Article

Q circles: why your match works at one frequency and nowhere else

Your match reads VSWR 1.0000. Move 5 MHz and it reads 2:1. Nothing is broken. You built a high Q network, and the Q circle overlay would have told you before you soldered anything.

What a Q circle is

Loaded Q, for a node on the chart, is just |X| / R: how much reactance there is compared with resistance. Every point with the same ratio forms a curve, and that curve is the Q circle. Points near the real axis have low Q. Points out near the rim have high Q.

Set Q circle in the Overlays panel and QuickSmith draws it. The useful thing is not the circle by itself, it is the circle next to your element path: the highest Q your path reaches is the Q of the network, and that is what sets the bandwidth.

Watch it happen

Same job in all three cases, 5 ohms to 50 ohms at 100 MHz. The only difference is how far out the path goes before it comes back to the centre.

Four times the Q, roughly a fifth of the bandwidth. Draw the Q circle for the number you asked for and the element path will just touch it at its furthest excursion, which is a satisfying thing to see and a quick way to check you got what you thought you asked for.

The load has a Q too

This trips people up. Loaded Q is measured along the whole path, and that path starts at the load. If the load itself is reactive, its Q is a floor you cannot get below.

A whip reading 25 - j40 has a Q of 40/25 = 1.6 before you have added anything. So every match of that antenna has a loaded Q of at least 1.6, no matter how clever the network is. If you need wider, the answer is not a better network, it is a less reactive antenna.

Why the bandwidth is not just f0 over Q

The textbook relation is BW = f0 / Q, and the numbers above do not obey it exactly: at Q of 3 it would predict 33 MHz and the real answer is 25 MHz.

Two reasons. The textbook relation is the 3 dB bandwidth of a single resonator, and a 2:1 VSWR limit is a different question. And a matching network is not a single resonator, it is a ladder whose parts each have their own frequency behaviour. QuickSmith does not use the formula. It walks outwards from the design frequency and bisects to find where VSWR actually crosses the limit, which is why the numbers above are not tidy multiples of anything.

Use the formula for a feel and the sweep for an answer.

Reading it off in the program

Three places tell you the same story at increasing resolution.

• Loaded Q in the readout is the number, computed from the path you have, not from the topology you think you have.
• BW (VSWR < 2) below it is the bandwidth that follows, found by sweeping rather than by formula. Change the VSWR circle overlay and the limit follows it.
• The Response plot is the shape. Two networks with the same bandwidth number can have quite different skirts, and if the network is also doing duty as a filter, the skirts are what you care about.

Designing to a bandwidth

Work backwards. Decide the bandwidth you need, turn it into a Q, and ask for that Q in the Auto-match dialog.

If the Q you want is lower than the L-network already gives, there is nothing on offer, and that is real rather than a limitation of the program: a Pi or a T can only ever narrow. Going wider means either reducing the impedance ratio, which usually means fixing the load, or a multi-section network built by hand through a virtual resistance that sits between the two ends rather than outside them.

And if the Q you want is high, remember it is not free. A high Q network circulates more current for the same delivered power, so the same imperfect inductor costs you several times more loss in a Q of 8 network than in a Q of 3 one.