Article

Single stub matching, worked end to end

Above a gigahertz or so, a length of line is a cheaper component than an inductor, and a lot more repeatable. Here is the standard two element distributed match, worked all the way through.

The problem

Match 10 - j15 ohms to 50 ohms at 1 GHz, using transmission lines only. This is Example 3 from the QuickSmith help, and it is the shape of a hundred textbook exercises.

Why lines behave differently

A lumped part changes one thing about the impedance. A length of line rotates the reflection coefficient around the centre of the chart, clockwise, at constant radius, one full turn per half wavelength.

That is the whole trick. A line does not change how badly matched you are, it changes where the mismatch sits. So you rotate until the mismatch is somewhere convenient, and then cancel it.

Convenient means the constant conductance circle through the centre, because from anywhere on that circle a single shunt element takes you to 50 ohms. Turn on Admittance grid in the Overlays panel and you can see it.

Step one: rotate

Drop a series transmission line, type 30 for its impedance, and walk the length up from zero. The dot swings round the chart. Stop when it crosses the circle you are aiming for. That happens at 56.4 mm, which at 1 GHz in air is 67.73 degrees, a bit under a fifth of a wavelength.

Why 30 ohms rather than 50? Because the load is a long way down and to the left, and a lower impedance line moves you across the useful part of the chart more directly. There is nothing sacred about it; try 50 and you will find a solution too, at a different length.

Step two: cancel

Now you are on the right circle but not at the centre: there is susceptance left over. Cancel it with a shunt element.

A stub is a short piece of line hanging off the main line, open or shorted at the far end, and it behaves as a pure reactance whose value you set by choosing the length. Open stubs are easier on a board because you do not need a via to ground; shorted stubs are shorter for the same job and give you a DC path, which is sometimes what you want for a bias.

An open 30 ohm stub, 38.5 mm long, which is 46.23 degrees, takes it to the middle.

Those lengths are for air

This is the step that catches people. Published stub lengths almost always assume a velocity factor of 1, which is to say free space. Put the same design on a board and it is wrong, because the wave travels slower.

What carries across is the electrical length in degrees. 67.73 and 46.23 degrees are the design. The millimetres are a consequence of whatever you build it in.

On 1.6 mm FR-4, a 30 ohm microstrip is 6.59 mm wide and has an effective dielectric constant of 3.559, so the wave travels at 0.53 c:

Half the length, same circuit. Data → Microstrip and coax does that conversion for you, and there is a whole article on what it does and where the numbers come from.

Single stub, double stub, and what to worry about

• A single stub can match anything, given a line long enough to rotate with. That is not true of a two element lumped network, which has regions it cannot reach.
• Double stub tuners exist because a single stub needs the stub at a specific distance from the load, which is awkward if the distance is fixed by mechanics. Two stubs at fixed positions with adjustable lengths give you the freedom back, at the cost of a region they cannot match.
• Bandwidth is worse than it looks. A line's electrical length is proportional to frequency, so both parts drift together and faster than lumped parts do. Sweep it before you believe it.
• Watch the proportions. A 30 ohm line on 1.6 mm FR-4 is 6.6 mm wide and 29.9 mm long. That is a quarter as wide as it is long, which is stretching what "line" means. A thinner board, or a higher impedance, keeps you honest.

If you want to see it built up one component at a time, with the reasoning at each step, Help → Guided examples has this design as a tour.