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I'm currently designing an RF board, and I'm trying to calculate losses in my transmission lines, I'd like someone to check my math and help me with my assumptions. What I'm doing here is trying to get a rough idea of the loss I could likely expect, a) so I can confirm that the system is likely to work reasonably, b) so I can look at a few different materials, and confirm which substrate I need to use (balancing cost vs loss)

Firstly, my TL properties image of TL properties It's a coplanar waveguide, W=0.21mm, H= 0.1mm, gap = 0.2032mm. With an Er of 3.92 (Isola 370HR dielectric), that gives me my Zo= 50 Ohm The length of the line is 47.7mm, frequency is 5.8GHz. These will all shift around a bit as I find and compare different stackup/substrate options.

Firstly, considering the dielectric loss tangent. From here I've found the formula loss tangent formlua

Using a loss tangent of 0.025 from the datasheet, that should work out as
0.9106 * sqrt(3.92) * 5.8 * 0.025 = 0.261 dB/cm
= -1.25 dB for my transmission line

Now for copper loss, which I'm a bit at a loss for. The resistivity of copper is 17.7 uOhm / mm
Saturn PCB toolkit tells me that the skin depth is 867nm. What I dont understand is how to approximate the area as a function of skin depth, width and height. If I crudely assume the current area could be calculated by skin depth x 2 x (height + width) then
resistance = resistivity x length / area
= 17.7u x 47 / 0.000867 x 2 x (0.21 + 0.035) = 1.987 Ohms.
I've really not got a good idea if this is a good approximation.

Expressing that in dB, loss = 1.987 Ohms / (2x50 Ohms) = 0.01987 dB

I am assuming that radiation losses and dielectric conduction losses are negligible.

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  • \$\begingroup\$ @ThePhoton I'm aware the top layer's ground basically does nothing, but it's there so I figure I might as well do my maths with it. I don't mind stitching vias. \$\endgroup\$
    – LordTeddy
    Apr 27, 2023 at 17:42
  • \$\begingroup\$ I think your math is OK. 0.02 dB (approximately) loss is probably way down in the noise for your overall system. Losses due to mismatches are probably going to be way more that that. \$\endgroup\$
    – SteveSh
    Apr 27, 2023 at 18:49

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There are a couple things going on here.

First, with H=0.1mm and G=0.2mm, you have microstrip with a slight adjustment for the nearby coplanar ground, not really CPW. Rather than fuss around with keeping the three different ground regions well-connected (which takes lots of vias) you might rather just make G even bigger and design this as proper microstrip.

What you have will give lower copper loss than true CPW because the whole bottom surface of the trace will be carrying current, rather than just two small regions near the edges.

What I dont understand is how to approximate the area as a function of skin depth, width and height.

I would just take skin depth times the trace width, again because the current is mainly flowing on the bottom surface of the trace facing the ground plane directly below it.

If you want a more accurate estimate, you'll need to do a full EM simulation to find the current distribution for your geometry and use that to determine the effective resistance of the trace.


If the loss is really critical for your application (maybe this is the input to an analog sampling circuit or something), I'd recommend:

  1. Make some test structures (longer than 50 mm) and measure it.

  2. Use a better dielectric than basic FR-4. Not only will it have lower loss tangent, but the loss tangent will be more consistent from batch to batch, less dependent on the fiberglass orientation, etc. For 5 GHz it doesn't need to be teflon material or anything, but it should be some kind of material that is specified for high-frequency performance.

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  • \$\begingroup\$ I don't have a better idea yet but, skin depth x width (which is basically what I've done) seems like a poor estimate, considering the Gaussian distribution of current. I probably could spend a few days calculating something, but I was hoping there was some kind of approximation to do with those two. I'm calculating using Esola 370HR, but I'd like to be able to do the maths and compare different dielectrics. It's not critical enough to measure, but some confidence the numbers are in the right order of magnitude would serve the purpose. Do you have any comments on dielectric loss? \$\endgroup\$
    – LordTeddy
    Apr 27, 2023 at 17:41
  • \$\begingroup\$ What are you referring to with "gaussian distribution of current"? \$\endgroup\$
    – The Photon
    Apr 27, 2023 at 17:42
  • \$\begingroup\$ When the skin effect occurs, the distribution of current isn't a brick wall, ie 100% of the current is inside 1 skin depth, uniformly distributed, but it's Gaussian. There's a nice graph of current density vs skin depth here microwaves101.com/encyclopedias/… \$\endgroup\$
    – LordTeddy
    Apr 27, 2023 at 17:46
  • \$\begingroup\$ It's not gaussian, it's an exponential fall-off. And I expect (but should probably do the math to re-check) that that conveniently results in the overall effective resistance being equivalent to having a uniform distribution with a thickness of one skin-depth. \$\endgroup\$
    – The Photon
    Apr 27, 2023 at 17:46
  • \$\begingroup\$ Oh yes, my mistake. Still, it's non-linear, hence why I don't think skin depth x width is a good approximation \$\endgroup\$
    – LordTeddy
    Apr 27, 2023 at 17:51

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