OPTIMAL RATIOS OF THE MAIN GEOMETRIC DIMENSIONS OF CYLINDRICAL DC ELECTROMAGNETS

A. V. KURNOSOV

(Presented by the scientific seminar of the Department of Electrical Machines and Apparatus)

Electromagnets of this type have found wide application for lifting steel scrap, as holding electromagnets, and as electromagnetic clutches. The calculation of these electromagnets can be considerably simplified if the conditions corresponding to the maximum utilization of active materials are known in advance. In the final result, they reduce to finding the ratios between the diameter of the core \(d\) and the diameter of the coil \(D\), and also between the length of the coil \(l_k\) and \(D\). These ratios completely determine the geometry of a cylindrical electromagnet. Figure 1 presents a sketch of the electromagnet.

Fig. 1. Sketch of an electromagnet.

As the basis for choosing the ratios between the main dimensions of the electromagnet, one may take the ratio of the greatest mechanical work \(W_m\) that the electromagnet can perform to the volume of active materials:

\[ \frac{W_{\mathrm{m}}}{V_{\mathrm{c}}+V_{\mathrm{o}}}, \]

where \(V_{\mathrm{c}}\) is the volume of steel of the magnetic core;
\(V_{\mathrm{o}}\) is the volume of the coil winding.

The greatest mechanical work that the electromagnet must perform is equivalent to the area of the loop \(S_{ocabo}\) (Fig. 2).

Fig. 2. Approximation of flux-linkage curves for two extreme positions of the armature.

In deriving the basic relations we shall make the following assumptions:

a) the steel cross section \(Q_{\mathrm{c}}\) along the length of the magnetic core is taken to be constant;

b) in the attracted position of the armature, the entire m.m.f. of the winding is expended on conducting the magnetic flux along the magnetic core;

c) in the non-attracted position of the armature, saturation of the steel is neglected;

d) after operation, the electromagnet remains energized for a long time and reaches a steady overheating temperature \(\tau\);

e) the current in the winding during motion of the armature remains unchanged.

The work performed by the armature of the electromagnet is proportional to the area of the “current–flux” loop enclosed between the curves \(\psi=f(i)\) constructed for the two extreme positions of the armature (attracted and non-attracted). We shall express the area \(S_{ocabo}\) through the electrical and magnetic parameters of the electromagnet. To do this, we approximate the curves \(\psi=f(i)\) in the closed and open positions of the armature.

Approximation by means of analytic formulas [1] leads to cumbersome expressions and does not make it possible to express the conditions for the maximum specific work in explicit form. As analysis of the problem shows, piecewise-linear approximation leads to simpler and more satisfactory results.

The actual curve “oc1” is replaced by the broken line “ok–k1.” The equations of the straight lines may be written in the form:

\[ 1)\quad \Psi=\Psi_{\mathrm{н}}+K_1 i,\quad \text{where } K_1=\frac{\Psi-\Psi_{\mathrm{н}}}{I},\quad \text{for } \Psi>\Psi_{\mathrm{н}}; \]

\[ 2)\quad \Psi=K_2 i,\quad \text{where } K_2=\frac{\Psi_{\mathrm{б}}}{I}. \]

The expression for the electromagnetic energy expended in performing mechanical work can be written as

\[ W_{\mathrm{m}}=\int_{0}^{1} K_{\mathrm{n}}\left(\psi_{\mathrm{n}}+K_{1}i\right)\,di-\int_{0}^{1}K_{2}i\,di . \tag{1} \]

Substituting their expressions for \(K_{1}\) and \(K_{2}\) and carrying out the corresponding transformations, we obtain the final expression for the work:

\[ W_{\mathrm{m}}=\frac{H V_{\mathrm{c}}}{2}\left[K_{\mathrm{n}}\left(B+B_{\mathrm{n}}\right)-\sigma_{\mathrm{p}}B_{\mu\delta}\right]. \tag{2} \]

(see Appendix 1).

In equality (2) we shall express \(H\) through the geometrical parameters of the electromagnet; in doing so we shall proceed from the permissible excess temperature of the electromagnet winding.

The losses in the winding of any direct-current electromagnet under steady-state conditions are determined by the expression:

\[ P=I^{2}\rho\,\frac{l_{\mathrm{m}}}{q_{\mathrm{m}}}\,w . \tag{3} \]

\(w\) — number of turns of the coil;
\(\rho\) — specific resistance of the electromagnet winding;
\(l_{\mathrm{m}}\) — mean length of a winding turn;
\(q_{\mathrm{m}}\) — cross section of the copper conductor.

Expression (3) can be written in the following form:

\[ j^{2}\rho V_{\mathrm{o}}K_{\mathrm{z}}=K_{\mathrm{t}}S_{\mathrm{o}}\tau, \tag{4} \]

where \(j\) — current density in the winding;
\(K_{\mathrm{z}}\) — filling factor;
\(K_{\mathrm{t}}\) — coefficient of heat transfer from the surface;
\(S_{\mathrm{o}}\) — external surface of the coil and magnetic core.

The magnetic-field intensity in a homogeneous magnetic circuit is

\[ H=\frac{jQ_{\mathrm{o}}K_{\mathrm{z}}}{1.65\,l_{\mathrm{c}}}, \tag{5} \]

\(Q_{\mathrm{o}}\) — cross section of the winding window;
\(l_{\mathrm{c}}\) — mean length of the magnetic circuit;
\(1.65\) — reserve factor for ampere-turns.

Squaring both sides of equality (5), expressing \(j^{2}\) from the resulting relation and substituting it into (4), we find an expression for \(H\) in terms of the geometrical parameters of the electromagnet:

\[ H=\frac{\sqrt{K_{\mathrm{o}}S_{\mathrm{o}}V_{\mathrm{o}}}}{l_{\mathrm{c}}l_{\mathrm{m}}}, \tag{6} \]

where

\[ K_{\mathrm{o}}=\frac{K_{\mathrm{t}}K_{\mathrm{z}}\tau}{1.65^{2}\rho}. \]

We substitute expression (6) into (2). Dividing both sides of the resulting equality by the sum of the volumes of the active materials, after some transformations we obtain the electromechanical energy per unit volume of active materials:

\[ \frac{W_{\mathrm{m}}}{V_{\mathrm{c}}+V_{\mathrm{o}}} = \frac{K_{\mathrm{o1}}V_{\mathrm{c}}\sqrt{V_{\mathrm{o}}S_{\mathrm{o}}}} {l_{\mathrm{c}}l_{\mathrm{m}}\left(V_{\mathrm{c}}+V_{\mathrm{m}}\right)}, \tag{7} \]

where

\[ K_{\mathrm{o1}}= \frac{K_{\mathrm{n}}\left(B_{\mathrm{n}}+B\right)-\sigma_{\mathrm{p}}B_{\mu\delta}}{2}\,K_{\mathrm{o}}. \]

Expression (7) is suitable for analyzing any type of tractive electromagnet. Let us apply expression (7) to the analysis of the cylindrical-type electromagnet in Fig. 1.

Expressing, in the right-hand side of expression (7), all geometric parameters of the electromagnet through its principal dimensions \((d, D, l_k)\), analogously to [2], and referring them to some one dimension (in our case \(D\)), we obtain:

\[ \frac{W_m}{K_{01}(V_c+V_o)\sqrt{D}} = \frac{ \sqrt{ \frac{ y(1-X)\sqrt{1+X^2}\left(1.5\sqrt{1-X^2}+y-1\right) }{ 1+X } } }{ y\left(\frac{1}{X^2}-1\right) + \left[ 1+2y+1.5\left(\sqrt{1+X^2}-1\right) \right] }; \tag{8} \]

where

\[ X=\frac{d}{D}; \qquad y=\frac{l_k}{D}. \]

We shall denote the right-hand side of expression (8) by \(\Gamma\) and call it the geometric factor. In Fig. 3, curves of \(\Gamma\) are plotted for various values of \(X\) and \(y\). As can be seen from the curves, the maximum of the geometric factor lies within the range of the ratios \(X=(0.65 \div 0.7)\) and \(y=(1.5 \div 2.5)\).

Fig. 3. Curves of the dependence of the geometric factor on \(X\) and \(Y\).

Conclusion: on the basis of an analysis of the curves of the geometric factor, the optimal ratios of the principal dimensions of a cylindrical-type electromagnet lie within the ranges

\[ \frac{d}{D}=0.65 \div 0.7; \qquad \frac{l_k}{D}=1.5 \div 2.5. \]

APPENDIX 1

\[ W_M = K_H \left[ \Psi_H + \frac{\Psi + \Psi_H}{2I} i \right] i - \frac{1}{2} i \left( \frac{\Psi_\delta}{I} i \right) = \]

\[ = K_H \left[ B_H + \frac{B - B_H}{2} \right] H V_c - \frac{w \Phi_{\mathrm{avg}} I}{2}, \qquad (1') \]

where

\[ K_H = \frac{S_{\mathrm{osado}}}{S_{\mathrm{okado}}} \]

is a correction factor taking into account the area \(S_{\mathrm{okass}}\);

\(B_H\) is the magnetic induction corresponding to \(\Psi_H\);

\(H\) is the magnetic-field strength in the closed position of the armature;

\(B\) is the magnetic induction chosen beyond the knee of the magnetization curve;

\(\Psi_\delta = W\Phi_{\mathrm{avg}}\), where \(\Psi_\delta\) is the flux linkage of the electromagnet in the un-attracted position of the armature.

\[ \Phi_{\mathrm{avg}} = \frac{1}{l_a}\int_{0}^{l_a} \Phi_x\,dx \ [3] \]

is the average flux along the length of the electromagnet core;

\(l_a\) is the length of the core.

\[ \Phi_x = \Phi_{\max} - \frac{Iwgx^2}{2l_a} \ [3] \]

is the value of the flux in section \(x\) of the core;

\(\Phi_{\max}\) is the maximum value of the magnetic flux in a section of the core.

\[ \Phi_{\mathrm{avg}} = \Phi_{\max} - \frac{1}{3}\left(\frac{Iwgl_a}{2}\right), \]

but

\[ \frac{Iwgl_a}{2} = \Phi_{\max} - \Phi_{\mu\delta}, \]

therefore

\[ \Phi_{\mathrm{avg}} = \Phi_{\mu\delta}\left(\frac{1 + 2\sigma}{3}\right) = \sigma_p \Phi_{\mu\delta}, \]

where \(\Phi_{\mu\delta}\) is the magnetic flux emerging from the core in the working air gap;

\[ \sigma = \frac{\Phi_{\max}}{\Phi_{\mu\delta}} \]

is the leakage coefficient.

Substituting the expression for \(\Phi_{\mathrm{avg}}\) into \((1')\), we obtain:

\[ W_M = \frac{H V_c}{2} \left[ K_H (B + B_H) - \sigma_p B_{\mu\delta} \right], \qquad (1'') \]

where \(B_{\mu\delta}\) is the magnetic induction in the working air gap.

REFERENCES

1. L. A. Bessonov. Electrical Circuits with Steel, Gosenergoizdat, 1948.

2. A. V. Kurnosov. “On the Design of Direct-Current Electromagnets with an Optimal Ratio of Copper and Steel,” Proceedings of MEI, Electromechanics, issue 56, 1964.

3. N. E. Lysov. Calculation of Electromagnetic Mechanisms, Oborongiz, 1949.