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Rule of thumb: Loop inductance/length in 50Ω lines

24 Mar 2014  | Eric Bogatin

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This rule of thumb estimates the loop inductance per length in all 50Ω transmission lines in FR4.

LLen = 8.3 nH/inch

At low frequency, a transmission line, shorted at the far end, looks like an inductor. After all, it is just a conductor in a funny shaped loop, current going down the signal path and getting shorted to the return path, making its way back.

This is illustrated in the figure which plots the input impedance of a shorted transmission line and the impedance of an ideal inductor. At low frequency, they predict exactly the same impedance.

Figure: Input impedance of a transmission line shorted at the far end and an ideal inductor.

To connect a transmission line with a total L and C value, we can approximate a real transmission line with an n-section LC model. Solving the circuit model, we get,

By taking the product, we get the total loop inductance as

This is very reasonable. If the length of the line increases, so should the total loop inductance. If the characteristic impedance increases, like the line width decreases, then the total loop inductance should increase.

The time delay is related to the speed of the signal in the material and the physical length:

For the special case of FR4 with Dk = 4 and the speed of light in air as 12 inch /nsec, the total loop inductance per length of any transmission line is

This says that ALL 50Ω transmission lines in FR4 have exactly the same loop inductance per length. If we make the line width wider, we have to make the dielectric thicker to preserve the 50Ω, and this keeps the loop inductance the same.

For example, a 50Ω line 2 inch long has a total loop inductance of about 16.6 nH.

Now you try it:

1. How much loop inductance does a surface trace have that is 0.2 inch long?

2. What is the total loop inductance in a long trace on a board, 5 inches long?

About the author
Eric Bogatin is Signal Integrity Evangelist at Teledyne LeCroy.

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