N. P. RYASHENTSEV, E. M. TIMOSHENKO

ON ENERGY CONVERSION IN AN ELECTROMAGNET

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

In designing electromagnetic mechanisms, the question of the efficiency of the electromagnet is not raised, since the mechanical work performed by various relays and contactors is negligibly small in comparison with the energy consumed from the network. The efficiency of mechanisms of the contactor type is practically equal to zero. Therefore, authors engaged in developing the theory and methods of calculation of electromagnetic mechanisms have had no need to devote much attention to questions of energy conversion. Energy conversion in an electromagnet has so far been considered only for the purpose of deriving a formula for determining the tractive force [1—7], and to an extent insufficient for a complete analysis of the working process, for identifying shortcomings, and for developing recommendations to improve electromagnetic machines of impact action.

Solving the problem of using an electromagnet as a drive for electrical machines of reciprocating action requires a more complete consideration of questions of energy conversion.

The processes occurring in an electromagnet when it is switched on to a constant voltage are described by the equation of electrical equilibrium:

$$ u = i^2 R + \frac{d\Psi}{dt}. $$

By multiplying the right- and left-hand sides of this equation by $i\,dt$ and then integrating, one obtains the energy balance in the electromagnet over the time interval from 0 to $t$:

$$ \int_0^t (u \cdot i - i^2 R)\,dt = \int_0^\Psi i\,d\Psi . \tag{1} $$

In the case when the external mechanical forces opposing the displacement of the armature remain equal to the tractive force of the electromagnet, no mechanical work is performed, and the energy supplied from the network, after subtracting the copper losses, goes to increasing the magnetic energy.

The magnetic energy of the entire volume of the electromagnet’s magnetic field is equal to $\int_0^\Psi i\,d\Psi$. Graphically, this energy is represented in Fig. 1 by the area $oac_1$.

We shall consider energy conversion in an electromagnet with a moving armature as applied to solenoid hammers with free striker run-out (switching on the electromagnet coil according to a half-wave rectification circuit), assuming, as is done in such cases [1–5], that the magnetic circuit is not saturated and that hysteresis and eddy-current losses are absent. The equation of electrical equilibrium with a moving armature may be written in the form

\[ u=iR+L\frac{di}{dt}+i\frac{dL}{dt}. \tag{2} \]

In order to obtain the energy balance in an electromagnet with a moving armature, we shall use the formula determining the magnitude of the tractive force [8]

\[ F_{\mathrm{e}}=\frac{1}{2}i^2\frac{dL}{d\delta}, \tag{3} \]

and the expression for determining the inductance

\[ L=\frac{\Psi}{i}, \tag{4} \]

where \(\Psi\) is the self-induction flux linkage;
\(i\) is the current of the electromagnet coil.

Multiplying (2) by \(i\,dt\) and substituting (3) and (4), after some transformations we obtain

\[ \int_{t_1}^{t_2}(u\cdot i-i^2R)\,dt = \frac{1}{2}\int_{t_1}^{t_2}(\Psi\,di+i\,d\Psi) + \int_{t_1}^{t_2}F_{\mathrm{e}}v\,dt, \tag{5} \]

where

\[ v=\frac{d\delta}{dt} \]

is the velocity of motion of the electromagnet armature;

\(\delta\) is the displacement of the armature.

Or one may write

\[ \int_{t_1}^{t_2} i\,d\Psi = \frac{1}{2}\int_{t_1}^{t_2}(\Psi\,di+i\,d\Psi) + \int_{t_1}^{t_2}F_{\mathrm{e}}v\,dt. \tag{6} \]

The left-hand side of expression (6) represents the amount of electrical energy converted into magnetic-field energy when \(u\cdot i-i^2R>0\), or released from it when \(u\cdot i-i^2R<0\), over the time interval under consideration \(t_2-t_1\). The first term on the right-hand side is the increment of magnetic-field energy, and the second term is the mechanical work of the electromagnet, or the amount of mechanical energy converted into magnetic energy over the same time interval.

Displacement of the armature, and consequently the process of conversion of magnetic energy into mechanical work, is possible when the tractive force of the electromagnet exceeds the opposing forces. Depending on the magnitude of these forces or of external forces directed in the same direction as the tractive force of the electromagnet, as well as on the mass of the armature and the applied voltage, there may be three qualitatively different cases of energy conversion.

1. The process of conversion of magnetic energy into mechanical work may proceed without conversion of magnetic energy into electrical energy and of electrical energy into magnetic energy, that is, at constant flux linkage. In this case the electrical energy is expended only on the losses—

to cover the heat losses in the winding, while the mechanical work performed by the electromagnet over the time interval \(t_2-t_1\) is equal to the decrease in the magnetic energy of the electromagnet.

\[ \int_{t_1}^{t_2} F_\mathrm{e}\, d\delta = \frac{1}{2}\int_{t_1}^{t_2}\Psi\, di . \]

1. The second qualitatively different case of energy conversion in an electromagnet is the conversion of magnetic energy into mechanical energy with the simultaneous conversion of electrical energy, drawn from the network, into magnetic energy. This process may proceed:

a) with partial restoration of the magnetic energy;
b) with its value constant;
c) with an increase in magnetic energy.

The magnitude of the magnetic energy will remain constant when

\[ \int_{t_1}^{t_2}(\Psi\,di+i\,d\Psi)=0, \]

that is,

\[ -\frac{d\Psi}{di}=\frac{\Psi}{i}=L. \]

If the process proceeds at a constant current, then from equation (6) we have

\[ \int_{t_1}^{t_2} i\,d\Psi = \frac{1}{2}\int_{t_1}^{t_2} i\,d\Psi + \int_{t_1}^{t_2} F_\mathrm{e}\,d\delta, \]

that is, the magnetic energy increases by exactly as much as is expended in performing mechanical work.

1. The process of energy conversion in an electromagnet may proceed with the simultaneous conversion of magnetic energy both into mechanical work and into electrical energy. The latter is then expended on heat losses and returned to the external energy source.

The energy conversion in an electromagnet can be represented visually by graphs plotted in the flux-linkage–current coordinates.

In Fig. 1 all three cases of energy conversion are presented. Here the straight lines 1 and 2 are the dependences \(\Psi=f(i)\), respectively for gaps \(\delta_1\) and \(\delta_2<\delta_1\); point \(a\) characterizes the state of the electromagnet before the beginning of motion.

Fig. 1. Electrical balance in an electromagnet.

The energy stored by the electromagnet by the beginning of armature motion is proportional to the area of triangle \(oac_1\). The straight lines \(ab_1\), \(ab_2\), and \(ab_3\) characterize the change in flux linkage during armature motion, while the areas of triangles \(ob_1c_1\), \(ob_2c_2\), and \(ob_3c_3\) are proportional to the energy stored by the electromagnet by the end of the considered time of motion—

... respectively in the first, second, and third cases of energy conversion.

The increase in magnetic energy at the expense of the electrical energy supplied from the external source (the second case of energy conversion—the flux linkage changes along the straight line \(ab_2\) from point \(a\) to point \(b_2\)) is determined by the area of the trapezoid \(ab_2c_2c_1\).

The magnetic energy converted into electrical energy during the motion of the armature in the third case of energy conversion is proportional to the area of the trapezoid \(ac_1c_3b_3\).

According to the law of conservation of energy, the mechanical work of the electromagnet in the first, second, and third cases of energy conversion is respectively proportional to the areas of the triangles \(oab_1\), \(oab_2\), and \(oab_3\).

If an external force is applied to the armature of the electromagnet—for example, one due to a spring, whose direction coincides with the tractive force of the electromagnet—then the potential energy of the springs is expended on increasing the kinetic energy of the armature and leads to a change in the tractive force; however, on the graph the corresponding area cannot be shown, since equation (6) does not directly reflect the work of external forces. The equation of the energy balance in the presence of external forces can be written using the equation of motion of the armature

\[ F_{\text{e}} = m \frac{dv}{dt} + F_{\text{c}} - F_{\text{p}}, \tag{7} \]

where \(m\) is the mass of the armature;
\(F_{\text{c}}\) are the forces resisting motion;
\(F_{\text{p}}\) are external forces directed in the same direction as the tractive force of the electromagnet.

Substituting (7) into equation (6), we obtain

\[ \int_{t_1}^{t_2} i\,d\Psi = \frac{1}{2}\int_{t_1}^{t_2}(\Psi\,di+i\,d\Psi) + \frac{m}{2}\left(v_2^2-v_1^2\right) - \int_{t_1}^{t_2} F_{\text{p}}\,dv + \int_{t_1}^{t_2} F_{\text{c}}\,dv . \]

In the presence of external forces directed in the same direction as the tractive force of the electromagnet, the energy-conversion process is accompanied by a decrease in the consumption, or by a more intensive release, of electrical energy.

Up to this point the motor mode of operation of the electromagnet has been considered, that is, a mode in which magnetic energy was converted into mechanical energy.

The energy-conversion process in an electromagnet is reversed if the external mechanical forces applied to the armature are greater than the tractive force of the electromagnet and the armature moves in the direction of increasing air gap. In this case, as is evident from equation (6), the mechanical energy expended in overcoming the tractive force of the electromagnet during displacement of the armature will be converted into magnetic energy. Three qualitatively different cases of conversion of mechanical energy into magnetic energy are possible.

1. Conversion of mechanical energy into magnetic energy without conversion of magnetic energy into electrical energy or electrical energy into magnetic energy. This process can occur only with constant flux linkage and an increase in the current of the electromagnet. In this case the electrical energy supplied from the external source is expended only on covering the heat losses in the winding.

1. Conversion of mechanical energy into magnetic energy with simultaneous conversion of electrical energy into magnetic energy. This process may be represented by Fig. 1 if it is assumed that the change in flux linkage proceeds along the straight line \(b_3a\) from point \(b_3\) to point \(a\).

2. Conversion of mechanical energy into magnetic energy with simultaneous conversion of magnetic energy into electrical energy. This process may proceed:

a) with an increase in magnetic energy;
b) with an unchanged value of the magnetic energy;
c) with a decrease in magnetic energy.

The process of converting mechanical energy into magnetic energy with a decrease in the latter means that magnetic energy is converted into electrical energy faster than it is replenished at the expense of mechanical energy.

One or another process of energy conversion depends on how, owing to external conditions, the flux linkage and the current of the electromagnet coil will vary with time, that is, what the derivative \(\frac{d\psi}{di}\) and the signs of the increments of current and flux linkage will be.

Let us illustrate the above-listed cases of energy conversion in one figure. Let point \(O\) (Fig. 2), lying on the straight line \(O_1 - O\), representing the dependence \(\psi=f(i)\) for some gap \(\delta\), characterize the state of the electromagnet at the moment under consideration. From point \(O\), as from a center, a circle is drawn, any of whose radii may be used to characterize a quite definite value of the derivative \(\frac{d\psi}{di}\) and the signs of the increments of current and flux linkage, and, consequently, one or another process of energy conversion in the electromagnet. Thus, with a braked armature the process will proceed along the radius \(0—4\) (Fig. 2) or \(0—8\), that is, with the conversion of electrical energy into magnetic energy or of magnetic energy only into electrical energy. With constant flux linkage there will be either conversion of magnetic energy into mechanical energy \((0—1)\), or of mechanical energy into magnetic energy \((0—5)\) without consumption or release of electrical energy. With a constant value of the current there will occur the process of conversion of magnetic energy into mechanical energy and of electrical energy into magnetic energy \((0—3)\), with an increase in magnetic energy by exactly as much as is expended on the creation of mechanical work; or the process of conversion of mechanical energy into magnetic energy and of magnetic energy into electrical energy \((0—7)\), with a decrease in magnetic energy by exactly as much as was obtained from the mechanical energy expended on moving the armature.

Fig. 2. Diagram of operating modes of an electromagnet.

If, in the process of displacement of the armature, \(\Psi=f(i)\) changes in such a way that the derivative \(\dfrac{d\Psi}{di}\) is negative and always equal to the inductance of the electromagnet, then energy conversion in the electromagnet will take place at a constant value of the magnetic energy, or in the case of conversion of magnetic energy into mechanical and electrical energy into magnetic energy (0–2), or in the conversion of mechanical energy into magnetic energy and magnetic energy into electrical energy (0–6).

It is evident from Fig. 2 that if the change \(\Psi=f(i)\) during displacement of the armature of the electromagnet is characterized by the radius of a circle located in sectors I, II, III, and IV, then a process of conversion of magnetic energy into mechanical energy takes place—the motor mode of operation of the electromagnet. Moreover, sector I is characterized by the conversion of magnetic energy into electrical energy, which erases the sharp boundary between the motor and generator modes. The processes taking place when the radius is located in sectors II, III, and IV differ in the degree of restoration of magnetic energy at the expense of electrical energy.

When the radius characterizing the change \(\Psi=f(i)\) is located in sectors V, VI, VII, and VIII, a process of conversion of mechanical energy into magnetic energy takes place—the generator mode of operation of the electromagnet. Sector V is characterized by the fact that the energy-conversion process is accompanied by an increase in magnetic energy also at the expense of electrical energy. The processes taking place when the radius is located in sectors VI, VII, and VIII differ in the intensity of the conversions of magnetic energy into electrical energy.

The process of energy conversion in an electromagnet is determined by: the applied voltage, the external forces applied to the armature, its mass, and the kinetic energy that the armature may have at the instant under consideration.

An electromagnet can operate in motor and generator modes, each of which has three qualitatively distinct cases of energy conversion. However, there is no sharp boundary between the motor and generator modes.

In the case of motion of the armature of an electromagnet in the direction of increasing the air gap, only that part of the mechanical energy which goes into overcoming the tractive force of the electromagnet is converted into magnetic energy.

A correct understanding of all possible cases of energy conversion in an electromagnet makes it possible to judge the harmfulness or usefulness of phenomena and processes occurring in reciprocating electromagnetic machines.

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