# "Rigorous" motivation behind the model of transmission line for microwave engineering

I am trying to understand under a rigorous enough point of view the model of transmission lines.

More specifically I am interested to this question under the context of microwave engineering.

Important precision: I am not an engineer so I might not understand technical temrs you might use. I thus would like to stick the closer to the vocabulary of physics in the answers.

If we consider a "long" transmission line, that current and voltage won't be uniform in it and we will be able to see propagation phenomenon.

Then, to model it we use the following model as explained here: https://www.microwaves101.com/encyclopedias/telegrapher-s-equations for example. For an length $dx$ of the line, we say that we have the following electrical circuit.

My general question is the following:

## What motivates rigorously this model for microwave engineering ?

This model has wider application than for microwave engineering but I would like to understand why it works in this context.

Thus, we will assume that the "+" line and the "-" line are composed of a single metallic layer. It is thus not a "big" cable in which you may have little ones inside (sorry for my limited vocabulary in electronics). Both + and - are really intrinsically single wires.

## More precise questions:

In understand the presence of the capacitor in the following way:

The + and - cable are separated by some insulator, we can model their connection as a capacitance (because metal - insulator - metal = capacitor behavior).

We can also include a resistor in this modelling (the air is a resistive material in a sense), which motivates G

Would you agree with what I say here ?

For the inductance now, I would say that it physically correspond to the magnetic flux between the + and - metallic "wires". Indeed as I said my wire are really single metallic layer so we cannot have some "intrinsic" inductance on those.

Would you agree with what I say here ?

For me the + and - wire should be symmetric with this description, so I don't see why we would'nt add as well a resistor and an inductance on the - wire ?

Why don't we add those extra resistor and inductance on the - wire ?

• G for air is very large. But for many solid dielectric materials, G becomes very important. Also, note that in real transmission lines, these parameters are also functions of frequency. Feb 5, 2020 at 4:46

For the inductance now, does it physically correspond to the magnetic flux between the wires ?

The inductance is what I’d call the leakage inductance i.e. that inductance that remains due to near full cancellation of fields from the opposing currents in the forward and return wires.

Why don't we add those extra resistor and inductance on the - wire ?

We could if we were considering twisted pair cable but, if we are considering coaxial cable it would be foolish to model the screen as an inductance because, the screen can be shown to possess zero inductance when the coax is driven correctly.

But why over complicate things if complications can be avoided.

Modification of question by op to say: -


Indeed as I said my wire are really single metallic layer so we cannot have some "intrinsic" inductance on those.

There sure is inductance in a sheet or layer.

More specifically I am interested to this question under the context of microwave engineering.

Well, if the RLGC model can be used then it applies to any frequency. However, the term "microwave" is not really clear enough to decide if the model is acceptable due to different modes of propagation.

• Maybe I should have started by this but I am more physics than engineering oriented so I need to make connection with maxwell equations to really understand the answers and connect to what I know. For me to define an inductance we need to talk at some point about a magnetic flux. We have L=Phi/I where I is the current of the wire, Phi the magnetic flux. Is the magnetic flux associated to the leakage inductance you talk about the one that is between the + and - wires ? Which area is associated to this flux physically ? Feb 4, 2020 at 18:57
• "if we are considering coaxial cable it would be foolish to model the screen as an inductance because, the screen can be shown to possess zero inductance when the coax is driven correctly." could you elaborate a little bit on this ? What do you call the screen ? Why do you mean by driving correctly ? Do you mean that you try to not impose too high current variation so that you can neglect this inductance ? Feb 4, 2020 at 19:03
• The net inductance is of a section of the pair of wires and not the individual wires. Half way between forward and return wires is zero flux. For coax, if you know physics then you know that the flux inside a tube of current is zero. You also know that at any distance beyond the shield (or outer screen), the magnetic fields from inner and shield cancel due to symmetry and equal but opposite currents. This means that flux exists only in the gap and is totally due to the inner core current hence the shield produces no net flux and therefore has zero inductance. Feb 4, 2020 at 19:28
• I am trying to understand your point. I agree that with a coaxial cable there will have a magnetic field only between inner and outer conductor (not outside the cable bc of symmetry). Now if I link the coaxial cable with my problem, you would say that the inner cable is the "+" and the outer is the "-" for example. Now I don't get your conclusion. For me we don't care "who" created the magnetic field in the "inside" zone of the coaxial cable. There will have a magnetic field between core and shield and it will induce an electromotive force. Thus why would there be 0 inductance ? Feb 4, 2020 at 19:55
• But in a more global vision, my problem is more related to study propagation of electric wave in long lines that are composed of single wire (it is for microwave physics). In such field we use the model of my question which only put an inductance on the + line. I do not get for which physical reason we wouldn't put as well an inductance on the - line. Feb 4, 2020 at 19:58

this model is what William Thomson could build using the differential equations of the day.

[ how does this simple sentence not explain the "motivation"?]