Article

50 ohm trace width on FR-4, and where that number comes from

If you came here for the number: on 1.6 mm FR-4, a 50 ohm microstrip is about 3.08 mm wide. That is 121 mil, if you work in those. The rest of this is why it is that number, why the third decimal place is not real, and what to do with it.

Why it is not one number

Two things set a microstrip's impedance: how wide the track is compared with how far it sits above the ground plane, and what the board is made of. Only the ratio matters, not the absolute size, which is why the same W/h of 1.925 gives you 50 ohms on any thickness of FR-4:

The material is the awkward half. FR-4 is not a specification, it is a flammability rating, and the dielectric constant of what your fabricator actually sends you lands somewhere around 4.2 to 4.6 depending on the glass weave, the resin content and the frequency you are asking about. That spread moves the answer:

About 0.2 mm from end to end. So when a calculator hands you 3.079 mm, the useful part of that is "a bit over three millimetres". Anyone quoting you four significant figures for an FR-4 track is quoting the arithmetic, not the board.

How much does being wrong cost you?

Less than people fear, which is the reassuring part. Etch a 3.08 mm track 0.1 mm too narrow and you get 50.99 ohms. Run it on a board that turned out to be er 4.6 rather than 4.4 and you get 49.00 ohms.

A 1 ohm error on a 50 ohm line is a reflection coefficient of about 0.01, or a VSWR of 1.02, which is 40 dB of return loss. Nothing on a short interconnect will notice. It starts to matter when the line is doing real work, such as a quarter wave transformer, where the impedance is the whole point rather than incidental.

The other substrates, for comparison

Notice the alumina line: 0.614 mm on a 0.635 mm substrate, a W/h just under 1. High permittivity buys you small circuits, which is exactly why microwave hybrids were built on ceramic.

Width is only half the job

A track width gives you the impedance. It does not give you the length, and for anything that is a transmission line rather than a wire, the length is the design.

A microstrip's field is partly in the substrate and partly in the air above it, so the wave sees something in between: the effective dielectric constant. On 1.6 mm FR-4 a 50 ohm line has an effective er of 3.332, so the wave travels at 0.548 c and a wavelength is that much shorter than in free space.

This is also why a narrow track is slightly faster than a wide one. Narrow keeps more of the field in the air, so its effective er is lower. It is a small effect, and it is the reason QuickSmith will not rewrite a design in millimetres unless all its lines are the same width: there is one velocity factor for the whole ladder, and lines of different widths do not share one.

Turning a match into a board

Here is the part a width calculator on its own cannot do. Take Example 3, which matches 10 - j15 ohms to 50 ohms at 1 GHz with a 30 ohm series line and a 30 ohm open stub. As published, the lengths are 56.4 mm and 38.5 mm, but those assume a velocity factor of 1, which is to say air. On a board they are wrong.

What carries across is the electrical length. Those two lines are 67.73 and 46.23 degrees at 1 GHz, and degrees are what the circuit actually cares about. Put them on 1.6 mm FR-4 and the same design comes out as:

Two things are worth noticing there. The lengths halved, near enough, because the board slows the wave down. And a 30 ohm line on 1.6 mm FR-4 is 6.59 mm wide, which is a quarter of the length of the stub it is feeding. At those proportions it is stretching the point to call it a line at all, and on a real board you would either go to a thinner substrate or pick a different topology. The tool will happily give you the number. Deciding whether it is a sensible number is still your job.

Doing it yourself

In QuickSmith, Data → Microstrip and coax. Enter the dielectric constant and the substrate thickness, and every transmission line in your ladder is listed with the track width and the physical length that would build it. There is a line at the top giving the 50 ohm width for that substrate whether you have any lines in the design or not, because that is the number people want first.

Apply to the design rewrites the ladder in millimetres and sets the velocity factor to match, so the circuit behaves exactly as it did but is now expressed in numbers you can hand to a layout tool. Switch the medium to Coax and it does the same job with the bore diameter and the inner conductor instead.

What this model does not do

The formulas here are Hammerstad's quasi-static approximation, good to about 1% across the range of W/h anyone builds in. They assume zero copper thickness and no dispersion, which means two things.

• Copper thickness makes a real track slightly wider than the model thinks, so a 1 oz finish will read a fraction of an ohm low. On FR-4 that disappears into the er spread.
• Dispersion means er falls with frequency, so above a few GHz the effective er drifts and the model drifts with it. FR-4 is a poor material up there anyway, and if you are working where dispersion matters you want a field solver, not a Smith chart.

For laying out a matching network on a board, in the frequency range where FR-4 is a reasonable thing to be using, it is well inside the tolerance of the material.

Every number on this page came out of the same code the program runs, and the substrate figures are pinned in the regression suite against the published 50 ohm widths for FR-4, RO4350B, RT/duroid 5880 and alumina. If the model ever drifts, the tests fail before the article does.