K. A. Connor

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Smith Chart

• Supplemental Information
• Fields and Waves I
• ECSE 2100

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Smith Chart

• Impedances, voltages, currents, etc. all repeat
  every half wavelength
• The magnitude of the reflection coefficient, the
  standing wave ratio (SWR) do not change, so they
  characterize the voltage current patterns on
  the line
• If the load impedance is normalized by the
  characteristic impedance of the line, the
  voltages, currents, impedances, etc. all still
  have the same properties, but the results can be
  generalized to any line with the same normalized
  impedances

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Smith Chart

• The Smith Chart is a clever tool for analyzing
  transmission lines
• The outside of the chart shows location on the
  line in wavelengths
• The combination of intersecting circles inside
  the chart allow us to locate the normalized
  impedance and then to find the impedance anywhere
  on the line

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Smith Chart
Imaginary Impedance Axis
Real Impedance Axis
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Smith Chart
Constant Imaginary Impedance Lines
Constant Real Impedance Circles
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Smith Chart

• Impedance divided by line impedance (50
  Ohms)
• Z1 100 j50
• Z2 75 -j100
• Z3 j200
• Z4 150
• Z5 infinity (an open circuit)
• Z6 0 (a short circuit)
• Z7 50
• Z8 184 -j900
• Then, normalize and plot. The points are plotted
  as follows
• z1 2 j
• z2 1.5 -j2
• z3 j4
• z4 3
• z5 infinity
• z6 0
• z7 1

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Smith Chart

• Thus, the first step in analyzing a transmission
  line is to locate the normalized load impedance
  on the chart
• Next, a circle is drawn that represents the
  reflection coefficient or SWR. The center of the
  circle is the center of the chart. The circle
  passes through the normalized load impedance
• Any point on the line is found on this circle.
  Rotate clockwise to move toward the generator
  (away from the load)
• The distance moved on the line is indicated on
  the outside of the chart in wavelengths

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Toward Generator
Away From Generator
Full Circle is One Half Wavelength Since
Everything Repeats
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Smith Chart References

• http//www.maxim-ic.com/appnotes.cfm/appnote_numbe
  r/742/
• http//www.ece.uvic.ca/whoefer/elec454/Lecture20
  04.pdf
• http//www.sss-mag.com/smith.html
• http//www.educatorscorner.com/index.cgi?CONTENT_I
  D2482 to download applet
• http//www.amanogawa.com/index.html Two examples
  from this page are shown in the following slides

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Smith Chart Example

• First, locate the normalized impedance on the
  chart for ZL 50 j100
• Then draw the circle through the point
• The circle gives us the reflection coefficient
  (the radius of the circle) which can be read from
  the scale at the bottom of most charts
• Also note that exactly opposite to the normalized
  load is its admittance. Thus, the chart can also
  be used to find the admittance. We use this fact
  in stub matching

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Note the cursor is at the load location
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Single Stub Matching (as in Homework)

• Load of 100 j100 Ohms on 50 Ohm Transmission
  Line
• The frequency is 1 GHz 1x109 Hz
• Want to place an open circuit stub somewhere on
  the line to match the load to the line, at least
  as well as possible.
• The steps are well described at
  http//www.amanogawa.com/index.html
• First the line and load are specified. Then the
  step by step procedure is followed to locate the
  open circuit stub to match the line to the load

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Smith Chart

• Now the line is matched to the left of the stub
  because the normalized impedance and admittance
  are equal to 1
• Note that the point on the Smith Chart where the
  line is matched is in the center (normalized z1)
  where also the reflection coefficient circle has
  zero radius or the reflection coefficient is
  zero.
• Thus, the goal with the matching problem is to
  add an impedance so that the total impedance is
  the characteristic impedance