Fundamental Equations of Synchronous Reluctance Reduction Motors (SRM-D)
Abstract Generated abstract
This paper formulates basic calculation equations for synchronous reluctance reduction gear motors, motivated by the need for stable low-speed drives and by the lack of a unified theory for their design. Using two-reaction theory and rotating magnetic-field representation, it derives voltage, flux linkage, torque, and motion equations in rotor-fixed d and q axes for motors with identical or different stator and rotor tooth pitches. The resulting equations are shown to be essentially analogous to those of ordinary synchronous reluctance machines, while incorporating the tooth-number relation governing speed reduction. The paper also outlines methods for determining the required parameters, especially the direct- and quadrature-axis synchronous reactances, using air-gap permeance calculations for salient-pole and distributed-winding designs.
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BASIC EQUATIONS OF SYNCHRONOUS RELUCTANCE GEAR MOTORS (SRD-R)
E. V. Kononenko, R. P. Laas, A. F. Fink
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
Recently, in various fields of science and technology, gear motors have found wide application, making it possible to reduce significantly the rotor speed at a given supply frequency. The basic requirement imposed on an SRD-R is linearity of rotational speed. This requirement determines the problem of creating highly stable micromotors and drives.
The broad introduction of gear motors is being restrained by the insufficient development of their theory, as well as by the absence of a wide range of motors and recommendations for choosing optimal designs.
Questions of the theory and calculation of gear motors were studied as early as the 1930s [1]; for example, G. G. Soroker calculated a single-phase motor for a gramophone at 78 rpm. However, because there were no fields of application for the motors, they remained little studied. Among works devoted to questions of theory and design, one should note the works of A. S. Kurakin [2], who, expanding the magnetic field in the gap into a series, identifies the principal harmonic waves of the slotted layer and uses them as working fields.
A common shortcoming in the works of various authors is the absence of a unified approach to the formulation of the basic calculation equations.
In the present work an attempt is made to formulate equations for reluctance-type gear motors on the basis of the application of the two-reaction theory, which has found wide use in the study of ordinary synchronous machines.
In terms of design, the motors can be made with identical tooth pitches of the stator and rotor and with different tooth pitches.
The stator of the first type of motors has clearly expressed poles (projections), on which a three-phase concentrated winding is placed. With six projections on the stator there is no mutual inductance between the windings of different phases. The winding of motors of the second type is distributed and placed in the stator slots.
The picture of the mutual arrangement of the stator and rotor teeth of a motor having identical tooth pitches can be represented by Fig. 1. Here \(\gamma\) is the angle of misalignment between the stator and rotor teeth. When \(\gamma = 0\), the axes of the rotor and stator teeth coincide and the permeance will be maximal. When the axis of a stator tooth coincides with the axis of a rotor slot, the permeance will be minimal.
Fig. 1
During one half-period of the current, the rotor turns through an angle corresponding to one tooth pitch. Consequently, one tooth pitch corresponds to 180 electrical degrees. The minimum permeance will occur when the rotor is displaced by \(1/2\) of a tooth pitch, i.e., when the “tooth” of the rotor is under the stator slot, which amounts to 90 electrical degrees.
Thus, by analogy with ordinary synchronous (reluctance) machines, the angle between the axes that determine the minimum and maximum permeance (\(d\) and \(q\)) is 90 electrical degrees. Similarly, it can be shown that for reluctance reduction motors having different tooth pitches, the angle between the axes determining the minimum and maximum permeance is also 90 electrical degrees. In this case the number of rotor teeth must be chosen as
\[ z_1 = z_2 \pm 2p_1 . \]
Since the rotor tooth pitch corresponds to 180 electrical degrees, it follows from this that the relation between the speed \(n\), the current frequency \(f\), and the number of rotor teeth \(z_2\) is
\[ n = \frac{120 f}{z_2}. \]
When the rotor turns relative to the stator, the magnetic permeance of the air gap and, consequently, the inductances and mutual inductances of the phase windings vary periodically with a period equal to the angle of rotation through one tooth pitch. Therefore, the frequency of variation of the alternating component of the magnetic permeance is twice the frequency of variation of the stator-winding voltage.
Then, by analogy with ordinary synchronous machines, the coefficients of self- and mutual inductance may be defined in the form
\[ \begin{aligned} L_a &= L_0 + L_2 \cos 2\gamma;\\ L_b &= L_0 + L_2 \cos 2(\gamma + 60);\\ L_c &= L_0 + L_2 \cos 2(\gamma - 60);\\ m_{ab} &= m_0 + m_2 \cos 2(\gamma - 60);\\ m_{ac} &= m_0 + m_2 \cos 2(\gamma + 60);\\ m_{bc} &= m_0 + m_2 \cos 2\gamma. \end{aligned} \tag{1} \]
At an arbitrary instant of time.
then the angle \(\gamma\) is determined as
\[ \gamma = \int_{0}^{t} \omega\,dt + \gamma_0 . \]
To relate the angle characterizing the position of the rotor in space in geometrical radians \(\gamma_{\mathrm{M}}\) to this same angle in electrical radians, one may write the relation
\(\gamma = p_2\gamma_{\mathrm{M}} = \dfrac{z_2}{2}\gamma_{\mathrm{M}}\); accordingly, the angular velocity in electrical radians will be
\(\omega = p_2\omega_{\mathrm{M}} = \dfrac{z_2}{2}\omega_{\mathrm{M}}\). Restricting ourselves to the consideration of reduction motors with laminated rotors, in deriving the basic equations we shall use the method of rotating magnetic fields. For this purpose, let us introduce the representative vectors
\[ \overline{i}_{\sigma} = \frac{2}{3}\left(i_a + a i_B + a^2 i_C\right); \]
\[ \overline{U}_{\sigma} = \frac{2}{3}\left(U_a + aU_B + a^2U_C\right); \tag{2} \]
\[ \overline{\psi}_{\sigma} = \frac{2}{3}\left(\psi_a + a\psi_B + a^2\psi_C\right). \]
Let the voltages of the phase windings vary according to the law
\[ \begin{aligned} U_a &= U_0 + U\cos(\omega_1 t+\alpha_0);\\ U_b &= U_0 + U\cos(\omega_1 t+\alpha_0-120^\circ);\\ U_c &= U_0 + U\cos(\omega_1 t+\alpha_0+120^\circ). \end{aligned} \tag{3} \]
Then, after some transformations,
\[ \overline{U}_{\sigma}=Ue^{j(\omega_1 t+\alpha_0)}, \tag{4} \]
i.e. \(\overline{U}_{\sigma}\) does not depend on the zero-sequence component of the voltage. Similarly,
\[ \overline{\psi}_{\sigma} = (L_0-m_0)\overline{i}_{\sigma} + \frac{3}{2}L_2 i_{\sigma}^{*}e^{j2\gamma}. \tag{5} \]
It is not difficult to show that, when the phase sequence or the sequence of inductances and mutual inductances is changed, the sign before the exponent \(\gamma\) in the second term of equation (5) changes; i.e., a change in the direction of rotation of the motor will take place.
In equation (5), the last term has a variable coefficient.
To pass to axes rigidly connected with the rotor, it is necessary to multiply equation (5) by \(e^{-j\gamma}\); then we obtain:
\[ \overline{\varphi}_{s} = (L_0-m_0)\overline{i}_{s} + \frac{3}{2}L_2 i_s^{*}, \quad \text{where} \]
\[ \overline{\psi}_{s} = \overline{\psi}_{\sigma}\cdot e^{-j\gamma}, \qquad \overline{i}_{s} = \overline{i}_{\sigma}\cdot e^{-j\gamma}, \quad \text{etc.} \]
The expression for the electromagnetic torque may be determined as
\[ M_{\mathrm{em}} = \frac{3}{2}\operatorname{Re}\!\left[j\,\overline{\psi}_{s} i_s^{*}\right]. \tag{6} \]
Solving complex equations on an analog computer is difficult. At the same time, the complexes and conjugate complexes of the quantities under consideration—
quantities can be represented on the complex plane coincident with the \(d\) and \(q\) axes, fixed relative to the rotor [3].
The nonconstancy of the angular velocity, by analogy with synchronous reluctance machines, is conveniently fixed by the change in the angle \(\Theta\), which is formed by the voltage vector with the transverse axis of the rotor.
From Fig. 2 we have:
\[ \theta=\alpha-\gamma-\frac{\pi}{2}, \]
Fig. 2.
where \(\alpha\) is the angle formed by the position of the representing vector of the voltage relative to phase “a.”
From the equations obtained using the representing vectors \(\bar{i}_s\), \(u_s\), \(\psi_s\), and so on, the equations in the \(d\) and \(q\) axes can be obtained by projecting them respectively onto the \(d\) and \(q\) axes. With the corresponding choice of the sign convention, these equations take the form:
\[ \begin{aligned} U_d&=p\psi_d-\psi_q(1-S)+r i_d;\\ U_q&=p\psi_q+\psi_d(1-S)+r\cdot i_q;\\ M_{\mathrm{cm}}&=Hps+(\psi_d\cdot i_q-\psi_q\cdot i_d), \end{aligned} \tag{7} \]
where
\[ H=\frac{4I\omega_1^3}{P_6 z^2}; \qquad \psi_d=x_d\cdot i_d; \qquad \psi_q=x_q\cdot i_q; \qquad x_d=\omega_1\cdot L_d; \qquad x_q=\omega_1\cdot L_q; \]
\[ L_d=L_0-m_0+\frac{3}{2}L_2; \qquad L_q=L_0-m_0-\frac{3}{2}L_2. \]
When considering a motor having six projections on the stator, the mutual inductance is absent; then
\[ L_d=L_0+L_2,\qquad L_q=L_0-L_2. \]
The equations obtained practically do not differ from the equations of an ordinary synchronous reluctance motor.
Losses in the rotor steel under remagnetization by the fundamental field wave can be refined additionally.
In order to use, in studies of geared motors, the system of equations (7), it is necessary to determine the parameters entering into them: \(r\), \(x_d\), and \(x_q\). The synchronous inductive reactances \(x_d\) and \(x_q\) may be represented in the form
\[ x_d = x_{ad} + x_s; \qquad x_q = x_{aq} + x_s, \tag{8} \]
where \(x_s\) is the leakage reactance of the stator winding.
As a first approximation, the method of specific magnetic permeance may be used as the basis for determining the parameters of the synchronous operating mode. The most convenient is the “substitution angle” method proposed by R. Pohl [4].
Let us first consider synchronous geared motors with identical tooth divisions on the stator and rotor. Having determined the permeances of the air gap per pole (\(\lambda_{nd}\) and \(\lambda_{nq}\)) by the method set forth in [4], and using the equivalent circuit of the magnetic circuit of such a motor [5], one can determine the maximum value of the magnetic flux linked with each phase, respectively along the longitudinal and transverse axes. For example, for phase “a” these fluxes may be represented as
\[ \Phi_{ad} = \frac{\sqrt{2}\, I_d W_1(2\lambda_0^2+\lambda_0\lambda_2-\lambda_2^2)} {3\lambda_0}\,\mu_0 l, \]
\[ \Phi_{aq} = \frac{\sqrt{2}\, I_q W_1(2\lambda_0^2-\lambda_0\lambda_2-\lambda_2^2)} {3\lambda_0}\,\mu_0 l, \tag{9} \]
where \(I_d\) and \(I_q\) are the rms values of the currents in the stator winding along the longitudinal and transverse axes,
\(W_1\) is the number of turns of the stator winding,
\[ \lambda_0=\frac{\lambda_{nd}+\lambda_{nq}}{2}; \qquad \lambda_2=\frac{\lambda_{nd}-\lambda_{nq}}{2} \]
are the first terms of the series expansion of the magnetic permeance of the air gap within one pole shoe [6].
\(\mu_0\) is the magnetic permeability of air,
\(l\) is the length of the machine.
Knowing the maximum magnetic fluxes of each phase, one can determine, by the usual formulas, the rms values of the emf along the longitudinal and transverse axes:
\[ E_{ad(q)} = 4.44 f W_1 \Phi_{ad(q)} \tag{10} \]
and the synchronous inductive reactances \(x_{ad}\) and \(x_{aq}\):
\[ x_{ad(q)}=\frac{E_{ad(q)}}{I_{d(q)}}. \tag{11} \]
Then, for the type of motors under consideration, the synchronous inductive reactances are determined in the following form:
\[ x_{ad}=2\pi f\mu_0 w_1^2\lambda_{ad}\cdot l; \]
\[ x_{aq}=2\pi f\mu_0 w_1^2\lambda_{aq}\cdot l; \tag{12} \]
where \(\lambda_{ad}\) and \(\lambda_{aq}\) denote the following expressions:
\[ \lambda_{ad}= \frac{\lambda_0^2+\lambda_0\lambda_2-\lambda_2^2} {3\lambda_0}; \qquad \lambda_{aq}= \frac{2\lambda_0^2-\lambda_0\lambda_2-\lambda_2^2} {3\lambda_0}. \]
More complicated is the determination of the synchronous inductive reactances in geared motors having different tooth—
... pole pitches and a distributed winding on the stator. To determine the magnetic permeance per pole pitch in this case, one may use the method set forth in [7]. The essence of the method is that the entire toothed air gap is divided into a number of elementary sections, in which, for determining the magnetic permeance, one may use R. Pohl’s formulas.
Taking into account only the first harmonics of the magnetomotive force created by the stator winding, the maximum values of the total magnetic fluxes along the longitudinal and transverse axes may be represented in the form (the \(d\) axis is taken as the origin)
\[ \Phi_{ad} = \int\limits_{-\frac{\tau}{2}}^{\frac{\tau}{2}} \mu_0 \lambda_x F_{ad}\cdot l \cdot \cos \frac{\pi}{\tau}x \cdot dx = \mu_0 l F_{ad} \int\limits_{-\frac{\tau}{2}}^{\frac{\tau}{2}} \lambda_x \cos \frac{\pi}{\tau}x \cdot dx = \mu_0 l F_{ad}\lambda_{ad}; \tag{13} \]
\[ \Phi_{aq} = \int\limits_{0}^{\tau} \mu_0 \lambda_x F_{aq}\cdot l \cdot \sin \frac{\pi}{\tau}x \cdot dx = \mu_0 l F_{aq} \int\limits_{0}^{\tau} \lambda_x \sin \frac{\pi}{\tau}x \cdot dx = \mu_0 l F_{aq}\lambda_{aq}, \]
where
\[
F_{ad},\ F_{aq}
\]
— the amplitude of the magnetomotive force per pole along the longitudinal and transverse axes, respectively,
\[
\lambda_x
\]
— the value of the magnetic permeance of the air gap at the point under consideration,
\[
\lambda_{ad},\ \lambda_{aq}
\]
— the total values of the magnetic permeance over the entire pole pitch along the longitudinal and transverse axes.
Using the formulas for the fluxes (13), as well as the expressions for the effective values of the emf (10), one can determine, by formula (11), the values of the synchronous inductive reactances
\[ x_{ad} = 2mf \frac{(W_1 \cdot k_{ob})^2}{p_1} \mu_0 \cdot l \cdot \lambda_{ad}; \tag{14} \]
\[ x_{aq} = 2mf \frac{(W_1 \cdot k_{ob})^2}{p_1} \mu_0 \cdot l \cdot \lambda_{aq}, \]
where \(p_1\) is the number of pole pairs of the stator winding.
The leakage inductive reactance \(x_s\), included in expression (8), can be determined by conventional methods.
REFERENCES
-
T. G. Soroker, P. P. Klobukov. Single-phase synchronous reluctance motors. Vestnik elektropromyshlennosti, No. 3, 1938.
-
A. S. Kurakin. Field in the gap of a reduction motor. Elektromekhanika, No. 2, 1963.
-
K. P. Kovács, I. Rácz. Transient processes in alternating-current machines. Gosenergoizdat, 1963.
-
Pohl R. Theory of pulsating-field machines, The Journal of the IEEEE, V93, 11, No. 31, 1946.
-
R. P. Laas, A. A. Brynko. Basic equations and synchronous mode of a parametric motor with capacitive compensation. Electronic and Electromechanical Devices, “Energiya,” 1969.
-
V. A. Ratmirov, B. A. Ivobotenko. Step motors for automatic-control systems, GEI, 1962.
-
L. E. Dombur. Magnetic field in the air gap of an axial inductor machine at no load with allowance for slotting. In: Contactless Electrical Machines, issue 4, Riga, 1965.
-
Lee C. H. Vernier motor and its calculation. Electrical Systems and Electrotechnical Equipment, No. 6, 1963.