# Calculating reluctance values for an electromagnet

Recently a customer asked how to use Magnum to calculate reluctance values for a magnet to insert into a motor design code. I’ll give some pointers in this article. To get started, it’s useful to review the meaning of terms like reluctance and magnetic circuit. The following material is adapted from Chap. 5 of my book Principles of Charged Particle Acceleration.

Figure 1. Function of iron in a magnetic circuit

Electromagnets are circuits in the sense that the lines of magnetic flux density circulate. Magnetic circuits have analogies with electric circuits where electrons circulate. The excitation windings provide the motive force (→ voltage), the vacuum gap is the load (→ resistance), and ferromagnetic material completes the circuit (→ conducting wire). Iron in magnetic circuits is useful primarily when there is a small, well-defined gap. Figure 1 illustrates the advantage of including ferromagnetic material in ordinary magnetic circuits. Assume that both the air core (a) and iron core (b) geometries produce the same field Bg in gaps of equal length. In order to compare the circuits directly, windings are included in the air core circuit so that the return flux is contained in the same toroidal volume. The magnetic flux in any cross section is a constant and is the same for both circuits. The gap has cross-sectional area Ag, and the core (or return flux coil) has area Ac. The length of the gap is g while the length of the core is l. The excitation coils have an ampere turn product given by the number of windings multiplied by the current input to the windings, NI. The wires that carry the current have resistivity in a non-super-conducting magnet; the power necessary to support the field is proportional to N (the length of the wire) and to I^2 . It is desirable to make NI as small as possible. The ampere turn products for the two circuits of Figure 1 can be related to the magnetic field in the circuit through Ampere’s law (Eq. 1). The constant circuit flux is given by Eq. 2. For the air core circuit, Eq. 1 may be written as Eq. 3. Equations 2 and 3 may be combined into Eq. 4, a relationship involving flux. Similarly, the Eq. 5 describes the ferromagnetic circuit. Comparing Eqs. 4 and 5, the ampere turn product for equal flux is much smaller for the case with the ferromagnetic core when μ » μ0 and g « l (small gap). An iron core substantially reduces power requirements. An alternate view is that the excitation coil need support only the magnetic field in the gap because the second term in Eq. (5) is negligible. The return flux is supported by the atomic currents of the iron. The excitation coils are located at the gap in Figure 1 to clarify this statement. In practice, the coils may be located anywhere in the circuit with about the same result. This follows from the principle that the field configuration minimizes the net field energy of the system. The field energy is a minimum if the flux flows in the ferromagnetic material.

Equation (5) has the same form as that for an electric circuit consisting of a power source and resistive elements with the following substitutions.

• Magnetic flux → Current. Although the magnitude of B may vary, the flux in any cross section is constant.
• Magnetomotive force → Voltage. The quantity (equal to the ampere-turn product, NI.) is the drive for magnetic flux.
• Reluctance → Resistance. Equation 5 two reluctances in series, Rg = g/(Ag*μ0) and Rc = l/(Ac*μ). The higher the reluctance, the lower the flux for a given magnetomotive force. The reluctance of the iron return flux core is much smaller than that of the gap (the load), so it acts in the same way as a low-resistivity wire in an electric circuit.

To illustrate a magnetic field calculation, consider the spectrometer magnet of Figure 2a. The components of reluctance already mentioned may be supplemented by additional paths representing fringing flux (magnetic field lines bulging out near the gap that don’t contribute to the application) and leakage flux (magnetic flux returning across the magnetic core without traversing the gap). These reluctances are combined in series and in parallel, just as resistances. The equivalent circuit is illustrated in Figure 2b.

Figure 2. Reluctance example

With this background, we can address the issue of how to find reluctance values with a numerical field-solver like Magnum. It’s obvious from the example of Fig. 2 that subjective judgements are necessary. There is no reluctance button where numbers pop out without effort by the user. There are two decision areas:

• What is a appropriate equivalent magnetic circuit model? With regard to original customer request, the model must be consistent with the motor design code.
• How does one separate different types of magnetic flux (e.g., useful gap flux versus fringing flux).

Once the decisions are made, it is relatively easy to use Magnum. By defining special diagnostic regions in the mesh, we can use the surface integral capabilities of MagView to calculate flux accurately across any surface in three dimension space. A previous article describes the technique. Given 1) the  circuit topology, 2) the magnetomotive force and 3) values of flux through each reluctance, it is possible to determine the individual reluctances. For example, the four reluctances in Fig. 2 can determined from four flux calculations (although the algebra is fairly involved). The solution is simple under the assumption that the series core reluctance is relatively small. In this case, each of the parallel reluctances is given by NI divided by the associated flux.