Line Characteristic Impedance (Zo) versus Reflection Coefficient Reference Base (Rho Zref)

Precise Calculations
Line Zo = 50 -j 2.4,  Rho Zref = 50 -j 2.4
Electrical
Length
Rho
Mag
Rho
Angle
Rho
Re
Rho
Im
0.667 +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
0.667 180.0° -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
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.

These plot inaccuracies are of little consequence. Computer generated charts are typically used to get a "feel" for the results rather than to obtain answers with a high degree of numerical precision. Of greater concern is the confusion that arises from the common practice of using the term "Zo" to refer to both the characteristic impedance of transmission lines and the rho reference base (also called the normalizing impedance) for Smith charts.

Back