An important secret about transmission lines

-December 08, 2013

In my So you think you understand transmission lines, I offered a seeming paradox about a property of transmission lines. If you look at the input impedance of a uniform transmission line, open at the far-end, it looks sort of like an LC circuit (Figure 1).

Figure 1. Input impedance of a transmission line alternatively open and shorted at the far end compared to the impedance of an L or C model.

The values of the inductor and capacitor that approximate the behavior of the transmission line so well are simply related to the characteristic impedance of the line, Z0, and the time delay of the line, TD, by

When we now take the impedance of the series LC circuit model, and compare it to the impedance of the transmission line, they don’t match so well. This result is shown in Figure 2.

Figure 2. Comparison of the input impedance of a single LC section model and an ideal transmission line.

Using exactly the same L and C values that worked so well at low frequency, we see that the self resonant frequency of the LC model does not match the minimum impedance dip of the transmission line. Why?

This is a bit of a trick question, but hits at the heart of a common mistake we often make when thinking about transmission lines.

A transmission line is NOT an LC model. It is sort of approximated by an LC model, but it is only an approximation. Of course, the more LC sections we use in the model, with the total L and total C being fixed, the better the approximation, but it is still only an approximation of a transmission line.

Figure 3 shows the comparison of the input impedance of a 16 section LC model and an ideal transmission line.

Figure 3. Impedance of a 16-section LC model and an ideal transmission line model. The approximation is better, but still is only an approximation.

The more sections we use in the LC model, the slightly better the approximation, but a transmission line is still not an LC lumped circuit model. It is a brand new, fundamentally different circuit element, in addition to R, L and C lumped circuit elements. We call it a distributed model to describe the property that there is a time delay for the signal to propagate between the input and output nodes of the circuit element.

If you continue to think of a transmission line as just a collection of LC elements, you'll miss the most important principle of how signals propagate on an interconnect. Those interconnects really see the distributed transmission line properties, not the LC properties of the interconnect.

Also see
So you think you understand transmission lines
Resistors aren't resistors
As edge speeds increase, wires become transmission lines
Analyze transmission lines with (almost) no math
A transmission line is always a transmission line
Smith chart and how to use it
The Impact of Conductor Surface Profile (Rrms) on Total Circuit Attenuation in Microstrip and Stripline Transmission Lines

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