Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 129

1965

N. P. Ryashentsev, E. M. Timoshenko

On the Calculation of the Traction Force of an Electromagnet

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

Despite the long history of practical use of electromagnetic mechanisms and the abundance of published works devoted to their calculation, the methods for calculating the traction force of an electromagnet are not substantiated with sufficient clarity. In the published works, opposing points of view are often defended. Thus, speaking of one and the same expression for determining the traction force of an electromagnet (see formula 5), some authors [1, 2] assert that it can be used only in the case when the magnetic characteristics are linear and the magnitude of the current does not change during the motion of the armature. Others [3, 4, 5, 6] extend this same expression also to the case of a current magnitude varying during the process of motion. A third group [7] say that this widely used formula for determining the traction force is inaccurate and attempt to refine it.

There is no single opinion as to whether the traction force of an electromagnet depends only on the energy of the flux in the working air gap or on the entire energy of the magnetic system. In work [8] the assertion of Yasse [9] that the electromagnetic force depends only on the energy of the flux in the air gap is refuted. From articles [5, 6, 10] the opposite conclusion follows.

All this provides grounds for dwelling in greater detail on certain works and on the analysis of the redistribution of magnetic energy in an electromagnet with a moving armature.

L. R. Neyman [3, p. 264] uses, for calculating the electromagnetic force, an expression that determines the energy of the magnetic field

$$ W = \frac{Li^2}{2}, $$

where \(L\) is the inductance of the circuit;
\(i\) is the current flowing in the circuit.

Assuming \(i = \mathrm{const}\) and differentiating with respect to displacement, the author [3] obtains

$$ F_{\mathrm{e}} = \frac{1}{2} i^2 \frac{dL}{d\delta} \tag{1} $$

and further says that “the last expression no longer requires any reservations about the constancy of the current. This reservation has been fully used in ...”

taking \(i^2\) outside the derivative sign. This expression is also suitable for calculating the instantaneous value of the electromagnetic force at a given instantaneous value of the alternating current. The author then shows that formula (1) is valid when the current varies and the flux linkage is constant. However, in our opinion, this is not proof that it is valid also for the general case, when both the current and the flux linkage vary during the motion. The applicability of expression (1) for calculating the instantaneous value of the electromagnetic force at a given instantaneous value of the alternating current remains unproved.

Nor is this proved by M. I. Vitenberg in [2, p. 134], where an expression similar to (1) is given, obtained under the assumption that the flux links with all turns of the winding, the magnetic characteristics are rectilinear, and the magnitude of the current does not change during the motion of the armature. In [1, p. 285] A. Ya. Builov emphasizes that only for constant current is the magnitude of the tractive force determined by formula (1), while for a current varying during the motion the electromagnetic force is

\[ F_{\mathrm{e}}=\frac{1}{2} i^2 \frac{dL}{d\delta}+\frac{1}{2} L \frac{d(i^2)}{d\delta}. \tag{2} \]

In order to see whether this is so, let us find an expression for determining the tractive force from the energy balance of an electromagnet having a series magnetic circuit, as M. A. Babikov does [4]. Suppose that the magnetic resistance of the magnetic core is absent, and that for some gap \(\delta_1\) the dependence of the flux linkage on the current of the electromagnet (the magnetic characteristic of the electromagnet) can be represented by straight line 1 (Fig. 1, \(a\)); for a gap \(\delta_2<\delta_1\), by straight line 2. Let, at the beginning of the armature motion, a current \(i\) flow in the electromagnet coil, and let during the armature motion \(\Psi=f(i)\) change along the straight line \(ab\), that is, the process proceeds with a simultaneous increase in both flux linkage and current. From Fig. 1, \(a\), the energy stored by the electromagnet at the beginning of the motion is

\[ A_1=\frac{1}{2}\Psi\cdot i; \]

the energy supplied from the network during the motion is

\[ A_2=i\Delta\Psi+\frac{1}{2}\Delta i\cdot\Delta\Psi; \]

the energy stored in the field of the electromagnet at the end of the armature motion is

\[ A_3=\frac{1}{2}(\Psi+\Delta\Psi)\cdot(i+\Delta i). \]

According to the law of conservation of energy, during the displacement of the armature by \(\Delta\delta=\delta_1-\delta_2\) the electromagnet performed mechanical work

\[ \Delta A_{\mathrm{m}}=A_1+A_2-A_3, \]

or, substituting the values of \(A_1\), \(A_2\), and \(A_3\), we obtain

\[ \Delta A_{\mathrm{m}}=\frac{1}{2}(i\Delta\Psi-\Psi\Delta i). \tag{3} \]

Let us transform expression (3), expressing the flux linkage through the inductance of the electromagnet coil

\[ \Psi=L\cdot i \]

and the increment of flux linkage

\[ \Delta \Psi = i \Delta L + L \Delta i + \Delta i \cdot \Delta L . \]

Substituting the values of \(\Psi\) and \(\Delta \Psi\) into expression (3), we obtain

\[ \Delta A_{\mathrm{m}} = \frac{1}{2} i (i \Delta L + \Delta i \Delta L). \]

The average magnitude of the tractive force during the displacement is

\[ F_{\mathrm{av}} = \frac{\Delta A_{\mathrm{m}}}{\Delta \delta} = \frac{1}{2} i \frac{i \Delta L + \Delta i \cdot \Delta L}{\Delta \delta}. \]

Passing to the limit, we obtain a formula for determining the instantaneous value of the tractive force for the given instantaneous value of the current varying during the motion:

\[ F_{\mathrm{e}} = \frac{1}{2} i^2 \frac{dL}{d\delta}. \]

This formula is identical to expression (1), obtained by differentiating the magnetic energy of the electromagnet with respect to displacement under the assumption \(i = const\). Consequently, the tractive force is determined only by the first term on the right-hand side of expression (2).

Let us now consider energy conversion with a moving armature in an electromagnet with a series magnetic circuit of the magnetic core, whose resistance is constant and does not depend on the current, in order to show what energy must be used in determining the tractive force and wherein lies the error of R. A. Agaponyants [7], who proves the inaccuracy of the widely used formula for determining electromagnetic force.

In Fig. 1,b the straight line \(L_{\mathrm{c}}\) represents the dependence of flux linkage on current, characterizing the magnetic core (the magnetic characteristic of the core); the straight line \(L_{\mathrm{v}_1}\) is the magnetic characteristic of the working air gap at gap \(\delta_2\); the straight line \(L_{\mathrm{v}}\) is the magnetic characteristic of the working air gap at gap \(\delta_1 > \delta_2\). The straight lines \(L_{\mathrm{ob}_1}\) and \(L_{\mathrm{ob}}\) are the magnetic characteristics of the electromagnet, respectively at gaps \(\delta_2\) and \(\delta_1\). We shall denote the inductance of the sections of the magnetic circuit by the same letter as the dependence of flux linkage on the current by which it is determined.

Let, at the instant of motion under consideration, the state of the electromagnet be characterized by point \(a\), that is, the inductance of the electromagnet is

\[ L_{\mathrm{ob}} = \frac{L_{\mathrm{c}} \cdot L_{\mathrm{v}}}{L_{\mathrm{c}} + L_{\mathrm{v}}}, \]

the current \(i_{\mathrm{ob}}\) flows in the coil, and during the motion of the armature \(\Psi = f(i)\) changes along the straight line \(ab\).

Reasoning in the same way as in the case of the absence of magnetic-core resistance, we find that the magnetic energy of the electromagnet proportional to the area of the triangle \(oab\) has gone into producing mechanical work, and the electromagnetic force for the state of the magnet characterized by point \(a\) is

\[ F_{\mathrm{e}} = \frac{1}{2} i_{\mathrm{ob}}^2 \frac{dL_{\mathrm{ob}}}{d\delta}. \]

Substituting its value for \(L_{\mathrm{ob}}\) and differentiating, we obtain

\[ F_{\mathrm{e}}=\frac{1}{2}i_{\mathrm{ob}}^{2}\frac{L_{\mathrm{c}}^{2}}{(L_{\mathrm{c}}+L_{\mathrm{v}})^{2}}\frac{dL_{\mathrm{v}}}{d\delta}. \tag{4} \]

Let us now consider how, in the process of displacement of the armature, the magnetic energy of the working air gap changes. At the beginning of the motion under consideration, the magnitude of the magnetic energy concentrated in the working air gap is determined by the point \(a_1\) (Fig. 2) and is equal to

\[ W_{\mathrm{v}}=\frac{1}{2}i_{\mathrm{v}}^{2}L_{\mathrm{v}}. \]

During the displacement of the armature, the change in the magnetic energy in the working air gap is characterized by the straight line \(a_1b_1\) (Fig. 1, b), and over the time interval under consideration, when the armature is displaced by \(\Delta\delta=\delta_1-\delta_2\), energy proportional to the area of the triangle \(oa_1b_1\) is released from the working air gap. Considering the triangles \(oab\) and \(oa_1b_1\) (Fig. 1, b), we find that their areas are equal, since the triangles \(a_1b_1c_1\) and \(abc\), \(oa_1c_1\) and \(oac\) are pairwise equal, having equal heights and bases (the segments \(aa_1\) and \(cc_1\) are equal by construction). This means that the mechanical work is performed only at the expense of the magnetic energy of the working air gap, and in determining the magnitude of the tractive force one may consider only its change, just as in the case where there is no magnetic resistance of the magnetic circuit.

Proceeding in this way, we find the magnitude of the tractive force for the state of the electromagnet characterized by the point \(a\):

\[ F_{\mathrm{e}}=\frac{1}{2}i_{\mathrm{v}}^{2}\frac{dL_{\mathrm{v}}}{d\delta}. \tag{5} \]

Expressions (5) and (4) have been obtained from the law of conservation of energy; therefore there should be no doubt as to their correctness, and the values of the electromagnetic force determined by (4) and (5) must be equal.

From Fig. 1, b we find that

\[ i_{\mathrm{v}}=i_{\mathrm{ob}}\frac{L_{\mathrm{c}}}{L_{\mathrm{c}}+L_{\mathrm{v}}}. \]

Thus, in expression (4) the coefficient \(\dfrac{L_{\mathrm{c}}}{L_{\mathrm{c}}+L_{\mathrm{v}}}\) shows how many times the current of the electromagnet coil is greater than the current determining the magnetizing force of the working air gap, and formulas (4) and (5) are identical.

Expressions (4) and (5) may also be obtained by differentiating with respect to displacement, respectively, the total magnetic energy of the electromagnet

\[ W_{\mathrm{m}}=\frac{1}{2}i_{\mathrm{ob}}^{2}L_{\mathrm{ob}} \]

and the magnetic energy of the working air gap

\[ W_{\mathrm{v}}=\frac{1}{2}i_{\mathrm{v}}^{2}L_{\mathrm{v}}, \]

taking \(i_{\mathrm{ob}}=\mathrm{const}\) and \(i_{\mathrm{v}}=\mathrm{const}\).

In determining the magnitude of the traction force of an electromagnet, R. A. Agaronian [7] expresses the energy of the working air gap through the current of the electromagnet coil

\[ W_{\mathrm{v}}=\frac{1}{2}\,i_{\mathrm{ob}}^{2}\, \frac{L_{\mathrm{c}}^{2}}{(L_{\mathrm{c}}+L_{\mathrm{v}})^{2}}\,L_{\mathrm{v}}, \tag{6} \]

but takes the derivative of this expression with respect to displacement, assuming \(i_{\mathrm{ob}}=\mathrm{const}\). Proceeding in this way, R. A. Agaronian arrives, according to the author’s own expression, at the absurd conclusions that the traction force is equal to zero for \(L_{\mathrm{c}}=L_{\mathrm{v}}\), and becomes negative for \(L_{\mathrm{c}}>L_{\mathrm{v}}\). From the above it is clear that, when determining the electromagnetic force as the derivative of the magnetic energy of the working air gap with respect to displacement, one should assume \(i_{\mathrm{v}}=\mathrm{const}\), or, equivalently,

\[ i_{\mathrm{ob}}\frac{L_{\mathrm{c}}}{L_{\mathrm{c}}+L_{\mathrm{v}}}=\mathrm{const}. \]

In order to derive an exact expression for determining the electromagnetic force, R. A. Agaronian [7] proposes representing the electromagnetic system in the form of two closed circuits located in space, and as a starting point writes the expression for the electromagnetic energy of the electromagnet in the form

\[ W_{\mathrm{m}}=\frac{L_{\mathrm{v}}i_{\mathrm{v}}^{2}}{2} +\frac{L_{\mathrm{c}}i_{\mathrm{c}}^{2}}{2} +k_{\mathrm{c}}i_{\mathrm{v}}i_{\mathrm{c}}\sqrt{L_{\mathrm{c}}L_{\mathrm{v}}}. \tag{7} \]

From the preceding it is evident that the magnetic energy of the electromagnet is equal to the sum of the first two terms on the right-hand side of expression (7).

Fig. 1. Redistribution of the magnetic energy of an electromagnet during displacement of the armature: a) magnetic resistance of the magnetic circuit is absent; b) resistance of the magnetic circuit is linear; c) resistance of the magnetic circuit is nonlinear; d) resistance of the magnetic circuit is linear, but changes during displacement of the armature of the electromagnet.

Expressions (4) and (5) were obtained for a linear magnetic characteristic of the magnetic circuit. Let us examine, referring to Fig. 1b, whether they are valid for a nonlinear magnetic characteristic of the magnetic circuit. In Fig. 1c the designations are the same as in Fig. 1b, but the magnetic

the characteristics of the magnetic circuit and of the electromagnet are nonlinear. According to the law of conservation of energy, the magnetic energy of the electromagnet expended on mechanical work is equivalent to the area of the curvilinear triangle \(oab\). Let us note that if, in this case as well, the mechanical work is performed only at the expense of the magnetic energy of the working air gap, then expression (5) is also valid for a nonlinear magnetic characteristic of the magnetic circuit.

Unlike the case shown in Fig. 1, b, when the dependence of the flux linkage on the electromagnet current changes along the straight line \(ab\) (Fig. 1, b), the change in the energy in the working gap during displacement of the armature will be characterized by the curve \(a_1db_1\), which is easy to construct if a series of intermediate magnetic characteristics of the working gap and of the electromagnet is drawn. The curve \(a_1db_1\) will pass in such a way that the areas of the figures \(abb_1da_1\) and \(cdbb_1c_1\) are equal. The magnetic energy released from the working air gap during the displacement is proportional to the area of the curvilinear triangle \(oa_1b_1\). We shall show that the areas of the curvilinear triangles \(oab\) and \(oa_1b_1\) (Fig. 1, c) are equal; for this purpose we consider the curvilinear triangles \(oac\) and \(oa_1c_1\), \(abc\) and \(a_1b_1c_1\). The areas of the figures \(abb_1da_1\) and \(cdbb_1c_1\) are equal by construction. If to each of them we add the area \(ca_1d\) and subtract the area \(dbb_1\), we find that the areas of the curvilinear triangles of interest to us, \(abc\) and \(a_1b_1c_1\), are equal. The areas of the curvilinear triangle \(oac\) and the triangle \(oa_1c_1\) are equal by construction. Consequently, the mechanical work is performed only at the expense of the magnetic energy of the working air gap, and the traction force is determined by formula (5), that is, by the derivative of the magnetic energy of the working gap with respect to displacement, taken at \(i_{\mathrm{v}} = \mathrm{const}\). This is proved in a somewhat different way also in works [5, 9]. In order to express the traction force, for a nonlinear magnetic characteristic, through the current of the electromagnet coil, it is necessary to use a coefficient showing how many times the coil current is greater than the current \(i_{\mathrm{v}}\) that determines the magnetomotive force of the working air gap. This coefficient in Fig. 1, c is equal to

\[ \frac{L_{\mathrm{c.st}}}{L_{\mathrm{c.st}}+L_{\mathrm{v}}}, \]

where \(L_{\mathrm{c.st}}\) is the static inductance of the magnetic circuit corresponding to that state of the electromagnet for which the traction force is determined.

For the state of the electromagnet characterized by point \(a\) (Fig. 1, c), the traction force is

\[ F_{\mathrm{e}}=\frac{1}{2} i_{\mathrm{ob}}^{2}\frac{L_{\mathrm{c.st}}^{2}}{(L_{\mathrm{c.st}}+L_{\mathrm{v}})^{2}}\frac{dL_{\mathrm{v}}}{d\delta}. \tag{8} \]

Expression (8) is convenient to use for determining the traction force of an electromagnet when the magnetic characteristic of the magnetic circuit is known and the law of variation of the inductance of the working air gap is known.

As is clear from the foregoing, the traction force of an electromagnet with a series magnetic circuit depends only on the magnetic energy of the working air gap, and the mechanical work of the electromagnet is numerically equal to the decrease in the magnetic energy of the working gap.

Let us dwell further on such a case of redistribution of energy in an electromagnet with a series magnetic circuit when, in the process of displacement of the armature, the length of the magnetic circuit changes; that is, new volumes of ferromagnetic material are introduced, in order to

to see at the expense of what energy the mechanical work is performed and what expression must be used to determine the tractive force of an electromagnet. Such a case is considered by Roters [11]; however, we have not found an answer to the questions posed.

Let, at the gap \(\delta_1\), the magnetic characteristic of the magnetic circuit be linear and be represented by the straight line \(L_c\) (Fig. 1, \(g\)). When the armature is displaced, owing to the introduction into the air gap of new volumes of ferromagnetic material, it changes, and at some gap \(\delta_2 < \delta_1\) it may be represented by the straight line \(L_{c_1}\). In Fig. 1, \(g\), the straight lines \(L_{\mathrm{v}}\) and \(L_{\mathrm{v}_1}\) are the magnetic characteristics of the working air gap, respectively at gaps \(\delta_1\) and \(\delta_2\); \(L_{\mathrm{em}}\) and \(L_{\mathrm{em}_1}\) are the magnetic characteristics of the electromagnet at gaps \(\delta_1\) and \(\delta_2\). Suppose that, at the instant of motion under consideration, the state of the electromagnet was characterized by point \(a\), and that, when the armature is displaced by \(\Delta\delta = \delta_1 - \delta_2\), the change of magnetic energy was characterized in the electromagnet by the straight line \(ab\), in the air gap by the straight line \(a_1b_1\), and in the magnetic circuit by the straight line \(a_2b_2\). Considering the energy conversions in the electromagnet in the same way as in the preceding cases, we find that the energy released from the working gap

\[ \Delta A_{\mathrm{v}} = \text{area } \Delta oa_1b_1 = \frac{1}{2}\left(i_{\mathrm{v}}\Delta\Psi - \Psi\Delta i_{\mathrm{v}}\right) \]

is expended not only on mechanical work

\[ \Delta A_{\mathrm{mech}} = \text{area } \Delta oab, \]

but also on increasing the magnetic energy in the magnetic circuit

\[ \Delta A_{\mathrm{mag}} = \text{area } \Delta oa_2b_2 = \frac{1}{2}\left(i_c\Delta\Psi - \Psi\Delta i_c\right). \]

Or one could say that not all the energy is released from the air gap. Part of it remains in the volume of ferromagnetic material introduced into the space previously occupied by the air gap.

The mechanical work performed by the electromagnet when the armature is displaced by \(\Delta\delta = \delta_1 - \delta_2\) is

\[ \Delta A_{\mathrm{mech}} = \Delta A_{\mathrm{v}} - \Delta A_{\mathrm{mag}} = \frac{1}{2}\left(i_{\mathrm{v}}\Delta\Psi - \Psi\Delta i\right) - \frac{1}{2}\left(i_c\Delta\Psi - \Psi\Delta i_c\right). \]

Replacing the flux linkage by inductance, after some transformations we obtain an expression for determining the tractive force:

\[ F_{\mathrm{e}} = \frac{1}{2} i_{\mathrm{em}}^2 \frac{L_c^2}{(L_c + L_{\mathrm{v}})^2} \frac{dL_{\mathrm{v}}}{d\delta} - \frac{1}{2} i_{\mathrm{em}}^2 \frac{L_{\mathrm{v}}^2}{(L_c + L_{\mathrm{v}})^2} \frac{dL_c}{d\delta}. \tag{9} \]

Up to this point we have considered an electromagnet with a series magnetic circuit and have arrived at the conclusion that mechanical work is performed only at the expense of the magnetic energy of the working air gap. The same conclusion can be reached by considering graphically the redistribution of energy in an electromagnet having leakage fluxes that link all the turns of the magnetizing winding. This was proved, relatively simply and without recourse to graphical material, by G. V. Mogilevskii [6] for electromagnets with complex branched magnetic circuits whose total magnetic flux links all the turns of the magnetizing winding.

In deriving the formula for determining the tractive force of an electromagnet, G. V. Mogilevskii proceeds from the fact that the electrical energy,

coming from the external source, after subtracting the copper losses, goes into the creation of mechanical work and into increasing the energy of the magnetic field

\[ i d\Psi = dA + dW \tag{10} \]

and the magnetic energy of the electromagnet is determined by the expression

\[ W = \int_{0}^{\Psi} i d\Psi, \]

where \(i = f(\Psi)\) and depends on the position of the armature of the electromagnet.

Thus, the magnetic energy of the electromagnet is a function of two variables—the displacement of the armature \(\delta\) and the flux linkage \(\Psi\). The total differential of the magnetic energy is determined by the expression

\[ dW = \frac{\partial W}{\partial \delta}\,d\delta + \frac{dW}{d\Psi}\,d\Psi = d\delta \int_{0}^{\Psi} \frac{\partial i}{\partial \delta}\,d\Psi + i d\Psi . \tag{11} \]

Comparing (10) and (11) and taking into account that

\[ dA = F_{\mathrm{e}} d\delta, \]

G. V. Mogilevskii [6] obtains the formula for the specified tractive force of the electromagnet

\[ F_{\mathrm{e}} = - \int_{0}^{\Psi} \frac{\partial i}{\partial \delta}\,d\Psi, \tag{12} \]

that is, the force of the electromagnet is determined by the partial derivative of the magnetic energy of the electromagnet with respect to displacement at constant flux linkage.

Using expression (12), G. V. Mogilevskii analytically shows that the widely used formula for determining the tractive force of an electromagnet

\[ F_{\mathrm{e}} = \left( \frac{\partial W_{\mathrm{v}}}{\partial \delta} \right)_{U_{\mathrm{v}}=\mathrm{const}} \tag{13} \]

is valid for electromagnets whose total magnetic flux embraces all turns of the magnetizing winding. He then extends the conclusions obtained to electromagnets with any number of coils having leakage fluxes, and, in particular, to a shell-type electromagnet. In this case the magnetic flux is represented as the sum of elementary tubes of induction, each of which is equivalent to a series magnetic circuit.

It should be noted that such a transfer of results obtained for a series magnetic circuit to electromagnets with a complex magnetic circuit, based in [6] on the fact that each elementary tube is equivalent to a series magnetic circuit, cannot be considered fully justified. In the general case, during the displacement of the armature of an electromagnet, not only the permeance of individual sections may change, but also the number of turns embraced by one or another elementary tube of induction. Therefore the conclusion that mechanical work is performed only at the expense of the magnetic energy of the air gaps cannot be considered correct. In order finally to resolve the question of the magnetic energy of which sections of the magnetic circuit can be converted into mechanical work, and what general formula can be used to determine the tractive force of an electromagnet of any type,

let us consider the redistribution of energy in a pot-type electromagnet.

Studies of the distribution of the magnetic flux of a pot-type electromagnet without a stop, carried out with the participation of engineer A. V. Frolov at the Department of Mining Machines and Mine Transport of TPI, showed that the leakage flux from the lateral surface of the armature is distributed nonuniformly, as shown in Fig. 2. Curves 1, 2, 3, 4, and 5 (Fig. 2) represent the distribution of the leakage flux along the length of the armature, respectively for gaps of 5, 10, 20, 30, and 40 mm. Such a distribution of the leakage fluxes makes it possible to represent the leakage permeance as concentrated at the end of the core. In this case the magnetic flux passing through this section will, as the magnitude of the working gap changes, embrace a greater or smaller number of turns. Sections of the magnetic circuit may be characterized by inductance, and in calculating the magnetic circuit the relation between the flux linkage and the current of the electromagnet may be determined. This makes it possible to depict on the diagram the magnetizing current as concentrated, which simplifies the equivalent circuit. Fig. 3 presents a simplified equivalent circuit of the magnetic circuit of a pot-type electromagnet. Here \(L_1\) is the inductance of the working gap and pole; \(L_2\) is the leakage inductance; \(L_3\) is the inductance of the “collar,” the pole, part of the armature, and the yoke; \(i\) is the current of the electromagnet. The magnetic flux \(\Phi_1\), passing through the working gap, links with all turns \(W_1\) of the winding and creates the flux linkage \(\Psi_1\). The leakage flux \(\Phi_2\) links with part of the winding turns \(w_2\) and creates the flux linkage \(\Psi_2\). The total magnetic flux and the total flux linkage of the coil are respectively equal to \(\Phi_0=\Phi_1+\Phi_2\) and \(\Psi_0=\Psi_1+\Psi_2\). The equivalent number of turns with which the total magnetic flux links is determined by the relation

\[ w_3=\frac{\Psi_0}{\Phi_0}. \]

Fig. 2. Distribution of leakage flux over the armature surface of a pot-type electromagnet.

Fig. 3. Equivalent circuit of the magnetic circuit of a pot-type electromagnet.

As the magnitude of the working gap decreases, the number of turns embraced by the flux \(\Phi_2\) increases; consequently, the inductance of this section also increases. With a decrease in the magnitude of the working gap, the ratio of the total flux linkage to the total flux also increases, since the equivalent number of turns \(W_3\), with whi-

the full magnetic flux is linked; consequently, the inductance \(L_3\) also increases. This means that, for the same flux linkage and a smaller working air gap, the magnetic energy concentrated in the magnetic core with inductance \(L_3\), and the magnetic energy of the leakage fluxes, decreases. In [12] it was shown that, at constant flux linkage, the increase in the mechanical work of an electromagnet is equal to the decrease in magnetic energy. In the present case, not only the magnetic energy of the working gap decreases, but also the energy of the leakage fluxes and of the magnetic core. Physically, the decrease in the energy of the magnetic core means that, in order to keep the total flux linkage unchanged as the armature is pulled in, the total flux must be reduced.

To determine the traction force, we shall use formula (12). The current of the electromagnet coil, according to the equivalent circuit (Fig. 3), is determined by the expression

\[ i=\frac{\Psi_0}{L_3}+\frac{\Psi_0}{L_2+L_1}, \tag{14} \]

where \(L_1\), \(L_2\), and \(L_3\) are the static inductances of the corresponding sections of the magnetic circuit, which in the general case may depend both on the magnitude of the air gap and on the flux linkage. The partial derivative of the current with respect to displacement at constant flux linkage is

\[ \left(\frac{\partial i}{\partial \delta}\right)_{\Psi=\mathrm{const}} = -\frac{\Psi_0}{L_3^2}\frac{\partial L_3}{\partial \delta} - \frac{\Psi_0}{(L_1+L_2)^2}\frac{\partial L_2}{\partial \delta} - \frac{\Psi_0}{(L_1+L_2)^2}\frac{\partial L_1}{\partial \delta}. \tag{15} \]

Proceeding from the equivalent circuit (Fig. 3), one may write

\[ \Psi_0=\frac{(L_1+L_2)\Psi_1}{L_1}; \quad \Psi_0=\frac{(L_1+L_2)\Psi_2}{L_2}; \tag{16} \]

\[ d\Psi_0=\frac{L_1+L_2}{L_1}\,d\Psi_1; \quad d\Psi_0=\frac{L_1+L_2}{L_2}\,d\Psi_2. \]

Substituting (15) and (16) into formula (12), we obtain

\[ F_{\mathrm{e}} = \int_{0}^{\Psi_0} \frac{\Psi_0}{L_3^2} \frac{\partial L_3}{\partial \delta} \,d\Psi_0 + \int_{0}^{\Psi_2} \frac{\Psi_2}{L_2^2} \frac{\partial L_2}{\partial \delta} \,d\Psi_2 + \int_{0}^{\Psi_1} \frac{\Psi_1}{L_1^2} \frac{\partial L_1}{\partial \delta} \,d\Psi_1 . \tag{17} \]

Each term on the right-hand side of formula (17) is the partial derivative of the magnetic energy of the corresponding section with respect to displacement at constant flux linkage. In the case where the static inductance of each section of the magnetic circuit is linear, that is, does not depend on the magnitude of the flux linkage, the traction force is determined by the expression

\[ F_{\mathrm{e}} = \frac{1}{2}\frac{\Psi_0^2}{L_3^2}\frac{dL_3}{d\delta} + \frac{1}{2}\frac{\Psi_2^2}{L_2^2}\frac{dL_2}{d\delta} + \frac{1}{2}\frac{\Psi_1^2}{L_1^2}\frac{dL_1}{d\delta}. \]

It follows from what has been said above that formula (13) is valid only for electromagnets in which the total flux links all turns of the magnetizing winding and, when the armature is displaced, the length of the magnetic core does not change, since only in this case is the mechanical work of the electromagnet equal to the decrease in the magnetic energy of the air gap.

Mechanical work may be performed by an electromagnet not only at the expense of the energy concentrated in the air gaps, but also at the expense of the energy concentrated in the magnetic core. Existing for-

formulas for determining the static traction force of a clapper-type electromagnet reflect the change in magnetic energy associated with the displacement of the armature not in all sections of the magnetic circuit. This, in our opinion, explains the fact that the most accurate formula, based on a more complete accounting of the magnetic field, may give an error on the order of 30–50% [13].

Mechanical work is not performed at the expense of the magnetic energy of those sections of the magnetic circuit whose permeance and the number of turns with which the flux passing through these sections is linked do not change when the armature is displaced.

Formula (12) is valid for electromagnets of all types, for any nonlinearities of the magnetic circuit, for determining the static and instantaneous values of the traction force.

REFERENCES

1. A. Ya. Builov. Fundamentals of Electrical Apparatus Engineering. Gosenergoizdat, 1946.

2. M. I. Vitenberg. Calculation of Electromagnetic Relays for Automation and Communication Apparatus. Gosenergoizdat, 1961.

3. L. R. Neiman and P. A. Kalantarov. Theoretical Foundations of Electrical Engineering. Part I, Gosenergoizdat, 1959.

4. M. A. Babikov. Electrical Apparatus. Gosenergoizdat, 1951.

5. N. A. Livshits. On the calculation of magnetic circuits of relays. Avtomatika i telemekhanika, No. 1, 1939.

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

7. R. A. Agaron’yants. Static traction characteristics of direct-current mechanisms. Vestnik elektropromyshlennosti, No. 7, 1958.

8. A. Ya. Builov. Method for calculating a direct-current electromagnet. Elektrichestvo, No. 3, 1949.

9. Yasse. Calculation of Electromagnets. Gosenergoizdat, 1934.

10. E. A. L’vov. Traction force in saturated electromagnets. Proceedings of MEI, issue XV, Gosenergoizdat, 1955.

11. Roters. Electromagnets. Gosenergoizdat, 1949.

12. N. P. Ryashentsev and E. M. Timoshenko. On energy conversion in an electromagnet. Article published in the present collection.

13. I. I. Pekker. Calculation of the traction force of clapper-type electromagnets with a flat end face of the core. Vestnik elektropromyshlennosti, No. 5, 1962.