N. P. RYASHENTSEV, A. V. FROLOV

CALCULATION OF THE LATERAL MAGNETIC RELUCTANCE OF ELECTROMAGNETIC HAMMERS

(Presented by the Department of Mining Machines and Mine Transport)

Electromagnetic hammers with a free striker travel [1, 2] are, in structural terms, similar to shell-type electromagnets; however, they differ from electromagnets in that the magnitude of the working air gap varies periodically within rather wide limits and, in absolute value, is comparable with the length of the magnetizing coil. Therefore, during one operating cycle, depending on the position of the striker relative to the fixed magnetic core, a redistribution of the magnetic flux occurs, since the main magnetic flux and the leakage flux in any section of the magnetic circuit depend on the magnitude of the working air gap.

Unlike electromagnets, in which a considerably larger part of the flux is concentrated between the inner parts of the poles and the pole shoes, in electromagnetic hammers with a free striker travel part of the magnetic flux closes through the outer side of the poles. Studies of the distribution of magnetic flux in electromagnetic hammers with a free striker travel have shown that the magnitude of the magnetic flux closing through the outer side of the poles constitutes a significant part of the total magnetic flux created by the coil (Fig. 1). The solid lines (Fig. 1) characterize the rear pole, and the dashed lines—the front pole. The magnitude of the relative flux \(\Phi^1\) is determined as

\[ \Phi = \frac{\Phi_{\mathrm{p}}}{\Phi_0}\cdot 100,\ \%, \]

where \(\Phi_{\mathrm{p}}\) is the flux passing through the outer lateral surface of the pole, and \(\Phi_0\) is the total flux created by the magnetizing coil.

In the absence of a working air gap, the lateral fluxes from the front and rear poles are equal to one another. As the air gap increases, the flux closing through the outer lateral surface of the front pole begins to decrease sharply and, at \(x = 2\) cm (Fig. 1 and Fig. 2, a), is practically equal to 0. The lateral flux of the rear pole increases with increasing air gap, reaches a maximum value, after which it remains practically constant. It has been experimentally established that the magnetic flux closing through the outer lateral surface of the rear pole reaches its maximum value at an air gap approximately equal to the incli-

of the pole height \(H\) (Fig. 2, a). Since the lateral flux constitutes a rather large part of the total magnetic flux (Fig. 1), it must obviously be taken into account when calculating the magnetic system of the hammer. The need to determine the lateral flux is also explained

![Figure 1 graph]

Fig. 1. Dependence of the lateral leakage flux on the magnitude of the striker overtravel for the hammer MS-16 (curve 1) and UÉ-1 (curve 2).

by the fact that the difference of magnetic potentials between the yoke and the striker, which determines the magnitude of the main leakage flux [3, 4, 5], is a function of the lateral magnetic reluctance. The magnitude of the lateral leakage flux depends on the magnetic reluctance of the section \(R\) through which this flux passes, and on the magnetic reluctance \(R_{\mathrm{v}}\) of the pole with the air gap between the pole and the striker \((d - d_{\mathrm{b}})\). In Fig. 2, a, a sketch is presented of part of the magnetic circuit of an electromagnetic hammer of the MS-16 type, which consists of the striker 1, the rear pole 2, the magnetizing coil 4, and the yoke 3. The same figure shows the cha-

![Figure 2 diagrams]

Fig. 2, a, b. Sketch of part of the magnetic circuit of an electromagnetic hammer (a) and the equivalent substitute circuit (b).

character of the magnetic-flux distribution in the rear pole and in the parts of the magnetic circuit adjoining it. For clarity, the magnetic flux is shown conventionally as concentrated, although in reality the magnetic lines of force encompass the entire space occupied by the magnetic system of the hammer, as well as some distance away from it. In Fig. 2, b an equivalent electrical substitute circuit is given for the part of the magnetic circuit under consideration, where \(R_y^I\) and \(R_y^{II}\) are the magnetic reluctances for the leakage fluxes \(\Phi_y^I\) and \(\Phi_y^{II}\); \(R_{\text{я}}\) is the reluctance of the yoke between the armature and the pole; \(R_{\text{в}}\) is the reluctance of the yoke between the pole and the striker; \(R_1\), \(R_2\), \(R_3\) are magnetic reluctances characterizing the magnitude of the lateral flux. The leakage fluxes from the outer lateral surface of the pole, like all leakage fluxes, have a very complex shape, and the fields formed by them are difficult to describe exactly mathematically. In those cases where the magnetic lines of force of leakage pass through the region of the magnetizing coil, the leakage fluxes cannot be calculated even by the method of graphical construction of the magnetic-field pattern [3]. Taking into account that the magnetic lines of force of the lateral leakage flux pass through a region not occupied by the magnetizing coil, it may be assumed that the magnetic field between the outer lateral surface of the pole and the striker is irrotational [4]. Consequently, to determine the magnetic reluctance to the lateral leakage flux, one can use the basic propositions of the method of the expected probable paths of the flux and of the method of graphical construction of the magnetic-field pattern [3, 4, 5]. In calculating the magnetic reluctance, we shall assume that the magnetic lines of force enter and leave the ferromagnetic sections of the magnetic circuit at a right angle.

We neglect the magnetic field outside the volume formed by rotation of a sector of radius \(H\) and aperture angle \(\alpha\) (Fig. 2, a) about the axis of symmetry of the hammer \(ox\). This assumption lies within the limits of the calculation accuracy.

On the basis of the general propositions of the method of the expected probable paths of the flux and the recommendations for dividing the field into simple geometric figures [3, 4, 5], the volume occupied by the lateral leakage flux is divided into three simple volumes, the magnetic reluctances \(R_1\), \(R_2\), and \(R_3\) of which can be determined rather simply. It is recommended that the magnetic reluctances \(R\) of volumes be determined as the quotient obtained by dividing the mean magnetic line of force \(l\) by the mean value of the cross section \(S\) along the path of the magnetic line of force

\[ R=\frac{l}{\mu_0 S}, \tag{1} \]

where \(\mu_0\) is the magnetic permeability of air.

When one of the elements \(l\) and \(S\) cannot be expressed analytically, it is determined by measurement from the drawing with subsequent approximation as necessary [5]. However, more accurate results in calculating the reluctance are obtained by using the volume \(V\) occupied by the magnetic field, since one of the components of expression (1) is then determined without error:

\[ R=\frac{V}{\mu_0 S^2}=\frac{l^2}{\mu_0 V}. \tag{2} \]

The volume \(V\) formed by rotation of an area \(S'\) about an axis lying in the plane of the figure and not intersecting it will be determined by Guldin’s rule [6]:

\[ V=2\pi S' R_0, \tag{3} \]

where \(R_0\) is the distance between the axis of rotation and the center of gravity of the rotating figure. When determining the magnetic reluctance \(R_1\) of the first section, we assume that its volume is formed by the rotation of an elliptical quadrant with semiaxes \((H-x)\) and \(\dfrac{d_\sigma}{2}\) about the axis \(ox\). Moreover, one of the semiaxes of the ellipse remains constant as the coordinate \(x\) changes, which corresponds to the assumptions adopted earlier.

According to [6], the volume occupied by the magnetic field of the first section is

\[ V_1 = 2\pi S'_1 R_{01}, \]

where \(S'_1\) is the area of the elliptical quadrant, proportional to the product of its semiaxes [6],

\[ S'_1 = \frac{\pi}{8} d_\sigma \cdot (H-x). \tag{4} \]

\(R_{01}\) is the distance from the center of gravity to the axis of rotation [6],

\[ R_{01} = \frac{d_\sigma}{2} - \frac{2d_\sigma}{3\pi}. \tag{5} \]

The notation adopted in expressions (4), (5) and below will correspond to Fig. 2.

We shall determine the cross-sectional area through which the magnetic lines of force pass as the mean area of entry and exit of the magnetic lines of force in the first section:

\[ S_1 = \frac{\pi d_\sigma}{2}(H-x) + \frac{\pi d_\sigma^2}{8} = \frac{\pi d_\sigma}{8}(4H + d_\sigma - 4x). \tag{6} \]

The notation adopted in expressions (4), (5) and below will [[unclear: beginning of phrase]] of the first section

\[ R_1 = \frac{4.6(H-x)}{\mu_0(4H+d_\sigma-4x)^2}. \tag{7} \]

We determine the volume of the second section as the product of the area of the part of the ring formed by the radii \(H\) and \((H-x)\) and the ordinate of the center of gravity relative to the axis of rotation, which is equal to [6]

\[ R_{02} = \frac{d_\sigma}{2} = \frac{(1-\cos\alpha)}{3\pi k_1} \cdot \left(\frac{H^3-x^3}{H^2-x^2}\right). \tag{8} \]

Then the volume of the second section is

\[ V_2 = 2\pi^2 k_1 \left[H^2-(H-x)^2\right] \left[ \frac{d_\sigma}{2} + \frac{(1-\cos\alpha)(H^3-x^3)} {3\pi k_1 (H^2-x^2)} \right], \tag{9} \]

where

\[ k_1 = \frac{\alpha}{360}. \]

When calculating the magnetic reluctance \(R_2\), it is more convenient to use the mean length of the magnetic line of force

\[ l_2 = 2\pi \frac{H+x}{2} \cdot \frac{\alpha}{360} = \pi k_1 (H+x); \tag{10} \]

Substituting (9) and (10) into (2) and carrying out elementary transformations, we obtain

\[ R_2 = \frac{k_1 (H+x)^2} {\nu_0 \left[ d_\sigma H^2 + \frac{2H^3}{3\pi k_1}(1-\cos\alpha) - d_\sigma x^2 - \frac{2(1-\cos\alpha)}{3\pi k_1}x^3 \right]} . \tag{11} \]

In a similar way we determine the reluctance of the third section, the volume of which is formed by rotating a sector of radius \(x\) about the axis \(ox\). The distance from the center of gravity of the sector to the axis of rotation [6] is

\[ R_{03} = \frac{d_\sigma}{2} + \frac{(1-\cos\alpha)x}{3\pi k_1} = \frac{3\pi k_1 d_\sigma + 2(1-\cos\alpha)x}{6\pi k_1}. \tag{12} \]

The area of the sector is

\[ S'_3 = \pi k_1 x^2 . \tag{13} \]

By solving (3, 12, 13) jointly, we determine the volume occupied by the magnetic field:

\[ V_3 = \frac{1}{3}\pi x^2 \left[3\pi k_1 d_\sigma + 2(1-\cos\alpha)x\right]. \tag{14} \]

It has been established by calculation that the equivalent cross-sectional area through which the magnetic lines of force of the third section pass can be determined with a sufficient degree of accuracy by the formula

\[ S_3 = \frac{1}{2}\pi x(d_\sigma + 4x\sin\alpha). \tag{15} \]

Substituting (14) and (15) into (2), we determine the magnetic reluctance of the third section:

\[ R_3 = \frac{4\left[3\pi k_1 d_\sigma + 2(1-\cos\alpha)x\right]} {3\pi \nu_0 (d_\sigma + 4x\sin\alpha)^2}. \tag{16} \]

The total lateral magnetic reluctance for the fluxes closing through the outer lateral surface of the rear pole is determined, according to the equivalent circuit of the magnetic circuit (Fig. 2, b), as

\[ R = \frac{(R_1+R_2)\cdot R_3} {R_1+R_2+R_3}. \tag{17} \]

When determining the total lateral reluctance in expressions (7), (11), and (16), we rewrite the numerator and denominator in the form of a power series:

\[ R_1 = \frac{H-x} {\nu_0(d_1-b_1x+3.46x^2)}; \]

\[ R_2 = \frac{H^2+2Hx+x^2} {\nu_0\left(a_2-\frac{d_\sigma}{k_1}x^2-b_2x^3\right)}; \tag{18} \]

\[ R_3 = \frac{a_3+b_3x} {\nu_0(c_3+x+2x^2\sin\alpha)}; \]

where

\[ a_1 = 0.218(4H+d_\sigma)^2; \qquad a_3 = \frac{k_1 d_\sigma}{2\sin\alpha}; \]

\[ b_1 = 1.74\,(4H + d_6);\qquad b_3 = \frac{1 - \cos \alpha}{12\pi \sin \alpha}; \]

\[ a_2 = \frac{H^2}{k_1}\left[d_6 + \frac{2H(1-\cos \alpha)}{3\pi k_1}\right];\qquad c_3 = \frac{d_6^2}{8\sin \alpha}. \]

\[ b_2 = \frac{2(1-\cos \alpha)}{3\pi k_1^2}. \]

Substituting (18) into (17) and carrying out the corresponding transformations, we obtain

\[ R = \frac{1}{\nu_0}\, \frac{a + bx + cx^2 + dx^3 + ex^4 + fx^5} {A + Bx + Cx^2 + Dx^3 + Ex^4 + Fx^5 + Gx^6}. \tag{19} \]

In expression (19), the constant coefficients for the variable have the following values:

\[ a = a_3H(a_2 + H); \]

\[ b = b_3\frac{a}{a_3} - a_3(a_2 - 2a_1H + b_1H^2); \]

\[ c = b\frac{b_3}{a_3} - a_3g - a\frac{b_3^2}{a_3^2}; \]

\[ d = a_3\left(\frac{d_6}{k_1} - b_1 + 6.92H\right) - b_3g; \]

\[ e = a_3(b_2 + 3.46) + b_3\frac{d}{d_3} + g\frac{b_3^2}{a_3}; \]

\[ f = b_3(b_2 + 3.46); \]

\[ g = \frac{d_6}{2}H + b_2H - a_1 + 2b_1H - 3.46H^2; \]

\[ A = a_1 \cdot a_2 \cdot a_3 + a\frac{c_3}{a_3}; \]

\[ B = \frac{a}{a_3} + \frac{c_3}{a_3}\left(b - a\frac{b_3}{c_3}\right) + a_2(a_1b_3 - b_1a_3); \]

\[ C = 2\frac{a}{a_3}\sin \alpha - c_3g - a_2b_1b_2 + a_3\left(3.46a_2 - a_1\frac{d_6}{k_1}\right); \]

\[ D = \frac{c_3}{a_3}(d + b_3g) - g + \frac{2\sin \alpha}{a_3}\left(b - \frac{x_3}{a_3}a\right) + a_3\left(b_1\frac{d_6}{k_1} - a_1b_2\right); \]

\[ E = \frac{f}{b_3}c_3 + \frac{1}{a_3}(d + b_3g) - 2g\sin \alpha - a_3\left(3.46\frac{d_6}{k_1} - b_1b_2\right) + \]

\[ {}+ b_3\left(b_1\frac{d_6}{k_1} - a_1b_2\right); \]

\[ F = \frac{f}{b_3} + \frac{2\sin \alpha}{a_3}(d + b_3g) - 3.46a_3b_2 - b_3\left(3.46\frac{d_6}{k_1} - b_1b_2\right); \]

\[ G = 2\frac{f}{b_3}\cdot \sin \alpha - 3.46b_2b_3. \]

Experimental verification of expression (19) for determining the lateral magnetic reluctance was carried out in the laboratories of the Department of Mining Machines of the Tomsk Polytechnic Institute. According to Fig. 2, a, b, the lateral magnetic reluctance can be determined as

\[ R = R_{\mathrm{v}}\frac{\Phi-\Phi_{\mathrm{l}}}{\Phi_{\mathrm{l}}}, \tag{20} \]

where \(R_{\mathrm{v}}\) is the magnetic reluctance for the flux passing along the rear pole in the direction of the striker through the air gap between the pole and the striker [3, 5]

\[ R_{\mathrm{v}}=\frac{\ln \dfrac{d}{d_{\mathrm{b}}}}{2\pi\mu_0 k}. \tag{21} \]

Substituting (21) into (20), we find

\[ R=\frac{\ln \dfrac{d}{d_{\mathrm{b}}}}{2\pi\mu_0 k} \left(\frac{\Phi}{\Phi_{\mathrm{l}}}-1\right). \tag{22} \]

Fig. 3. Experimental (1, 3) and calculated (2, 4) dependences of lateral magnetic reluctance on the striker overhang for the MS-16 hammer (curves 1, 2) and the UE-1 hammer (curves 3, 4).

The magnetic flux \(\Phi_{\mathrm{p}}\), closing through the outer lateral surface of the rear pole, and the flux \(\Phi\), are measured by the induction method with the aid of measuring turns \(W_{\mathrm{pv}}\) and \(W_{\mathrm{b}}\) (Fig. 2, a). The magnitude of the e.m.f. induced in the measuring turns is determined with the aid of an electronic voltmeter with high input resistance. The results of calculating the lateral magnetic reluctance, carried out for electromagnetic hammers of the MS-16 type (\(\alpha=56^\circ\), \(d_{\mathrm{b}}=2.8\ \mathrm{cm}\), \(D=9.3\ \mathrm{cm}\), \(k=1.0\), and \(d=3.0\ \mathrm{cm}\)) and with impact-unit power \(\alpha=90^\circ\), \(d_{\mathrm{b}}=6.8\ \mathrm{cm}\), \(D=20.0\ \mathrm{cm}\), \(k=2.4\ \mathrm{cm}\), \(d=7.0\ \mathrm{cm}\), are presented in Fig. 3. It is seen that the calculated values of the lateral magnetic reluctance (curves 2 and 4) differ only slightly from these same dependences obtained experimentally (curves 1 and 3). This provides grounds for recommending the obtained expression (20) for practical use in determining the lateral magnetic reluctances of electromagnets of similar designs.

References

1. N. P. Ryashentsev, E. M. Timoshenko. Solenoid Hammers. Proceedings of TPI, vol. 108, Metallurgizdat, 1959.

2. V. V. Ivanov, N. P. Ryashentsev. New electric hammers. Journal “Mechanization of Construction,” No. 7, 1961.

3. A. V. Gordon, A. G. Slavinskaya. Direct-Current Electromagnets. GEI, 1960.

4. V. A. Tovorkov. Electric and Magnetic Fields. GEI, 1956.

5. Roters. Electromagnetic Mechanisms. GEI, 1949.

6. Machine Designer’s Handbook, vol. 1, Moscow, 1954.

7. A. I. Markushevich. Theory of Analytic Functions. State Publishing House of Technical-Theoretical Literature, 1950.