we add
  le/2 25.65.014830.06 mm
  
•  the stub susceptance Bs must be negative of B to
  cancel the imaginary part of the admittance

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Design of a Microstrip Line

• From the equation,
• For an open stub
• For a short circuit stub,

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Design of a Microstrip Line

• if the length given by these equations is
  negative, l/2 can be added to give a positive
  result
  

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Design of a Microstrip Line

• attach an open-stub matching circuit to the
  transmission line and obtain the S11 response
  

48
Design of a Microstrip Line
49
Design of a Microstrip Line

• this result may be slightly different from
  measurement, the open stub has some end
  capacitance that is being ignored in addition to
  the frequency dependence of the characteristic
  impedance and effective dielectric constant

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Multisection Transformer

• consider the reflection from a segment of a
  transmission line discontinuity depicted below
  

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Multisection Transformer

• If the line impedances are only slightly
  different, and , Eq. (1) becomes
  
• as

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Multisection Transformer

• now let us consider a multisection transformer
  with N sections, each segment has a
  characteristic impedance slightly different from
  the adjacent ones, the reflection coefficient can
  be written as

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Multisection Transformer

• assuming that the segments are symmetry so that
• and so
  on
  
• ?

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Multisection Transformer

• it should be noted that this does not imply ,
  etc.
  
• for N even
• ? ? N/2

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Multisection Transformer

• For N odd,
• 
  ?
  
• ?

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Multisection Transformer

• this is a Fourier cosine series which implies
  that we can synthesize any desired reflection
  coefficient response (vs frequency) by properly
  choosing the with enough number of sections as
  the Fourier series can match any arbitrary
  function if enough terms are used

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Binomial Multisection Matching Transformers

• for a given number of sections N, the binomial
  matching transformer yields a flat response as
  much as possible near the design frequency
  
•  
• this is achieved by setting the first N-1
  derivatives of the reflection coefficient equal
  to zero at the center frequency fo

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Binomial Multisection Matching Transformers

• Let then the
  magnitude of the reflection coefficient will be
  

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Binomial Multisection Matching Transformers

• recall that at the design frequency fo, q p/2
  (quarter wavelength), the above criteria are
  therefore satisfied
  
• the constant A can be obtained by letting f goes
  to zero at which q 0

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Binomial Multisection Matching Transformers

• the reflection coefficient will be determined by
  the characteristic impedance of the line and the
  load impedance, the matching transformer has no
  electrical length

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Binomial Multisection Matching Transformers

• or

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Binomial Multisection Matching Transformers

• according to the binomial expansion, the
  reflection coefficient reads
  
• Where

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Binomial Multisection Matching Transformers

• compare this equation with Eq. (2), we have
• the characteristic impedance of each segment can
  be found as
  
• and we can start from n 0

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Binomial Multisection Matching Transformers

• note that we assume there is only slight change
  in impedance among the segment and its adjacent
  neighbors, we can approximate Gn by
  
• knowing that when x - 1

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Binomial Multisection Matching Transformers

• the fractional bandwidth of the binomial
  multisection transformer is given by
  
• note that is the maximum allowable reflection
  coefficient, not the reflection efficient at the
  junction between the mth section and (m1)th
  section

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Chebyshev Multisection Matching Transformers

• the Chebyshev transformer optimizes bandwidth at
  the expense of passband ripple
  
• the bandwidth of the Chebyshev transformer is
  substantially better than that of the binomial
  transformer if such ripple is tolerable
  
• the Chebyshev transformer is designed by matching
  the coefficients of the Chebyshev polynomial

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Chebyshev Multisection Matching Transformers

• the nth order Chebyshev polynomial is a
  polynomial of degree n which is denoted by Tn(x),
  e.g.,
  
• for x

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Chebyshev Multisection Matching Transformers

• the first few Chebyshev polynomial is plotted
  below
  

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Chebyshev Multisection Matching Transformers

• it can be seen that between -1 and 1, the
  Chebyshev polynomials oscillate between -1 and 1,
  this is the equal ripple property
  
• for x 1, the region will be mapped to the
  frequency range outside the passband
  
• for x 1, the higher the order of the
  polynomial, the faster the polynomial grows

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Chebyshev Multisection Matching Transformers

• The passband is between and
• , therefore we will map to
• x 1 and to x -1

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Chebyshev Multisection Matching Transformers

• Consider
• For 1, the Chebyshev
  polynomial can be written as

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Chebyshev Multisection Matching Transformers

• or for the first few polynomials
• 1

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Design of Chebyshev Transformer

• Using the previous equations, we have
• 
  
  
• ? ? ?

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Design of Chebyshev Transformer

• once we have chosen the order of the Chebyshev
  polynomial, the coefficients s can be found

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Design of Chebyshev Transformer

• the constant A can be found by setting q0
  corresponding to zero frequency
  
• note that is specified for the passband which is
  not the length of section m

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Design of Chebyshev Transformer

• if the maximum allowable reflection coefficient
  magnitude is , then
  
• Or

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Design of Chebyshev Transformer

• once the bandwidth is known, we can determine the
  fractional bandwidth as