Non-90deg mitered bends

Question

Is there an approximate equation giving the optimal miter for a non-90degree mitered bend? In particular, I'm trying to determine the correct miter for a 45degree bend. There are a number of resources giving the correct miter for 90degree bends (e.g. Microwaves101), but I haven't found anything for other angles. Is this because angles less than 90degrees have a negligible Z0 discontinuity effect?

I can simulate this in OpenEMS (and I probably will), but it's nice to start from an approximate equation.

I've come across a design that does miter 45degree bends. The result looks like this.

The original trace width is 0.34mm (corner to miter here, as shown in picture, is 0.3mm). The height to the ground plane is 0.17mm. My max signal frequency is 6GHz. Again, I could use this as a starting point for an openems simulation, but it would be nice if there was an easy way to make a rough calculation first.

Answer 1

At 6 GHz you really don't have to worry about 45 deg corners. I'm going to look at this like a signal integrity (SI) problem.

A square corner is going to add a bit of capacitance to the trace at the corner, which impacts Zo of the line. How much this affects your signal depends on the amount of capacitance added (which depends on the line width) and the frequency of the signal. Now Dr. Eric Bogatin (an industry recognized SI expert) has a rule-of-thumb that says "A [square] corner will affect the signal when the (line width in mils) is greater than (5× the rise time in picoseconds)."

6 GHz translates to 160 ps in time. An estimate for the rise or fall time of such a signal (I know it's a sine wave) is 1/3 that, so around 53 ps.

5x53 = 265. Your line (trace) width is 13.6 mils. 13.6 << 265. Therefore you don't have to worry about a square corner at 6 GHz.

If you want to tidy things up a bit, just do what you can to reduce the added capacitance at the corner.

Answer 2

I wasn't able to track down an equation, but the Transmission Line Design Handbook (Wadell, 1991) (sec.5.5.3) contains a table of optimal arbitrary angle miters for microstrip width to height ratios and bend angles. Here's a table giving the values (taken from here):

and here's the bend diagram and corresponding equation, taken from Wadell:

\$M=\frac{x}{d}\$

You can use bilinear interpolation to estimate the optimal miter for other width-height ratios between 0.5 and 2 and angles between 0 and 120 degrees. For example, this gives an optimal miter for the OP case of 19%, which corresponds to the measured image dimensions.

Unfortunately, I don't know of a way to calculate miters for width-height ratios outside this range. Still, hopefully this can be a useful starting point for a proper EM simulation.