An Analytical Model of an Electrical Machine with Internal Permanent Magnets

Abstract

This paper presents an analytical model of the electrical machine with permanent magnets submerged in ferromagnetic of rotor. This model has been prepared as a circuit model of electrical machine. The model takes into account a strong, local saturation of the magnetic circuit of the rotor. In the bibliography lacks a comprehensive development, enabling a transparent transition from the geometrical dimensions of the machine to a model useful to simulate its dynamic states. This mathematical formalism can be created only as a circuit model. Created a model not to be competitive against the finite element method. It allow to understand the interrelationship between design parameters and their effect on the properties and parameters of the machine. The verification of the model related to electromotive force induced in the various circuits of the armature. Verification of the parameters of the armature is planned in the future.

Appendices

Supplement 1

An important, but controversial assumption of the proposed model is to introduce the concept of magnetic bridges. Magnetic bridges are fragments of the rotor, where the magnetic field H _b is constant and dependent on the dimension l _b . It was assumed that the product of these two values is constant and equal to magnetic tension of the magnet H _m  · l _m . For comparison’s sake, the real situation versus the model calculations were determined using the FEM 2D field distribution in the machine cross section with only permanent magnet excitation. Figures 13 and 14 shows:

• field distribution on the axis of the “internal” bridge—Fig. 13. Position of magnet on the x-axis corresponds to the interval 0–4 mm (see Fig. 5),

• distribution of the tangential component of the field strength on a piece of a circle containing centers of the “external” magnetic bridges, for four different positions of the stator tooth, to the bridge—Fig. 14. “External” bridge predicted position is within the range 35°–38°.

Fig. 13

Field distribution on the axis of the internal bridge (FEM)

Fig. 14

Distributions of the tangential component of the field strength on a piece of a circle containing centers of the external magnetic bridges

As a result of integration for magnetic tension fragments of the magnetic circuit the following is obtained: for “internal” bridge: U _bw  = 540 A, for “external” bridge: 548 A. Magnetic tension of permanent magnet is U _m  = 552 A, the field strength H _m  = 13.2e⁴ A/m. Magnetic tensions are therefore similar, but the distribution of the magnetic field strength depends on the shape of the edges of a constriction of the magnetic circuit, that creates the bridge.

The graphs and values allow us to assess the level of approximation to reality and the importance of the assumptions. It would be unacceptable if the purpose of the calculations was mapping of the fields in the machine’s cross section. However, the degree of discretization of space is limited to a single slot of the stator, and the comparison value is the average value of induction in the teeth. So it seems, that the problem boils down to the choice of criteria for the selection of H _b field strength and length l _b bridges, especially the “external” ones, and establish the position of the radial axis of the magnet (angle γ). In this study, the major criterion was to make the best representation of the tooth-coil EMF waveform, i.e. changes in the magnetic flux passing through the tooth.

Supplement 2

For the magnetic circuit, as shown in Fig. 15, we have:

$${\left\{ {\begin{array}{*{20}l} {H_{m} \;l_{m} + H_{b} \;l_{b} = 0} \hfill \\ {H_{g} \;\delta + H_{m} \;l_{m} = 0} \hfill \\ {\varPhi_{m} = \varPhi_{b} + \varPhi_{g} } \hfill \\ \end{array} } \right\}}$$

where:

$$\begin{aligned} &{\varPhi_{m} = B_{m} b_{m} l_{Fe} } \hfill \\ &{\varPhi_{g} = B_{g} \alpha_{ 1} R_{sw} l_{Fe} } \hfill \\ &{\varPhi_{b} = B_{b} b_{b} l_{Fe} } \hfill \\ &{B_{m} b_{m} = B_{b} b_{b} + B_{g} \alpha_{ 1} R_{sw} } \hfill \\ &{B_{g} = \mu_{o} H_{g} \to H_{g} = \frac{{B_{g} }}{{\mu_{o} }}} \hfill \\ &B_{b} = B_{bo} + \mu_{b} H_{b} \to H_{b} = \frac{1}{{\mu_{b} }}\left( {B_{b} - B_{bo} } \right);\mu_{b} = \frac{{dB_{b} }}{{dH_{b} }}H_{b} \hfill \\ &B_{m} = B_{r} + \mu_{m} H_{m} = B_{r} + \frac{{B_{r} }}{{ - H_{C} }}H_{m} \to H_{m} = \frac{ 1}{{\mu_{m} }}\left( {B_{m} - B_{r} } \right) \hfill \\ &{\mu_{m} = \frac{{B_{r} }}{{ - H_{C} }}};H_{C}<H_{m} \le 0\hfill \\ \end{aligned}$$

Fig. 15

Analyzed magnetic circuit

Combining above dependences we get the circuit equation as shown in Fig. 15. Their matrix form is presented below:

$${\left[ {\begin{array}{*{20}l} {\frac{{l_{b} }}{{\mu_{b} }}B_{bo} + \frac{{l_{m} }}{{\mu_{m} }}B_{r} } \hfill \\ {\frac{{l_{m} }}{{\mu_{m} }}B_{r} } \hfill \\ 0\hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\frac{{l_{b} }}{{\mu_{b} }}} \hfill & 0\hfill & {\frac{{l_{m} }}{{\mu_{m} }}} \hfill \\ 0\hfill & {\frac{\delta }{{\mu_{o} }}} \hfill & {\frac{{l_{m} }}{{\mu_{m} }}} \hfill \\ {b_{b} } \hfill & {\alpha_{ 1} R_{sw} } \hfill & { - b_{m} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {B_{b} } \hfill \\ {B_{g} } \hfill \\ {B_{m} } \hfill \\ \end{array} } \right]}$$

Hence:

$$\begin{gathered} {\left[ {\begin{array}{*{20}l} {B_{b} } \hfill \\ {B_{g} } \hfill \\ {B_{m} } \hfill \\ \end{array} } \right] = \frac{ 1}{\varDelta }\left[ {\begin{array}{*{20}l} { - \frac{{\delta b_{m} }}{{\mu_{o} }} - \frac{{\alpha_{ 1} R_{sw} l_{m} }}{{\mu_{m} }}} \hfill & {\frac{{\alpha_{ 1} R_{sw} l_{m} }}{{\mu_{m} }}} \hfill & { - \frac{\delta }{{\mu_{o} }}\frac{{l_{m} }}{{\mu_{m} }}} \hfill \\ {\frac{{l_{m} b_{b} }}{{\mu_{m} }}} \hfill & { - \frac{{l_{b} b_{m} }}{{\mu_{b} }} - \frac{{l_{m} b_{b} }}{{\mu_{m} }}} \hfill & { - \frac{{l_{b} }}{{\mu_{b} }}\frac{{l_{m} }}{{\mu_{m} }}} \hfill \\ { - \frac{{\delta b_{b} }}{{\mu_{o} }}} \hfill & { - \frac{{\alpha_{ 1} R_{sw} l_{b} }}{{\mu_{b} }}} \hfill & {\frac{\delta }{{\mu_{o} }}\frac{{l_{b} }}{{\mu_{b} }}} \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {\frac{{l_{b} }}{{\mu_{b} }}B_{bo} + \frac{{l_{m} }}{{\mu_{m} }}B_{r} } \hfill \\ {\frac{{l_{m} }}{{\mu_{m} }}B_{r} } \hfill \\ 0\hfill \\ \end{array} } \right]} \hfill \\ {\varDelta = \; - \frac{{\delta b_{m} }}{{\mu_{o} }}\frac{{l_{b} }}{{\mu_{b} }} - \frac{{\alpha_{ 1} R_{sw} l_{m} }}{{\mu_{m} }}\frac{{l_{b} }}{{\mu_{b} }} - b_{b} \frac{{l_{m} }}{{\mu_{m} }}\frac{\delta }{{\mu_{o} }} = - \frac{{\mu_{m} \delta b_{m} l_{b} + \mu_{o} \alpha_{ 1} R_{sw} l_{m} l_{b} + \mu_{b} \delta l_{m} b_{b} }}{{\mu_{o} \mu_{b} \mu_{m} }}} \hfill \\ \end{gathered}$$

After arranging the equations we obtain:

$$\begin{gathered} {B_{m} = - \frac{{l_{m} }}{\delta }\frac{{B_{r} - \frac{{b_{b} }}{{b_{m} }}B_{bo} }}{{\frac{{\alpha_{ 1} R_{sw} }}{{b_{m} }}\frac{{l_{m} }}{\delta } + \mu_{mr} + \mu_{br} \frac{{l_{m} }}{{l_{b} }}\frac{{b_{b} }}{{b_{m} }}}}} \hfill \\ {B_{g} = - \frac{{l_{m} }}{\delta }\frac{{B_{r} - \frac{{b_{b} }}{{b_{m} }}B_{bo} }}{{\frac{{\alpha_{ 1} R_{sw} }}{{b_{m} }}\frac{{l_{m} }}{\delta } + \mu_{mr} + \mu_{br} \frac{{l_{m} }}{{l_{b} }}\frac{{b_{b} }}{{b_{m} }}}}} \hfill \\ \end{gathered}$$

The B _g induction corresponds to the maximum value of induction in the air gap between the rotor and the stator, according to formula (51). The differences in signs arise from the assumed character of customization. The formulas may be helpful in the selection of substitute parameters of “external” magnetic bridges.