Electrical length versus physical length
A radio wave travels more slowly in a coaxial cable than in free space, because the insulation between the center conductor and the shield slows it down. The ratio between the two speeds is the cable’s velocity factor (VF). A wavelength inside the cable is therefore shorter than in air, and a “quarter-wave” section of cable is physically shorter than a quarter of the free-space wavelength.
The formulas
- Free-space wavelength (m) = 299.79 ÷ frequency (MHz)
- Wavelength in the cable = free-space wavelength × VF
- Physical length of a section = fraction (1/4, 1/2 or 1) × wavelength in the cable
At 145 MHz, the free-space wavelength is 2.068 m. In RG-58 with solid polyethylene insulation (VF 0.66) it shrinks to 1.365 m, so a quarter-wave stub is 34.1 cm long.
| Dielectric | Typical velocity factor | Examples |
|---|---|---|
| Solid polyethylene | 0.66 | RG-58, RG-213, RG-8 |
| Foam polyethylene | 0.78 to 0.85 | RG-6, LMR-400 (0.85) |
| PTFE | 0.70 | RG-142, RG-316 |
| Air or ladder line | 0.95 or more | Hardline, open wire |
What the sections are used for
- A quarter-wave stub open at the far end looks like a short circuit at its design frequency; shorted at the end, it looks like an open circuit. Hams use them as notch filters and to suppress harmonics.
- A half-wave section repeats the impedance at its far end, whatever the cable impedance.
- A quarter-wave transformer matches two impedances: Z = √(Z1 × Z2).
- Phasing lines for antenna arrays need precise electrical lengths.
Delay and phase
The calculator also gives the delay and phase shift of any length of cable. Ten meters of 0.66 cable delay a signal by about 50.5 ns; at 145 MHz that is 7.33 wavelengths, or a phase shift of 118°.
Accuracy
Velocity factors vary by a few percent between manufacturers. For critical stubs, cut slightly long and trim while measuring with an antenna analyzer or a NanoVNA.