Selection of the Shape of Star-Shaped Permanent Magnets for Power Stepper Motors
Abstract Generated abstract
This paper addresses the choice of pole geometry for star-shaped permanent magnets in high-torque stepper motors, where effective use of rotor volume is critical. It develops a method for selecting the pole-arc coefficient from the condition of maximum static synchronizing torque, using the magnetic energy expression for torque and a dimensionless permanent-magnet state diagram to relate air-gap induction, stator magnetomotive force, and pole height. The analysis indicates that, under the stated assumptions of negligible saturation and pole leakage, the optimized product of stator magnetomotive force and air-gap induction is governed mainly by magnet material properties and is nearly independent of pole overlap. The resulting simplified torque expression leads to a graphical selection procedure, showing that the optimum pole-overlap coefficient depends on the number of magnet poles and slot-root geometry, but not on magnetic permeability.
Full Text
Selection of the Shape of Star-Shaped Permanent Magnets for Power Stepper Motors
O. N. Savidov
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
Stepper motors with permanent magnets on the rotor are among the most promising designs and may find wide application in various areas of automated electric drive.
In comparison with motors intended for operation in indicator mode, the design of high-torque power stepper motors with permanent magnets on the rotor presents considerable difficulty. Therefore, from the standpoint of maximum utilization of the rotor volume, it is necessary to take a well-founded approach to the choice of the pole geometry of the star-shaped magnet.
Recommendations for selecting the optimal geometrical ratios of magnets in the calculation of synchronous machines, as set forth, for example, in [3], are in general valid and applicable to stepper motors as well. The exception is the choice of the pole-arc coefficient, because of a number of features of the operating mode of these motors.
Below one possible method is presented for determining the pole-arc coefficient of star-shaped permanent magnets.
For stepper motors, the pole-arc coefficient may be chosen on the basis of the condition of obtaining the maximum value of the static synchronizing torque on the shaft.
The static synchronizing torque of a stepper motor, found by differentiating the magnetic energy with respect to displacement [1], is expressed as a function of the angle of misalignment of the rotor with the axis of the resultant magnetomotive force of the stator:
\[ M(\theta_e)=M_1\left(\sin\theta_e+0.31\,\kappa_2\frac{\Delta b_0}{b_0}\sin2\theta_e+ \right. \]
\[ \left. +\frac{1}{3}\kappa_3\sin3\theta_e+0.04\,\kappa_4\frac{\Delta b_0}{b_0}\sin4\theta_e- \right. \]
\[ \left. -\frac{1}{5}\kappa_5\sin5\theta_e-0.085\,\kappa_6\frac{\Delta b_0}{b_0}\sin6\theta_e-\ldots\right). \tag{1} \]
Here and below:
\[
\theta_e
\]
— the misalignment angle, electrical radians;
\[
b_0
\]
— the average value of the induction in the air gap in relative units;
\(\Delta b_0\) — the component of the magnetic induction in the air gap due to remagnetization of the rotor by the magnetomotive force of the stator, in relative units;
\(k_2,\ k_3,\ k_4,\ldots\) — coefficients of the series, for the \(\nu\)-th harmonic
\[ k_\nu=\frac{\sin \nu \frac{\pi}{2}\alpha_r}{\sin \frac{\pi}{2}\alpha_r}; \]
\(\alpha_r\) — pole-arc coefficient;
\(p\) — number of pairs of rotor poles;
\(D_r\) — rotor diameter, cm;
\(l_r\) — rotor length, cm;
\(\delta\) — air gap, cm;
\(i\) — reduced magnetomotive force of the stator;
\(H_c\) — coercive force of the magnet material, A/cm;
\(B_r\) — residual induction, G;
\(M_1\) — maximum synchronizing static torque
\[ M_1=1.8\cdot 10^{-9}\cdot p\cdot D_r\cdot l_r\cdot B_r\cdot b_0\cdot H_c\cdot i\cdot h_m\cdot \sin \frac{\pi}{2}\alpha_r \quad [\mathrm{kgf\cdot m}] \tag{2} \]
Replacing in expression (2) the quantities that do not depend on the pole-arc coefficient by the constant coefficient \(A\), we obtain
\[ M_1=A\cdot b_0\cdot i\cdot h_m\cdot \sin \frac{\pi}{2}\alpha_r . \tag{3} \]
Fig. 1.
We shall next investigate the influence of the pole-arc coefficient on the product \((b_0\cdot i)\). For this purpose it is convenient to use the state diagram of a permanent magnet, shown in Fig. 1 in dimensionless coordinates.
In the diagram:
\(\rho=\operatorname{tg}\beta\) is the reduced recoil coefficient;
\(b_1\) is the coordinate of the demagnetization curve corresponding to the maximum value of the energy of the magnet material,
\[ b_1=\sqrt{\frac{(B\cdot H)_{\max}}{B_r\cdot H_c}}\,; \]
\[ \operatorname{tg}\alpha=1.256\,\frac{h_m}{\delta}\,\frac{H_c}{B_r} \]
is the magnetic conductance of the air gap reduced to dimensionless coordinates.
The state diagram of the permanent magnet makes it possible to relate the product \((i\cdot b_0)\) to the pole-overlap coefficient. In doing so, we neglect saturation of the magnetic circuit and magnetic scattering of the poles, in view of the small air gaps of active stepper motors.
For the diagram in Fig. 1, let us write the equation of the recoil straight line \(AC\)
\[ \frac{i}{b_0-b}=\operatorname{ctg}\beta; \tag{5} \]
the equation of the straight line \(OD\), characterizing the conductance of the air gap,
\[ \frac{b_0}{h-i}=\operatorname{tg}\alpha; \tag{6} \]
the equation of the demagnetization curve of the magnet material, according to [2],
\[ h=\frac{b_1^2(1-b)}{b_1^2-b(2b_1-1)}. \tag{7} \]
Solving the system of equations (5), (6), and (7) jointly, we obtain
\[ i\cdot b_0=\left[i^2m+i\cdot\kappa(\operatorname{ctg}\alpha+1)\right] -i\left[i^2m-i\cdot2\kappa m\cdot(\operatorname{ctg}\alpha-3)+\right. \]
\[ \left.+\kappa^2(\operatorname{ctg}\alpha+1)^2+4\kappa\cdot\operatorname{ctg}\alpha\right]^{\frac12}, \tag{8} \]
where
\[ \kappa=\frac{b_1^2}{2b_1-1}; \]
\[ m=\frac{\operatorname{tg}\alpha+\operatorname{tg}\beta}{\operatorname{tg}\alpha}. \]
The reduced conductance of the air gap entering into equation (8) is related to the height of the rotor pole. The latter, for a sufficiently large number of poles on the rotor \((2p\geqslant 6)\), can be determined [3] as
\[ h_m=\frac{D_r}{2\sin\frac{\pi}{2p}} \left[(1-\alpha_r)\frac{\pi}{2p} -2\frac{r}{D_r}\left(1-\sin\frac{\pi}{2p}\right)\right]. \tag{9} \]
From the state diagram of the permanent magnet (Fig. 1) and equation (8), it follows that each value of the pole-overlap coefficient will correspond to one recoil straight line for which the product \((i\cdot b_0)\) is maximal.
By differentiating equation (8) with respect to the variable quantity—the coefficient of the pole arc—it is possible to determine the magnetomotive force of the sta-
…of the rotor corresponding to the optimal return line. The solution is represented by a fourth-degree equation and is not given here.
Analysis of equation (8) showed that the product \((i \cdot b_0)\) corresponding to the optimal return line is practically independent of the value of the pole-overlap coefficient and is determined by the magnetic characteristics of the magnet material.
The nature of the variation of the product \((i \cdot b)_{\max}\) as a function of the pole-overlap coefficient is shown in Fig. 2. As the magnetic permeability of the magnet material increases, the influence of the value of the pole-overlap coefficient on the product \((i \cdot b_0)_{\max}\) increases.
Fig. 2.
In connection with the foregoing, the problem of selecting the pole geometry of the star-shaped magnet is considerably simplified.
Expression (3), taking (9) into account, can be represented in the form
\[ M_1 = C [k - N\alpha_r]\sin \frac{\pi}{2}\alpha_r, \tag{10} \]
where \(C\), \(k\), \(N\) are coefficients independent of \(\alpha_r\).
The static synchronizing torque (10) is maximal at a certain value of the pole-overlap coefficient, determined by the relation
\[ \frac{\pi}{2}(k - N\alpha_r) - N \cdot \tg \frac{\pi}{2}\alpha_r = 0. \tag{11} \]
Equation (11) was solved graphically for different numbers of poles and ratios \(r/D_r\). The magnitude of the radius at the bottom of the slot substantially affects the utilization of the magnet in terms of magnetizing force. Therefore, reducing the radius after casting the magnet by mechanical machining increases the utilization of the rotor volume [3].
The investigations carried out make it possible to note that the value of the static synchronizing torque on the shaft of a stepper motor reaches a maximum at a certain pole width and height, determined by \(\alpha_r\). The optimal value of the pole-overlap coefficient depends on the number of poles of the magnet and does not depend on the magnetic permeability of the magnet material.
To select the optimum value of \(\alpha_r\), which determines the utilization of the rotor volume of a stepper motor, the curves presented in Fig. 3 may be proposed.
Fig. 3.
REFERENCES
-
V. A. Ratmirov, B. A. Ivobotenko. Stepper Motors for Automatic Control Systems. GEI, 1962.
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R. Pick, G. Weigtar. Calculation of Commutation Relays. GEI, 1961.
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N. N. Pleshchunov. Selection of the optimum parameters of star-shaped permanent magnets. Vestnik elektropromyshlennosti, No. 1, 1961.