# Balanced, Twin-Conductor Transmission Lines, 20Hz to 1GHz

Author: R.J.Edwards G4FGQ © 19th May 2001The input to this program is a line specification, including line length, and a terminating impedance, Zt = Rt+jXt. Line input impedance, Zin = Rin+jXin, is computed together with other performance characteristics and line parameters.

In addition to line design, the program can be used to design transmission line matching transformers when both Zt and Zin are complex. For given Zt the process of finding line length and Zo for the desired impedance match is facilitated by fast variation of length and Zo in small increments, up or down, by pressing one or the other of two adjacent keys. See data screen for key menu.

To assist searching for critical line lengths, when Zin angle changes sign the angle data changes colour. When another line, Zo=50 ohms is terminated with Zin the reflection coefficient and VSWR on the 50-ohm line are computed. Frequency can also be swept in small intervals from 20 Hz upwards.

By hitting a key line length can be increased by 1/4-wave. To set one exact 1/4 wavelength first set length to zero. To set line loss to a precise value while keeping Zo constant, vary conductor diameter in small increments.

Much of the data applies to coaxial lines with the same Zo, attenuation and VF.

General

This program analyses the performance of a variety of balanced lines
from lines with very wide-spaced, large-diameter, bare conductors for
power distribution, via HF ladder-line antenna feeders, to figure-of-8
and oval plastic-insulated types. There are also the various overhead
open-wire and paper-insulated underground forms of construction used in
telephone and digital networks. At the UHF end of the spectrum there are
receiver input tuned lines 1 or 2 inches long. Large diameter tubes are
used as tank circuits in high power VHF transmitters.

The most uncertain of program input parameters are propagation velocity and the dielectric loss factor. A problem arises when the dielectric material consists of 2 materials one of which is air and the other is a spacer material which has permittivity and loss factor greater than air. The effective loss factor and permittivity can be estimated very crudely by considering the fraction of total volume surrounding the conductors which is occupied by spacers. The effects are not accurately calculable. Surface leakage is also involved. But VF is easily measured on a sample length. Dielectric loss up to a few hundred MHz is usually small and effective dielectric loss factor guesstimates are good enough.

Exact classical transmission line formulae are used in this program. Also skin and proximity {of the line wires} effects are accounted for. Compare AC with DC wire resistance as frequency increases, and as Zo and wire spacing decrease.

Line Specification

As frequency increases, Zo of practical lines decreases asymptotically
towards a value which at 10 or 20 MHz may be considered to be a steady
value. Zo at HF is usually referred to as a line's nominal value of Zo
and for most purposes it can be considered to be purely resistive. It
is this nominal HF value which is entered in the program. It is a function
of the wire spacing/diameter ratio.

Wire diameter is an input. Annealed copper wire at 20 degrees C is assumed. If the wires are stranded a fair approximation is obtained by entering a wire diameter 90 percent of overall diameter. This approximation is OK at MF and HF but below 50 KHz AC resistance and attenuation may be underestimated. The DC resistance temperature coefficient for copper is +0.4% per degree C. The temperature coefficient of attenuation, decibels per kM, is +0.2% per degree C.

Line length should be consistent with intended application and frequency else numerical overflow may occur. A warning is displayed. Initially, if in doubt, enter a very short length - say 1 metre and change it later.

The most uncertain input item will be dielectric loss factor (DLF). But for all normal, clean, dry dielectrics, especially on ladder-lines below 100 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, Conductance,
Leakance

The above are some of the many ways of quantifying/defining losses
in spacers and other insulating and supporting materials used in transmission
lines. Names change with frequency. At DC and power frequencies there is
insulation resistance.

The situation is further complicated in twin lines because there are always at least two materials involved, one being air. Outdoors there are surface pollutants such as acid rain, soot, salt, bird droppings, spider-webs, etc.

The sum effect of all losses associated with the line's capacitance in this program is called the Dielectric Loss-Factor = DLF. It is the power factor. It is equivalent to a high value leakage resistance shunting the line which decreases inversely proportional to frequency as does capacitive reactance. But the DLF itself is assumed constant over the whole frequency range for any particular combination of the composite wire insulating/supporting materials.

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 75-ohm and lower impedance lines should insulants be assumed to occupy 100% of the space surrounding the wires. The reciprocal of the DLF can be thought of as the Q factor of the line capacitance.

VF = Velocity Factor = Propagation Velocity

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 0.998 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 close-spaced wires embedded
in thick oval-shaped plastic 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 the computed spacing is correct.

Terminating (or Load) Impedance

The terminating impedance can be specified in two forms:- as a resistance
and 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 input 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 propagation velocity is shown. Velocity always decreases with frequency until at power frequencies it may fall to a small fraction of the HF value.

Input Impedance Zin for a Given Termination -

Line Transformers

Computed Zin is displayed as Rin +/-jXin. The equivalent parallel
form is also displayed with the angle of Zin. When using the program to
design transformers the designer is interested in the manner in which line
input impedance varies with line length and with line impedance Zo. In
particular, antenna designers may be interested in the reflection coefficient
of Zin with respect to 50 ohms. The magnitude of this coefficient is computed,
also the VSWR on a 50-ohm line which may be terminated by Zin.

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.
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 due to line imperfections 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. But in a
practical situation Zt will also change with frequency.

Any pair of complex impedances can be matched by choice of Zo and line length. Note that maximum power transfer occurs with conjugate matches at the transformer ends and it may be desirable, for example, for Zin to be the conjugate of the generator impedance to which it is connected rather than the same impedance.

Computed Performance Data

The reflection coefficient is the fraction of the forward voltage
wave which is reflected by a mismatched termination. Its phase angle can
vary between zero and +/- 180 degrees. 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. Resonant Q is a measure of quality when
a length of line, either short or open circuited, is used as a capacitor,
inductor, or a tuned circuit.

As with ordinary tuned circuits Q is the ratio of reactance to loss resistance. But the resonant increase in voltage at the open end of a 1/4-wave line when a small voltage is applied at the input end is 4/Pi times greater than Q. It is equal to the VSWR at the 1/4-wave line input with the other end open circuit.

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, VSWR's greater than 1 can 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, between a transmitter and its antenna, between one telephone and another, what really matters is power transmission efficiency. This program answers the question directly by computing the power lost in the feedline as a percentage of the line's power input.

No account is taken of radiation loss which may occur when the spacing between conductors exceeds a few percent of the free-space wavelength. It may become important at UHF.

Miscellaneous Notes

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

When a zero value for resistance or reactance is entered, to avoid numerical overflow the value actually used in calculations is 1 micro-ohm. Extremely high "open-circuit" values are truncated and the value actually used is displayed. To enter an "open circuit" hold down a numerical key for >12 digits.

When on the data screen, hitting key "A" will add exactly 1/4-wavelength of line to what already exists. To obtain a total length of 1/4-wavelength first set length to zero and then hit "A". To vary length rapidly in fine steps use "H" & "J". For finer steps use "N" & "M". To vary Zo in 0.2 ohm steps use "Y" and "U". Vary conductor diameter using "5" & "6".

When velocity decreases, e.g., due to a thicker spacing web, this results is an increase in attenuation for the same wire diameter and spacing. This important effect cannot be demonstrated directly because, when a lower VF is entered, to maintain the same Zo the program considerably increases spacing. So decrease Zo and then decrease VF to restore the spacing. Then note the higher loss.

Hint: determine VF at >10 MHz. It will then be practically constant vs frequency. But if known only at LF, use the program to determine the constant VHF value.

More on Insulation Resistance (IR),

Dielectric Loss Factor (DLF) and Q

Since IR is normally so high and the power lost in it is so small
in comparison with other line losses, the program assumes a fixed value
of 10,000 megohms for a 1 kilometre length of line. The user is not required
to enter a value for it. If in unusual circumstances the actual IR is much
lower, perhaps due to wet and dirty line separators or supports, it can
be approximated at a particular frequency by entering a larger than normal
value for the DLF.

Dielectric Q = (Shunt loss resistance)/(Reactance). Loss resistance decreases with increasing frequency and so Q for most dielectrics tends to be independent of frequency. The DLF can be considered to be the reciprocal of Q.

Dielectric loss can also be represented by a small shunt conductance G. For a given DLF the program computes the equivalent conductance G in units of micro-mhos per metre length of line.

**Typical DLF's for Various Insulating Materials**

Material | DLF |
---|---|

Dry wood |
0.02 |

Glass |
0.01 |

Porcelain |
0.007 |

PVC |
0.001 |

Nylon |
0.0005 |

Polystyrene |
0.0003 |

Teflon |
0.0001 |

Polyethylene |
0.00005 |

Air |
0.0 |

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