Determination of the Magnetic Permeances of the Air Gaps of an E-Shaped Electromagnet with a Transversely Moving Armature
Abstract Generated abstract
This paper addresses the difficulty of calculating magnetic permeances for the leakage and fringing air gaps of an E-shaped electromagnet with a transversely moving armature, used as an element of a four-winding linear motor. It introduces relative geometric parameters based on armature displacement and describes an experimental method in which coil current measurements under unsaturated conditions are separated into iron-path and air-path components to derive equivalent inductance and magnetic permeance. Families of permeance curves are obtained for specified ranges of relative stack height, tooth length, displacement, and air gap, with a scaling rule proposed for geometrically similar electromagnets. Verification on a similar electromagnet showed deviations within about 5 percent for the nominal relative core length and 7 percent over the tested core-length range.
Full Text
DETERMINATION OF THE MAGNETIC PERMEANCES OF THE AIR GAPS OF AN E-SHAPED ELECTROMAGNET WITH A TRANSVERSELY MOVING ARMATURE
V. N. Gurnitskii
(Presented by the scientific seminar of the Electromechanical Faculty)
Existing methods for calculating the magnetic permeances of the air gaps of electromagnets, owing to the three-dimensional nature (in the general case) of the magnetic field and to the cumbersomeness of the calculation methods themselves, do not give full assurance that the data obtained as a result of the calculation have sufficient agreement with their true values. If, in the case of simple geometric forms of air gaps, the calculation of their permeances is relatively uncomplicated [1—4], then, when it is necessary to calculate the permeances of the leakage and fringing gaps for an electromagnet, such calculations often prove practically infeasible. In a number of cases, however, one can make use of certain experimental curves of the magnetic permeances of air gaps for calculating the magnetic circuit of an electromagnet.
Let us consider here the case of an E-shaped electromagnet with a transversely moving armature, which is an element of a four-winding linear motor [5].
The sliders of two-winding and three-winding linear motors, under overloads, possess the property of self-reversal. The self-reversal of the slider is considerably reduced as the number of stator windings of the motor increases, but at the same time the weights of the active materials per unit motor power also increase; a four-winding design is optimal in this respect. It is expedient to relate the geometrical dimensions of the yoke and armature of the electromagnet to the magnitude of the armature displacement \(\tau\) from the initial position (Fig. 1, a) to the final position (Fig. 1, b). Then the relative height of the iron stack of the electromagnet is
\[ b_*=\frac{b}{\tau}, \]
the relative length of the armature tooth is
\[ l_*=\frac{l}{\tau}, \]
the relative length of the yoke core is
\[ h_*=\frac{h}{\tau}, \]
relative magnitude of the armature displacement
\[ x_\tau=\frac{x}{\tau}, \]
relative magnitude of the electromagnet air gap
\[ \delta_\tau=\frac{\delta}{\tau}. \]
The introduction of relative units for expressing the geometrical dimensions of the given electromagnet makes it possible:
Fig. 1. E-shaped electromagnet with a transversely moving armature.
a) to relate to one another the values of the permeances of the air gaps for different \(b_\tau, l_\tau, x_\tau, \delta_\tau\), by constructing a family of curves of the equivalent magnetic permeances of the air gaps
\(G_\tau=f(b_\tau, l_\tau, x_\tau, \delta_\tau)\) for some definite value of \(\tau\);
b) to compute from the constructed curves, by simple recalculation, the magnitudes of the permeances of the air gaps for any \(\tau'\);
c) to trace the influence of the geometry of the magnetic circuit, \(\delta_\tau\), and \(x_\tau\) on the quantities \(G_\tau\). The construction of the family of curves \(G_\tau=f(b_\tau, l_\tau, x_\tau, \delta_\tau)\) was carried out in the following way. An alternating current \(I\) of specified voltage \(U\) was passed through the electromagnet coil (all measurements were carried out with an unsaturated magnetic circuit). At \(x_\tau=0\) (the final position of the armature) and \(\delta_\tau=0\) (\(b_\tau=\mathrm{const},\ l_\tau=\mathrm{const}\)) the current was minimal; as \(x_\tau\) and \(\delta_\tau\) increased, the current increased. We represent the current \(I\), for any \(b_\tau=\mathrm{const}\), \(l_\tau=\mathrm{const}\), \(x_\tau=\mathrm{const}\), \(\delta_\tau=\mathrm{const}\), in the form of two components (Fig. 2):
\(I_{\mathrm{m}}\)—the current proportional to the magnetic resistance of the magnetic circuit;
\(I_{\mathrm{a}}\)—the current proportional to the equivalent magnetic resistance of all paths of the induction flux through the air.
The current \(I_{\mathrm{ж}}\) was measured at \(x_z=0\), \(\delta_z=0\), \(b_z=\mathrm{const}\), \(l_z=\mathrm{const}\), and was considered constant for all other values of \(x_z\) and \(\delta_z\). This assumption proved justified, since when \(I_{\mathrm{ж}}\) and \(I_{\mathrm{B}}\) are commensurable, i.e., \(x_z\) and \(\delta_z\) are close to zero, the error \(\Delta I_{\mathrm{ж}}\) is also close to zero; when \(\Delta I_{\mathrm{ж}}\) begins to change appreciably, \(I_{\mathrm{B}}\gg I_{\mathrm{ж}}\). Thus, according to Kirchhoff’s first law,
\[ I_{\mathrm{B}} = I - I_{\mathrm{ж}} . \tag{1} \]
The equivalent inductance \(L_{\mathrm{B}}\) (Fig. 2), proportional to the equivalent magnetic permeance of the air gaps \(G_z\), for various \(b_z\), \(l_z\), \(x_z\), \(\delta_z\), is equal to (the active resistance of the winding is neglected)
Fig. 2. Equivalent circuit of the magnetic circuit of the electromagnet.
\[ L_{\mathrm{B}}=\frac{U}{2\pi f\cdot I_{\mathrm{B}}}, \tag{2} \]
where
\(f\) is the frequency of the supply voltage \(U\).
The quantities \(L_{\mathrm{B}}\) for various \(b_z\), \(l_z\), \(x_z\), \(\delta_z\) were converted into values of the equivalent magnetic permeances of the air gaps \(G_z\) by the formula
\[ G_z=\frac{L_{\mathrm{B}}}{w^2}, \tag{3} \]
where \(w\) is the number of turns of the electromagnet winding.
The curves \(G_z=f(b_z,l_z,x_z,\delta_z)\) were plotted for \(b_z=2\div 5\), \(l_z=1\div 3\), \(x_z=0\div 2\), \(\delta_z=0\div 0.1\), \(h_z=6\) (Fig. 3a, b, c, d). When \(h_z=5\div 7\) was varied, the values of \(G_z\) deviated only slightly from the mean value (\(h_z=6\)).
The calculated values of the equivalent magnetic permeances \(G\) for an electromagnet with any \(z'\) (in meters) are obtained by multiplying the values of \(G_z\) taken from the curves by \(\tau_0\):
\[ G=G_z\cdot \tau_0, \tag{4} \]
where \(\tau_0=\dfrac{z'}{0.01}\); \(0.01\) is the value of \(z\) (in meters) of the electromagnet on which the curves were obtained,
\[ G_z=f(b_z,l_z,x_z,\delta_z). \]
Indeed, formula (4) is valid, since the permeance of a similar electromagnet with \(z'\ne z\) changes by a factor of \(\dfrac{z'}{z}\). The permeance \(G_z\), for example, at large values of \(b\), \(l\), small \(\delta\), and \(x=0\), can be calculated as
\[ G_z=4\pi\cdot 10^{-7}\frac{b\cdot z}{\delta}. \tag{5} \]
If
\[ b=5z,\qquad \delta=0.025z, \]
then
\[ G_z=4\pi\cdot 10^{-7}\frac{5z\cdot z}{0.025z}=8\pi\cdot 10^{-5}\cdot z, \]
i.e., \(G_z\) depends linearly on \(z\).
Fig. 3 a, b, c, d. Curves of the equivalent magnetic permeances of the air gaps of an E-shaped electromagnet with a transversely moving armature.
Formula (4) is exact for homogeneous fields and approximate for nonhomogeneous ones; moreover, the approximation is the cruder the more \(\tau'\) differs from \(\tau\). When plotting the curves \(G_\tau=f(b_\tau,l_\tau,x_\tau,\delta_\tau)\), the magnetic circuit was laminated, and the surfaces bordering the paths of the magnetic flux through the air were machined with a finish of at least \(\nabla 4\) and with a deviation from flatness of no more than \(0.025\) mm \((0.0025\tau)\).
When using the curves, the values of the quantities \(G_\tau\) and \(b_\tau, l_\tau, x_\tau, \delta_\tau\) may be interpolated within the limits of their variation. Verification of the curves for calculating \(G\) of a similar electromagnet with \(\tau'=0.015\) \((\tau_0=1.5)\) showed that the deviation of the calculated permeance values from the true values in all cases did not exceed \(5\%\) for \(h_\tau=6\) and \(7\%\) for \(h_\tau=5\div 7\).
Conclusions
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The described method for finding the equivalent magnetic permeances of the air gaps of an E-shaped electromagnet with a transversely moving armature, while simple, makes it possible to obtain sufficiently accurate results.
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The constructed curves give a clear picture of the variation in the permeance values \(G_\tau\) as a function of the geometrical dimensions of the magnetic circuit \((b_\tau,l_\tau)\), the air gap \((\delta_\tau)\), and the tangential displacement of the armature \((x_\tau)\).
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The presented procedure for determining the permeances of the induction flux through the air can be applied to a series of similar electromagnets of any type.
LITERATURE
- Roters. Electromagnetic Mechanisms, GEI, 1949.
- A. V. Gordon, A. G. Slivinskaya. Direct-Current Electromagnets, GEI, 1960.
- Yu. S. Rusin. Calculation of magnetic permeance. Izv. vuzov “Elektromekhanika”, No. 12, 1963.
- B. K. Bul. Fundamentals of the Theory and Calculation of Magnetic Circuits, Energiya Publishing House, 1964.
- A. I. Zaitsev, V. N. Turnitskii. Development of a slow-speed linear drive. Abstracts of papers of the Third Scientific and Technical Conference on Problems of Production Automation, Tomsk, 1964.