Magnetic Circuit Design:

In any application for a permanent magnet material, we are interested in the quantity of magnetic flux that it can provide for us in a particular region of interest within a complete magnetic circuit, which is usually an air gap. To this end, any given magnet material can be designed to provide an air gap with a certain level of flux density within a reasonable range. This range, and the exact flux density, are determined by the physical dimensions of the magnetic circuit, most specifically those of the permanent magnet and air gap(s).

A permanent magnet material is characterized by its demagnetization curve, but this second quadrant of the B vs. H curve gives magnetic properties per unit magnet volume. We need a comparable characteristic for the magnetic circuit, i.e. the "load" this imposes upon the magnet, whose solution together with the demagnetization curve will yield the magnet's operating point on this curve. In this Section, we will derive this load line for a simple magnetic circuit, and see that it incorporates the necessary dimensions of the principal components.

Consider the simple magnetic circuit shown below (left), in which a permanent magnet (whose magnetization M is oriented right to left) provides flux into a single air gap via two steel pole pieces. The pole pieces have a sufficiently high permeability and are so shaped as to direct most of the magnetic flux from the magnet (area A_m , length l_m) into the air gap (area A_g , length l_g). To illustrate this, the lines of magnetic flux are plotted below (right).

Flux Conservation

Flux must always be conserved in any complete magnetic circuit. If this were not so, it would mean that there must exist a point source of magnetic flux, such as a magnetic monopole, which is not possible; all magnetic north poles must be paired with comparable south poles. Just look at the simple magnetic circuit above to verify that all the lines of magnetic flux contained within the device form continuous, complete loops. This also means that for any closed region which is within part of a magnetic circuit, the total flux leaving this region must equal the total flux which entered it; flux must be conserved through any region, which also cannot contain a point source of magnetic flux. To illustrate this fact, look at the air gap region (of area A_g and length l_g) from the magnetic circuit above, which is highlighted in green in the drawing at right. All the magnetic flux lines which enter this region also leave it. This principal of flux conservation is expressed mathematically as follows, the integration being over the closed surface area bounding any region; this is one of the two fundamental equations which form the basis for magnetic circuit design:

Ampère's Law

In Section 1A, we described a simplified form of Ampère's Law for the field B rotating around a conductor carrying current density J. We then derived a more general form of Ampère's Law to incorporate magnetic materials with magnetizations M, which only required that H be used in place of B (of course in an air gap, B = µ_oH).

Look at the "B,J" diagram (click the button above) and imagine one of the B loops replaced by H. The general differential form of Ampère's Law we derived before describes the relationship between the magnetizing force H flowing around a closed loop and the current density J that flows through the surface spanning that loop. Now, integrate both sides of the Ampère's Law equation over an area dA, and we get an equivalent integral form of Ampère's Law which happens to be more useful for magnetic circuit design:

To eliminate the rotation operation in this equation we use Stokes' Theorem, which is best understood by applying the H vector to a surface S spanning a closed loop L. If S comprises a number of sub-areas as shown in the diagram here, when the rotation of H is summed for all sub-areas, the internal components of H between the sub-areas cancel out, leaving only the H flowing around the boundary L. Thus, Stokes' Theorem for H (or any other vector quantity) is expressed as:

Combining these last two equations and noting that the integral of current density simply yields the total real current i through the surface S, we get the second fundamental equation which forms the basis of magnetic circuit design:

Magnetic Circuit Equations

Flux conservation gave us the integral of flux density through a closed surface, while Ampère's Law yielded the integral of magnetizing force around a closed loop. Both of these will be applied to the simple magnetic circuit shown at the beginning of this Section, to derive its load line and hence find the flux density levels which exist throughout the circuit. Any convenient surface and loop may be chosen to achieve this. We have used the suffixes m to denote the permanent magnet and g for the air gap, but we should also include s for both of the steel pole pieces, and l for the surrounding air regions (not the main air gap) into which some magnetic field does leak.

We choose to integrate the magnetizing force H in the direction shown around the closed loop shown in red. In this simple example, there are no real currents passing through any surface spanning this loop, so i=0. It can be seen from the diagrams that the closed integral has the following components:

Since the pole pieces have a very much higher permeability than either the magnet or air gap, their contribution to this integral will be small. It is therefore a common simplification to replace this term with a loss factor k₂, which has a typical value a fraction over unity. Each individual term in these equations is called the magnetomotive force (m.m.f.) of the respective component, H_ml_m being the m.m.f. of the permanent magnet, H_gl_g the m.m.f. of the air gap.

We choose to integrate the flux density B through the closed surface shown in blue (which closes outside the device beyond the plane of this sectional view). Notice that the magnet flux passes out of this surface, while the air gap and leakage fluxes pass into it, so:

Each individual term in these equations is the flux in the respective component, B_mA_m being the flux in the permanent magnet, B_gA_g the flux in the air gap, and B_lA_l the total leakage flux. While the magnet and air gap volumes are well defined in this example, the leakage regions are not, yet the magnetic flux plots show that only a small amount of the magnet flux does not reach the air gap. It is therefore a common simplification to replace the B_lA_l term with a leakage coefficient k₁, which has a typical value a little over unity.

The permanent magnet's demagnetization curve (shown in red in the Figure below) gives its magnetic properties per unit volume as a characteristic relationship between its B_m and its H_m. The load line gives the characteristics of the magnetic circuit, also in terms of B_m and H_m. From the foregoing equations, we already have:

In the air gap, B_g = µ_oH_g, so these terms can be eliminated to give the load line (of slope -µ_oS) as:

The intercept between the demagnetization curve and the load line, the two characteristic equations for this magnet in this magnetic circuit, gives a unique magnet operating point with a specific B_m and H_m for the magnet in this situation. The values of flux density and magnetizing force elsewhere in the magnetic circuit can be deduced from the foregoing equations, such as the air gap flux density B_g.

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