On an Analytical Dependence for the Magnetization Characteristic of DC Machines

L. I. Gandzha

It is known that the magnetization characteristic of an electrical machine, at the present state of the science concerning it, cannot be expressed analytically; yet the need for an equation expressing, in mathematical form, the dependence of electromotive force on the excitation current is, in a number of cases, extremely acute. What has been said is especially true in the theory of electric drive, where this dependence is needed in solving the differential equations of transient processes in an electric drive.

Let us explain this by an example.

When a DC motor, supplied from a series generator according to the Ward Leonard system (Fig. 1), is operating, its transient processes are described by the differential equation:

Fig. 1

\[ e_i = R \cdot i + L \frac{di}{dt} + e_g, \]

where \(e_i\) is the instantaneous value of the e.m.f. of the series generator,
\(i\) is the same current in the transient period, flowing through the armature windings of the generator and the motor,
\(e_g\) is the instantaneous value of the counter-e.m.f. of the motor,
\(t\) is time,
\(R\) and \(L\) are the ohmic resistance and the coefficient of self-induction of the power circuit of the Ward Leonard system.

In order to solve equation (1), it is necessary to reduce the number of variables in it \((e_i, i, e_g, t)\) to two. This can be done if equation (1) is solved jointly with two others:

\[ e_g = c_e \cdot n, \tag{2} \]

where \(c_e\) is the coefficient of proportionality between \(e_g\) and the motor speed \(n\) at constant magnetic flux of the motor, and

\[ M = M_{cm} + \frac{GD^2}{375}\frac{dn}{dt}, \tag{3} \]

which is the equation of moments, where, in turn, \(M\) is the torque of the motor, equal, at constant magnetic flux,

\[ M = c_M \cdot i, \]

where \(c_M\) is the coefficient of proportionality between torque and current, \(M_{cm}\) is the static resisting torque on the motor shaft, and \(GD^2\) is its flywheel moment.

In addition, into equation (1) there must be substituted the functional dependence \(e_g\) on the current \(i\).

As a result of these calculations, differential equations for \(i\) and \(n\) may be obtained, whose solutions will lead to the functions:

\[ n=f(t), \]

\[ i=f(t), \]

describing the transient process of the motor speed and current.

We see that a mathematical solution of the transient process is possible only in the case when the analytical dependence \(e_g=\psi(i)\), representing the generator magnetization curve, is given (Fig. 2).

Wishing to study transient regimes in analytical form, a number of authors neglect the phenomenon of generator saturation and consider the function \(e_g=\psi(i)\) to be rectilinear. The solutions obtained in this case are comparatively simple, but they do not correspond to reality; moreover, their use is possible only when the generator operates on the unsaturated part of its magnetization characteristic.

Fig. 2

Other authors, wishing to take into account the phenomenon of generator saturation, replace the curve \(e_g=\psi(i)\) (Fig. 2) by a series of straight-line segments and solve the transient processes for each segment separately. The solutions obtained in this case are likewise comparatively simple, but are inconvenient in the sense that they divide the single transient process of the electric drive into parts corresponding to the segments replacing the magnetization curve.

One may also point to examples where the magnetization curve is expressed by Froelich’s equation, replaced by a parabola, etc. [1; 2].

It seems possible to us to express the magnetization curve by the exponential function

\[ e_g=e_m\cdot\left(1-e^{-\frac{i_{\mathrm{в}}}{T_i}}\right)+e_0\cdot e^{-\frac{i_{\mathrm{в}}}{T_i}}, \quad \text{(Fig. 3),} \]

where \(i_{\mathrm{в}}\) is the excitation current of the machine,

\(e_0\) is the e.m.f. due to residual magnetism,

\(e_m\) is the maximum e.m.f. of the machine, which the magnetization curve approaches asymptotically as the excitation current increases,

\(T_i\) is the “current constant,” and

\(e\) is the base of natural logarithms.

If residual magnetism may be neglected, then \(e_0=0\) and

\[ e_g=e_m\cdot\left(1-e^{-\frac{i_{\mathrm{в}}}{T_i}}\right). \tag{5} \]

The quantities \(e_m\) and \(T_i\) are unknown for the machine and must be determined from experiment. Thus, if for the machine under test the magnetization curve has been taken ...

magnetization (Fig. 3), then the quantity \(T_i\) can be preliminarily determined from the known formula characterizing the exponential:

\[ T_i=\frac{\Delta i_v}{\ln \dfrac{e_{i_2}-e_{i_1}}{e_{i_3}-e_{i_2}}}, \tag{6} \]

where \(\Delta i\) is the distance along the abscissa axis between points 1, 2, and 3, taken equal, and \(e_{i_1}, e_{i_2}, e_{i_3}\) are the values of the machine e.m.f. corresponding to these points of the curve.

The value of \(T_i\) calculated in this way, when substituted into equation (4) or (5), written for some point of the curve (for example, point 2),

\[ e_{i_2}=e_m\cdot\left(1-e^{-\frac{i_v}{T_i}}\right), \]

makes it possible to determine a preliminary value of \(e_m\). Since points 1, 2, and 3 do not lie on an ideal exponential, the exponential constructed according to equation (4) or (5) using the preliminary values found for \(T_i\) and \(e_m\) will differ to one degree or another from the actual magnetization curve. By varying the quantities \(T_i\) and \(e_m\) in one direction or the other, it is possible, by repeated calculations, to bring the exponential defined by equations (4) or (5) closer to the actual curve. This is usually accomplished quickly.

Fig. 3

To confirm what has been said, we carried out the corresponding calculations for a number of machines. The objects of the experimental investigation were the following direct-current machines.

1. A series-excited machine with nameplate data:

manuf. No. 182765,
220 V,
32.5 A,
1300 rpm,
8 hp.

1. A shunt two-commutator machine with data:

No. 2319548,
type 66,
1500 rpm,
220/8 V,
1.14/31.2 A.

1. A shunt machine with nameplate data:

No. 1376, type NGEO,
dynamo, motor,
115 V, 110 V,
30.5 A, 31 A,
1730 rpm, 1180 rpm,
3.5 kW, 3.7 hp.

1. Shunt machine with nameplate data:

\[ \begin{gathered} \text{plant No. } 41115,\\ 17\ \text{kW},\\ 120\ \text{V},\\ 780\ \text{rpm},\\ 142\ \text{A}. \end{gathered} \]

1. Shunt machine

\[ \begin{gathered} \text{No. } 40147,\\ 1500\ \text{rpm},\\ 110\ \text{A},\\ 43\ \text{A},\\ 5.5\ \text{hp}. \end{gathered} \]

1. The same machine as machine 5, with the same nameplate data.

Fig. 4-a

Fig. 4-b

For the machines listed, no-load characteristics \(e_2 = f(i_{\mathrm{в}})\) (also the magnetization curves) were taken in all cases with independent excitation.

Fig. 5

Fig. 6

Fig. 7

The results of the experiments are summarized in the tables below: Tables 1-a, 1-b, 2, 3, 4, 5, and 6 (the first two rows), whose numbering corresponds to the numbering of the machines listed above.

From the data in these tables, the magnetization curves shown in Figs. 4-a, 4-b, 5, 6, 7, 8, and 9 were constructed.

Further, on the basis of the results of experiments given in the indicated tables, replacing exponentials were selected for them according to the calculation procedure described above. The data of these exponentials are entered in the same tables (rows 1 and 3).

Figure 8

Figure 9

Table 1-a (machine 1, Fig. 4-a)
\(T_i = 21.6\ \mathrm{A},\ e_m = 285\ \mathrm{V}\)

Table 1-b (machine 1, Fig. 4-b).
\(T_i = 18\ \mathrm{A},\ e_m = 265\ \mathrm{V}\).

Table 2 (machine 2, Fig. 5).
\(T_i = 0.18\ \mathrm{A},\ e_m = 360\ \mathrm{V}\).

Table 3 (machine 3, Fig. 6)
\(T_i = 0.65\ \mathrm{A},\ e_m = 154\ \mathrm{V}\)

Table 4 (machine 4, Fig. 7)
\(T_i = 1.18\ \mathrm{A},\ e_m = 138\ \mathrm{V}\)

Table 5 (machine 5, Fig. 8).
\(T_i = 1.2\ \mathrm{A},\ e_m = 138\ \mathrm{V},\) taken at \(n = 1200\ \mathrm{rev/min}\)

Table 6 (machine 6, Fig. 9)
\(T_i = 1.2\ \mathrm{A},\ e_m = 156\ \mathrm{V},\) taken at \(n = 1300\ \mathrm{rev/min}\)

The last two rows of the tables give an indication of the absolute and relative, in percent, difference between the emf obtained from equations (4) or (5) and the emf from the actual magnetization curve.

The magnetization curves expressed by an exponential are also shown in Figs. 4-a, 4-b, 5, 6, 7, 8, and 9.

In Fig. 4-a the actual magnetization curve and the exponential replacing it according to equation (4), taking into account the emf of residual magnetism, are shown.

In this case the magnetization curve was obtained experimentally by decreasing the excitation current from a maximum value equal to 50 amperes.

Fig. 4-b presents the same magnetization curve and the exponential replacing it, calculated from equation (5) without taking into account the emf from residual magnetism.

In this case the magnetization curve was obtained by increasing the excitation current, but not up to 50, rather up to 36.5 amperes.

A comparison of these figures shows that, within smaller limits of variation of the excitation current and when the emf from residual magnetism is neglected, the exponential more closely approaches the actual magnetization curve and, consequently, replaces it better.

Fig. 5 depicts the replacement of the magnetization curve by an exponential when the emf from residual magnetism is neglected (equation 5) for machine 2; Fig. 6 shows the same, but with allowance for the emf from residual magnetism for machine 3; Fig. 7 shows the same for machine 4; and, finally, Figs. 8 and 9 show the same for machines 5 and 6 without taking \(e_0\) into account. As Tables 5 and 6, corresponding to these figures, show, the value \(T_i\) for both machines remains one and the same, equal to

\[ T_i = 1.2\,a, \]

which was to be expected, since these machines are of identical design. The difference in the values of \(e_m\) for them is explained by the different speeds at which the magnetization characteristic was taken.

The results of the experiments performed show that magnetization curves can quite well be replaced by corresponding exponentials and, thus, can be expressed analytically. Such a replacement should be considered especially successful for the curves presented in Figs. 4-b, 7, 8, and 9.

Conclusion

In most cases, the magnetization curve of a DC machine can be represented analytically by an exponential, which can facilitate a number of investigations associated with the need to account for saturation of the machine.

References Used

1. Jouillart E. — Automatic regulation of electrical machines.
2. Nitusov E. V. — On the energetics of braking series-excitation motors, Elektrichestvo, Nos. 11–12, 1944.