portable-antennas - Extras, Coax Data

Coaxial cables for portable use
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Chart of additional line losses due to SWR

Use this to chart to estimate additional losses due solely to SWR in a transmission line.

Colors:

Hint: to isolate a single SWR curve in the chart, use the "Toggle all" button to hide all the curves, then click on the legend item on the RH side to activate the curve for the desired SWR value.

The chart shows additional losses due solely to SWR, in a transmission line.

To use the chart, you need first to know the SWR at the antenna feed-point - this you can calculate if you know the impedance of your feed-line (usually 50 Ω for coax cable), and the impedance at the antenna feed-point.

For instance, if your coax cable has a characteristic impedance of 50 Ω, and the impedance at the antenna feed-point is 100 Ω, then the SWR presented at the antenna is:

                    SWR = (feed-point impedance) / (coax impedance)
                    
                    SWR = 100 / 50
                    
                    SWR = 2:1

Next estimate the loss in dB to be present in the coax cable of your choice: you can calculate this from the cable manufacturer's data sheet, or use the tools available in the section "Coax charts and data" in this page. This loss is known as the "matched line loss."

Using this chart, trace upward from the matched line loss on the horizontal axis to intersect with the SWR you have already calculated - now trace across from that point to the vertical axis and read the value there. This figure represents the additional loss due to the SWR in your system - this loss is then added to the matched loss to give the total loss in the system.

For example, if the matched line loss in your cable is 1.30 dB, and the SWR is 2:1, the chart gives an additional loss of 0.24 dB. The total loss is then 1.30 + 0.24 = 1.54 dB.

This chart is an interactive version of one found in several editions of the ARRL Handbook. It can be used for most antennas to give good results, but it does not give accurate results in every single case. See e.g. https://www.ac6la.com/swrloss.html for a detailed discussion of the chart's merits.

Curves of SWR at TRX against SWR at ANT

Use this to chart to estimate the SWR at the antenna, given the SWR at the transmitter.

Chart dB-loss curves at: dB steps Colors:

Hint: to isolate a single dB loss curve in the chart, use the "Toggle all" button to hide all the curves, then click on the legend item on the RH side to activate the curve for the desired dB loss value.

When using a transmission line "in the real world", the line will present a loss depending on the coax type, length and operating frequency.

Due to these losses, the SWR measured at the transmitter end of the line will be less than the SWR at the antenna feed-point - the line acts to "dampen" or reduce the actual feed-point SWR, and masks the true value of the feed-point SWR.

Use this chart to estimate the feed-point SWR from both the line losses and the SWR measured at the transmitter end of the line.

Example: if a SWR of 2.3 : 1 is measured at the transmitter, and the line has a loss of 3.2 dB, then the chart shows a SWR at the antenna feed-point of about 10.5 : 1. This may come as something of a surprise to many!

This means that a poorly-matched antenna, fed by a lossy line, CAN present a "reasonable" SWR at the transmitter end of the line, which may mean that the antenna can actually be used despite the high degree of mismatch.

This "advantage" comes at a cost, however: considerably less power will be actually radiated from the antenna than is provided by the transmitter - the rest will be lost as heat in the line.

There are a number of commercial OCFD antennas available, at least two of which rely on the "advantage" afforded by the use of thin and lossy coax cable (e.g. RG-174 and/or RG-316) - either permanently attached to the antenna, or recommended as an option - in order to achieve acceptable VSWR values on WARC bands, in addition to those bands on which the antenna is nominally resonant. We leave it to the reader to do their own research to identify such products.

Calculation of losses due to transmission line

Real HF transmission lines, in particular coaxial cable, are not perfect conductors of HF energy, and will therefore lead to a certain amount of RF signal loss along their length. Depending on the type and length of transmission line, and the operating frequency, these losses can be quite substantial.

For single-band antennas, such losses are negligible around the resonant frequency. For multi-band antennas - such as OCFD, EFHW antennas, and half-square antennas configured as voltage-fed EFHWs - the losses are also negligible at the various resonant frequencies, but can become very significant at frequencies where the antenna is not resonant:

40-meter EFHW antenna - chart of VSWR at the feed-point

In this example, showing only VSWR values at the antenna feed-point, very high VSWR values are seen at the frequencies where the antenna is not resonant, i.e at the 30m, 17m and 12m bands.

By introducing a configurable length and type of coax cable into the analysis, however, it is possible to transform such high values to ones more accessible to the average antenna tuner:

40-meter EFHW antenna - chart of VSWR at feed-point, and at the transmitter end of a length of coax

Here, the green curve shows the VSWR values to be expected at the transmitter end of a length of coax (in this example, 12 meters of RG-316 U coax). It is apparent that the resulting VSWR values at the 30m, 17m and 12m bands are much more amenable to tuning below 3:1 by a decent antenna tuner; it should be understood that this comes at the penalty of higher signal loss.

The following section shows how such losses, and hence VSWR curves at the transmitter end of a length of coax, are calculated.

The input impedance of a real, lossy transmission line is computed using the Transmission Line Equation which can take several forms. Here we use a variation using hyperbolic trigonometric functions, after ARRL Antenna Book, 20th Edition, p. 24-12:

$Z_{in} = Z_{0} \frac{Z_{L} cosh (γ l) + Z_{0} sinh (γ l)}{Z_{L} sinh (γ l) + Z_{0} cosh (γ l)}$

We expand this expression in order to separate out the real and imaginary parts in the sinh and cosh arguments.

$Z_{in} = Z_{0} \frac{Z_{L} cosh (l (α + j β)) + Z_{0} sinh (l (α + j β))}{Z_{L} sinh (l (α + j β)) + Z_{0} cosh (l (α + j β))}$

This second expression is then expanded fully, using standard identities for hyperbolic sines and cosines of complex numbers. This is the basis of the code used to calculate the value of Z_in .

$Z_{in} = Z_{0} \frac{Z_{L} (cosh (l α) * cos (l β) + j sinh (l α) * sin (l β)) + Z_{0} (sinh (l α) * cos (l β) + j cosh (l α) * sin (l β))}{Z_{L} (sinh (l α) * cos (l β) + j cosh (l α) * sin (l β)) + Z_{0} (cosh (l α) * cos (l β) + j sinh (l α) * sin (l β))}$ Eq. (1)

where
$Z_{in} =$ complex impedance at input of coax line
$Z_{L} =$ complex load impedance at end of coax line, i.e. at the antenna $= R_{a} \pm j X_{a}$
$Z_{0} =$ characteristic impedance of coax line $= R_{0} + j X_{0}$
$l =$ physical length of coax line in meters
$γ =$ complex loss coefficient $= α + j β$
$α =$ matched line loss attenuation constant, in nepers per unit length
(1 neper = 8.68589 dB; cables are rated in dB per 100 meters)
$β =$ phase constant of coax line in radians per unit length
$= \frac{2 π}{F_{v} L_{w}}$ for $F_{v} =$ velocity factor of coax line, and $L_{w} =$ wavelength $= \frac{300}{f_{MHz}}$

Since, for a given setup, all of these values except frequency are constant, $Z_{in}$ will vary with frequency.

Once the value of $Z_{in}$ has been established for a particular combination of antenna, coax feed-line and frequency, we use the following expression (from a letter in the Technical Correspondence section of QST magazine, November 1997, p70-71) to calculate the loss in dB due to VSWR:

$L_{VSWR} = 10 {log}_{10} (\frac{(1 - | ρ_{in} |^{2})}{(1 - | ρ_{L} |^{2})})$ Eq. (2)

where
$L_{VSWR} =$ loss in dB due to VSWR
$| ρ_{in} | = the magnitude of (Z_{in} - Z_{0}^{*}) / (Z_{in} + Z_{0})$
$| ρ_{L} | = the magnitude of (Z_{L} - Z_{0}^{*}) / (Z_{L} + Z_{0})$
$Z_{0}^{*} = complex conjugate of the feed-line characteristic impedance R_{0} + j X_{0}$

This expression avoids problems associated with possibly negative values of VSWR by using complex impedances and thus eliminating VSWR from the calculation.

In calculating losses for a particular combination of antenna and coax feed-line over several frequencies, it is necessary to calculate both equations 1 and 2 for each frequency. The number of frequencies covered for a VSWR curve over a single band is on the order of 40 or 50.

Extras - Coax cable data and charts

Coax cables - charts and data

Convert dB attenuation to power loss

Chart of losses in dB against coax length, for multiple coax types, at a single frequency

Chart of additional line losses due to SWR

Curves of SWR at TRX against SWR at ANT

Calculation of losses due to transmission line