Power in transmission lines
An antenna with SWR 3 at the end of 20 m of RG-58 looks like SWR 1.5 from the transmitter. The equipment is satisfied, but less than a third of the power reaches the antenna.
01The power that travels along the line
In a transmission line the power travels in two waves: the incident one, which goes from the generator to the load, and the reflected one, which comes back. Since the characteristic impedance is real in a low-loss line, each wave carries its own power and what the load receives is the difference:
With an SWR of 2, 11% of the incident power is reflected; with an SWR of 3, 25%. That power is not necessarily lost: it goes back toward the generator and, if the generator is not matched, it is reflected again toward the load. What is lost is what the line itself dissipates on each trip.
02The losses of the line
A real line attenuates the wave as it advances. The attenuation is given in decibels per 100 m, with the line matched, and it has two causes with different dependence on frequency:
| Cable (50 Ω) | Diameter | At 146 MHz | At 435 MHz | At 2.4 GHz |
|---|---|---|---|---|
| RG-58 | 5 mm | about 20 dB/100 m | about 36 dB/100 m | about 100 dB/100 m |
| RG-213 | 10 mm | about 9 dB/100 m | about 16 dB/100 m | about 44 dB/100 m |
| LMR-400 (foam) | 10 mm | about 5 dB/100 m | about 8.5 dB/100 m | about 22 dB/100 m |
The values are typical; each manufacturer publishes its own table. A thicker cable loses less because the current is spread over more surface, and a foam dielectric loses less than a solid one. With 3 dB of loss, half of the power arrives; with 10 dB, one tenth.
03Mismatch and losses together
If the line has losses and the load is also mismatched, the reflected wave is attenuated too on its trip back. This has two consequences. The first: the total loss is greater than that of the matched line, because the power that goes and comes back travels along the cable more than once. With \( a \) the attenuation of the matched line as a power ratio (\( a = 10^{A/10} \)):
The second consequence is misleading: the cable improves the SWR that the transmitter measures. An antenna with an SWR of 3 at the end of 20 m of RG-58 at 146 MHz looks like SWR 1.5 from the equipment. The transmitter is satisfied, but less than a third of the power reaches the antenna. That is why the SWR is measured, when possible, at the antenna feed point.
A transmitter delivers 100 W net to a 50 Ω cable that ends in an antenna. The graph shows the power of the incident wave and of the reflected one along the cable, from the transmitter (on the left) to the antenna. The attenuations are typical catalog values.
04How much power a cable can take
A cable has two power limits. The thermal one: what it dissipates heats it, and the dielectric softens above a certain temperature; since the losses grow with frequency, the power rating drops with it. The voltage one: with standing waves, the voltage maxima are \( (1 + |\Gamma_L|) \) times those of the matched line, and the dielectric can break down. Manufacturers give the power rating with the line matched and a derating factor for each SWR.
For a transmitter, mismatch is also a risk: the reflected power returns to the output stage and the transistors can receive voltages higher than the design values. That is why solid-state transmitters automatically reduce their power when the SWR exceeds a threshold.
05Where it appears in electronics
| Case | What decides the useful power |
|---|---|
| Amateur radio station | The length and type of cable up to the antenna, more than the SWR at the equipment |
| Cellular base station | Thick foam cables, or the radio mounted next to the antenna to eliminate the cable |
| Television antenna with amplifier | The amplifier goes on the mast, ahead of the cable losses |
| Broadcast transmitter | SWR protection and a permanent directional meter |
The complete link, with the gain of the antennas and the path loss, is put together in the unit on antennas; the cable losses enter that budget by subtracting decibels, just as in the microwave links of the technician program.
06In the lab
With a generator and a power meter, measure the output power of a roll of RG-58 loaded with 50 Ω at several frequencies. Plot the decibels per 100 m against frequency and separate the square-root term from the linear one.
Terminate the same roll in 150 Ω and measure the SWR at its input with a directional meter. Repeat with a short section and compare with the SWR of the load.
For an antenna 30 m from the equipment at 435 MHz, compare in catalogs the cost per meter and the power that arrives with RG-58, RG-213 and a foam cable.
07Common mistakes
- Trusting the SWR measured at the equipment. A very lossy cable artificially improves it.
- Believing that all the reflected power is lost. What is lost is what the line and the generator dissipate, not the reflected power itself.
- Using the attenuation at one frequency for another. It grows with frequency, and at 2.4 GHz a thin cable is useless.
- Ignoring the voltage limit at high SWR. A cable that withstands the matched power can break down with a badly mismatched load.
08Self-assessment
What fraction of the incident power is reflected at SWR 3?
\( |\Gamma| = 0.5 \), so \( |\Gamma|^2 = 25 \)%.
How much power reaches the matched load through a cable with 6 dB of loss if 100 W go in?
\( 100 / 10^{0.6} \approx 25 \) W.
An antenna with SWR 3 is at the end of a cable with 4 dB of loss. What SWR does the transmitter measure?
\( |\Gamma_{in}| = 0.5 / 10^{0.4} \approx 0.2 \), and the SWR is \( 1.2 / 0.8 = 1.5 \).
Why does the attenuation of a coaxial cable grow faster than the square root of frequency at microwaves?
Because the dielectric loss grows in proportion to frequency and comes to dominate over that of the conductors.
09Further reading
- H. Ward Silver (ed.). The ARRL Antenna Book. ARRL, periodic editions. Its chapter on transmission lines covers losses with SWR using examples from real stations.
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. Power in lossy lines and the rigorous definition of return loss and mismatch loss.
- Walter C. Johnson. Transmission Lines and Networks. McGraw-Hill, 1950. A classic on real lines, with the attenuation of conductors and dielectrics derived step by step.