SELECTION OF RATIOS BETWEEN THE PRINCIPAL DIMENSIONS OF A MINIMUM-WEIGHT DC ELECTROMAGNET

A. V. KURNOSOV

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

For electromagnets operating in air and ground transport, and in various mobile devices, the requirements of reducing their overall dimensions and weights are highly pressing. The approach to satisfying these requirements may vary depending on the conditions imposed on the electromagnetic mechanism. However, in most cases it is expedient to choose the principal dimensions so that the ratio of the mechanical work performed by the electromagnet to the weight of its active materials is maximal.

The variety of existing electromagnet designs does not allow all the features of each type to be taken into account. Let us consider the ratios between the principal geometric dimensions as applied to a valve-type electromagnet (Fig. 1). We shall adopt the following assumptions: the cross section of the steel along the length of the magnetic circuit is constant; in the unoperated position of the electromagnet armature the steel is unsaturated; when the electromagnet armature is attracted, the entire m.m.f. of the winding is expended on conducting the magnetic flux through the magnetic circuit; after operation of the electromagnet its winding remains under current for a long time and reaches the steady heating temperature; the current in the winding during the motion of the armature remains unchanged. The work performed by the armature of the electromagnet is proportional to the area bounded by the flux-linkage curve constructed for the two extreme positions of the armature (attracted and unattracted) and by the straight line of the steady-state current value (Fig. 2) [1].

Fig. 1. Sketch of an electromagnet.

Having transformed the general expressions describing the operation of dc electromagnets, as applied to a valve-type electromagnet in the same way as was done for a cylindrical electromagnet [1], we find the expression for the mechanical work of the electromagnet under consideration:

\[ W_{\mathrm{m}}=\frac{K V_{\mathrm{c}}\sqrt{S_0 V_0}}{l_{\mathrm{c}}l_{\mathrm{m}}}. \tag{1} \]

where

$$ K=\frac{K_{\mathrm{n}}(R+B_{\mathrm{n}})-K_{\mathrm{v}}\sigma_{\mathrm{p}}B_{\mu\mathrm{t}}}{2}\sqrt{K_0}, $$

$$ \sigma_{\mathrm{p}}=\frac{1+2\sigma}{3} $$

is the reduced leakage coefficient for a valve-type electromagnet;

\(\sigma\) is the leakage coefficient of the valve-type electromagnet [2];

$$ K_{\mathrm{n}},\ B,\ B_{\mathrm{n}},\ B_{\mu\mathrm{t}},\ K_0 \quad [1]; $$

$$ V_{\mathrm{c}}=Q_{\mathrm{c}}l_{\mathrm{c}} $$

is the volume of steel in the magnetic circuit;

$$ Q_{\mathrm{c}}=\frac{\pi}{4}D^2X^2 $$

is the cross section of the magnetic circuit; \(X=\dfrac{d}{D}\);

$$ l_{\mathrm{c}}=D\left(\frac{3\pi}{4}X^2+2Y+1\right) $$

is the mean length of the magnetic circuit;

$$ Y=\frac{l_{\mathrm{k}}}{D};\qquad V_{\mathrm{o}}=Q_{\mathrm{o}}l_{\mathrm{m}} $$

is the volume of the winding;

$$ Q_{\mathrm{o}}=D^2\cdot Y\cdot \frac{1-X}{2} $$

is the winding cross section;

$$ l_{\mathrm{m}}=\frac{\pi}{4}D(1+X) $$

is the mean turn length;

$$ S_{\mathrm{o}}=D^2\left[\pi(Y+X^2)+\left(1+\frac{\pi}{2}X^2(1+\pi X^2+X+Y)\right)\right] $$

is the heat-emitting surface of the electromagnet, which consists of the outer surface of the magnetic circuit and the outer lateral surface of the coil. The surfaces of the coil flanges are neglected.

$$ K_{\mathrm{v}}=\frac{G_{\mu\delta}}{G_{\mu T}} $$

is a coefficient taking into account the bulging of the magnetic lines of force. Figure 2 presents experimental curves for the dependence of \(K_{\mathrm{v}}\) on the armature rotation angle \(\varphi\). As is seen from the curves, the value of \(K_{\mathrm{v}}\) changes only very slightly within the range of optimal values

Fig. 2. Dependence of the bulging coefficient on the armature rotation angle \(\varphi\).
1 — for \(X=0.7\); 2 — for \(X=0.6\).

\(X\). In addition, experiment showed that \(\dot K_{\mathrm{v}}\) does not depend on the magnitude \(Y\).

\(G_{\mu\delta}\) is the permeance for the magnetic flux in the working air gap,

\(G_{\mu T}\) is the permeance for the magnetic flux leaving the end face of the core in the working air gap.

To find the ratios \(X\) and \(Y\) corresponding to an electromagnet of minimum weight when it performs a specified mechanical work, we divide [1] by the total weight of the active materials:

\[ \frac{W_{\mathrm{m}}}{G} = \frac{K V_{\mathrm{c}}\sqrt{V_{\mathrm{o}}S_{\mathrm{o}}}}{l_{\mathrm{m}}l_{\mathrm{c}}G}. \tag{2} \]

The weight of the active materials, reduced to the weight of the conductor, can be expressed as

\[ G=\gamma_{\mathrm{e}}\left(K_{\mathrm{v}}V_{\mathrm{o}}+\varepsilon V_{\mathrm{c}}\right), \tag{3} \]

where

\[ \varepsilon=\frac{\gamma_{\mathrm{c}}}{\gamma_{\mathrm{e}}}, \]

\(\gamma_{\mathrm{c}}\) is the specific weight per unit volume of steel;

\(\gamma_{\mathrm{e}}\) is the specific weight per unit volume of the conductor;

\(K_{\mathrm{z}}\) is the winding space factor of the wire.

Expressing in the right-hand side of expression (2) all geometric parameters through the ratios \(X\) and \(Y\), and transferring all absolute parameters to the left-hand side, we obtain

\[ \frac{W_{\mathrm{m}}} {K\sqrt{\dfrac{D}{\pi}\left(K_{\mathrm{z}}V_{\mathrm{o}}+\varepsilon V_{\mathrm{c}}\right)}} = \frac{ X^3 \sqrt{ \dfrac{ \left[\pi\left(Y+X^2\right)+\left(1+\dfrac{\pi}{2}X^2\right)\left(1+\pi X^2+X+Y\right)\right]Y(1-X) }{ 1+X } } }{ K_{\mathrm{z}}Y(1-X^2)+\varepsilon X^2\left(2.36X^2+2Y+1\right) }, \tag{4} \]

The right-hand side of expression (4) is a geometric factor \(\Gamma_{\mathrm{v}}(X,Y)\), characterizing the geometry of the electromagnet with respect to the mechanical work per unit weight of active materials. Figure 3 presents the curves \(\Gamma_{\mathrm{v}}(X,Y)\) for a copper conductor and \(K_{\mathrm{z}}=0.6\) as functions of \(X\) and \(Y\). The optimal values of \(X\) lie within the range \((0.5\div 0.6)\); as the optimal value of \(Y\) it is recommended to take \((1.5\div 3)\). Analysis of \(\Gamma_{\mathrm{v}}(X,Y)\) as a function of \(K_{\mathrm{z}}\) shows that, as \(K_{\mathrm{z}}\) decreases, the optimum value of \(X\) shifts to the left; thus, for \(K_{\mathrm{z}}=0.3\), \(X=(0.4\div 0.5)\), with the optimum value of \(Y\) unchanged; for \(K_{\mathrm{z}}=0.7\), \(X=(0.55\div 0.7)\). Investigation of \(\Gamma_{\mathrm{v}}(X,Y)\) with respect to \(\gamma_{\mathrm{e}}\) shows that the lighter the conductor material is in comparison with copper, the more the optimum value of \(X\) shifts to the left, and conversely, with the optimum value of \(Y\) unchanged.

From expression (4), the basic dimension can be found as

\[ D=\sqrt[7]{ \frac{\pi W_{\mathrm{m}}^2} {K^2\left[\Gamma_{\mathrm{v}}(X,Y)\right]^2 b} }, \tag{5} \]

where

\[ b= \frac{\pi}{4} \left[ K_{\mathrm{z}}(1-X^2)Y+\varepsilon X^2\left(2.36X^2+2Y+1\right) \right]. \]

The remaining values of the geometric quantities of the electromagnet are determined from the basic dimension \(D\) and the principal ratios \(X\) and \(Y\).

Fig. 3. Dependence of the geometrical factor on the ratios between the principal dimensions.

Conclusions

1. The ratios of the principal geometrical dimensions corresponding to an electromagnet of minimum weight are in direct dependence on \(K_3\) and \(\gamma_3\), and must be selected on the basis of experience in manufacturing coils for known \(K_3\) and \(\gamma_3\).

2. The optimal values of \(X\) corresponding to the maximum of \(\Gamma_{\mathrm{B}}(X,Y)\) for a copper conductor lie within the range \((0.5 \div 65)\); the optimal values of \(Y\) in this case are recommended to be taken within the range \((1.5 \div 3)\).

3. The overall dimensions and weight of the electromagnet will be smaller for that electromagnetic system in which \(X\) and \(Y\) are optimal.

REFERENCES

1. A. V. Kurnosov. Most advantageous ratios between the principal dimensions of cylindrical-type DC electromagnets. Izvestiya TPI, vol. 160, 1966.

2. F. A. Stupel. Calculation and Design of Electromagnetic Relays. Gosenergoizdat, 1950.