Smith Chart & RF Black Magic Lab

The Smith chart is a picture of a single complex number. Γ = (Z_L − Z_0)/(Z_L + Z_0) takes every load that exists and folds it into a disc of radius one, so the position of the dot is the answer. The five rows underneath are not five measurements. They are that one number, written five ways.

What each symbol means

Γ

the reflection coefficient — the fraction of the incoming wave that comes back, as a magnitude and an angle. Everything else is derived from it.

VSWR

(1+|Γ|)/(1−|Γ|). Runs from 1 at a perfect match to infinity at total reflection, so it magnifies the bad end of the range and compresses the good one.

RL

return loss, −20 log₁₀|Γ|, in decibels. The same quantity turned upside down: bigger is better, and a perfect match is infinite.

y

the normalised admittance 1/z. It exists in the readout because shunt components add to admittance while series components add to impedance.

Where the formula comes from

1. One measurement starts it all. Feeding 36 − j20 Ω into a 50 Ω system gives Γ = 0.276 ∠ −111.9°. Nothing else on the page is measured; the rest is arithmetic.
2. The magnitude alone drives four rows. 0.276 gives VSWR 1.76, return loss 11.17 dB, mismatch loss 0.345 dB and 92.4% of the power delivered. The angle −111.9° appears in none of them.
3. So the angle has exactly one job: it says where on the circle you are, and therefore what to add. The two normalised rows are how you cash it in — z = 0.720 − j0.400 for series parts, which change impedance, and y = 1.061 + j0.590 for shunt parts, which change admittance.
4. Auto-match solves that geometry. It picks a series inductor of +j42.45 Ω — 7.38 nH at 915 MHz — then a shunt capacitor of +j12.5 mS, or 2.17 pF, and the readout drops to |Γ| = 0.000.

How to read what you see

Put the presets side by side and watch the five names disagree. The 915 MHz antenna sits at VSWR 1.76, which sounds like a problem, and delivers 92.4% of its power, which does not. The 2.4 GHz amplifier input reads VSWR 4.56 and 59.0%. The short circuit reads VSWR 5000 and 0.1%. VSWR ran over three orders of magnitude while the delivered power went from most of it, to some of it, to none — because VSWR has infinity at one end and percentage cannot. Read in decibels the same three loads cost 0.345, 2.293 and 30.971 dB, which is the scale that behaves itself.

Assumes

That the matching elements are lossless, that only one frequency matters, and that a component is small enough to have no length. The nomograph checks the last of those and will tell you when it fails: at 915 MHz a wavelength is 327.6 mm, so a 10 mm trace is 3.05% of it and the lumped model holds. Carry that same 10 mm to 2.4 GHz and it becomes 8.0%, and the λ/10 warning appears. The single-frequency assumption is never checked at all: an L-network is solved at the design frequency and nothing on this page shows how fast it falls apart either side of it.

Breaks when

A mismatch loss of 0.345 dB is, by itself, nothing. No link budget would notice a third of a decibel, and 92.4% delivered is a fine number. Yet the page also warns that a standing-wave ratio like this can damage a transmitter output stage, and that warning is not an exaggeration. The two statements answer different questions. Mismatch loss asks how much power fails to arrive. The hazard asks where the reflected wave goes — back down the line into the amplifier, as heat it was not sized for and as a voltage maximum somewhere along the cable that the insulation was not rated for. A load can waste almost none of your power and still be the thing that destroys the radio. That is why RF parts are specified in VSWR and return loss rather than in percent delivered: the percentage answers the question nobody was worried about.