Balanced-Pair Transmission Lines, 20Hz to 1GHzAuthor: R.J.Edwards G4FGQ © 20th February 2000
- Specifying the Line
- Loss Factor, Tan-Delta, Power Factor, Conductance,
- VF = Velocity Factor = Propagation Velocity at
HF Relative to Light
- Terminating (or Load) Impedance
- Computed Line Characteristics
- Input Impedance for a Given Termination - Line Transformers
- Design of Transmission Line Impedance Transformers
- Computed Performance Data
- Miscellaneous Notes
- More on Insulation Resistance (IR), Dielectric Loss Factor (DLF), and Q
- Run this Program from the Web or Download and Run it from Your Computer
This program is a modified version of rjeLine2. Essentially, program input is a line specification including Zo and length, and a user-defined terminating impedance Zt = Rt+jXt. Line input impedance Zin = Rin+jXin is then computed.
Therefore in addition to line design it can be used directly to design matching transformers. It is specially useful when the input and terminating impedances are both required and complex.
The process of converging to optimum values of Zo and length has been made less tedious by allowing rapid variation of these input parameters in small increments, either up or down, by depressing one or the other of adjacent keys while observing computed data for required values of Rin, Xin, etc.
Single increments or decrements are made by single key strokes. To assist with determining critical line lengths, when the angle of Zin changes sign the text and background of the data change colour. For the special but common case of a target of Zin = 50+j0 ohms, the SWR on a 50-ohm line from Zin back to the transmitter is computed. The facility to vary frequency in small increments is also available.
This program analyses the performance of a variety of balanced lines from very wide-spaced bare conductors to figure-of-8 plastic insulated types, from power frequencies up to UHF, when terminated with any impedance between an open or short circuit. To specify the line the only dimensions required are wire diameter and line length. The effects of insulators and spacers are incorporated indirectly by specifying the HF line impedance (Zo) and relative 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 and the program computes the wire spacing necessary to comply with the specified Zo and velocity factor.
A value can always be assigned to Zo. If not known VF is easily and accurately determined by measuring the resonant frequency of a known length. The computed conductor spacing will then provide a cross-check on data self-consistency.
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.
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 is then defined for all other frequencies.
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 actual diameter. This approximation is OK at MF and HF, but at ELF and AF resistance and attenuation may 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, 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 often unwanted unwanted materials present such as rainwater, acid rain, soot and salt.
The sum effect of all losses associated with the line's capacitance in this program is named the Dielectric Loss-Factor. It is a material's 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 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 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 of 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.
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.
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.
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, radio engineers are interested in the reflection coefficient of Zin with respect to 50 ohms. The magnitude of this coefficient is computed together with the equivalent VSWR on a 50-ohm line between a radio transmitter and the feedline input.
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.
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.
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 a string of digits.
To enter frequently used standard lengths of line with one key operation, hit "Q" to enter exactly one quarterwave and "A" to Add another quarterwave. To vary length rapidly in fine steps use keys "H" or "J". For finer steps use keys "N" or "M". To vary Zo up or down in 1 ohm steps use "Y" or "U".
When velocity decreases, e.g., due to a thicker spacing web, 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 >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.
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|
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