Impedance Matching Calculator: Stubs, Transformers and Line Lengths

A length of transmission line is not a neutral pipe. It transforms whatever impedance sits on its far end into something else at its near end, and the transformation depends on how many wavelengths long the line is. That behaviour is a nuisance when you are trying to measure an antenna and a genuinely useful tool when you deliberately choose the length.

What a transmission line does to impedance

Put an impedance on the far end of a line and the near end shows you something else, and what it shows depends on the electrical length. Three cases are worth carrying in your head because everything else interpolates between them.

• Half a wavelength repeats. Whatever is at the far end appears unchanged at the near end, regardless of the line's own impedance.
• A quarter wavelength inverts. The near-end impedance is the line impedance squared, divided by the far-end impedance. A short becomes an open, an open becomes a short, and a low impedance becomes a high one.
• Everything between rotates. The impedance moves around a circle on a Smith chart, one full revolution per half wavelength.

That quarter-wave inversion is the entire basis of stub matching. Choose the line impedance and you choose the transformation, and choosing it as the geometric mean of the source and load gives you an exact match.

The quarter-wave transformer, worked

Suppose an antenna measures 112 ohms and you want to feed it with 50 ohm coax. Untreated, that is an SWR of 2.24 to 1, which is usable but will make a solid state radio fold back slightly.

The transformer impedance needed is the square root of 50 times 112, which is 74.8 ohms. Seventy-five ohm cable is manufactured in quantity for television and is sold everywhere, so this particular match is close to free. Its physical length at 14.175 MHz is a free-space quarter wave of 17.35 feet multiplied by the cable velocity factor, so 11.45 feet in solid polyethylene cable.

Insert that 11.45 foot section between the antenna and the 50 ohm run, and the transmitter sees 50 ohms. There is no tuner, no adjustment, no loss beyond the cable's own attenuation, and nothing to go wrong. The limitation is that it is exact at one frequency and degrades as you move away, which is fine for a monoband antenna and useless for a multiband wire.

Note the trick in rows three and four. Two identical lines in parallel halve the characteristic impedance and two in series double it, which gives you 25, 37.5, 100 and 150 ohm sections out of the two cable types anyone can buy. That covers most of the matches an amateur actually needs.

Stubs used as filters

The shorted quarter-wave stub is the one worth knowing about in a crowded station. A stub cut for the second harmonic of your operating frequency, connected as a T at the transmitter output, presents a near short to that harmonic and almost nothing to the fundamental. Multi-transmitter contest stations use banks of them to keep one radio out of another's receiver, and the same trick will notch a local broadcast transmitter out of your receive path.

Build them with an SO-239 barrel couplers, 6-pack$9.99 T-piece or a proper tee connector, cut two percent long, and trim against a NanoVNA H4 vector network analyzer$89.90 sweep. A stub is exact enough that trimming by ear is hopeless and trimming against a sweep takes five minutes.

Velocity factor is not optional here

Every stub calculation depends on getting the physical length right for the electrical length you want, and that means multiplying by the cable's velocity factor. Solid polyethylene cables run about 0.66 and foam dielectric cables about 0.85, which is a difference of nearly 30 percent in physical length for the same electrical length.

Datasheet velocity factors are close enough to start with, and they are not exact for a specific reel. If a stub matters, measure it: a NanoVNA in time-domain mode or a simple open-stub resonance sweep tells you the real velocity factor of the cable in your hand. Cable that has been in the sun for a decade is not the cable in the datasheet. The full velocity factor table is on the coax loss calculator and the underlying arithmetic on the wavelength calculator.

The half-wave measurement trick

This one saves climbing. A half wavelength of line repeats whatever is on its far end, so if you connect a half-wave jumper between your analyser and the antenna, the analyser sees the true feedpoint impedance rather than a version transformed by the feedline.

That lets you measure an antenna at the top of a mast from the ground, which is otherwise a genuinely difficult problem, because an arbitrary length of feedline transforms a bad match into something that looks entirely different. The catch is that it works on one band only, since a half wave at 14 MHz is a full wave at 28 MHz and neither is a half wave at 21 MHz. For a monoband antenna it is exact and free.

When to use a tuner instead

A matching section is fixed, lossless and monoband. A tuner is adjustable, slightly lossy and works everywhere. The choice is not close in most amateur situations.

• Use a stub when you have one antenna on one band with a known impedance that is stable, such as a loop, a folded dipole or a phased array. Cut it once and forget it.
• Use a tuner for anything multiband, anything whose impedance moves with weather or with band segment, and anything you have not measured. An LDG Z-11ProII automatic antenna tuner$225.44 handles 125 watts across 1.8 to 54 MHz with memory, so a band change retunes instantly, and an ATU-100 EXT automatic antenna tuner$109.99 does much the same for less.
• Use both at a serious station: a matching section at a monoband beam to get the feedpoint right, and a tuner in the shack for everything else.

What neither does is reduce the standing wave on the feedline between the tuner and the antenna, which is the point most often missed. A tuner in the shack makes the transmitter happy and leaves the feedline loss exactly as it was. The honest treatment is on antenna tuner basics and tuner versus resonant antenna.

Measuring the load in the first place

Everything on this page needs a measured load impedance, and an SWR meter cannot give you one. An SWR of 2.24 to 1 could be 112 ohms or 22 ohms, and it could have any amount of reactance attached, which a meter also cannot see.

A NanoVNA H4 vector network analyzer$89.90 gives resistance and reactance separately across a sweep, plots the Smith chart that makes the transformation visible, and costs under a hundred dollars. A RigExpert AA-650 Zoom antenna analyzer$746.73 does the same in a ruggedised package you can carry up a mast without worrying about it. Either one turns matching from guesswork into arithmetic, and the arithmetic is the easy part. Full field on antenna analyzers compared.