Complex Line Characteristic Impedance (Zo)

Line Zo = 50 -j 2.4
Electrical
Length
Rho
Mag
Rho
Angle
0.667 +178.9°
45° 0.619 +88.9°
90° 0.574 -1.1°
135° 0.532 -91.1°
180° 0.494 +178.9°
Rho Zref = 50 -j 2.4
Calculations for this example were made at 3.75 MHz. At this frequency the characteristic impedance (Zo) of RG-174 is approximately 50-j2.4 ohms. This value for Zo was used in the hyperbolic tangent transmission line equation to calculate the impedance at various points along the line. Then this same Zo was used to calculate the reflection coefficient.

Results are shown at left for several points. It is the reactive component of the line's characteristic impedance (-2.4 ohms in this case) that is responsible for the slight variation in the rho angle from what you might have expected.

Line Zo = 50 +j 0
Electrical
Length
Rho
Mag
Rho
Angle
0.667 180°
45° 0.618 +90°
90° 0.573
135° 0.532 -90°
180° 0.493 180°
Rho Zref = 50 +j 0
If the reactive component of the line's characteristic impedance had been ignored the results would look like this. Note that the rho magnitude still decreases as the line length increases, due to the loss in the line. Also note that in both cases the rho angle changes by 90° for every 0.125 wavelength (45 electrical degrees) change in the line length.

This example is somewhat extreme (for Amateur Radio purposes) in that lossy lines and low frequencies combine to produce a relatively large reactive component for Zo. For instance, a less lossy cable like RG-213 at a higher frequency like 14 MHz would have a Zo of approximately 50-j0.3 ohms.

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