ON THE SELECTION OF THE MAIN PARAMETERS OF CYLINDRICAL DC ELECTROMAGNETS

A. V. KURNOSOV

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

If, in designing a cylindrical electromagnet, the initial data specify the mechanical work that the electromagnet must perform, then, under the condition of maximum utilization of active materials, one should take \(X = 0.65—0.7\) and \(Y = 1.5—2.5\) [1].

From expression (8) [1] the basic dimension can be found as

\[ D = \sqrt[7]{\frac{W_{\mathrm{m}}^{2}}{[\Gamma(X,Y)]^{2} K_{01}^{2} a^{2}}} , \tag{1} \]

where \(W_{\mathrm{m}}\) is the mechanical work of the electromagnet;

\[ a = \frac{\pi}{4} \left[ X^{2}\left(2Y + 1.5\sqrt{1 + X^{2}} - 0.5\right) + Y(1 - X^{2}) \right] — \]

a coefficient depending on the relationships between the principal dimensions (see Appendix 1).

\[ K_{01} = \frac{1}{2} \left[ K_{\mathrm{n}}(B + B_{\mathrm{n}}) - \sigma_{\mathrm{p}} B_{\delta 0} \right] \sqrt{\frac{K_{\mathrm{T}} K_{\mathrm{z}} \tau}{\rho}} \ [1]. \]

\(K_{\mathrm{n}} = 0.92—0.96\) [1], with the larger value of the coefficient \(K_{\mathrm{n}}\) corresponding to annealed steel, and the smaller to unannealed steel.

The coefficients \(K_{\mathrm{T}}, K_{\mathrm{z}}, \rho\) may be taken according to [2], with subsequent refinement during the calculation.

The value of \(\tau\) is taken depending on the adopted insulation class.

The values of the inductions \(B_{\mathrm{n}}\) and \(B\) for ARMCO steel may be taken, respectively, as \((1.4—1.5)\ \mathrm{Wb/m^{2}}\) and \((1.7—1.75)\ \mathrm{Wb/m^{2}}\), and \(\Gamma(X,Y)\) is a geometric factor determined from Fig. 3 [1].

The calculation of the geometric parameters of the electromagnet is carried out by the following formulas:

1) volume of the magnetic circuit, Fig. 1 [1],

\[ V_{\mathrm{c}} = \frac{\pi d_{2}}{4} l_{\mathrm{c}} = \frac{\pi}{4} D^{3} X^{2} \left[ 2Y + 1.5\sqrt{1 + X^{2}} - 0.5 \right]; \]

\[ l_{\mathrm{c}} = D\left(2Y + 1.5\sqrt{1 + X^{2}} - 0.5\right) — \]

the mean length of the magnetic circuit;

2) coil volume

\[ V_o = Q_o l_M = \frac{\pi}{4} D^3 Y(1 - X^2); \]

\[ Q_o = \frac{1}{2} D^2 Y(1 - X) \quad \text{— cross section of the winding window;} \]

\[ l_M = \frac{\pi}{2} D(1 + X) \quad \text{— mean turn length;} \]

3) blown surface of the electromagnet

\[ S_o = \pi D^2 \sqrt{1 + X^2}\,(1.5\sqrt{1 + X^2} + Y - 1). \]

Usually [2—3], when selecting the principal dimensions of electromagnets, the values of the induction in the working gap and the leakage coefficient are specified on the basis of experience in designing electromagnet structures. With the introduction into the theory of electromagnet geometry of dimensionless quantities, it became possible to choose the above-mentioned parameters specifically in each case.

The leakage coefficient for cylindrical-type electromagnets can be written as

\[ \sigma = 1 + \frac{G_{\mathrm{p}}}{K_{\mathrm{v}}G_{\mu T}}, \tag{2} \]

\[ G_{\mathrm{p}} = \frac{g l_k}{2} \quad \text{— leakage-flux permeance;} \]

\[ g = \frac{2\pi\mu_o}{\ln \dfrac{D}{d}} \quad \text{— specific leakage-flux permeance [4],} \]

\(G_{\mu T}\) — permeance of the flux in the region of the end part of the core and yoke in the working air gap;

\(K_{\mathrm{v}} = 1.2—1.5\) — coefficient allowing for the bulging of the magnetic lines of force in the working air gap.

Expressing all geometric dimensions in (2) through dimensionless quantities, we obtain the final expression for the leakage coefficient:

\[ \sigma = 1 + \frac{8Y}{K_{\mathrm{v}}X^2 \ln \dfrac{1}{X}} \cdot \frac{\delta}{D}. \tag{3} \]

Knowing \(\delta\), one can prescribe, in a definite way, the values of the magnetic induction in the working gap. In order that the electromagnet be utilized in terms of steel and not consume a large mmf in the unpulled-in position of the armature, it is expedient to take the induction at the base of the core \(B_o\) at the knee of the magnetization curve of the magnetic-circuit material. Then, neglecting the mmf losses in the steel of the magnetic circuit, the induction in the working air gap is determined as

\[ B_{\mu\delta} = \frac{B_o}{\delta}. \tag{4} \]

The mmf of the winding is determined by the formula

\[ IW = \sqrt{\frac{K_T K_З Q_o S_o \tau}{\rho l_M}}. \tag{5} \]

The further calculation of the electromagnet is carried out in the usual manner [2].

Conclusion

The introduction of the dimensionless quantities \(X\) and \(Y\) into the theory of the geometry of cylindrical DC electromagnets makes it possible to find the parameters of electromagnets corresponding to the maximum utilization of active materials, and also to determine unambiguously the magnetic induction in the working air gap.

APPENDIX 1.

The total volume of active materials in the left-hand side of expression (8) [1] can be expressed through the ratios between the principal dimensions \(X\) and \(Y\) as

\[ V_o + V_c = Q_o l_m + Q_c l_c = \]

\[ = \frac{\pi}{4} \left[ X^2 (2Y + 1.5)\sqrt{1+X^2} - 0.5) + Y(1-X^2) \right] D^3 = aD^3. \tag{1′} \]

LITERATURE

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

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

3. Roters. Electromagnetic Mechanisms, Gosenergoizdat, 1949.

4. A. V. Gordon, A. G. Slivinskaya. DC Electromagnets, Gosenergoizdat, 1960.

5. M. G. Yudelin. Selection of optimal ratios of copper and steel in DC apparatus, Vestnik elektropromyshlennosti, No. 11, 1961.