# Coaxial Transmissions Lines, 50Hz to 1GHz

Author: R.J.Edwards G4FGQ © 5th October 1998This program allows the user to specify/design a coaxial line, to compute the line's principal characteristics, to terminate the line with any complex impedance between open and short circuit and obtain the line's input impedance. Performance is computed in terms of reflection coefficients, VSWR and actual transmission efficiency. The line's matched loss is displayed for comparison.

The line is specified in terms of nominal HF Zo, diameter of inner conductor, thickness of the outer conductor, velocity factor, dielectric loss factor and length of line. Diameter over the insulant (outer conductor diameter) is a computed dimension. To use it as an input parameter would force use of non-existent types of insulant to give the required Zo, whereas velocity is easily and accurately measured. The actual insulant diameter, if known, will provide a check on input data reliability by comparing it with the computed value.

The terminating or load impedance can be specified as R and X in series, or in the equivalent shunt form. Whichever is chosen its equivalent is automatically displayed. Input impedance is computed in three forms: series, shunt, magnitude plus angle. Further information on input and output data is given in the notes.

Entering Data

All data is entered on line 25 and is then displayed at the top of
the screen. When a full set of data has been entered and results obtained,
any input item can be changed on a "what-if" basis and a new
set of results computed. Line 24 shows information which may be useful
when entering data. It allows the option of changing line length to frequently
needed values by one key stroke.

Key "Q" sets line length precisely to 1/4-wavelength."A" adds another 1/4 wave. "3" sets length to 3.0103 dBs. "=" increases length by 0.1%. "-" decreases it by 0.1%. "6" sets length to 60 dB at which high value of attenuation, Zin = Zo = Ro + jXo regardless of the value of the terminating impedance.

To prevent numerical overflow overall line loss is limited to 160 dB which may occur due to excessive length or at too high a frequency for a particular application. A warning is given. If during initial data entry length is uncertain, then enter 0 metres. If Rtn is entered in error then follow with correct data.

If the thickness of a braided outer conductor is uncertain enter a value which would have the same DC resistance if it were a plain copper tube. But thickness affects performance only at frequencies below which the RF skin depth in copper is greater than about half the outer conductor thickness. Skin effect on solid inner conductors extends to lower frequencies due to radius > thickness.

Nominal Zo, Nominal Velocity Factor, Dielectric
Loss

The magnitude of Zo = Ro -jXo of all lines decreases with increasing
frequency until at VHF it converges on a sensibly constant value which
in this program is referred to as Nominal Zo. It may be 2 or 3% less than
Zo at 1 MHz but this is usually of little consequence if only because
batch to batch manufacturing variations are of the same order. Also at
VHF, the angle of Zo will have fallen to 0 degrees from the AF value of
approximately -43 degrees. So if a line is to have Zo = 50 ohms at
2 MHz, enter a nominal of 49 ohms, or thereabouts, at VHF.

Propagation velocity and hence the Velocity Factor VF increases with frequency. At VHF it will have converged to a value referred to as the Nominal VF which is the value to be entered in the program. But if unknown, a value measured at any frequency above a few MHz will be sufficiently accurate. It is not necessary for the program user to know the permittivity of the insulant or the volume ratio of insulant-to-air as in foam or disk-insulated lines. But it may be useful to remember: for solid polythene VF = 0.665 and for solid Teflon VF = 0.69

The dielectric loss in plastic insulating materials is negligible compared with conductor loss below say 300 MHz. If unknown, enter a Loss Factor = 0.0003. For solid polythene or Teflon enter 0.00003 If a material occupies 25% of the space between conductors then, crudely, the effective loss factor will be 25% of that of the solid material - provided the remaining space is very dry, clean, air.

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.

When a zero value of resistance or reactance is entered the program actually uses 1 micro-ohm. The line never internally dissipates 100% of its input power.

Computed Diameter of Insulant

This important dimension is assumed to be equal to the inside diameter
of the outer conductor. Because inner and outer conductor diameters are
mathematically locked to Zo and VF, given any 3 of these parameters the
4th is predetermined. So if a sample of the line being modeled is available,
measurement of insulant diameter provides a means of checking reliability
of input data. If there is a discrepancy it is not possible to identify
which of the four is incorrect. Most likely the measurement of Zo will
have the largest uncertainty.

Computed Line Characteristics

Zo is shown in the form of impedance magnitude and angle. It may
be found in in the form Ro +jXo, or the equivalent shunt form, by lengthening
the line to 50 dB or more when Zin will then equal Zo. On practical
lines the angle of Zo will always be negative because dielectric loss
is always less than loss in the conductors. When the two are artificially
made equal the angle of Zo is 0.

R, L, C and conductance G are in units per metre length of line. G is the equivalent shunt conductance in micro-mhos of the dielectric loss factor. Attenuation is in dB/kilometre. Attenuation due to dielectric loss is separated out for interest. The velocity factor at LF may be compared with its HF value. The overall length of the line in wavelengths is also computed.

Line Input Impedance for a Given Termination

Zin is shown in three forms: as Rs +jXs in series, the equivalent
parallel form of Rp and jXp, and in the form of magnitude and angle. No
R or X is allowed to exceed 10 megohms. A long string of 9's, if one
should occur, may be taken to be an open circuit. Input Z will be needed
for designing a network to match it to an RF generator, or when the line
itself is the matching transformer and Zin is required to be purely resistive
of a specific value.

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.

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 transferred between generator and a load, e.g., between radio transmitter and antenna, what really matters is power transmission efficiency.

*This program answers the question directly by computing
the power lost in the line as a percentage of input power accepted from
the generator.*

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.

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.

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 frequently used standard lengths of lines with one key operation, hit Q to enter exactly one quarterwave, A to Add another quarterwave, etc. The 3 dB line lengths are actually 3.0103 dB which corresponds to a power ratio of 2 to 1. The 60 db length corresponds to a power ratio of 10^6. A fast way of finding Zo in terms of Ro + jXo is to hit "6". Regardless of the terminating impedance, the line input impedance, Rin + jXin, will then be equal to Zo.

When velocity decreases, e.g., due to more, thicker, spacers, the result 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 >3 MHz where it will have become constant vs frequency.

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.

The Q of capacitor C is the ratio (Parallel loss resistance)/(Reactance). The parallel loss resistance decreases with frequency and so Q for most dielectrics materials remains fairly constant with frequency. DLF is the reciprocal of Q.

Instead of a large shunt resistance, loss can be represented by a small conductance in shunt with C. For an entered DLF the program computes the equivalent conductance, G, in units of micro-mhos (micro-Siemens) per metre of line.

Typical DLF's for Various Insulating Materials

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

Multiply DLF's by the fraction of volume between conductors occupied by spacers.

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 CoaxLine 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.

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