Velocity factor and electrical length
A signal travels more slowly in a cable than in free space, so a quarter-wavelength of cable is physically shorter than a quarter-wavelength in air. The velocity factor (VF) is the ratio of the two speeds, and it's set by the insulation between the conductors. Solid polyethylene, used in RG-58 and RG-213, gives 0.66. Foam cables such as LMR-400, Ecoflex and Aircell are 0.8 to 0.88, and ladder line is about 0.9.
How a stub filter works
A length of cable connected across the line behaves like a tuned circuit. A quarter-wave stub with its far end open looks like a short circuit at its input, on the frequency where it's a quarter-wave. A quarter-wave stub with the far end shorted looks like an open circuit there. Connected with a T-piece, the open stub shorts out the frequency you want to block. The shorted stub does the opposite: it passes its own frequency but shorts out twice that frequency, so a shorted quarter-wave stub at your operating frequency removes the second harmonic (and every even harmonic) from your transmitter.
The table checks what each option does at your operating frequency. The stub adds a reactance across the line, and a reactance much larger than 50 Ω has little effect. A stub that's a half-wave at your operating frequency is invisible to it, which is why the shorted stub at the fundamental works so well for even harmonics. Stubs are commonly used to stop two stations at a contest or field day interfering with each other, and to cut the second harmonic of a 7 MHz transmitter landing on 14 MHz.
Cutting a stub accurately
Velocity factors vary by a few percent between batches, so cut the stub slightly long and trim it while watching with an analyser. Include the T-piece and connector in the length, as they add a few centimetres at HF and a lot at VHF. An open stub resonates lower than its calculated length because of the capacitance at the open end, so trim it shorter to move the notch up.
Quarter-wave transformers
A quarter-wave of line transforms one impedance into another. The line's impedance needs to be the geometric mean of the two:
Two 50 Ω loads in parallel give 25 Ω, and two 75 Ω cables in parallel make a 37.5 Ω line. That's how two antennas are fed from one 50 Ω cable: each antenna's quarter-wave of 75 Ω coax transforms 50 Ω up to 112.5 Ω, and two 112.5 Ω loads in parallel make 56 Ω.
Questions
How long is a quarter-wave of RG-213 at 14 MHz?
About 3.48 m at 14.2 MHz, using a velocity factor of 0.66.
What's the difference between an open and a shorted stub?
At the frequency where it's a quarter-wave, an open stub looks like a short circuit and a shorted stub looks like an open circuit. Connected across a line, an open stub blocks its own frequency, while a shorted stub passes its own frequency and blocks twice that frequency.
How do I measure the velocity factor of my coax?
Leave the far end open, sweep the cable with an analyser and find the lowest frequency where the impedance dips to near zero. The velocity factor is 4 × length (m) × frequency (MHz) ÷ 299.79.
Can a stub remove the third harmonic?
Yes. Use an open stub that's a quarter-wave at the third harmonic. At the fundamental it's a twelfth of a wave long and looks like a capacitive reactance, so check the SWR it adds in the table.