Where does definition B come from? Charge density and Voltage don't even have the same units. [C/m3] vs [J/C]
Yes the units don't quite match up because they are not the same thing.
It comes from removing the effects of external fields. As you have seen in this thread circuit meshes don't handle external fields. The components that make up a mesh model act as if they have zero physical size (This includes all wires being 0m long). This means that no matter how strong a magnetic or electric field the circuit is exposed for there will never be any observable field gradient across a component. A loop with a area of 0 m2 can't have EMF induced in it no matter what you do.
So when external fields have no effects the only thing remaining pushing electrons around is the internal electric field caused by charge separation and this is basically different charge density between two points.
This section of the Wiki page for voltage explains the conflicting definitions:
https://en.wikipedia.org/wiki/Voltage#DefinitionThe kind of voltage i am talking about in definition (B) is what they call "Definition via decomposition of electric field". You can read up on the details by flowing the links. It uses a different way of treating the magnetic field to make charge separation via EMF visible as a voltage. This solves all the multiple result ambiguities of definition (A) and always gives a single number for every point in the circuit. By doing this it also causes a voltage to appear on a wire when in a changing magnetic field.
This is the source of argument for most of the recent posts in this thread. The answer to the above diagrams depends on what definition you are using.
I disagree. You can split the total mutual inductance M of the loop into two strings of as many inductors as you want in spice. The value you measure in spice will not be the actual scalar voltage potential between the ends of the resistors (which is approximately zero as measured by the voltmeter). Lumping can't be done in this kind of circuit in spice without creating false outcomes.
Well i was comparing my spice simulation results to experimental data made on youtube and on my own bench, they seamed to match pretty closely. So it does appear to work fine for circuits discussed here. If it is wrong then it looks like it takes a different or more complex circuit to cause it to break. Lumping is very commonly used even in RF circuits, its just a matter of lumping it correctly.
Do note that SPICE like all other circuit analysis uses the second definition (B) for voltage. The equation used for that is seen on the Wiki article above.
I see now that you're trying to model the mutual inductance of the "outer loop" i.e. the path formed by the two measurement loops, but not going through R1 and R2. You've arranged the coupling dots in a way that the inner inductors and outer inductors cancel each other out in a way that satisfies there being no flux coupling in the two measurement loops.
Yes the coupling dots are very important. The dots indicate that both loops are going clockwise around the center. So if you go around the loop you will always see them pointing the same way. If the dot points the other way that would signify the wire turning around and going counterclockwise, its allowed to do that if it wants and will still give correct results, however this circuit does not have the wire changing direction around the loop so they all point the same direction.
You have indeed identified correctly how this model works. Because the wire going to the resistor and the wire going to the voltmeter flow the same path this means they get the same voltage induced in them. Since you need to go clockwise to get to the circle midpoint and then back counterclockwise to get back to the meter the voltages are opposite in sign and they subtract out. This is called bifilar winding and is widely used for removing inductance when its not desired.
If you use the second definition (B) of voltage and as described in the Wiki article use the "magnetic vector potential" to think about the magnetic field you will find the exact same behavior in the real life circuit.
Its just a different way of thinking about it. Results are identical in the end, just how you get to them is slightly different.