## Lossy, Mismatched Transmission Lines

### by Brian Austin GØGSF and Adam Farson VA7OJ/AB4OJ

Figure 1: P_{L}/P_{S} vs. reflection coefficient, with intrinsic line loss α as parameter. |

#### Notes:

The family of curves in Figure 1 is derived from a mathematical expression (Equation 1, below) relating the ratio of output to input power in a lossy, mismatched transmission line. The effect of both the total mismatch loss (intrinsic line loss α + augmented loss due to standing waves on the line) and the reflection coefficient (|ρ|) is clearly visible.

The curves illustrate clearly how these various parameters interact, with P_{L} (the power delivered to the load) decreasing dramatically with severely mismatched loads even when the intrinsic line loss is minimal (other than when P_{L }= 0, when the degree of mismatch makes no difference at all, of course).

It should be noted that that ρ (rho) represents |ρ|, the **magnitude** of reflection coefficient in all expressions and resulting calculations, and on the X-axis of Figure 1.

Intrinsic line loss α is expressed as a numerical value 0 < α < 1, where α = 1 for a lossless line. Curves for the range 0.25 ≤α ≤ 0.9 are shown here.

Attenuation A is specific to each type of transmission line. A in dB = attenuation in dB/unit length X length = 10 log α.

The ratio of power at load P_{L }to_{ }power at source P_{S }is given by Equation 1.

| | Equation 1 |

where α = intrinsic line loss and ρ = reflection coefficient.

Also,

| | Equation 2 |

For convenience, a VSWR scale is shown below the |ρ| scale of Figure 1**.**

Copyright © 2014 B.A. Austin GØGSF and A. Farson VA7OJ/AB4OJ. Last updated: 09/25/2019