Question 5.13.2: Develop a three-dimensional field solution for the gap regio...
Develop a three-dimensional field solution for the gap region of an axial-field motor (Fig. 5.42). Assume that the magnet has a second quadrant demagnetization curve
B = μ_{0}(H ± M_{s} \hat{z}) , (5.276)
where the ± term takes into account the alternating polarity of adjacent poles [22, 23].
For the three-dimensional model we approximate the magnetic structure of the motor in terms of the magnetic circuit shown in Fig. 5.43. Specifically, we assume that the upper and lower flux plates have infinite permeability (μ = ∞), and are of infinite extent in both the vertical and horizontal directions. The field in the gap region of this idealized geometry can be determined using the method of images (Section 3.7). That is, the field in the gap can be expressed as a superposition of the fields from a double infinite sum of image magnets. Let B_{mag,z}(r, Φ, z) denote the field due to the motor magnet in free space. We studied this geometry in Example 4.2.8. The axial field B_{mag,z}(r, Φ, z) is given by Eq. (4.163). Notice that B_{mag,z}(r, Φ, z) can be considered to be a function of the spatial coordinates (r, Φ, z), and the axial position of the magnet,
B_{mag,z}(r, Φ, z) = B_{mag,z}(r, Φ, z; z_{1}, z_{2}), (5.277)
where z_{1} = z_{s}(1) and z_{2} = z_{s}(2) are the positions of the bottom and top of the magnet on the z-axis, respectively.
Image magnets: To obtain a field solution we need to take into account the fields from the image magnets. These are located in the regions occupied by the flux plates. For example, the images for an infinitesimal magnetic charge ΔQ on the bottom surface of the motor magnet are shown in Fig. 5.44. Each image magnet has a field component B_{mag,z}(r, Φ, z; z_{1}, z_{2}), and the positions z_{1} and z_{2} are chosen so that the tangential component of H is zero and the normal component of B is continuous at the surface of the flux plates. Each image magnet in one flux plate gives rise to a mirror image magnet in the other flux plate. Therefore, there is a double infinite set of image magnets. We write the total field as a superposition of all the magnets, real and image,
B_{z}(r, Φ, z) = \sum\limits_{n=1,3,5,…}^{∞} [B_{mag,z}(r, Φ, z; Z_{1}(n), Z_{2}(n)) \\ +B_{mag,z}(r, Φ, z; \hat{Z}_{1}(n), \hat{Z}_{2}(n))]. (5.278)
The axial positions are given by the following recursive relations:
Z_{1}(n+2) = Z_{1}(n) + 2(h+t_{m})
Z_{2}(n+2) = Z_{1}(n+2) + 2t_{m}
and
\hat{Z}_{1}(n) = – Z_{2}(n)
\hat{Z}_{2}(n) = – Z_{1}(n),
where Z_{1}(1) = h, Z_{1}(1) = h + 2t_{m} (n = 1, 3, 5, . . .). According to these relations, the first few terms are as follows:
Z_{1}(3) = h+2(h+t_{m})
Z_{2}(3) =(3h+2t_{m}) +2t_{m}
Z_{1}(5) =(3h+2t_{m}) + 2(h+t_{m})
Z_{2}(5) =(5h+4t_{m}) +2t_{m}
and
\hat{Z}_{1}(1) = – (h+2t_{m})
\hat{Z}_{2}(1) = -h
\hat{Z}_{1}(3) = -(3h+4t_{m})
\hat{Z}_{2}(3) = -(3h+2t_{m})
\hat{Z}_{1}(5) = -(5h+6t_{m})
\hat{Z}_{2}(5) = -(5h+4t_{m}).
The first and second terms in Eq. (5.278) correspond to the images in the upper and lower flux plates, respectively. If the lower flux plate is absent, the field reduces to
B_{z}(r, Φ, z) = B_{z}(r, Φ, z; h,h+2t_{m}). (5.279)
Calculations: We apply Eq. (5.278) to a motor geometry with the following parameters:
M_{s} =7.2 × 10^{5} (A/m)
N_{pole} = 4
R_{1} = 0.5 cm
R_{2} = 2.5 cm.
First, we compute B_{z} at the midpoint of the gap (r = (R_{1}+ R_{2})/2, Φ = 0°, and z = h/2) for a range of gap heights h = 4, 5, . . . , 10 mm, and magnet heights t_{m} = 4, 6, 8 and 10 mm. These data are shown in Fig. 5.45. Notice that the field drops off sharply as the gap h increases. This analysis is performed with N_{Φ} = 40 and N_{r} = 20, and there is no improvement in accuracy beyond the first five terms in Eq. (5.278). From these initial calculations, one can select a gap height h and magnet thickness t_{m} that render the desired field strength. For example, h = 6 mm and t_{m} = 10 mm, which yields a field of 0.428 T at the midpoint of the gap. Next, we fix h = 6 mm and t_{m} = 10 mm, and then perform an analysis to determine the variation of B_{z} in the radial direction. Specifically, B_{z} is computed along two different radial lines located midheight in the gap (z = h/2) and spanning the surface of the magnets from R_{1} ≤ r ≤ R_{2} (Fig. 5.46). These lines are positioned at Φ = 0° (center line of a pole) and Φ = 30°, respectively. Notice that at Φ = 0, B_{z} ≥ 0.35 T for 10 mm ≤ r ≤ 23 mm, and that it drops off rapidly outside of this interval. There is also a considerable drop in field strength as one moves in an angular sense from the center of the pole Φ = 0. To see this, the field is computed along three different circumferential arcs spanning an angular measure of 90°, that is, from the center of one pole to the center of the neighboring pole at radii r = 10, 15 and 20 mm, respectively, with (z = h/2) (Fig. 5.47). This analysis shows that the field drops off sharply for Φ > 30°. In summary, we find that the region of maximum field strength for this motor geometry is 10 mm ≤ r ≤ 23 mm with -30° ≤ Φ ≤ 30°.
