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Analysis & design software

# Compute Load Impedance from Measured Input Impedance for any Length & Zo

Author: R.J.Edwards G4FGQ © 19th April 2001

The input impedance of an inaccessible radio antenna can by obtained by calculating what it must be to give the impedance measured at the input end of the antenna's feed line.

Depending on its characteristic impedance, length, propagation velocity and attenuation the line behaves as a complex impedance transformer. But it is important to be aware calculation results may be no more accurate than accuracy of knowledge of the line's characteristics. The attenuation of very short lines does not have a great effect on input impedance but its value at 10 MHz should be known fairly accurately because the program uses it to estimate Zo = Ro+jXo at other frequencies. If the exact value of ¦Zo¦ at 10 MHz is not known then enter the manufacturer's stated HF nominal value. Or measure it.

This program will be useful when the test lead between an R+jX meter and an unknown impedance can be represented as a transmission line. For example, a pair of 1mm wires with alligator clips spaced apart about 100mm has Zo = 550 ohms, VF 0.95. This program accepts values of Zo from 20 to 800 ohms, either of coaxial or balanced-pair types, and at frequencies from 0.1 to 1000 MHz.

Operating and Other Notes
An open-circuit impedance, either entered or computed, is represented by a value of 99,999,999.9 ohms. To enter an open circuit hold down any numerical key for long enough to enter a string of 9 or more digits. Then hit 'Return.'

Line attenuation is specified in dB per kM at 10 MHz - 32.8 times larger than the American unit of dB per 100 feet at 10 MHz. The sensitivity of a computed result to uncertainty in attenuation, Zo, or VF is easily tested by "what-if" variation of input data. See menu near bottom of input/output data screen.

Line input impedance may be entered in the form of either series or parallel values of R and X. On changing previously entered values of R or X, parallel or series equivalents are immediately displayed. And vice-versa. First time round it is possible to enter only the series R and X components of input Z. The load (terminating) impedance is computed and displayed in three equivalent forms: magnitude & angle, series and parallel R & X.

The exact length in wavelengths of the antenna feeder is important. When constructing the system, before connecting the line to the antenna, its 1/4-wave open-circuit resonant frequency should be determined in situ. Its exact propagation velocity and wavelength at any frequency will then be known for future antenna adjustments and maintenance.

Coaxial Line dB/Km, 10MHz dB/100ft, 10MHz
RG-174 119 3.63
RG-58 37 1.13
RG-62, 71 28 0.85
RG-8, 9, 11, 12, 13 20 0.61
50-ohm, 14AWG inner 18 0.55
75-ohm, 14AWG inner 11 0.34
Rigid 1/2" 75-ohm 7.5 0.23
Rigid 1/2" 50-ohm 6.6 0.20
Rigid 7/8" 75-ohm 4.6 0.14
Rigid 7/8" 50-ohm 4.0 0.12

Open-wire & Ladder Line dB/Km, 10MHz dB/100ft, 10MHz
300-ohm, 1mm 7.9 0.241
300-ohm, 2mm 3.9 0.119
450-ohm, 1mm 5.2 0.158
450-ohm, 2mm 2.6 0.079
600-ohm, 1mm 3.9 0.119
600-ohm, 2mm 1.9 0.058

American Wire Gage Diameter
18 AWG 1.0mm
14 AWG 1.6mm
12 AWG 2.0mm

Attenuation of a coaxial line = 1150*(1/d + 1/D)/Zo dB per Km at 10 MHz, where d and D are inner and outer conductor diameters in millimetres. Attenuation of a balanced-pair line = 2300/d/Zo dB per Km at 10 MHz, where d is conductor diameter in mm. These two equations may be used to provide accurate estimates of attenuation when nothing is known except Zo and wire diameters can be measured.

A negative value of the resistive component of the load is computed when an impossible value of line input impedance has been entered. In practical situations if a negative value occurs it indicates either a measurement error has occurred or the line has been incorrectly specified. When a negative value of load resistance is computed a warning notice is displayed on the screen.

To make clear how impossible values of Zin can arise consider a long line with high attenuation. Whatever the value of a real resistive load it is impossible for the input impedance to deviate far from Zo. It is instructive to model a very long line with overall loss greater than 15 dB and see the large changes in load impedance needed to compensate small changes in Zin. This demonstrates that the greater the line loss the more uncertain is our knowledge of what's at the other end. It is also instructive to ask what type of termination is needed to give an effective s/c or o/c input Z on short, low-loss lines.

Electrical characteristics of lines and cables sold for domestic or amateur radio purposes are subject to +/- 5% or more variation between one reel and another. Manufacturers state typical values of Zo, attenuation and velocity factor without providing tolerances. Professional grade cables are specified in greater detail and manufacturing tolerances are more tightly controlled.

In the present context the characteristic which needs to be known most accurately is the line's length in wavelengths at a particular frequency. This can be determined at HF, within +/- 0.3 percent, on an antenna feeder in situ with an open circuit at the antenna end. Just measure the frequency at which the line is 1/4-wave resonant as indicated by the very sharp minimum in the input impedance at that frequency. The line length in wavelengths at any other frequency F is then equal to F/Fr/4. The physical length of the line then need be known with an accuracy sufficient only to estimate overall line attenuation.

Zo can be measured at any odd numbered 1/4-wave resonant frequency near to 10 MHz with an instrument capable of measuring the magnitude of Zin. Terminate the line with Zt = 1/2-watt, metal film resistor of value near to the expected expected Zo and measure Zin. Zo is then given by Sqrt(Zin*Zt). Another method is to measure o/c and s/c reactances on an odd numbered length of 1/8th waves. At the frequency where the two reactances are equal but of opposite signs, Zo equals the reactance value. Line velocity can also be deduced from this test.