Calculation of the Traction Force of an E-Shaped Direct-Current Electromagnet with a Transversely Moving Armature

V. N. Gurnitskii

(Presented by the scientific seminar of the Electromechanical Faculty)

A distinctive feature in calculating the traction force of an E-shaped direct-current electromagnet with a transversely moving armature is the presence of two electromagnetic forces that vary in magnitude as the armature moves. This electromagnet is an element of a four-winding linear electromagnetic motor. Calculation and construction of the static traction characteristic of the electromagnet make it possible to determine the dependence of the maximum developed force on the total displacement of the armature, and also to select a rational segment of the electromagnet armature travel.

The forces acting on the armature will be directed as shown in Fig. 1, where the following are denoted:

Fig. 1. Diagram of the action of static forces on the armature of the electromagnet.

$F_{\mathrm{t}}$ — the traction force developed by the electromagnet;
$F_{\mathrm{n}}$ — the normal force attracting the armature to the yoke;
$F_{\mathrm{tr}}$ — the friction force arising as a result of the action of the normal force and equal to the product of the normal force by the coefficient of friction $f_{\mathrm{tr}}$:

\[ F_{\mathrm{tr}} = F_{\mathrm{n}} \cdot f_{\mathrm{tr}}, \tag{1} \]

\(F_{xx}\) — no-load force;
\(F_p\) — prescribed function of the useful force.

Taking into account the directions of action of the forces considered, the equation of motion of the electromagnet armature may be written as follows:

\[ F_t - F_p - F_{tr} - F_{xx} = m \frac{\partial^2 x}{\partial t^2}, \tag{2} \]

where \(m\) is the mass of all moving parts;

\[ \frac{\partial^2 x}{\partial t^2} \]

is the acceleration with which the motion occurs in the direction \(x\).

The solution of equation (2) in statics, that is, when the current in the electromagnet winding is constant throughout the motion, \(I = \mathrm{const}\), and the motion occurs with an infinitely small velocity, while the right-hand side of equation (2) becomes zero, reduces to determining the forces \(F_t\) and \(F_p\) (the quantities \(F_{xx}\) and \(f_{tr}\) are determined by the design of the electromagnet).

The traction force \(F_t\), which is a function of the armature position \(x\) (for constant values of \(\delta\), \(\tau\), \(b\), \(l\), \(h\)), is equal to [1]:

\[ F_t = - \frac{(I w)^2}{2} \cdot \frac{\partial G_m}{\partial x}, \tag{3} \]

where \(w\) is the number of turns of the electromagnet winding;

\(G_m\) is the equivalent permeance of the magnetic core, of the paths along which the fringing flux passes, and of the air gap.

To determine this permeance, let us consider the equivalent substitution circuit of the magnetic circuit of the electromagnet (Fig. 2), where

\(G_{zh}\) — equivalent permeance of the magnetic core;

\(G_{\delta}\) — equivalent permeance of the air gap;

\(G_v\) — equivalent permeance of the paths along which the fringing flux passes;

\(G_z\) — equivalent leakage permeance.

(The arrows indicate permeances that vary when the armature is displaced and the current in the electromagnet winding remains unchanged.)

Let us denote the variable permeance by

\[ G_{\delta} - G_v = G_v, \tag{4} \]

then the equivalent permeance of the air gaps is

\[ G = G_{\delta} + G_v + G_z = G_v + G_z, \tag{5} \]

and the total equivalent permeance of the entire magnetic circuit of the electromagnet is

\[ G_m = \frac{1}{\dfrac{1}{G_{zh}} + \dfrac{1}{G}} . \tag{6} \]

Fig. 2. Equivalent-substitution circuit of the permeances of the magnetic circuit of the electromagnet.

The permeance \(G_z\), for an unchanged value of the current \(I\), does not depend on the position of the electromagnet armature and may be found for our,

case analytically [2]; after simple transformations we obtain

\[ G_{\sigma}=\mu_0 h\left(\frac{b}{3\tau}+0.396\right)\simeq \mu_0 h\left(\frac{b}{3\tau}+0.4\right). \tag{7} \]

This permeance does not enter under the derivative sign in the expression for the traction force as a constant quantity (3).

The permeance \(G_{\mathrm{m}}\) is equal to

\[ G_{\mathrm{m}}=\frac{1}{\dfrac{1}{G_v}+\dfrac{1}{G_{\mathrm{zh}}}}, \tag{8} \]

whence

\[ \frac{\partial G_{\mathrm{m}}}{\partial x} = \frac{G_{\mathrm{m}}^{2}}{G_v^{2}}\cdot \frac{\partial G_v}{\partial x} \tag{9} \]

and

\[ \frac{\partial G_{\mathrm{m}}}{\partial \delta} = \frac{G_{\mathrm{m}}^{2}}{G_v^{2}}\cdot \frac{\partial G_v}{\partial \delta}. \tag{9a} \]

Fig. 3a. Graphical method for finding the derivative

\[ \left(\frac{\partial G_v}{\partial x}\right)_{\delta=\mathrm{const}} \]

We find the derivative \(\dfrac{\partial G_v}{\partial x}\) from the curves \(G=f(x)\) given in [3] in the following way.

Compute, by formula (7), the value \(G_{\sigma}\) for the specified \(\tau, b, h\) and, subtracting it from the ordinates of the curves \(G=f(x)\) for the same \(\tau, b, h\), construct the curves \(G_v=f_1(x)\). From the constructed curves we obtain the derivative \(\dfrac{\partial G_v}{\partial x}\) graphically, drawing a tangent to the curve \(G_v=f_1(x)\) (Fig. 3a) and finding

\[ \left(\frac{\partial G_v}{\partial x}\right)_{x=x_1} =\tg\alpha=\frac{G_1}{x_2-x_1}. \tag{10} \]

The force of normal attraction of the armature to the yoke of the electromagnet (in the direction of the other possible displacement \(\delta\), which is, however, fixed by the structural elements of the electromagnet) is determined by

\[ F_{\mathrm{n}}=-\frac{(Iw)^2}{2}\cdot\frac{\partial G_{\mathrm{m}}}{\partial \delta}. \tag{11} \]

For finding \(\dfrac{\partial G_{\mathrm{m}}}{\partial \delta}\) and, first of all, \(G_{\mathrm{m}}=\varphi(\delta)\), we carry out the following operations.

We transform the curves \(G=f(x)\), constructed for fixed values of \(\delta\) and constant \(\tau, b, l, h\), into curves \(G=\varphi(\delta)\) for certain fixed values of \(x\) and the same \(\tau, b, l, h\).

Fig. 3b. Graphical method for finding the derivative

\[ \left(\frac{\partial G_v}{\partial \delta}\right)_{x=\mathrm{const}}. \]

Next, in accordance with equations (4), (5), and (7), we construct the curve \(G_{\nu}=\varphi_1(\delta)\) analogously to the preceding one. We find the derivative \(\dfrac{\partial G_{\nu}}{\partial \delta}\) graphically, drawing a tangent to the curve \(G_{\nu}=\varphi_1(\delta)\) (Fig. 3, b) and finding

\[ \left(\frac{\partial G_{\nu}}{\partial \delta}\right)_{\delta_i=\delta_1} =\tan\beta=\frac{G_2}{\delta_2-\delta_1}. \tag{12} \]

The conductance of the magnetic circuit \(G_{\mathrm{zh}}\) for a constant current \(I\) and a known grade of steel is found from the formula

\[ G_{\mathrm{zh}}=2\mu_{\mathrm{zh}}\frac{b\cdot \tau}{l_{\mathrm{cp}}}, \tag{13} \]

where \(l_{\mathrm{cp}}\) is the length of the mean line of force of the magnetic circuit (shown by a dashed line in Fig. 1), equal to

\[ l_{\mathrm{cp}}=2h+2l+8\tau, \tag{14} \]

\(\mu_{\mathrm{zh}}\) is found from the curves \(\mu_{\mathrm{zh}}(B)\), and
\(B\) from the curves \(B(H)\).

The accuracy of the calculation of the static forces \(F_{\mathrm{t}}\) and \(F_{\mathrm{n}}\) depends on the accuracy of the graphical constructions \(\dfrac{\partial G_{\nu}}{\partial x}\) and \(\dfrac{\partial G_{\nu}}{\partial \delta}\), and lies within the range of 5–10%, as well as on the errors introduced by the curves \(G_{\nu}=f(b_z,l_z,x_z,\delta_z)\) [3].

The theoretically constructed sample curves of the static forces \(F_{\mathrm{t}}\) and \(F_{\mathrm{n}}\) at \(\delta=\mathrm{const}\) (Fig. 4) were compared with their experimental values, measured by means of strain gauges, which gave well-matching results (the error did not exceed 10–15%).

NOTES

The curves \(G_z=f(b_z,l_z,x_z,\delta_z)\) were obtained for \(h_z=6\); for other values of \(h_z\), the indicated curves can be corrected by means of formula (7), which can take into account the leakage conductance \(G_\delta\) for various \(h\).

The curves \(G_z=f(b_z,l_z,x_z,\delta_z)\) were constructed with the electromagnet winding supplied at \(U_{\sim}=\mathrm{const}\), although, from the standpoint of applying formula (7) in calculations, such constructions should have been made at \(I_{\sim}=\mathrm{const}\). The practically coinciding curves \(G_z=f(b_z,l_z,x_z,\delta_z)\) for electromagnet-winding supply at \(U_{\sim}=\mathrm{const}\) and supply at \(I_{\sim}=\mathrm{const}\) [4], and the difficult experimental technique in the second case (the range of applied voltages reaches 100), led to the choice of the first variant.

Fig. 4. Static-force characteristics \(F_{\mathrm{t}}\) and \(F_{\mathrm{n}}\) of the electromagnet.

Conclusions

1. The proposed method makes it possible, in a simple way, to construct the static characteristics of the forces \(F_{\mathrm{t}}\) and \(F_{\mathrm{n}}\).

2. For an armature travel of the electromagnet equal to \(\tau\) and an overall change in the forces \(F_{\mathrm{t}}\) and \(F_{\mathrm{n}}\) within \(2\tau\), the curves \(F_{\mathrm{t}}\) and \(F_{\mathrm{n}}\) make it possible to choose a rational section of the electromagnet travel.

References

1. G. V. Mogilevskii. On the calculation of tractive forces in electromagnets, Vestnik elektropromyshlennosti, No. 4, 1960.

2. M. I. Vitenberg. Calculation of Electromagnetic Relays, GEI, 1961.

3. V. N. Turnitskii. Determination of the magnetic permeances of the air gaps of an E-shaped electromagnet with a transversely moving armature, Izvestiya TPI, vol. 139, Tomsk, 1965.

4. A. Ya. Builov. Fundamentals of Electrical Apparatus Engineering, GEI, 1946.