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Chapter 2
Part 2: Smith chart
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Introduction
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The primary objective of the Smith chart is to graphically compute the
input impedance at any position along a transmission line terminated with
an arbitrarily load.
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Powerful graphical tool for computing the complex impedance of
transmission line circuits (Originally called transmission line calculator).
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Useful CAD presentation tool for displaying the performance (noise, gain,
stability, …) of microwave circuits.
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Still a good way to design matching networks.
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Useful for lossless or lossy lines.
Components of the chart
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Polar plot for the reflection coefficient at the load (Fig. 2-20, P. 67).
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Lines of a pair of parametric equations, one for the real (resistive) part and
the other for the imaginary (reactive) part, are plotted on the chart to
indicate the normalized load impedance (zL = ZL / Z0) (Fig. 2-21, P. 69).
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Three concentric scales are on the perimeter of the chart (Fig. 2-22, P. 70).
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The innermost scale indicates the phase angle of the reflection coefficient.
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The other two scales indicate the distance, in terms of electrical length (),
of the movement of the measurement plane.
o The outermost scale WTG ( toward generator) indicates the
movement of the measurement plane toward the generator (away from
the load).
o The middle scale WTL ( toward load) indicates the movement of the
measurement plane toward the load (away from the generator).
Graphical calculations
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The reflection coefficient  along the line: The magnitude of  is constant
along the line. The only other unknown is the phase angle r.
1) Plot the normalized load impedance (zL = ZL / Z0) on the chart.
2) Draw a radial line through zL until it intersects the r scale.
3) r is at the position where the radial line intersects the r scale.
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VSWR:
1) Plot the normalized load impedance (zL = ZL / Z0) on the chart.
2) Starting at zL, draw a complete circle centering at the origin. This is
called the constant-|| circle, or more commonly known as the SWR
circle since S is a function of || only.
3) S equals to the value where the SWR circle intersects the  = 0
radial line.
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The first voltage minimum equals to the angular displacement between zL
and the  = 180 radial line using the WTG scale (Fig. 2-24, P. 74).
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The first voltage maximum equals to the angular displacement between zL
and the  = 0 radial line using the WTG scale. You can also use lmin 
¼  to obtain lmax.
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The normalized input impedance (zin = Zin / Z0) along the line:
1) Plot the normalized load impedance (zL = ZL / Z0) on the chart.
2) Draw a radial line through zL until it intersects the WTG scale. This is
the starting position WTGstart.
3) From the physical distance between the measurement plane and the
load, compute the equivalent electrical length  (in ).
4) From the WTG scale compute the equivalent angular movement.
WTGfinal = WTGstart + 
5) Draw a radial line through WTGfinal.
6) Starting at zL, draw a complete circle centering at the origin. This is
called the constant-|| circle, or more commonly known as the SWR
circle since S is a function of || only.
7) zin is at the position where the WTGfinal radial line intersects the the
SWR circle.
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Zin is real at  = 0 and is maximum (= Z0 S); Zin is real at  = 180
and is minimum (= Z0 / S)
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Z-Y conversion: It is often more convenient to work with Y since parallel
components are easier to insert into a transmission line circuit (???) and
the result can be obtained by simply adding the Y’s.
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In a Smith chart z(= Z/Z0) and y(= Y/Y0 = Y Z0) are 180 apart.
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Example 2-10, P. 76-78 and 2-11, P. 78.
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Examples 1-5
Impedance matching
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Whenever a transmission line is not terminated with a matched load (ZL =
Z0, portion of the incident energy will be reflected back. This usually
causes two problems: 1) energy is wasted; 2) the reflected energy could
upset/damage the transmitter.
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Components are added between the generator and the load to achieve
matching. This is called an impedance-matching network.
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The match point in a Smith chart is at the origin where zin = yin = 1 + j 0
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The single-stub matching technique:
1) Since the stub, which is a section of transmission line with a sliding
short at the end, is connected in parallel to the line, it is more
convenient to work with admittance than impedance. First transform zL
into yL by rotating the zL 180 along the SWR circle.
2) Determine where to attach the stub by rotating yL in the CW (WTG)
direction along the SWR circle until it intersects the real circle = 1. At
this location the yin has a real part = 1.
3) There will be two points on the chart that satisfy the above condition
(real{yin} = 1).
4) Measure the electrical length (in ) for the above rotation and convert
it into physical distance d. This should be the distance between the
load and the stub.
5) Determine the imaginary part (susceptance) of yin (= j B) from the
chart.
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6) Starting from the short circuit point (where and why), rotate in the CW
(WTG) direction along the SWR =  circle (outer rim) until it
intersects the imaginary line = -j B. At this location the yin of the stub
has a susceptance = -j B.
7) Measure the electrical length (in ) for the above rotation and convert
it into physical distance l. This should be the length of the stub.
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Example 2-12, P. 80-84.
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Example 6
Discrete conjugate matching: Handout and RF Design article