Quantitative Consideration of the Effect of Transverse Reaction in DC Micromotors with Anisotropic Magnets

V. F. Kostin, F. A. Serdyuk, S. T. Popov

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

When the transverse reaction of the armature (TRA) acts in magnetoelectric machines (MEM) on an unstabilized magnet, its irreversible demagnetization (stabilization) occurs. If, under normal operating conditions of an MEM, the effect of TRA may be neglected, then under nonstationary operating conditions of the motor this effect must be taken into account.

Known methods for accounting for and analyzing TRA in MEM [1, 2] proceed from diagrams of the state of the magnet using the limiting demagnetization curve, i.e., the demagnetization curve obtained if a field opposite in sign to the magnetizing field is applied to the magnet.

When anisotropic magnets are used in MEM, such an approach to accounting for and analyzing TRA cannot reflect either the qualitative or the quantitative aspect of the process, since in essence it proceeds only from the action of longitudinally demagnetizing fields.

If a longitudinally demagnetizing field is applied to an anisotropic magnet having residual magnetization \(I_0\) in the direction of the texture, then the demagnetization processes will be determined by the usual limiting demagnetization curve (Fig. 1, curve 1). If, to the same magnet having residual magnetization \(I_0\) in the direction of the texture, an external field is applied perpendicular to the axis of the principal magnetization, then the demagnetization processes will be determined by the curve

Fig. 1

2 in Fig. 1. Under the action of the TRA field, the lines of force cannot enter normally to the surface of the magnet because of the difference in magnetic permeabilities along \((\mu_{\parallel})\) and across the texture \((\mu_{\perp})\), and also because of their refraction at the interface between two media (air gap—magnet body). As a result, the resulting vector of the TRA field will have two components, one of which, \(H_{\parallel}\), is directed along the texture axis, while the other, \(H_{\perp}\), is perpendicular to it. Therefore the stabilization (demagnetization) processes will be determined by a certain curve 3 lying between curves 1 and 2 in Fig. 1.

Since \(H_{\parallel}\) will always keep one half of the pole magnetized, curve 3 in Fig. 1 cannot drop to the abscissa axis. Because of this, remagnetization of the magnet in the perpendicular direction will have the character of an open hysteresis loop.

It is obvious that quantitative accounting for all these complex phenomena in a DCM will cause a number of difficulties and special features in the calculation, and the accuracy of accounting for all these features will be reflected in the accuracy of calculating the magnetic system of the DCM as a whole.

The quantitative aspect of the stabilization of the magnet by the TRA field will depend on the armature-reaction flux passing through the body of the magnet. When two-pole magnetic systems are used in micromotors, the simplified equivalent circuit for the TRA flux may be represented as shown in Fig. 2.

Fig. 2

Using Kirchhoff’s second law for the magnetic circuit, one may write

\[ F = \frac{\Phi}{G_{\delta}} + \frac{\Phi}{G_{\mathrm{M}}}, \tag{1} \]

where

\(F\) — m.m.f. of the transverse armature reaction;
\(\Phi\) — flux of the transverse armature reaction;
\(G_{\delta}\) — permeance of the air gap;
\(G_{\mathrm{M}}\) — permeance of the magnet body to the TRA flux.

The permeance of the magnet body to the TRA flux may be expressed as

\[ G_{\mathrm{M}} = \int_l \frac{S_{\mathrm{m}}}{L_{\mathrm{m}}}\, d\mu_{\mathrm{M}} = \frac{S_{\mathrm{m}}}{L_{\mathrm{m}}}\int_l d\mu_{\mathrm{M}} . \]

Consequently, the TRA flux cannot be determined from equation (1), since the quantity \(\mu_{\mathrm{M}}\) is not known (the permeability along the line of the TRA flux in the magnet body).

Determination of

\[ \int_l d\mu_{\mathrm{M}} \]

is associated with great difficulties, since \(\mu_{\mathrm{M}}\) depends on the ratio of the permeabilities along and across the texture of the magnet, the ratio of the vectors \(H_{\parallel}\) and \(H_{\perp}\), the state of the magnet, the magnitude of the TRA field, etc.

By means of an experimental study of DCMs with an external annu-

with an external ring magnet and a toothed armature, as well as for DC micromotors with an internal cylindrical magnet and a smooth armature, the dependences were obtained

\[ \frac{B'}{B}=f(I_a), \]

where \(B\) is the induction in the bore of the micromotor with a non-stabilized magnet;

\(B'\) is the same with a stabilized magnet;

\(I_a\) is the armature current.

If these dependences are represented as

\[ \frac{\Phi'}{\Phi} = f\!\left(F\,\frac{\lambda\mu_0}{\lambda'\mu_m}\right) = f\!\left(I_a N\,\frac{\lambda\mu_0}{\lambda'\mu_m}\right) = f(I_a K), \]

where

\[ \Phi' \]
is the flux in the bore of the micromotor with a stabilized magnet;

\[ \Phi \]
is the same with a non-stabilized magnet;

\[ N \]
is the number of armature conductors;

\[ \lambda\mu_0 \]
is the permeance of the air gap to the TAR flux;

\[ \lambda'\mu_m \]
is the permeance of the magnet body to the TAR flux.

\[ K=N\,\frac{\lambda\mu_0}{\lambda'\mu_m} \]
is the parameter of the electromagnetic system,

where

\[ \lambda=\frac{2\pi l}{4\ln \dfrac{D}{D'}}, \]

where

\(l\) is the armature length;

\(D\) is the internal diameter of the magnet bore for DC micromotors with an external ring magnet;

\(D'\) is the reduced armature diameter, taking Carter’s coefficient into account.

For a DC micromotor with an internal magnet:

\(D'\) is the external diameter of the armature;

\[ D \]
is the internal diameter of the housing bore,

\[ \lambda'=\frac{S_m}{L_m}, \]

where

\(S_m\) is the magnet cross section;

\(L_m\) is the magnet length.

On the basis of the above reasoning, generalized curves can be obtained for taking into account the influence of TAR in DC micromotors with the given configuration of the magnetic system.

For DC micromotors, DPM, and DPR the dependences

\[ \frac{\Phi'}{\Phi} = f\!\left(F\,\frac{\lambda\mu_0}{\lambda'\mu_m}\right) \]

are presented in Figs. 3, 4.

The ratio

\[ \frac{\mu_0}{\mu_m} \]

entering the expression

\[ \left(F\,\frac{\lambda}{\lambda'}\,\frac{\mu_0}{\mu_m}\right) \]

need not be determined, since this ratio is already incorporated in the experimental curves of Figs. 3, 4.

In quantitatively accounting for TAR in DC micromotors with the above-mentioned magnetic systems, the curves of Figs. 3, 4 may be used. For this purpose, from the calculated parameter \(K\) and the specified operating mode, determined by

Fig. 3

Fig. 4

...with the armature current \(I_a\), the product \(I_a K\) is found, and from it, using the curves of Figs. 3 and 4, the ratio \(\Phi'/\Phi\). It should be noted that, in passing from one set of dependences to another, the calculated permeances used are not reduced (to full mmf).

Quantitative allowance for transverse armature reaction in DC micromotors can also be carried out analytically. For this purpose the curves of Figs. 3 and 4 must be approximated.

Approximating the curves of Figs. 3 and 4 by analytical expressions, we obtain:

a) for micromotors with an external ring magnet made of YuNDK-24 alloy and a toothed armature,

\[ \frac{\Phi'}{\Phi} = \left[ 1 + 0.013x - 0.065x^2 \right], \tag{2} \]

where

\[ x = 0.26\, K I_a\, 10^{-5}. \]

b) for micromotors with an internal cylindrical magnet made of YuNDK-24 alloy and a hollow smooth armature,

\[ \frac{\Phi'}{\Phi}=\left[\,1-0.078x-0.112x^2\,\right], \tag{3} \]

where

\[ x=1.15KI_{\mathrm{a}}10^{-5}. \]

REFERENCES

1. V. A. Balagurov et al. Electrical Machines with Permanent Magnets. “Energiya,” 1964.

2. Yu. S. Zakharov. Armature Reaction of Magnetoelectric Machines. Author’s abstract, 1959.

3. A. S. Kanter. Taking into account the demagnetizing effect of short-circuit currents in determining the dimensions of permanent magnets of a machine. “Elektrichestvo,” 1940, No. 8.