Effective Permittivity and Trace Delay

Effective permittivity is the number a microstrip actually travels at. It sits between 1 and εr, because part of the field is in the air above the strip. On the 50 Ω, 1.6 mm FR-4 line, 118.43 mil wide, the static εeff is 3.330 and the value at 1 GHz is 3.345. Delay is $\sqrt{\varepsilon_{\mathrm{eff}}}/c$, which is 154.96 ps/in, or 6.101 ps/mm. A symmetric FR-4 stripline does not get the air. This model sets its εeff equal to εr, 4.400, and the delay is 177.72 ps/in. Fifty millimetres of the microstrip at 2.4 GHz is 306.6 ps. Fifty millimetres of the stripline is 349.8 ps.

What is effective permittivity?

The laminate’s εr is the dielectric. The wave on a microstrip is not entirely in the dielectric. The Hammerstad–Jensen static result mixes the two:

$$\varepsilon_{\mathrm{eff}}(0) = \frac{\varepsilon_r+1}{2} + \frac{\varepsilon_r-1}{2}\left(1+\frac{10}{u}\right)^{-ab}$$

with $u = W/h$ after the thickness correction, and $a$ and $b$ the usual functions of $u$ and εr. You do not need the polynomials to use the number. You need to know it is not εr, and it is not $(\varepsilon_r+1)/2$ either, except as a limit for a very narrow strip. On this line εr is 4.4 and εeff(0) is 3.330. A naive $(\varepsilon_r+1)/2$ would be 2.700, and it would make the delay about 10 percent short.

Stripline in this calculator skips the formula. Both sides of the strip are the same dielectric, so εeff is εr at every frequency. That is the model. A real asymmetric strip, or a strip near a plane opening, is not that ideal.

How much delay is that on 1.6 mm FR-4?

The width is the 1 GHz 50 Ω solve from 50 Ω microstrip on 1.6 mm FR-4, held fixed in the rows below. Phase velocity over $c$ is $1/\sqrt{\varepsilon_{\mathrm{eff}}}$. Multiply by $c$ if you want metres per second. Divide a length by that velocity if you want time. The per-millimetre figure is the one I use on a board.

At 2.4 GHz the microstrip quarter wave is 16.99 mm. The stripline quarter wave, εeff 4.400, is 14.89 mm. A stub copied from one onto the other is 2.10 mm off. The 50 mm comparison above is $50 \times 6.132\,\mathrm{ps}$ and $50 \times 6.997\,\mathrm{ps}$. Same copper length, 43.2 ps apart, because one field is partly in air. At 2.4 GHz, 43.2 ps is 37°.

At 0.1 GHz the dispersed εeff is already 3.331, against the static 3.330. Z0 reads 50.11 Ω. That is the static line, for practical purposes. The movement that matters starts further up the table.

Why does the same trace speed up or slow down with frequency?

It slows down. Kirschning and Jansen slide εeff from εeff(0) toward εr as frequency rises. The argument is frequency in GHz times height in millimetres, so a thick board feels it sooner. On this 1.6 mm core, εeff is 3.345 at 1 GHz, 3.380 at 2.4 GHz, and 3.627 at 10 GHz. Z0 falls with it, because $Z_0(f) \propto 1/\sqrt{\varepsilon_{\mathrm{eff}}(f)}$, from 50.00 Ω at 1 GHz to 49.74 Ω at 2.4 GHz and 48.02 Ω at 10 GHz. The width was solved at 1 GHz. It was not re-solved.

Delay follows the same slide: 154.96 ps/in at 1 GHz, 155.76 ps/in at 2.4 GHz, 161.35 ps/in at 10 GHz. The quarter wave on that same width is 3.94 mm at 10 GHz. A clock-distribution rule that uses one picoseconds-per-inch number for every frequency is using the static column. On a thin core the slide is smaller, because $f \times h$ is smaller. On this thick FR-4 it is visible at a few gigahertz.

The TE1 estimate on this board is 50.81 GHz. Ten gigahertz is still under it. I would still not take the third digit of a 10 GHz FR-4 delay to a length match without a measurement. The closed form is a smooth correction, not a mode solver, and FR-4’s Dk is already a round 4.4.

Why is stripline slower?

There is no air path. εeff cannot drop below εr, and in this model it does not rise above it either. 177.72 ps/in is $\sqrt{4.4}/c$, at 2.4 GHz and at 1 GHz. The topology note’s stripline is the same ideal: 0.8 mm of FR-4 on each side of a 1 oz strip, solved to 50 Ω, 26.69 mil wide.

That extra delay is sometimes what you want. A buried run radiates less, and the length is set by εr, which you can read off the laminate datasheet without an effective-permittivity formula. It is the wrong choice when the line is a phase match to a microstrip on the same board and someone assumed one velocity. The 37° on a 50 mm run at 2.4 GHz is enough to miss a length match that was drawn in copper millimetres.

The microstrip’s speed advantage is the air. Cover that air with a flooded solder mask and εeff ticks up. On a thick board the tick is small. On a thin prepreg it is large enough to retune the width. The impedance side of that correction is copper, solder mask, and ENIG.

What this delay is not

It is not group delay through a connector, a via, or a bond. It is not the electrical length of a line with a mask, a finish, or a different Dk than 4.4. Stripline dispersion is absent on purpose. Microstrip dispersion is Kirschning–Jansen, not a full-wave εeff(f).

Loss does not enter the delay. The 1 GHz microstrip on this board is about 0.098 dB/in in the loss model, and that attenuation is not a slowing. If you need time of flight, use the picoseconds. If you need the line to stay 50 Ω at 10 GHz on 1.6 mm FR-4, solve the width at 10 GHz. This width will not be 50.00 Ω there. It will be 48.02 Ω.

References

• E. Hammerstad and Ø. Jensen, “Accurate Models for Microstrip Computer-Aided Design,” IEEE MTT-S International Microwave Symposium Digest, 1980. εeff(0).
• M. Kirschning and R. H. Jansen, “Accurate model for effective dielectric constant of microstrip with validity up to millimetre-wave frequencies,” Electronics Letters, vol. 18, no. 6, pp. 272–273, 1982. εeff(f).
• S. B. Cohn, “Characteristic Impedance of the Shielded-Strip Transmission Line,” IRE Trans. Microwave Theory Tech., vol. 2, no. 2, pp. 52–57, 1954. The stripline Z0. εeff = εr is the embedded-line assumption, not a separate dispersion paper.
• IPC-2141. The finite-thickness correction on the stripline width is the Wheeler form cited with that guide.

Related

• 50 Ω microstrip width on 1.6 mm FR-4
• Microstrip vs GCPW vs stripline
• Microstrip calculator

FAQ

What is effective permittivity of a microstrip?

εeff is the permittivity the wave actually travels at. It sits between 1 and εr because some of the field is in the air above the strip. On the 50 Ω, 1.6 mm FR-4 microstrip, εeff is 3.330 in the static Hammerstad–Jensen result and 3.345 at 1 GHz. The delay is 154.96 ps/in, 6.101 ps/mm.

How do I convert effective permittivity to propagation delay?

Delay per length is √εeff / c. At 2.4 GHz the same fixed width has εeff 3.380, Z0 49.74 Ω, and delay 155.76 ps/in. A quarter wave is 16.99 mm. Fifty millimetres of that line is 306.6 ps.

Why is stripline slower than microstrip?

This stripline model sets εeff equal to εr and does not disperse it. On FR-4 that is 4.400 and 177.72 ps/in. Fifty millimetres is 349.8 ps, and a quarter wave at 2.4 GHz is 14.89 mm. The field is in dielectric on both sides of the strip, so none of the speedup from air is available.

Does a 50 ohm microstrip stay 50 ohms as frequency rises?

Not on a thick board. Kirschning–Jansen moves εeff from 3.345 at 1 GHz to 3.627 at 10 GHz on this 1.6 mm line, and Z0 falls from 50.00 Ω to 48.02 Ω because the width was solved at 1 GHz. The TE1 estimate on that board is 50.81 GHz. Above it, do not trust the closed form.