1. Supplemental Information Fields and Waves I ECSE 2100
Smith Chart
Supplemental Information
Fields and Waves I
ECSE 2100
3 February 2004
K. A. Connor
RPI ECSE Department

2. 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

3. 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

4. Smith Chart Imaginary Impedance Axis Real Impedance Axis
3 February 2004
K. A. Connor
RPI ECSE Department

5. Smith Chart Constant Imaginary Impedance Lines Impedance Z=R+jX
Normalized
z=2+j for
Zo=50
Constant Real Impedance Circles
3 February 2004
K. A. Connor
RPI ECSE Department

6. Smith Chart Impedance divided by line impedance (50 Ohms)
Z1 = 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 = j18S
3 February 2004
K. A. Connor
RPI ECSE Department

7. 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

8. Constant Reflection Coefficient Circle
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

9. Smith Chart References
to download applet
Two examples from this page are shown in the following slides
3 February 2004
K. A. Connor
RPI ECSE Department

10. 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

11. 3 February 2004
K. A. Connor
RPI ECSE Department

12. Note – the cursor is at the load location
3 February 2004
K. A. Connor
RPI ECSE Department

13. Single Stub Matching (as in Homework)
Load of 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
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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K. A. Connor
RPI ECSE Department

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K. A. Connor
RPI ECSE Department

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K. A. Connor
RPI ECSE Department

17. 3 February 2004
K. A. Connor
RPI ECSE Department

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K. A. Connor
RPI ECSE Department

19. 3 February 2004
K. A. Connor
RPI ECSE Department

20. 3 February 2004
K. A. Connor
RPI ECSE Department

21. 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