Input Impedances of Two Cascaded Transmission Lines with a Load

Transmission lines are impedance transformers. For a given terminating or load impedance this program calculates the input impedance of a line. Line lengths are specified in terms of wavelengths at the line's own velocity. To obtain greatest accuracy when calculating it is desirable to take line loss into account. There is no need to enter loss or attenuation with great accuracy. The right ballpark is good enough. The following figures are for guidance.

A 50 ohm coaxial line with an inner conductor diameter of 1mm has an attenuation of 0.34 dB per wavelength at 30 Mhz, 1.8 dB per wavelength at 1 MHz, and 5.6 dB per wavelength at 100 KHz. Loss decreases inversely proportional to square root of frequency.

A 600 ohm openwire line with conductor diameters of 1mm has an attenuation of 0.08 dB per wavelength at 30 MHz, 0.4 dB per wavelength at 1 MHz, and 1.4 dB per wavelength at
100 KHz. Loss decreases inversely proportional to square root of frequency.

At LF to VHF, doubling conductor diameters roughly halves the attenuation. Doubling the line impedance for constant conductor diameter halves attenuation. Impedance transforming lines need not be 1/4-wavelength long.

Efficiency is stated in terms of the percentage of line input power lost in the line. The remaining power is dissipated in the resistive component of the load or termination. It is unnecessary to consider SWR or reflections.

Overall attenuation of a line is decibels per wavelength times the number of wavelengths. Attenuation is a property of the line itself. It is the line loss which occurs when the line is terminated in its characteristic impedance Zo. If the termination is not Zo, above or below, efficiency always deteriorates.

Line 1 can be considered to be the main transmission line from the transmitter, with Line 2 being a Line 1-to-load impedance matching section, and the load being the input impedance of the antenna. There are other applications.

The program deals with design of the impedance transforming section, choice of Zo and line length, and efficiency of the transmission system as a whole.

The program can be used to calculate line loss on any length of line for any terminating impedance. Use Line 2 for this. There is no involvement with SWR. Insert in the program the attenuation in dB per wavelength which automatically takes frequency into account. The user will soon become accustomed to thinking of transmission lines in terms of wavelengths rather than in feet and inches.

Transforming a Load Impedance to Something Else

1. Set the parameters of Line 1 to any sensible values.
2. Use Line 2 as the transforming line.
3. Enter Zo2 of the transforming line.
4. Set line length initially to 0.25 wavelengths.
5. Enter attenuation per wavelength or initially enter 0 dB.
6. Enter R and jX of the load impedance which is to be transformed.
7. Computed line input impedance, Zin, is the new transformed impedance.
8. Vary Zo2 until Zin, the transformed load, is at or near to its required value.
9. If the load impedance is greater than Zo2 then the transformed impedance will be less than Zo2. And vice-versa. Transformed Z = Sqr(Zo2)/(Load impedance).
10. If the load has a reactive component, jX, then vary length of Line 2 until the transformed impedance is purely resistive with jX = 0. It will be necessary to return to readjusting Zo2 to obtain the required purely resistive value.
11. It may be necessary to alternate between adjusting Zo2 and length of Line 2. Check that attenuation per wavelength is approximately correct. All calculations conform to an exact Classical Transmission Line analysis.

Run this Program from the Web or Download and Run it from Your Computer
This program is self-contained and ready to use. It does not require installation. Click this link TwoLines then click Open to run from the web or Save to save the program to your hard drive. If you save it to your hard drive, double-click the file name from Windows Explorer (Right-click Start then left-click Explore to start Windows Explorer) and it will run.