From 69675cb1768c59ee1ebcc1d7a3551220084d6072 Mon Sep 17 00:00:00 2001 From: David Preiss <davepreiss@gmail.com> Date: Thu, 12 May 2022 16:59:11 +0000 Subject: [PATCH] Update README.md, images/diySim1.png --- README.md | 68 ++++++++++++++++++++++++++++++++------------- images/diySim1.png | Bin 0 -> 177196 bytes 2 files changed, 48 insertions(+), 20 deletions(-) create mode 100644 images/diySim1.png diff --git a/README.md b/README.md index a8b2468..808cca9 100644 --- a/README.md +++ b/README.md @@ -1,3 +1,51 @@ +# Electromagnetic FEA Simulations + +We can create rudimentary 2D and 3D FEA simulations of current carrying wires and permanent magnets in free space using only two equations: + +For current carrying wires we use [Biot-Savart](https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law): + + + +Current carrying wires get boundary conditions, and in this case a single value representing the _Idl _term. It is assumed that all wires are pointing directly into or out of the page, so the cross product _Idl_ x r_hat' term will always use an angle of 90 deg. Also note that we are calculating magnetic field strength H and not flux density B, despite the equation above, so no u_0 term. + +For permanent magnets we can use the equations for point source [magnetic dipoles](https://en.wikipedia.org/wiki/Magnetic_moment): + + + +Magnetic dipoles get boundary conditions and a magnetization vector m, which can be applied to a magnetic region at any angle and magnitude within the XY plane. The dirac delta term is applied only if calculating magnetic field strength [inside a magnetic material](https://en.wikipedia.org/wiki/Magnetic_moment#Internal_magnetic_field_of_a_dipole). + +The rest is surprisingly simple and can be implemented in about 100 lines of code in Python. We can make a numpy array that computes the effect of any dipoles or current carrying wires on every location of a uniformly spaced grid of "elements." Each element is assigned a field strength in X and Y, and a material property such as air, magnet, or wire. By iterating across every field-producing element's effect on every other element, we can create vector fields representing magnetic field strength H like the one below, which shows a permanent magnet (red bounding box) with m pointing in -X, and a current carrying wire (green bounding box) with current flowing out of the page. + + + +1) First let's note that there appears to be divergence on the left and right bounds of the permanent magnet where the field changes direction. I initially assumed that this was incorrect, but it turns out [H fields are indeed divergent](https://en.wikipedia.org/wiki/Magnetic_field#H-field_and_magnetic_materials) at the boundary of a magnet, and [Gauss's law for magnetism](https://en.wikipedia.org/wiki/Gauss%27s_law_for_magnetism) applies to magnetic flux density B, but not field strength H. This should really be apparent from the fact that H = B/u_0 - M, where the magnetization vector M is just a uniform vector field that sources from one side of the magnet and sinks into the other, as shown in the image linked to above. + +2) The field strength inside the permanent magnet appears to be several orders of magnitude stronger than outside the magnet, and the field strength vectors are all extremely parallel (this is the result of that dirac delta term noted earlier). This is actually incorrect as noted by Erik, and the reason is that the magnetic dipole equation is not valid for calculating the field strength contribution of an element to itself (r must be significantly greater than the radius of the Bohr magneton or else the dipole equation suggests a field strength approaching infinity as we divide by r=0). With that said, the above simulation should be valid for field strength outside of permanent magnets + +3) Solving for systems with ferritic or "magnetizable" materials requires iteration and annealing, and therefore this simulation tool is limited to materials with permeability of free space. A magnetized ferromagnetic material proceeds to produce an H field of its own, which in turn changes the H field generated on the first solve from any wires or permanent magnets. This process must then be iterated through until the solution converges to some acceptable degree. + +4) Finally one additional thing worth addressing is that we have not yet seen a reason for commercial EM FEA modeling systems to require an infinite ["air domain bound"](https://www.comsol.com/blogs/course-modeling-electromagnetic-coils-in-comsol/) to guarantee accuracy. This suggests that their underlying physics engine is doing something completely different, and it would be interesting to try and address that moving forward. + +## Attempt 2 - Magnets as Current Sheets + +Lots of improvements to make after several very productive conversations and whiteboarding sessions with Erik⦠+ +# Supplementary Materials + +## 2D Simulations + +#### Infinite Wire Approximation + +Things get more complex when trying to extrapolate these 2D simulation results to 3 dimensions. For example, Biot-Savart tells us the contribution of the infinitesimal wire length dl, but what if the wire extends 5cm into the screen? In that case the sum of contributions from each element dl all contribute to the field strength at point A, but with increasing effective radii, and decreasing angle theta, which results in the cross product dl x r_hat also decreasing in magnitude. It turns out that if we integrate the contribution of an infinite wire on a single point A, we scale Bio-Savart by a factor of 2r. For small distances r relative to the length of the wire, this is a valid assumption, where true 3D FEA would likely be necessary for better resolution. + +#### Coil Approximation + +In the case of a coil, the distance r remains constant for an entire circumference of wire (assuming it isn't spiraling), and therefore we can scale Biot-Savart by + +<p id="gdcalert4" ><span style="color: red; font-weight: bold">>>>>> gd2md-html alert: equation: use MathJax/LaTeX if your publishing platform supports it. </span><br>(<a href="#">Back to top</a>)(<a href="#gdcalert5">Next alert</a>)<br><span style="color: red; font-weight: bold">>>>>> </span></p> + +. This is also true for points not at the center of the coil, as the sum of distance to the edge of a coil is constant*** not sure about this. + # Motor Questions This project attempt to answer some non-intuitive motor design questions through simulation, fundamental equations, and hopefully some homemade discrete 2D EM (static) FEA. @@ -37,23 +85,3 @@ With that said, 3.5mm corresponds with the ID lying almost exactly on the center Again a sweep for optimal torque producing rotor displacement was run, analogous to load angle (max when 90 deg but unclear for this single pole linear displacement case). See CAD above for exact dimensions. Interestingly with this new ID, the optimal torque producing location has shifted up to 3.5mm. This suggests that iteration after tweaking certain parameters will be necessary to really converge on an optimal solution.  - -# EM Simulations - -Towards making a simple 2D simulation tool to answer some of these questions, it seems as though the best place to begin would be to simply model conductors with [Biot-Savart](https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law): - - - -and (point source) [magnetic dipoles](https://en.wikipedia.org/wiki/Magnetic_moment) with: - - - -Although for the latter it is not clear if a monolithic magnet can indeed be modeled as an array of infinitesimal magnetic dipoles, or if more complex math is required to represent what happens at the corners of a rectangular magnet for example. - -Note that it's necessary to model everything with permeability of free space, and if different material permeability are included it will be necessary to converge on a solution with annealing as discussed below. - -### A note on including materials with changing permeability - -The math tells us that B = H x u, which makes it tempting to think that only the magnitude of our B field will change in comparison to H. 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