Comment by em3rgent0rdr
1 year ago
When spice simulation isn't deep enough... Very educational to show how circuit elements work "under the hood"...for example the LC example doesn't use an L element and a C element as building blocks, but rather it is the two metal plates in close contact which form the bulk of the circuit's capacitance and it is the loop of metal itself which form the inductance.
I wonder how they simulate EM in only 2 dimensions.
I also wonder why the simulator only allows to show E and D fields, and not H and B.
I don't pretend to know what this simulation is doing, but for the record, electromagnetism works just fine in 2D. You might be thinking "but magnetic fields are intimately tied to cross products, which only work in three dimensions." But you can set up the equations of electromagnetism just fine either using differential forms or bivector magnetism (https://arxiv.org/abs/2309.02548), and it works in any dimension you'd like. (The cross product version is really a narrow and sometimes misleading special case.)
Possibly related: there are options to "View B" and "View H" in the scalar dropdown, not in the vector one. That may be closely related to the fact that in two dimensions, the magnetic field has just a single component. Whether you describe is as a 2-form or a bivector, the magnetic field is an antisymmetric rank-2 tensor: an antisymmetric matrix. In 3D, that means 3 independent components, and there's a one-to-one mapping to vectors (more or less). But in 2D, an antisymmetric matrix has just one independent component. (And in 4D, it's got six: this is precisely the relativistic electromagnetic field tensor, that in 3D splits into an electric part and a magnetic part. My paper has more details.)
That's exactly right! In my simulation quantities like E and J are vectors with x and y components. In contrast B can be thought of as a vector (or bivector, technically) pointing in the z direction, but since it it only has one component it's simpler to just lump it in with the other scalars. (Aside: Having the simulation be in 2D brings in some interesting toplogical restrictions on circuits).
- Brandon
Thanks but I was thinking more about how fields drop off in 2D space versus 3D space. Simple electrostatic example: consider a 1D string of identical resistors. Voltage drops linearly as you go along this string. Now consider a 2D grid of resistors: voltage does not change linearly anymore if you move between two points (current will move in a more complicated spread-out pattern). So the dimensionality changes how fields behave.
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