Precise Calculations
Line Zo = 50 -j 2.4, Rho Zref = 50 -j 2.4 |
Electrical
Length |
Rho
Mag |
Rho
Angle |
Rho
Re |
Rho
Im |
| 0° | 0.667 | +178.9° | -0.667 | +0.013 |
| 45° | 0.619 | +88.9° | +0.012 | +0.619 |
| 90° | 0.574 | -1.1° | +0.574 | -0.011 |
| 135° | 0.532 | -91.1° | -0.011 | -0.532 |
| 180° | 0.494 | +178.9° | -0.494 | +0.010 |
As Plotted
Line Zo = 50 -j 2.4, Rho Zref = 50 +j 0 |
Electrical
Length |
Rho
Mag |
Rho
Angle |
Rho
Re |
Rho
Im |
| 0° | 0.667 | 180.0° | -0.667 | 0 |
| 45° | 0.586 | +88.8° | +0.013 | +0.586 |
| 90° | 0.576 | -2.7° | +0.575 | -0.027 |
| 135° | 0.563 | -91.0° | -0.010 | -0.563 |
| 180° | 0.493 | -179.0° | -0.493 | -0.008 |
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The top table at left shows the reflection coefficients at several points along the RG-174 transmission line that was used in this example.
Modern digital computers make short work of the complex number equations that are required to accurately compute rho.
However, there is still a slight problem
when it comes to presenting the results on a Smith chart (or any reflection coefficient chart).
Recall that the reflection coefficient must be calculated for a particular reference base. Using the line Zo
as this base may make sense when dealing with transmission lines at a single frequency.
But the line Zo changes with frequency, so if the Smith
chart is used to show simultaneous results at several different frequencies no one Zo would be "the right one."
Also, Smith charts are frequently used to represent circuit components other than transmission lines.
The usual solution is to choose a pure resistance value as the chart reference base, typically 50+j0 ohms,
leading to results as shown in the bottom table.
This may produce slight inaccuracies in the plotted positions, especially for extreme cases like
lossy RG-174 at relatively low frequencies.
For example, compare the rho magnitude at 45° in the top table with the same information in
the bottom table. The precise magnitude is 0.619, while the plotted magnitude is 0.586.
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