Mitered Bends

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New for February 2014: we have more information on an alternative to the "Sharkbite mitre".

New for February 2011! We have some content on the equivalent electrical length of mitered bends in microstrip, thanks to Kevin, who introduced us to the "sharkbite mitre"!


Here we will review some ways to minimize the effects of bends in transmission lines, by mitering or curving transmission lines. What's the best way to bend a microstrip or stripline transmission line? There is no one single answer, and this causes a lot of disagreements at design reviews. In truth, it isn't the big deal that some engineers make it out to be, if you understand the two problems that bends create.

The first problem is that the discontinuity changes the line characteristic impedance, without compensation the bend adds shunt capacitance. But in reality the small capacitance that is usually a result doesn't change the circuit's performance very much.

The other problem associated with bends is can cause far more damage to the intended performance of a highly tuned circuit: the effective length of the transmission line becomes shorter than the centerline length. Electromagnetic waves like to take shortcuts!

Time for a Microwaves101 Rule of Thumb!

Mitered BendsWhenever you bend a transmission line, to model the length of the line you should simply ignore the extra length that is added by the bend. We'll cover our butts by saying this is just an approximation, if the effective length of a line is critical to the design success, you'd better simulate it in Sonnet!

Example 1: if you use a curved bend of ninety degrees, the effective length of the line is approximately the centerline length minus Mitered Bends w/4.

Mitered Bends


Example 2: to model the length through a corner bend, simply ignore the length of L2.

Mitered Bends

Sorry, we have little experience with the length calculation of mitered bends, so we're not going out on a limb and claim this rule of thumb works in that case! Why don't we have experience here? Because we almost never bother to use them! But wait, we have a new rule of thumb for the case of mitered bends, see below.

Corner bends

More on this later...

Radiused bends

From Harlon Howe's book book on Stripline (see our book page), we can arrive at a rule of thumb for curving transmission lines:

Mitered BendsIf you use a radius greater than three times the line width, you will have a transmission line that is almost indistinguishable in impedance characteristics from a straight section.

Mitered bends

Before we continue, let's review the many ways the word "miter" (or "mitre") is (are) used. In the good old U.S. we prefer the "miter" spelling, in the more ancient tea-sipping, bowler-wearing U.K. they use "mitre". In both cases, if you look up the definition in a dictionary, you will see only two meanings, neither of which is what microwave engineers are talking about when they say "mitered bends". Miter can mean the ridiculous fishhead-shaped hat that a bishop wears (think about a chess set), or the manner in which two rectangular pieces of material (boards, tiles, shingles, etc.) are beveled so they can be joined together to create an angle with no gaps. That being said, pay attention below to see how we use the word, and maybe someday this use will be added to the dictionary where it belongs. Note that what started out as an adjective (mitered bend) is now an accepted (at least in the microwave community) noun.

When you make a ninety degree bend in a transmission line you add a small amount of capacitance. "Mitering" the bend chops off some capacitance, restoring the line back to it's original characteristic impedance. The image below shows the important parameters of a mitered bend.


Microstrip miter compensation

The "optimum" mitered bend equations for microstrip were found empirically way back in the 1970s. Here's two references:

R.J.P. Douville and D.S. James, Experimental Characterization of Microstrip Bends and Their Frequency Dependent Behavior, 1973 IEEE Conference Digest, October 1973, pp. 24-25.

R.J.P. Douville and D.S. James, Experimental Study of Symmetric Microstrip Bends and Their Compensation, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-26, March 1978, pp. 175-181.

Now it's time for the math that Douville and D.S. James came up with: For a line of width W and height H,

D = W* SQRT(2) (the diagonal of a "square" miter)

X= D* (0.52 + 0.65 e ^ (-1.35 * (W/H)))

A = ( X- D/2) * SQRT(2)

A missing bracket was added to the equation for "X" on November 16, 2011, thanks to John. Notice the result that the miter is NOT a function of substrate dielectric constant. Who would have guessed that? But the range that the accuracy of this calculation is valid is limited to:

0.25<=W/H<=2.75 see (Thanks for the correction, Kevin! and also for pointing out that the upper limit can be exceeded without worries.)


There's a spreadsheet in our download area that does this math for you, go check it out! Our spreadsheet download does all of this for you, and even makes a plot of the results. Here's an example for H=10, W=10. The higher the W/H ratio, the more drastic the miter becomes.

Equivalent length of a microstrip mitered bend - introducing the Sharkbite miter

The following was supplied by Kevin in February 2011... many thanks!

I noticed that you had a statement on the "mitered_bends" page: "Sorry, we have little experience with the length calculation of mitered bends, so we're not going out on a limb and claim this rule of thumb works in that case! Why don't we have experience here? Because we almost never bother to use them!" Well, I am designing a very small module where board space is at a premium, so I don't want to use big gradual curves to avoid using miters. I also was interested in the precise delay through a miter. So I whipped up a Sonnet simulation and found something interesting that I thought you might want to share with your readers!

So the first thing I did was simulate (on a fairly fine grid) the classic Douville and James miter. It worked very well. (A useful simulation check.) I then was curious to find a coarser grid pattern that would be a good approximation of the classic miter, because I wanted to put the miter together with a bunch of other stuff in a bigger simulation, but the fine grid would not allow much else within the computational restrictions of the freeware version of Sonnet. So I started puttering around and found a pattern that appears to work better than the classic miter, and on a very coarse grid! The figure below shows the results at 7.8 GHz (click to enlarge it). I also simulated (but did not include the results) from 1 to 15 GHz and the return loss improvement was consistent over the range. (I realize that >30 dB return loss is plenty for all but the most finicky applications, but it's intriguing that this crude pattern is so close to perfect!)

A common rule of thumb for feature size being relevant to the situation is lambda/20. With this criterion, the blocky features should have a fairly minor effect up to about 32 GHz. This style of miter needs a name. I propose "sharkbite miter". (Second place was "stealth fighter miter" which has a catchy rhyme to it. Third place was "Batman miter", no panache.)

From "The actual range of parameters for which Douville and James present evidence is 0.25 < W/H< 2.75". The equation approaches an asymptote and the calculated result is virtually the same for any W/H>2.75. It seemed to work pretty well in my case where W/H=7.1

I used the Rogers propagation simulator to estimate the delay of a similarly sized (straight) microstrip. I adjusted the length in the Rogers simulator until the phase delay equaled the phase delay observed in the Sonnet simulation. The excess delay beyond the inside corner path length is conveniently almost exactly equal to half the width of the trace.

Time for another Microwaves101 Rule of Thumb!

Mitered Bends From the above simulation, it appears that both the classic and sharkbite miters cause an extra delay equal to W/2 (W is the trace width), so the total effective path length is L1+L2+W/2. The excess delay beyond the inside corner path length is conveniently almost exactly equal to half the width of the trace.

An alternaitive sharkbite miter... and a free EM tool!

This came from Darrell in January 2014... also thanks!

I came to your site to get the mitered bend formulas, saw the sharkbite miter and decided to try simplifying it even further. I was surprised how well simply taking a square out of the corner worked. That makes it easy to simulate as well as get into a PCB without doing the whole DXF import thing.

I did later find that with thinner dielectrics the remaining width in the corner gets too small for the usual PCB design rules and ended up going with the regular miter. I actually wrote a script that builds the OpenEMS model and creates a PCB footprint file avoiding the DXF conversion.

I am a former HFSS user, but have not wanted to put out that much cash since starting out on my own. OpenEMS is a plug-in for Matlab and you build models by writing Matlab scripts with commands like AddBox, AddCylinder and AddPolygon. That is a bit painful for a new user but I now find it faster to work with than the HFSS modeler. The port implementation is somewhat lacking, and the absorbing boundaries aren't quite right.

I feel like OpenEMS is accurate given the right inputs, but have not yet built anything from it. I did model one of my old bandpass filter designs and it matches reasonably to production units. I'll find out in a few weeks when I get a board fabricated which has 7 different interdigital bandpass filters on it...

Here's an image of the square corner, for reference, drawn in OpenEMS. In this case and the next case, ER=3, H=20 mils (508um), and the straight fifty ohm line is 1.1mm wide.

Mitered Bends


For reference, here is what a square corner looks like (assuming ER=3, H=20 mils). Blue is S21 and green is S11.

Mitered Bends

The line is 1.1 mm wide; the square removed is 0.9 mm.

Mitered Bends

Here's the performance of the miter as shown:

Mitered Bends

Just for fun, I dropped it to a 5 mil substrate and it does work decently at 110 GHz. Of course, it's not readily manufacturable with that tiny corner. Thin film on quartz, perhaps.

Mitered Bends

Note from Unknown Editor: There are many ways to skin this cat, all that is necessary is to remove enough metal at the corner to get rid of its offending capacitance. If you are interested in very high peak power handling, you might want to consider compensating the corner without creating sharp corners...

Stripline miter compensation

Here's references for optimum stripline miters:
Harlan Howe, Jr. Stripline Circuit Design, Artech House Inc., 1982.

G. Matthaei, L. Young and E.M.T. Jones, Microwave Filters, Impedance-Matching Networks and Coupling Structures, Artech House, 1080, pp. 203, 206.

Coming soon!


Author : Unknown Editor