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# Behaviour of Coaxial Transmission Lines at Low Frequencies

Author: R.J.Edwards G4FGQ © 12th April 2006

At decreasing frequencies, when conductor inductive reactance becomes less than its resistance, the magnitude of line impedance Zo increases rapidly and the angle of Zo becomes progressively more negative. At power frequencies Zo is an order of magnitude greater than its HF value and its angle nears -45 degrees. From its constant value at HF the propagation velocity decreases to a low value which complicates fault-locating techniques which depend on a knowledge of the velocity factor. In general, things begin to happen at frequencies less than several hundred kilohertz. The Smith Chart makes no provision for a reactive component of Zo and becomes useless. It results in errors in the use of coaxial cables at frequencies as high as 2 or 3 MHz.

Input data to the program is very simple. The line is defined by its nominal HF impedance and the diameter of its inner conductor, plus line length and frequency. A load (terminating) impedance is specified in the form of R+jX ohms.

First Zo = Ro+jXo is calculated. Then line input impedance Zin = Rin+jXin. Then transmission loss when the line is terminated with Zo, and actual loss when the line is loaded and standing waves occur on it. There is other data of interest.

The program apples only to solid polyethylene insulated cables which at HF have a constant velocity factor of 0.665. As may be seen, the VF at low frequencies decreases. But this occurs on all types of line at sufficiently low frequencies from coax to open-wire. There is nothing unusual about coax.

The program takes into account the increase in skin effect as frequency increases from power frequencies to HF. Skin effect begins to show itself at the high audio frequencies. See conductor loss resistance versus increasing frequency from 1 KHz.

Input data can be varied in small increments from the keyboard. By hitting key "T" the line can be terminated with its own Zo. The effects on SWR and the reflection coefficient can be seen immediately. Calculating accuracy can be judged by comparing the re-calculated line input impedance with Zo. Line Zin = Zo. Zo can also be determined by increasing line length and attenuation to greater than 35 or 40 dB. In this condition line input Zin again equals line Zo regardless of terminating impedance.

An interesting experiment is to set line length to 1/2, 1/4 and 1/8 wavelengths at 10 KHz and compare the two reflection coefficients (in dB) and their angles.

If line attenuation much exceeds 150 dB the program may abort due to numerical overflow. Avoid long line lengths at high frequencies with small wire diameter.

How to Obtain a Reflection Coefficient Much Greater than Unity

• Clear the screen by going back to S(tart again) and E(nter data).
• Enter a frequency of 10 Hertz. (just hit zero + return).
• Enter nominal HF Zo = 50 ohms.
• Enter a line length of 50,000 metres = 50 kilometres = 31 miles.
• Enter inner conductor diameter = 1 mm.
• Enter a load resistance of zero ohms. (it's actually limited to 1 micro-ohm).
• Enter a load reactance of 2117 ohms. (note the +ve value).

You will now have a load reflection coefficient = 2.4128 which is very near to the greatest possible value of 1+Sqrt(2) = 2.4142 and which occurs only as the frequency ultimately approaches DC. Values of RC>1.00 can be obtained at HF, but only when the load resistance is small and the load reactance is positive.

It will be noticed that the load reactance has been set equal to the value of the line impedance Zo. Use keys 9,0 to vary load reactance to verify that this value of load reactance maximises the reflection coefficient. Also notice that the line input SWR is unusually very much greater than the load SWR.

Use your pocket calculator to show that 10*log of the ratio of the two computed RC's gives the line attenuation in decibels. Which is what it ought to be.