16.3 Transmission Lines, Velocity Factor & SWR

16.3 Transmission Lines, Velocity Factor & SWR

Quick Answer: Z0 (e.g. 50 Ω coax, 75 Ω video, ~300 Ω twin-lead) is set by geometry + dielectric, not length. Velocity factor = (v_{\mathrm{line}} / c), set by dielectrics (solid PE ≈ 0.66). Mismatch → reflections → SWR. Loss in dB/100 ft stacks with length (6 dB/100 ft × 200 ft → 12 dB → 100 W → ~6 W). Nitrogen keeps moisture out of pressurized lines.

Key topic 3-J-066 (Transmission Lines) is the cable plant between the PA wattmeter and the antenna feedpoint. GROL service work lives here: wrong Z0 adapters, waterlogged coax, ignored VF when cutting phasing lines, and unmeasured dB losses that “steal” ERP.

Why use a transmission line?

The transmitter is rarely at the antenna. A transmission line guides RF with controlled impedance so power arrives with predictable loss and phase. At low HF with short runs, almost any cable “works”; at VHF/UHF and microwave, line choice and length dominate system performance.

RG-58 and most transmitter jumpers: 50 ohms. Never assume a “TV” 75 Ω cable is interchangeable in a 50 Ω PA path without checking match and connectors.

Characteristic impedance Z0

For coax, Z0 depends on the ratio of shield diameter to center conductor diameter and on the dielectric constant of the insulator between them—not on how many feet you buy. Cutting a 50 Ω cable in half does not make it 25 Ω.

When the load impedance equals Z0, the line is matched: energy travels one way (aside from normal attenuation); no reflection. When the load ≠ Z0, part of the wave reflects, interfering with the forward wave to form standing waves.

Velocity factor and electrical length

Waves travel slower in cable than in vacuum because fields exist partly in the dielectric.

Velocity factor (VF):

[ \mathrm{VF} = \frac{v_{\mathrm{line}}}{c} = \frac{\text{velocity of the wave on the line}}{\text{velocity of light in vacuum}} ]

What determines VF? The dielectrics in the line—not termination impedance, not length, not center-conductor resistivity alone.

Electrical length:

[ \ell_{\mathrm{electrical}} = \frac{\ell_{\mathrm{physical}}}{\mathrm{VF}} ]

A physical 10 m of VF 0.66 coax is about 15.2 m of free-space electrical length. Phasing harnesses, stub lengths, and “quarter-wave” matching sections must be cut using electrical wavelength on that specific cable.

Worked example — physical length for λ/4 at 150 MHz on VF 0.66 coax:

[ \lambda_0 = 2,\mathrm{m},\quad \lambda_{\mathrm{line}} = \mathrm{VF},\lambda_0 = 1.32,\mathrm{m},\quad \lambda/4 = \mathbf{0.33\ m} ]

Reflections, SWR, and matching networks

VSWR (Voltage Standing Wave Ratio) is the ratio of maximum to minimum RF voltage along the line. Equivalently from reflection coefficient magnitude |Γ|:

[ \mathrm{SWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|} ]

Matching networks (antenna couplers, LC networks, transformers, stubs) transform antenna Z to Z0 so the transmitter sees a happy load. They do not create RF power; they reduce reflections. A detuned coupler, as in Section 16.1, is a leading cause of high SWR.

High SWR consequences for GROL maintainers:

1. Reflected power heats the line and PA.
2. Voltage maxima can arc connectors or dry coax.
3. Solid-state transmitters reduce power (foldback).
4. Common-mode currents may radiate from the feed line itself (RFI).

Line losses in decibels

Manufacturers specify attenuation in dB per 100 feet (or per 100 m) at a stated frequency. Loss rises with frequency and with water in the dielectric.

Worked example (pool): loss 6 dB per 100 feet, run 200 feet, transmitter 100 W:

[ \text{Total loss} = 6,\mathrm{dB}/100,\mathrm{ft} \times 2 = 12,\mathrm{dB} ]

[ P_{\mathrm{out}} = 100 \times 10^{-12/10} = 100 / 15.85 \approx \mathbf{6\ W} ]

(Pool answer 6 watts.) −3 dB halves power; −10 dB is a decade; −12 dB is about 1/16.

Another pool-style ratio: a perfect (no-loss) line described with 7 dB of reflected/returned level relative to 5 W input yields about 1 W at that −7 dB figure (5 × 10^(−0.7) ≈ 1 W)—practice reading dB power ratios fluently either as attenuation or as reflection levels.

Nitrogen in transmission lines

Large rigid coax and waveguide runs are often pressurized with dry nitrogen. Purpose on Element 3: prevent moisture from entering the line. Water raises loss, corrodes inner conductors, and can cause arcing. Nitrogen is not primarily for “improving skin effect,” “reducing SWR by magic,” or replacing a proper match—it is a dry dielectric / purge practice.

Harmonic stubs

A stub is a short section of line (open or shorted) teed across the main feed to create a frequency-selective short or open.

Pool item: a shorted stub attached to absorb even harmonics, referred to the fundamental, can be 1/4 wavelength long (at the fundamental). At the second harmonic that physical length is λ/2, transforming the short into a short again at even harmonics in the classic stub-filter picture the pool expects—recognize λ/4 shorted stub as the stock answer for even-harmonic absorption, not 1/2 or 1.41 wavelength distractors.

Putting the line plant together

1. Choose Z0 compatible with radio equipment (50 Ω typical).
2. Minimize length and use low-loss cable at VHF/UHF.
3. Keep connectors dry; use N-type outdoors where appropriate; install PL-259 with proper center and shield bonds (no braid shorts to the pin).
4. Cut phasing/matching sections with VF.
5. Measure forward/reflected power and SWR after any antenna or coupler change.
6. Budget line loss before you promise ERP (next section).

Exam-day transmission-line checklist (3-J-066)

1. VF = (v_{\mathrm{line}}/c); set by dielectrics (PE coax ≈ 0.66).
2. Z0 from geometry/dielectric; 50 Ω radio coax standard.
3. Mismatch → reflections → SWR > 1; match with coupler/network.
4. 6 dB/100 ft × 200 ft @ 100 W → ~6 W at the antenna.
5. Nitrogen → keep moisture out.
6. Shorted λ/4 stub class answer for even-harmonic absorption.
7. Electrical length = physical / VF when cutting stubs and harnesses.

With power at the feedpoint understood, Section 16.4 multiplies that power by antenna gain (and subtracts remaining losses) to get ERP, and links bandwidth and loading-coil tradeoffs from 3-J-065.