Losses in mismatched line

The literature describes approximate formulas for calculating additional heat losses in uncoordinated transmission lines (cables, waveguides) due to standing waves. The limitation of such formulas is that the own losses of the matched cable should be up to 2 dB. Today, high-loss cables are used more often for microwave frequencies above 1 GHz. Broadband antennas often have SWR >> 1. To find a compromise between the length of the line (for example, the antenna's suspension height), its cost, estimate the loss in a deliberately mismatched line (for example, 75 Ohm cable in 50 Ohm systems) - it is desirable to have a loss estimate for such a line .

To estimate the losses in the line with arbitrary attenuation and arbitrary CWS, we derive a formula and create an Excel calculator.

The reflected voltage is equal to (SWR-1) / (SWR + 1) The
reflected power is r = ((SWR-1) / (SWR + 1)) ^ 2
The transmission coefficient of the line in times is equal to d = 10 ^ (- loss / 10) where loss is the passport attenuation matched in line expressed in decibels.

When a wave reaches the end of a line (for example, an antenna), d * (1-r) goes into the antenna.
r kicks back and comes to the generator d * r. Fights back and reaches the load d * d * r

After an infinite number of reflections, the transfer coefficient of such a line (the remainder of the unscattered energy) will be the sum of a row:

The sum of such a row has a solution:

The balance of energy will be:

p / d is an additional loss in excess of d associated with the presence of standing waves in the line.

To calculate all this instantly for any d and CWS, type these formulas in Excel:

In the yellow cells we substitute the input data, in blue we get the answer

UPD:
To understand the nature of these increased heat losses, we will demonstrate the electric field strength along the transmission line for two cases: SWR = 1 (perfectly matched line) and SWR = 6 (very poorly matched line) , on the same scale Volt / meter: The

red spots of maximum tension in the mismatched line are above both the area and the duration. These areas of heightened strength affect the dielectric and cause it to warm up, which is proportional to the tangent of the loss angle of the material.