Smith Chart
Supplemental Information
Fields and Waves I
ECSE 2100
3 February 2004
K. A. Connor
RPI ECSE Department
1
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
3 February 2004
K. A. Connor
RPI ECSE Department
2
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
3 February 2004
K. A. Connor
RPI ECSE Department
3
Real Impedance
Axis
3 February 2004
Smith Chart
K. A. Connor
RPI ECSE Department
Imaginary
Impedance Axis
4
Smith Chart
Constant Imaginary
Impedance Lines
Impedance
Z=R+jX
=100+j50
Normalized
z=2+j for
Zo=50
Constant Real
Impedance
Circles
3 February 2004
K. A. Connor
RPI ECSE Department
5
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
• z8 = 3.68 -j18S
3 February 2004
K. A. Connor
RPI ECSE Department
6
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
3 February 2004
K. A. Connor
RPI ECSE Department
7
Toward
Generator
Constant
Reflection
Coefficient Circle
Away
From
Generator
Scale in
Wavelengths
Full Circle is One Half
Wavelength Since
Everything Repeats
3 February 2004
K. A. Connor
RPI ECSE Department
8
Smith Chart References
• http://www.maximic.com/appnotes.cfm/appnote_number/742/
• http://www.ece.uvic.ca/~whoefer/elec454/Lectur
e%2004.pdf
• http://www.sss-mag.com/smith.html
• http://www.educatorscorner.com/index.cgi?CON
TENT_ID=2482 to download applet
• http://www.amanogawa.com/index.html Two
examples from this page are shown in the
following slides
3 February 2004
K. A. Connor
RPI ECSE Department
9
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
3 February 2004
K. A. Connor
RPI ECSE Department
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3 February 2004
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RPI ECSE Department
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Note – the cursor is at
the load location
3 February 2004
K. A. Connor
RPI ECSE Department
12
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
3 February 2004
K. A. Connor
RPI ECSE Department
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RPI ECSE Department
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RPI ECSE Department
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RPI ECSE Department
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RPI ECSE Department
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RPI ECSE Department
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RPI ECSE Department
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K. A. Connor
RPI ECSE Department
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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 z=1)
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
3 February 2004
K. A. Connor
RPI ECSE Department
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