2D magnetics

Currents in coils and the magnetization of permanent magnets set up a magnetic field, which iron guides along its paths and air gaps resist. The flux density decides whether the iron saturates, the field in an air gap the force on an armature, and the flux through a coil its inductance. Formulas give these for a few shapes, such as a long solenoid or a magnetic circuit with a narrow gap, as long as the iron is far from saturation. For a real cross-section with its leakage flux, its fringing at the gaps and its saturation, they follow from a solution of the magnetic field equation. The tool meshes the part and solves that equation with the finite element method, and shows the flux density, the field intensity and the flux lines.

Planar or axisymmetric

A planar model stands for a long part, such as a laminated core, a motor section, a cable or a bus bar, in which the field lies in the plane of the drawing and the currents run across it. The Length gives the part's length, the depth of a laminated stack for example, and the inductances and forces hold for that length and also per meter. An axisymmetric model stands for a part that is round about an axis, such as a solenoid, a pot magnet or a coil on a round core. The drawing is then the half-section right of its y axis, which is the axis of rotation, the currents run around the axis, and the results hold for the whole part. With 3D on, the drawing shows the part revolved through half a turn, seen at an angle, with the field and the flux lines on its cut face on both sides of the axis.

Static or time-harmonic

The static analysis solves for direct currents and permanent magnets, with iron on its B-H curve. The time-harmonic analysis solves for currents that alternate at one frequency, with the eddy currents they induce in conductors. See the section on the time-harmonic analysis below for what changes there.

Regions and materials

Every closed region of the drawing gets a number, F1, F2 and so on, and a material from the list. A loop inside another on the same layer is a hole, and a loop on a layer of its own is an insert, a region of its own material. Regions that touch must lie on layers of their own, and holes are filled with air.

• Air and the non-magnetic conductors, copper, aluminum, brass and stainless steel, have a relative permeability μr of 1. Their electrical conductivity σ counts in the time-harmonic analysis.
• Electrical steel, structural steel, cast iron and soft ferrite follow typical B-H curves, so their permeability falls as they saturate. B-H curve from a file takes the curve of a data sheet as a CSV file with the columns B in tesla and H in A/m. Linear takes a constant μr.
• The permanent magnets have their typical remanence Br and recoil permeability μr, which the Magnets table shows and lets you change. Each magnet is magnetized along an angle from the x axis, or radially out of or into the drawing's origin, as in the rotor of a motor.

Circuits and windings

A circuit carries a current, and its windings are the regions that the current flows through. Each winding has its circuit, its number of turns and the direction of its current, out of the drawing or into it, and a coil's current density is the turns times the current over the region's area. In a planar model a coil appears twice, as the side where its turns go into the drawing and the side where they come back, and both belong to the same circuit, so that its flux linkage and inductance come out right. A circuit whose currents do not add up to zero in a planar drawing has its return path outside, and its inductance is left out. In an axisymmetric model a coil is a single region, since its turns run around the axis. With several circuits the results also give the mutual inductances.

Edge, hole and interface numbers

The outline's edges get E1, E2 and so on from the lowest left corner around the part, every hole one number for its whole boundary, H1, H2 and so on from the left, and every edge between two regions I1, I2 and so on. The Flux parallel table refers to these numbers, and the drawing shows them, with each edge's length on hover. The Propagation column carries a row on along the outline, with planex2d's selection propagation.

• Minimal keeps the edge alone.
• Coincident adds the edges joined to it end to end, which on a closed outline is all of it.
• Tangent adds the edges that continue it tangentially, such as the lines on either side of a fillet.
• Curvature adds the edges that continue its curvature as well.

Boundaries and open space

With Open space on, the part sits in air that reaches to infinity, as a core or a coil does on a test bench. The tool adds an air region around the drawing, and around that a thin layer whose coordinates a map stretches out to infinity, where the field vanishes. The result is then that of all the space outside and does not depend on the size of a box. Without it, the drawing's outline bounds the field. Edges in the Flux parallel table hold the vector potential at zero, so the flux runs along them and none crosses, as at the edge of a magnetic shield. The other edges are open to the flux, which crosses them at right angles, as on a plane of symmetry between two poles. In an axisymmetric model the axis is always flux parallel.

Forces and torque

The Force column marks the regions of a moving part, such as the armature of an electromagnet, the plunger of a solenoid or a plate held by a magnet. The tool integrates the Maxwell stress tensor over the air around the part, weighted by a function that falls from 1 on the part to 0 away from it, and gets the force on the whole part from it. In a planar model the result is the force in x and y and the torque about the drawing's origin, and in an axisymmetric model the force along the axis. The moving part needs air around it, a region of air or the open space, and must not touch iron that does not move with it.

Measurements

The Measurements group, folded below the tables, adds values for the regions and edges you choose to the results.

• A region gives its mean flux density and mean field intensity and its highest flux density, for iron with the relative permeability there, and in the time-harmonic analysis its losses and skin depth. A region drawn in an air gap thus gives the field in that gap.
• An edge, a hole or an interface, carried on with the Propagation column, gives its length, the flux across it, the mean normal flux density and the magnetic voltage ∫H·dl along it. The flux across is per the Length in a planar model and through the surface of revolution in an axisymmetric one, and ∫H·dl around a closed loop is the current inside it.

Results

For each circuit the results give its flux linkage and its inductance. With iron on a B-H curve the inductance is the flux linkage over the current at that current, so it falls as the iron saturates. They also give the force and the torque on the moving part, the energy stored in the field and the highest flux density with its place. The PDF and the CSV file also list every region with its mean and highest flux density. The drawing colors the field by the flux density, its components, the field intensity, the flux function, the relative permeability, the current density or the energy density with the list beside Results. The flux lines are lines of the flux function, as many as the number beside Results, with the same flux between each pair of them, so they crowd where the field is strong.

At a sharp inner corner of iron the flux density of a magnetic solution grows without limit, so the highest flux density there depends on the mesh. The accuracy estimate in the Solver group shows this as a large change, and the mean values of the regions are not affected.

The time-harmonic analysis

In the time-harmonic analysis all currents alternate at the frequency given, and the currents are their amplitudes. Every conductor with a conductivity carries eddy currents, and a winding is either stranded or solid.

• A stranded winding is a coil of thin wire, whose current spreads evenly over its area. Its own wire resistance depends on the wire and is not part of the results.
• A solid winding is one conductor, such as a bus bar or a cable core, whose current the skin and proximity effects push to its surface and away from its neighbors. Its turns are 1.
• A conductor in no circuit carries eddy currents only, with no net current in a planar model, and in an axisymmetric model as a ring around the axis.

The results give each circuit's resistance and inductance at that frequency, the losses in every conductor and in total, and the skin depth of its material. The drawing adds the loss density, and the fields are their amplitudes. Iron is linear in this analysis, with the initial permeability of its curve, electrical steel counts as laminated with no eddy currents across the stack, and magnets count with their recoil permeability only, since a constant magnetization adds nothing at the frequency.

Simplify outline

Drawings often carry arcs and holes as chains of short lines, which give the outline many edge numbers. With Simplify outline on, planex2d fits arcs to such chains, within a ten-thousandth of the part's size, before the part is numbered and meshed. Turn it off to analyze the outline exactly as drawn. Where the fit misses an arc, the contour checker fits arcs with a tolerance and an edge length of your choice, closes gaps and removes tiny edges, and saves the cleaned drawing for the magnetics tool.

Units

Lengths are in millimeters, currents in amperes, frequencies in hertz, conductivities in MS/m, the flux density and the remanence in tesla, the field intensity in A/m and kA/m, the current density in A/mm², the energy density in J/m³ and the loss density in W/cm³. The flux function is in mWb/m in a planar model and in mWb, the flux through the circle about the axis, in an axisymmetric model. A file is read in the unit it names and converted to millimeters.

1. Load

   The page loads Python and planex2d into this tab, and NGSolve with the first solve, about 23 MB the first time. Your browser keeps them in its cache, so later visits start faster.

2. Read

   planex2d reads the file, joins its closed contours into regions, simplifies the outline and numbers its regions, edges and holes.

3. Solve

   NGSolve meshes the part and solves the magnetic field equation, again whenever a setting changes after the first solve.

4. Save

   The results are saved as a PDF report or a CSV table, made in the tab as well.

The finite element model

The unknown is the magnetic vector potential, the component along the part in a planar model and around the axis in an axisymmetric one, whose lines of constant value are the flux lines. The mesh has elements of about a thirtieth of the part's size, finer along arcs and short edges and within a skin depth of a conductor's surface, and it is curved to the exact outline. The hp refinement in the Solver group refines the mesh geometrically toward every vertex and edge of the part, as many levels deep as given, since the field changes fastest there, and it integrates the refined elements with a higher order. It is off by default, and 3 levels sharpen the field at sharp corners. NGSolve solves with second-order elements, set from 1 to 5 in the Solver group, and a sparse Cholesky factorization. With iron on a B-H curve the tool minimizes the field's energy with Newton's method, and it finds the inductances with the permeability of that solution held fixed. In the time-harmonic analysis it solves once for the given currents and once for each solid conductor, and combines these solutions so that each conductor carries its current.

How it is tested

The tests solve parts whose field, inductance or force is known exactly and compare the tool's values with those, with second-order elements, and third-order elements for the field outside the magnetized sphere. The saturated steel sleeve is a coaxial cable with a sleeve of structural steel in its insulation, where the field intensity is I/(2πr) in any material, so the flux density follows from the B-H curve alone. A further test compares the force on an electromagnet's armature with the stress tensor on a contour around it, which agrees within 1.2 %.

Supported files

Run it in Python

The same analysis takes a few lines with planex2d and NGSolve. The example is a coaxial cable with a copper core of 2 mm diameter and a copper tube from 7 to 8 mm, with 1 A going out through the core and back through the tube. planex2d draws the section as three glued regions and names the outer edge, and NGSolve meshes it and solves for the vector potential with A = 0 outside. The integral of A times the current density then gives the inductance per meter, L = ∫A·J for 1 A. See Meshing with netgen in the docs, and the surface magnet motor example for a magnetic field in a motor's lamination.

$ pip install "planex2d[occ]" ngsolve

import math

import ngsolve as ngs
import planex as px

core = px.Circle((0, 0), 1).Face()
gap = px.QuickSketch().Circle(3.5).Circle(1).Face()
tube = px.QuickSketch().Circle(4).Circle(3.5).Face()
core.name, gap.name, tube.name = "core", "gap", "tube"
cable = px.Glue([core, gap, tube])
for e in cable.edges:
    if abs(e.mass - 8 * math.pi) < 1e-6:
        e.name = "outside"
mesh = ngs.Mesh(px.OCCGeometry(cable).GenerateMesh(maxh=0.2))
mesh.Curve(2)
mu0 = 4e-7 * math.pi
J = mesh.MaterialCF({"core": 1 / (math.pi * 1e-6), "tube": -1 / (math.pi * (4**2 - 3.5**2) * 1e-6)}, default=0)
fes = ngs.H1(mesh, order=2, dirichlet="outside")
u, v = fes.TnT()
a = ngs.BilinearForm(ngs.grad(u) * ngs.grad(v) * ngs.dx).Assemble()
f = ngs.LinearForm(mu0 * 1e-6 * J * v * ngs.dx).Assemble()
A = ngs.GridFunction(fes)
A.vec.data = a.mat.Inverse(fes.FreeDofs(), inverse="sparsecholesky") * f.vec
L = ngs.Integrate(A * J * 1e-6, mesh)
b, c = 3.5, 4
exact = mu0 / (2 * math.pi) * (0.25 + math.log(b) + c**4 * math.log(c / b) / (c**2 - b**2) ** 2
                               - (3 * c**2 - b**2) / (4 * (c**2 - b**2)))
print("L", round(L * 1e9, 3), "nH/m, exact", round(exact * 1e9, 3))

It prints an inductance of 310.058 nH per meter, against 310.059 nH from the closed form.

Privacy

Limits

• A planar model assumes a field that does not change along the part, with no end effects of a finite stack, and an axisymmetric model a part and a field that are round about the y axis.
• The circuits are driven by their currents. Voltage sources, moving parts and transients are not computed, and the part is solved in the position drawn.
• Magnets follow their recoil line and do not demagnetize. Hysteresis and the core losses of laminated steel are not computed, and iron is linear in the time-harmonic analysis.
• The B-H curves of the list are typical values. Load the curve of the data sheet as a CSV file where the exact steel matters.
• The flux density at a sharp inner corner of iron depends on the mesh, and a file must be planar, and its text, dimensions and hatches are not read.
• A part that fails or a result that looks wrong? Open an issue on GitHub and attach the file if you can.