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Propagation of travelling waves on cables may sometimes be difficult to apply and understand. There is a method called in french "abaque de Bewley" or "diagramme de Bergeron" (in hydraulics). When I try to google it, I don't find a lot of information.

So, I would share this calculation method which can be made easily also by hand. It is widely used in "reflectometry" for finding some defaults on cables ... and where they are. Used for controlling Zc in continuous production of cable manufacturing.

Used also in optics fiber with interesting applications.

This method can be used everywhere in physics where propagation is concerned. Where there is forward wave and reverse wave propagation demonstrated in mathematics but also in other departments. Laplace transforms can also be used for calculation. What is needed to understand is only the definition of "characteristic impedance" of a line, when it is a resistance calculation is easy, and some important definition (speed of wave in that line).

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    \$\begingroup\$ From a quick google search, it seems that the name in English is "Bewley lattice diagram". \$\endgroup\$
    – The Photon
    Jul 1 '21 at 20:22
  • \$\begingroup\$ Thanks for that information ... I did not found in french :) Only in wikipedia where i made an insert. Have you noted an error in my text ? \$\endgroup\$
    – Antonio51
    Jul 1 '21 at 20:25
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    \$\begingroup\$ They are travelling waves until steady state then become “standing waves” \$\endgroup\$ Jul 1 '21 at 20:40
  • \$\begingroup\$ Yes. It is the case also with sinusoidal waves, after a transient wich can be "long" or "short". Ohm's law must be verified, at end. \$\endgroup\$
    – Antonio51
    Jul 1 '21 at 20:42
  • \$\begingroup\$ What you have written is basic transmission line theory and fairly widely available and understood. I realize that you believe it to be some kind of revolution in your acquisition of knowledge but, it is really quite common-place to many engineers. If you are going to talk about lattice diagrams why don't you embed a picture of one there are plenty of them on the internet. \$\endgroup\$
    – Andy aka
    Jul 2 '21 at 7:55
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Simplified method of using line transmission theory, "abaque de Bewley".

The schematic is simple : Generator Ugen with internal Zin -> coaxial line of lenght=L, Zc -> Load ZL.

Zc: characteristic impedance of the line (generally resistive, example 50 Ohm coaxial)

Zin: operational impedance of the generator (for simplicity, resistance)

ZL: operational impedance of the load (for simplicity, resistance)

When we have this type of circuit, we can study it in the following way.

We first define an emission coefficient Ke at the generator. Ke = Zc / (Zc + Zin)

A generator reflection coefficient is also defined. Kgr = (Zin-Zc) / (Zc + Zin)

We define a reflection coefficient on the load. KLr = (ZL-Zc) / (Zc + ZL)

These are "voltage coefficients" definition.

examples:

for generator Zin = Zc, (matching Zc), Kgr = 0

for Load = short circuit, ZL = 0 KLr = -1

for Load = open circuit, ZL = infinity KLr = +1

for Load ZL = Zc, (matching Zc), KLr = 0

These coefficients can also be Laplace transforms. But the calculation can get complicated quite quickly if you want a complete diagram. It helps in any case for the explanation of the first waves. Simulation can be made for verifying what happened very easily using what is named a Tline.

We then proceed as follows: The generator wave will meet the line ... what is sent in the line is Uin = Ke * Ug

The Uin wave travels along the line, until it finds the charge. Propagation time.

The wave meets the charge and sees the "nature" of it. A return wave begins.

This starts again in the opposite direction after multiplication with an amplitude of U1r = Ug * Ke * KLr

Then U1r wave returns to the load ... or it undergoes a Kgr reflection and the generator therefore generates a wave whose value is U2g = Ug * Ke * KLr * Kgr which goes back to the load ...

etc ... U2r, U2g, U3r, ...

The visible voltage is then the “sum” of the wavelets generated over time… ..

Examples: case1 of an adapted load, case2 of an open circuit, etc ...

Case1 : the simpliest :) Ugen (step) = 2V*Heaviside(0) , Zin = Zc ; ZL= Zc

Wave sended in line : Uinput_coax = Ugen * Ke = 1V -> step 1 V sended in line.

Travelling at 2/3*c (c = speed of light, 20 cm / 1ns, calculate for length L=20 cm ...)

After 1 ns, wave meet load ... KLr = 0 -> no wave reflected -> end of travel, nomore calculation needed.

Volage at line input = 1V (t=0) ; Voltage at line output or on Load = 1 V (delayed 1ns) -> End of transient : Ohm's law must be verified. Zc not used ! Simple divider Ug * ZL / (Zin + ZL) = 1 V : Ok

Case2 : Ugen (step) = 2V*Heaviside(0) , Zin = Zc ; ZL= infinity (open)

Wave sended in line : Uinput_coax = Ugen * Ke = 1V -> step 1 V sended in line.

Travelling at 2/3*c (c = speed of light, 20 cm / 1ns, calculated for length L=20 cm ...)

After 1 ns, wave meet charge ... KLr = 1 -> 1 V wave reflected -> voltage on load : 1V + 1 V = 2V ( incident wave + reflected wave at t= 1ns)

The wave reflected return to generator -> travel 1 more ns

Meet generator : Kgr = 0 -> end of travel, Uin line = 1 V (start t= 0) + 1 V (t= 2 ns) = 2 V -> nomore calculation needed.

Voltage at line input = 2V (after t=2 ns) ; Voltage at line output or on Load = 2 V (after t=1n) -> End of transient : Ohm's law must be verified. Open circuit: Ok

Some questions, more examples ? I hope no errors in text :)

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    \$\begingroup\$ Hi Antonio, nice answer. If you want to, you can beautify math here by using MathJax. Here are some examples to help you get started. \$\endgroup\$
    – rdtsc
    Jul 1 '21 at 20:59
  • \$\begingroup\$ Thanks for MathJax ... I did not know ... If I have time ..., I will probably try with MapleSoft Software (licenced) because it can also make calculations on line, I think :) \$\endgroup\$
    – Antonio51
    Jul 1 '21 at 21:05

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