Bends in transmission lines

Updated December 16, 2006

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New for December 2006! Here we will review some ways to minimize the effects of bends in transmission lines, by mitering or curving transmission lines.

Overview

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!

Whenever 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 w/4.

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

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!

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:

If 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!

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)

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:

0.5<=W/H<=2.75

2.5<=Er<=25

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.

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!