Relation between transmission line and circuit impedance

Question

What is the relation between the impedance of lumped components and the characteristic impedance of transmission lines?

I have noticed that in some cases transmission lines are loaded with lumped components (e.g. for matching) and in some other cases they are loaded with other transmission lines (characteristic impedances).

I don't understand how the two impedances can be interchangeble, when, for example, in circuit theory two series impedances are summed up, while in transmission lines only the reflection coefficent is considered.

Answer 1

I have noticed that in some cases transmission lines are loaded with lumped components (e.g. for matching) and in some other cases they are loaded with other transmission lines (characteristic impedances).

In that case, you have seen cases where the designers were trying to do specific things, and did them with the most appropriate components that were to hand.

Ideal components are generally considered to act 'at a point'. It's a fiction of course, but a useful one until they get large, or the frequencies get large.

You can make a transmission line (up to a certain frequency) out of a ladder of series Ls and shunt Cs.

You can make a small L (up to a certain frequency) with a shorted length of transmission line. You can make a small C (up to a certain frequency) with a short length of open circuit transmission line.

Depending on what's actually needed, designers can flip between lumped and distributed components. All have terminals, and create some sort of relation between the voltage across them and the currents through them. At the end of the day, all the designer wants is a network that provides the right V and I performance, however that is implemented.

Answer 2

trivially, Zo = [sqrt ( L / C)

which comes from constants needed in the wave equations.

In early work (the 1880s) Oliver Heaviside and his brother showed loading coils, placed in parallel between the two wires of a telegraph system, would usefully expand the bandwidth.

Answer 3

Impedance is nothing more than the ratio of voltage to current, in the frequency domain, of a two-terminal circuit element. In a transmission line, even though the impedance properties that govern it are distributed, the interactions are all local. That is, one point of the transmission line only interacts with the point immediately next to it. There is no long-distance interaction between one part of the line and another.

The effects of a length of transmission line section with some termination all come together to have a net effect on the voltage-current relationship at whatever point you are examining as the 'input' to that section. This relationship is expressed as a complex impedance, just as you would express the voltage-current relationship of a lumped element.

Answer 4

The relation between the two "models" can be made simply, by simulation, for one section (1 meter of classic coaxial cable 50 Ohm, 101 pF/m, v=2/3*c) RG-xx.

Here is an example of a relative comparison in the matched case. What is done can also be evaluated with a tolerance precision if mathematical calculus is done.

Example : for 5 % error on output voltage, use lumped until 10 MHz is ok. So for f < (200 MHz/20).