N. P. RYASHENTSEV, E. M. TIMOSHENKO

ON THE EXPERIMENTAL DETERMINATION OF THE DYNAMIC TRACTIVE FORCE AND PERMEANCE OF THE WORKING GAP OF AN ELECTROMAGNET

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

An analysis of methods for calculating the dynamic characteristics of electromagnetic mechanisms, carried out by A. K. Ter-Akopov [1], shows that, of all existing methods, the most rigorous and general is the graphoanalytic method of N. E. Lysov [2]. However, it does not take into account the influence of eddy currents on the dynamic characteristics of electromagnets and assumes that the forces opposing the motion of the armature, and the dependences of flux linkage on current at various armature positions, are known. Therefore, the inaccuracy of the calculation may be determined both by the failure to take into account the influence of eddy currents and by inaccuracies in determining the dependence of flux linkage on current and by incomplete accounting of the forces opposing the motion of the armature. In order to develop a method for a more accurate calculation of the dynamic characteristics of an electromagnet, it is necessary to establish experimentally the causes of the inaccuracies in existing calculations.

One of the most important dynamic characteristics of an electromagnet, from which one can judge the causes of calculation errors, is the tractive force. The change in tractive force during the motion of the armature can be determined comparatively easily when the forces of resistance to motion are known and when the armature travel curve has been obtained experimentally. In this case the dynamic tractive force is determined by the sum of the forces of resistance to motion and the resultant force acting on the armature during motion. The resultant force is obtained by double differentiation of the armature travel curve.

The problem of experimentally determining the dynamic tractive force becomes insoluble when the influence of eddy currents is unknown and the forces of resistance to the armature motion are unknown. In the literature there is only a general indication that the dynamic tractive force, because of the presence of eddy currents under real conditions, is not equal to the static force. Before speaking about the influence of eddy currents on the tractive force and about the possibilities of taking them into account, let us consider the concept of the inductance of an electromagnet.

In the book by A. Ya. Builov [3, p. 280] it is stated that inductance (the coefficient of self-induction) “is not a fully definite quantity, but a conditional one, depending on the definition adopted for it” (what is meant is that, depending on the definition, inductance may be static, dynamic, or energetic), and “therefore cannot

be regarded as a physical parameter that completely determines the process.

However, it follows from [4] that the static inductance can be used to determine the tractive force of an electromagnet.

In [3, p. 279; 5, p. 250] the distinction between the static and dynamic inductance of a coil with steel is shown, but nowhere is the relationship between them indicated; this relationship can be established if one uses the equations of electrical equilibrium for a circuit containing inductive-active resistance. The equation of electrical equilibrium for such circuits can be represented in the form

\[ u=iR+\frac{d\Psi}{dt}, \tag{1} \]

or, for a known dynamic inductance \(L_{\mathrm{d}}\),

\[ u=iR+L_{\mathrm{d}}\frac{di}{dt}, \tag{2} \]

or, for the case of a moving armature with linear magnetic characteristics of the magnetic system,

\[ u=iR+L_{\mathrm{st}}\frac{di}{dt}+i\frac{dL_{\mathrm{st}}}{dt}. \tag{3} \]

The last expression is valid for a stationary armature and a nonlinear dependence of the flux linkage on the electromagnet current, if \(L_{\mathrm{st}}\) is understood as the static inductance.

Indeed, if the dependence of the flux linkage on the electromagnet current is given, \(\Psi=f(i)\), then the static inductance as a function of current is determined by the expression

\[ L_{\mathrm{st}}=\frac{f(i)}{i} \]

and the time derivative of the static inductance is

\[ \frac{dL_{\mathrm{st}}}{dt} = \frac{1}{i^{2}} \left[ \frac{df(i)}{dt}\, i - \frac{di}{dt}\, f(i) \right]. \]

Substituting \(L_{\mathrm{st}}\) and \(\dfrac{dL_{\mathrm{st}}}{dt}\) into equation (3), we obtain

\[ u=iR+\frac{df(i)}{dt}, \]

i.e., we obtain equation (1). Consequently, the static inductance can also be used to calculate transient processes in circuits containing inductive resistance.

By static inductance in the present case we mean not the ratio of the self-induction flux linkage, determined at constant current, to the current in the circuit, as is customary [4], but the ratio of the instantaneous value of the self-induction flux linkage to the instantaneous value of the alternating current, determined for one and the same instant of time. The relationship between the dynamic and static inductances from equations (2) and (3) is determined by the expression

\[ L_{\mathrm{d}}=L_{\mathrm{st}}+i\frac{dL_{\mathrm{st}}}{di}. \tag{4} \]

All this indicates that the inductance, whether static, dynamic, or energetic, is not a conditional but a quite defi-

quantity. To determine the magnitude of the magnetic energy of an electromagnet at a given current value, it is more convenient to use the energy inductance. When determining the magnitude of the e.m.f. induced in the circuit, it is more convenient to use the dynamic inductance. Static inductance, however, reflects most fully the processes occurring in an electromagnet, and can be used for calculating transient processes, for determining the magnitude of the magnetic energy, the e.m.f. induced in the circuit, the flux linkage, and the tractive force of the electromagnet.

It follows from the adopted definition of static inductance that if the resultant flux caused by the presence of short-circuited contours in which eddy currents arise is formally regarded as the flux of self-induction, then under real conditions the presence of eddy currents affects not only the magnitude of the dynamic inductance, as is stated in [5, p. 249], but also the magnitude of the static inductance.

If it were possible to take into account the change in the static inductance of the magnetic circuit of an electromagnet due to the phenomena of hysteresis and eddy currents, then it could be used to determine the tractive force and to calculate a real electromagnet. In other words, the presence of eddy currents could be taken into account by changing the dependence of flux linkage on current for particular sections of the magnetic circuit. For electromagnets with complex circuits this question requires special elaboration.

Let us consider simpler cases and the practically useful conclusions that follow from them. Suppose we have an electromagnet with a series magnetic circuit whose total static inductance at some flux linkage \(\Psi\) is equal to

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

where \(L_c=f(\Psi)\) is the static inductance of the magnetic circuit, taking eddy currents into account;

\[ L_{\mathrm{v}}=f(\delta) \]
is the inductance of the working air gap.

Passing to the determination of the tractive force of the electromagnet and using the general formula (12) given in [4], we write

\[ i=\frac{\Psi\,[f(\Psi)+f(\delta)]}{f(\Psi)+f(\delta)}, \]

\[ \left(\frac{\partial i}{\partial \delta}\right)_{\Psi=\mathrm{const}} = -\frac{\Psi}{f^2(\delta)}\cdot\frac{df(\delta)}{d\delta}, \]

\[ F_{\mathrm{e}}=\int_0^{\Psi}\frac{\Psi}{f^2(\delta)}\cdot\frac{df(\delta)}{d\delta}\,d\Psi = \frac{1}{2}\frac{\Psi^2}{L_{\mathrm{v}}^2}\cdot\frac{dL_{\mathrm{v}}}{d\delta}. \tag{5} \]

The result obtained permits the conclusion that, regardless of whether direct current flows in the coil or varies according to some law, and regardless of whether eddy currents arise in the magnetic circuit or not, the tractive force at identical positions of the armature of an electromagnet with a series magnetic circuit and at the same flux linkage has the same magnitude. Only the magnitude of the current that provides the same flux linkage changes.

This conclusion can also be extended to electromagnets with a more complex magnetic circuit, in which, over a section of the magnetic circuit,

in which eddy currents may arise, the entire magnetic flux passes through it, and this section is connected in series with all the other sections of the magnetic circuit. Indeed, in this case \(L_c\) may be regarded as the inductance of such a section, and \(L_{\mathrm{в}}\) as the total inductance of all the remaining sections of the magnetic circuit.

The equality of the static and dynamic tractive forces at identical armature positions and identical flux linkage of the coil makes it possible to determine experimentally the magnitude of the tractive force acting on the armature during its motion, even when the forces opposing the motion of the armature are unknown. Observance of this equality makes it possible, in determining the dynamic tractive force, to avoid the inaccuracy of the method given in [6], where eddy currents are taken into account when recording the dependence of flux linkage on the current of the electromagnet, and where, in calculating the tractive force, the heat losses in the steel are neglected.

In order to establish whether, for a given electromagnet, the static force may be considered equal to the dynamic force at the same gap and flux linkage, it is necessary, at fixed values of the gap—which is considerably simpler than with motion of the armature and unknown forces of resistance to motion—to determine experimentally the ratios of the tractive force to the square of the flux linkage at different rates of change of current. This equality is satisfied if, for all the air-gap values used, when a current flows in the coil whose magnitude and rate of change correspond to the working current of the electromagnet, the condition following from expression (5) is fulfilled:

\[ \frac{2F_{\mathrm{э}}}{\Psi^2}=\mathrm{const} \]

at a fixed armature position and identical flux linkage, independently of the rate of change of current and flux linkage.

Investigations carried out with the participation of engineer A. P. Tronov at the Department of Mining Machines of TPI showed that, for an electromagnet of the shell type used as the drive of the MS-15 hammer, this condition is satisfied. The deviations of the ratio of the tractive force to the square of the flux linkage from a constant value were within the limits of experimental error (5–10%). This makes it possible, with known static dependences of the tractive force on the flux linkage of the coil and static dependences of the flux linkage on the electromagnet current for various fixed values of the working gap, to determine from oscillograms of the current, voltage, and displacement of the striker the work performed by the electromagnet, the influence of eddy currents on the working process of the hammer, and the work of friction forces in dynamics.

Another useful conclusion follows from expression (5). Since the tractive force depends only on the magnetic energy of the working air gap and the condition

\[ \frac{2F_{\mathrm{э}}}{\Psi^2}=\mathrm{const}\quad \text{at } \delta=\mathrm{const} \]

is satisfied independently of the magnitude and rate of change of the flux linkage, there is the possibility of using expression (5) for the experimental determination of the permeance of the working gap of an electromagnet with a series magnetic circuit. For this purpose the dependence \(F_{\mathrm{э}}=f(\Psi^2)\) is determined for various fixed values of the air gap. The dependence \(F_{\mathrm{э}}=f(\Psi^2)\) can be determined by connecting the coil to a constant voltage while oscillographing the current, the magnitude of...

of the tractive force of the electromagnet and the voltage applied to the coil. The change in the magnitude of the tractive force can be recorded by means of a strain-gauge transducer or determined by a simpler method (for example, by suspending a load) at a steady-state coil current. On the basis of the oscillograms obtained, with the ohmic resistance of the coil known, the curve is plotted

Fig. 1. Dependences of the tractive force on flux linkage for different values of the air gap.

Fig. 2. Dependence of the tractive force on the magnitude of the air gap at constant flux linkage.

Fig. 3. Dependence of the inductance of the working gap on the magnitude of the air gap.

\[ \frac{d\Psi}{dt}=u-iR. \]

By graphical integration of this curve, one obtains

\[ \Psi=\int_{0}^{t}(u-iR)\,dt+c_1. \]

If, before each oscillographic recording, the magnetic circuit of the electromagnet is demagnetized, then the constant \(c_1=0\). From the force and flux linkage found, the straight lines \(F=f(\Psi^2)\) are constructed (Fig. 1).

In order to obtain an expression for determining the inductance of the working gap, multiply the right- and left-hand sides of expression (5) by \(d\delta\),

\[ F_\delta d\delta=\frac{1}{2}\Psi^2\frac{1}{L_{\mathrm{B}}^2}\,dL_{\mathrm{B}}. \]

If the flux linkage remains constant during displacement of the armature, then

\[ \int_{0}^{\delta_i} F_{\Psi=\mathrm{const}}\,d\delta = \frac{1}{2}\Psi^2\frac{1}{L_{\mathrm{B}i}}+c_2. \]

Since at \(\delta = 0\), \(L_{\mathrm{v}} = 0\), the constant of integration \(c_2 = 0\), and the inductance of the working gap at a certain gap \(\delta_i\) is determined by the expression

\[ L_{\mathrm{v}i} = \frac{\Psi^2} {2 \displaystyle \int_{0}^{\delta_i} F_{\mathrm{t}\,\Psi=\mathrm{const}}\, d\delta} . \]

The points of the curve \(F_{\mathrm{e}} = f(\delta)_{\Psi=\mathrm{const}}\) (Fig. 2) are determined by the points of intersection of the straight line \(A\!-\!A\) (Fig. 1), drawn parallel to the ordinate axis, with the corresponding straight lines \(F_{\mathrm{e}} = f(\Psi^2)_{\delta_i}\). Thus, determining the inductance of the working gap \(L_{\mathrm{v}i}\) at a certain gap \(\delta_i\) reduces to determining the area enclosed between the curve \(F_{\mathrm{e}} = f(\delta)_{\Psi=\mathrm{const}}\) and the abscissa axis over the interval from \(\delta = 0\) to \(\delta_i\). Based on the results of the calculations, the curve \(L_{\mathrm{v}} = f(\delta)\) (Fig. 3) is constructed, and, for a known number of turns of the coil, the permeance curve \(G_{\mathrm{v}} = f(\delta)\) is constructed.

REFERENCES

1. A. K. Ter-Akopov. Analysis of methods for calculating the dynamic characteristics of electromagnetic mechanisms. Proceedings of MAI, issue 85, Oborongiz, 1957.

2. N. E. Lysov. Calculation of electromagnetic mechanisms. Oborongiz, 1949.

3. A. Ya. Builov. Fundamentals of electrical-apparatus engineering. Gosenergoizdat, 1946.

4. N. P. Ryashentsev, E. M. Timoshenko. On the calculation of the tractive force of an electromagnet. The article is published in the present collection.

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

6. A. G. Slivinskaya, A. A. Chunikhin. On the calculation of the dynamic tractive characteristic of a direct-current electromagnet. Proceedings of MEI, issue XV, Gosenergoizdat, 1955.