Balanced-Pair, Open-Wire, Transmission Lines, 16Hz-1GzAuthor: R.J.Edwards G4FGQ © 2nd August 1998
- Line Specification
- Terminating (or Load) Impedance
- Line Secondary Constants
- Computed Input Impedance for a Given Terminating Impedance
- Reflection Due to a Mismatched Terminating Impedance
- Design of Transmission Line Impedance Transformers
- Computed Performance Data
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This form of transmission line has been in use since the earliest days of the electric telegraph. When the telephone was invented a long-distance network of low attenuation, balanced pair, overhead lines was already in existence. There was a rapid world-wide telephone expansion. Simultaneously the distribution of electric power within towns and into the countryside began using lines of the same type. They are still widely used. This program will investigate behaviour of lines at power frequencies. It will also show that a pair of wires, diameter = 4mm (0.16"), spaced at 300mm (12"), Zo = 600 ohms, allowed phone calls at distances of 1000kM (620 miles) or more, long before the age of electronic amplifiers.
For 75 years open-wire lines have been used at radio transmitting and receiving stations between antennas and equipment. Such lines are also used as RF transmitters and as high-Q tuned circuits. Whatever the function, a design objective is to maximise power transmission efficiency for a given cost. This program performs a full performance analysis with any complex load impedance from s/c to o/c.
A simplifying assumption: line spacers and supports cause negligible increase in line capacitance above the theoretical air-spaced value. As a consequence actual resonant line lengths may be 1 or 2 percent less than computed values.
As frequency increases the magnitude of Zo of a practical line decreases but at VHF it converges on a sensibly steady value and may be considered to be purely resistive, depending only on the line's physical dimensions. This VHF value of of Zo is referred to in this program as the line's nominal Zo. It is the value required to be initially entered. The actual value of Zo and its angle at any lower specified frequency can then be computed.
Conductor diameter is also required. A solid copper round wire is assumed. The program then computes centre-to-centre conductor spacing corresponding to the nominal Zo. Also conductor resistance including skin and proximity effects.
The line length entered should be consistent with the intended application and operating frequency. Otherwise numerical overflow may occur. E.g.; 100kM of line at 30 MHz will result in an enormous in/out volts ratio. A warning is given. If in doubt enter a very short length. Any individual data item input to this program may be changed at any time on a "what-if" basis.
Leakage between conductors is assumed to be 10 megohms/kM and independent of frequency. In practice it is very variable depending on weather. This value has no significant effect except over long distances at power and audio frequencies. An allowance for leakage maintains program modeling validity down to 50 Hz.
The terminating impedance can be specified in two forms: as a pair of series resistive and reactive components, or as a pair of such components in parallel (in shunt) with each other, whichever is more convenient for the user.
On start-up, the initial entry of terminating impedance must be in series form. Subsequent entries may be in either form. Which ever is used the program always calculates and displays its equivalent. This facility to transform from one to the other may be useful in other applications.
Zo is shown in the form of impedance magnitude and angle. The angle is always negative but never exceeds 45 degrees. Zo may found as Zin = Ro + jX by terminating a very long line with an approximation of Zo.
Because spacer capacitance is neglected, at VHF propagation velocity is almost the speed of light and VF = 1. Attenuation is in the form of dB per kilometre.
Electrical length of the line is given in wavelengths and the frequently needed length in metres of 1/4-wavelength of line is also displayed.
It is often necessary to tune out the series reactive component with a series variable capacitor or inductor to leave the pure series resistance. Similarly the shunt reactance can be tuned out to leave the pure shunt resistance.
In the design of quarterwave impedance matching transformers the facility to predict the effect of an imperfect line on the transformation ratio is useful. It may be found that a line of somewhat different length or impedance provides a better match. Note that this type of transformer is not restricted to matching pure resistances. Conjugate matches between complex Z's are possible.
Due to a Mismatched Terminating Impedance
Line impedance Zo is never a pure resistance. To eliminate reflections it is necessary to terminate the line with Zo = Ro + jXo. This frequently leads to unexpected results such as reflection coefficients being greater than unity and line loss not being at a minimum when VSWR is 1-to-1. At HF these effects are small and are mentioned here to remove suspicion of computing errors. In fact, least loss occurs when a long line is terminated with its conjugate Z.
of Transmission Line Impedance Transformers
To match two pure resistances, R1 and R2, to each other, use one resistance to terminate the line. Set conductor diameter. Set frequency. Press P to set line length to exactly one quarterwave and vary line Zo until line input resistance is equal to the the other resistance to be matched. A small reactive component of line input impedance can be removed by a small change in line length. For a very precise match it may be necessary to go back and forth between adjusting line impedance Zo and line length several times.
Two complex impedances can be matched by a similar but more lengthy process. Useful bandwidth can be found by observing the effects of changing frequency.
The reflection coefficient is the fraction of the forward voltage wave which is reflected by a mismatched termination. It is not of immediate practical use but it can be used to calculate VSWR and reflected power. It has a phase angle which is of use when line input impedance is to be calculated. The phase angle implies that the reflected wave suffers a phase shift at the reflecting point.
VSWR is a useful measurable indication of conditions on the line when actually in use. But it is not a direct indication of power transmission efficiency.
VSWR at the input to the line is smaller than that at the mismatch which causes it. It decreases with line length at a rate governed by twice the line overall attenuation. In practice, VSWR at line input is of the greater interest.
Overall transmission loss when "correctly" terminated, i.e. when VSWR is 1-to-1, is displayed for comparison with actual loss with the selected termination.
In all engineering applications, when energy is to be transported between generator and load, between transmitter and antenna, what really matters is power lost in the transmission line. This program deals with the matter directly and and computes power lost in the line as a percentage of line input power.
Q or Quality Factor applies to resonant lengths of line. As with corresponding coil and capacitor equivalents, Q is the ratio of inductive reactance to conductor resistance. High Q means high selectivity when lines are used as tuned receiving circuits and as frequency-filters. In the vicinity of resonance the relationship between Q and bandwidth is identical to that of L & C circuits.
This program does not take into account radiation loss which may occur when the spacing between conductors exceeds a few percent of the free-space wavelength.
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