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Feed-line analysis & design software

# Balanced-Pair Transmission Lines, 20Hz-500MHz

Author: R.J.Edwards G4FGQ © 12th September 1998

Introduction
This program analyses the behaviour/performance of a variety of balanced lines from very wide-spaced bare conductors to figure-of-eight plastic insulated types, from power frequencies up to UHF, when terminated with any impedance between a short and open circuit. To specify the line the only dimensions required are conductor diameters and length. Shapes and sizes of insulators and spacers are incorporated indirectly via the resulting impedance (Zo) and velocity (VF).

These components replace the same volume of air with materials having higher permittivity and greater dielectric loss. Their effects on line capacitance and conductance (leakance) are not directly calculable. Instead, the program user is asked to enter the more familiar HF or VHF values of Zo and VF.

A value can always be assigned to Zo. If not known VF can easily and accurately be determined by measuring a resonant frequency of a known length. The program also computes conductor spacing and so provides a cross-check with reality.

From nominal Zo, wire diameter, VF, dielectric loss factor, all other primary and secondary characteristics are computed using exact, classical transmission line formulae. Performance for a given line termination can then be analysed.

Specifying the Line
As frequency increases, the magnitude of Zo of a practical line decreases until >10 MHz it converges to a steady value which can be considered purely resistive and depends on the line's dimensions. Zo at HF is usually referred to as the nominal impedance and is required to be initially entered into the program. The value of Zo and impedance angle can then be computed for any other frequency.

Conductor diameter is input. Solid copper round wires are assumed. The program then computes centre-to-centre conductor spacing corresponding to nominal Zo, followed by wire resistance including skin and proximity effects. For stranded conductors enter 90% of actual overall diameter. This approximation will be OK at MF and above but, at ELF, resistance and attenuation will be underestimated.

Line length should be consistent with intended application and frequency else numerical overflow may occur. A blinking warning is given. If in doubt, enter initially a very short length - say 1 metre. Any item of input data may be re-entered later, on a "what-if" basis.

The most uncertain input item will be dielectric loss factor (DLF). But for all normal, clean, dry dielectrics, especially on ladder-lines below 200 MHz, loss is so small in comparison with that in the conductors, entering a guess of DLF = 0.00003 will not result in significant errors.

Loss Factor, Tan-Delta, Power Factor, Q, Conductance, Leakance, Insulation
There are many ways of quantifying/defining internal and surface losses of insulating/ supporting materials used in transmission lines. Names may change with frequency. In the case of twin conductors the situation is complicated further because the materials surrounding the conductors are composite - air is always involved. Unwanted substances are acid rain, atmospheric pollution, etc.

The combined effect can be represented on an equivalent circuit of a short line by a pair of high-value resistors in shunt with line capacitance. One resistor represents a loss which is independant of frequency and relates to insulation resistance and surface leakage which can be assumed independent of frequency. The second resistor represents the dielectric hysterisis loss which increases directly proportional to frequency. ( R = 1/F ). It is convenient to quantify the effect in terms of the ratio of the capacitive reactance to the very high shunt resistance. This ratio, the Dielectric Loss Factor DLF, is a constant for the composite insulating/supporting material and is input to the program.

IMPORTANT: If a material occupies 5% of the volume of the space between the conductors then, very crudely, the effective loss factor will be 5% of that of the material itself. Only for a 75-ohm line should the insulant be assumed to occupy 100% of the space. Consider 1/DLF equal to Q of the line capacitance.

VF = Velocity Factor = Propagation Velocity at HF Relative to Light
At HF and above, VF depends entirely on the quantity, disposition and permittivity of the line's supporting/spacing/insulating materials. The highest values of approximately 0.997 correspond to bare wires supported at intervals much greater than conductor spacing. The lowest velocities, VF approximately 0.45, occur on 75-ohm lines with very close-spaced conductors within oval-shaped pvc sheaths.

A figure-of-8 pvc line typically has Zo = 135 ohms and VF = 0.67. A continuous-web 300-ohm line has a VF approximately 0.72 A 450-ohm ladder line with a sparce web may have VF = 0.95 or more. Uncertainty in the entered value of VF results in errors in computed resonant lengths and in the computed conductor spacing. When spacing is already known, enter VF such that computed spacing is correct.

Terminating (or Load) Impedance
The terminating impedance can be specified in two ways - as a resistance and a reactance in series, or as an equivalent pair of such components in parallel. On start-up, the initial entry of terminating impedance must be in series form. Subsequent entries may be in either form. Whichever is used the program always calculates and displays its equivalent. To prevent numerical overflow a truly open-circuit termination is not allowed. The program will accept values up to 10,000 megohms. Higher values are truncated. So an open circuit = 10^10 ohms.

Computed Line Characteristics
Zo is shown in the form of impedance magnitude and angle. With practical low loss lines the angle is always negative and never exceeds 45 degrees. Zo may be found as Ro - jXo by terminating a long line with an approximate value of Ro. Positive angles of Zo will occur only when dielectric loss or leakage over the spacers exceeds conductor loss. When both are equal the angle of Zo is zero.

Resistance, inductance, capacitance and conductance are in units per metre of line but attenuation is shown as decibels per kilometre. It is common to report phase shift as radians/metre but in this program the more practical value of the relative propagation velocity is shown. On all lines the velocity decreases with frequency until at power frequencies it is a fraction of the HF value.

Computed Input Impedance for a Given Termination
Input impedance is shown as Rin +/-jXin. The equivalent parallel form is also displayed with the angle of Zin. For both forms the type and value of component L or C, needed to transform Zin to a pure resistance is shown. At radio frequencies these components will usually be incorporated in an impedance matching network located between radio transmitter and the input end of the line. Also at this point the matching network may transform from the balanced pair configuration to an unbalanced coaxial line back to the transmitter/receiver.

Design 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 Q to set line length to exactly 1/4 wave and vary line Zo until line input resistance equals the other resistance to be matched. A small reactive component of Zin which may occur can be removed by a small change in line length. The bandwidth over which a transformer is usable can be investigated by observing the effects of changing frequency. In practice the termination will also change vs frequency.

Transformers are not restricted to purely resistive terminations. Any pair of complex Z's can be matched - including a conjugate match. Note that minimum loss on a long line occurs with a conjugate termination and not with minimum VSWR. As with a Smith Chart, using this program for transformer design improves with practice.

Computed Performance Data
The reflection coefficient is the fraction of the forward voltage wave which is reflected by a mismatched termination. It has a phase angle which can vary between 0 and +/- 180 degrees. At audio frequencies its magnitude can exceed 2.0.

VSWR is a useful monitor of conditions on an HF line when actually in use, but it is not a direct indication of power transmission efficiency.

At HF, the VSWR at line input is less than at the reflection which causes it. It decreases with line length at twice normal line attenuation rate. At audio frequencies, where Zo may have a large -ve angle, unusual values of VSWR may occur.

Overall transmission loss when "correctly" terminated, i.e., a "matched" line, is computed for comparison with the actual loss using the specified termination.

In all engineering applications, when energy is to be transported between generator and load, e.g., between transmitter and antenna, what really matters is power transmission efficiency. This program settles the question directly by computing the power lost in the feedline as a percentage of input power.

Q or Quality Factor applies to resonant lengths of line. As with corresponding coil and capacitor equivalents, Q is the ratio of line reactances to loss resistances. High Q = high selectivity when lines are used as tuned circuits and in frequency-filters. In the vicinity of resonance the relationship between Q and bandwidth is identical to that of lumped 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. It may be important at UHF and is avoided by using coaxial lines or cavities.

Miscellaneous Notes
"Attenuation" in dB per unit length or overall, is an intrinsic property of the line. It is the "matched" loss, i.e., that with no standing waves present.

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