COMMUTATION ARMATURE REACTION IN SERIES II DC MACHINES WITH ADJUSTABLE ROTATIONAL SPEED

E. G. Chebotkov, Yu. P. Galishnikov

(Recommended by the seminar of the Department of Electrical Machines and General Electrical Engineering)

In dc machines of ordinary design, the magnetizing force of the commutation armature reaction \(F_k\) is small and amounts to about \(3\text{–}6\%\) of the m.m.f. of the main pole. However, with deep regulation of the motor rotational speed by weakening the main field, the influence of \(F_k\) on the output characteristics of the machine is quite significant. Thus, Fig. 1 shows the change in rotational speed as a function of the commutation setting for a normal main field (curves I) and a weakened main field (curves II), with unchanged armature current. The curves were obtained for a motor of frame size 4 of series II when varying the nature of commu-

Fig. 1. Influence of the m.m.f. of the commutation armature reaction on the rotational speed of the motor.
I — with a normal field of the main poles.
II — with a weakened field of the main poles.

tation within the sparkless-operation zone. As we see, the change in rotational speed under a weakened field reaches 40%. Hence it is clear how important it is to determine and take into account the m.m.f. of the commutation armature reaction when regulating by field weakening.

The existing calculation methods for determining \(F_k\) have been developed mainly for special electrical machines, for example, transverse-field EMU machines [1, 2]. As for analytical methods of determining \(F_k\) in DC machines, they give a considerable error. For example, the calculated values of \(F_k\) for a four-pole motor of small power, determined by the method of V. T. Kas’yanov [3] and N. M. Nemchenko [4], differ from the experimental values by 30–70% (Table 1). This is explained by the fact that the named methods do not make it possible to take into account a number of essential factors, including the rotational speed of the machine.

Table 1

The influence of rotational speed on the magnitude of \(F_k\) is taken into account by an expression that was proposed in its time by M. S. Mikhailov-Mikulinskii [5] and has the following form:

\[ F_k=\pm \frac{e_r-e_r'}{\sum R_{\mathrm{sh}}} W_{c.a} =\pm i_{d\,\mathrm{av}}\cdot W_{c.a}, \tag{1} \]

where \(\pm(e_k-e_r)\) is the uncompensated e.m.f. of the section during commutation,
\(W_{c.a}\) is the average number of turns of an armature section,
\(i_{d\,\mathrm{av}}\) is the average additional commutation current,
\(\sum R_{\mathrm{sh}}\) is the resistance of the brush–commutator contact, determined as a function of the brush overlap.

It should be noted that the assumption \(R_{\mathrm{sh}}=\mathrm{const}\), adopted by the author of [5] in determining the additional commutation current, corresponds very poorly to present-day ideas about sliding contact and, as is seen from Table 1, leads to a considerable discrepancy between calculated and experimental data.

Below another method is presented for calculating the average value of the additional commutation current and the m.m.f. of the commutation armature reaction as applied to Series II DC machines of the smaller frame sizes.

Analysis of oscillograms of the commutation current shows that, without any particular error, one may assume a sinusoidal law of variation of the additional current. Then, for a simple loop winding, the variation of the additional current under the brush, for example at a brush overlap

\[ \frac{b_{\mathrm{sh}}}{b_k}=3, \]

may be represented as in Fig. 2. In this case, the mean value over one revolution of the commutator of the additional current, calculated per one pole of the machine, is determined as follows:

\[ i_{d\,\mathrm{av}} = \frac{K}{t_{\mathrm{ob}}}\,2\int_{0}^{\pi/2} i_{dm}\sin\omega t\,dt = 2\frac{b_{\mathrm{sh}}-b_m}{\pi b_k}\,i_{dm} = m_1\cdot i_{dm}, \tag{2} \]

where \(i_{\mathrm{dm}}\) is the amplitude value of the additional current;
\(K\) is the number of commutator segments;

\[ \omega = \frac{\pi}{T} \]

is the angular frequency corresponding to the commutation period;

\[ T = \frac{b_{\mathrm{sh}} - b_{\mathrm{m}}}{100 v_{\mathrm{k}}} \]

is the commutation period of a section;

\(b_{\mathrm{sh}}\) is the brush width;
\(b_{\mathrm{m}}\) is the interlamellar distance;
\(v_{\mathrm{k}}\) is the commutator speed.

\[ t_{\mathrm{rev}} = \frac{K \cdot b_{\mathrm{k}}}{100 v_{\mathrm{k}}} \]

is the time of one revolution of the commutator.

\(b_{\mathrm{k}}\) is the commutator pitch.

Fig. 2. On determining the average value of the additional commutation current.

An analogous expression for a simple wave winding has the form:

\[ i_{\mathrm{d\,av}} = 2 \frac{b_{\mathrm{sh}} - b_{\mathrm{m}} + (b_{\mathrm{k}} - \tau + y_{\mathrm{k}} b_{\mathrm{k}})} {\tau b_{\mathrm{k}}} \, i_{\mathrm{dm}} = m_2 i_{\mathrm{dm}}, \tag{3} \]

where

\[ \tau = \frac{\pi D_{\mathrm{k}}}{P} \]

is the pole pitch along the commutator circumference,

\(y_{\mathrm{k}}\) is the winding pitch on the commutator.

Thus,

\[ F_{\mathrm{k}} = i_{y_{\mathrm{k}}\mathrm{\,av}} \cdot W_{\mathrm{c}} = m \cdot i_{\mathrm{dm}} W_{\mathrm{c}} . \tag{4} \]

In the general case, the value \(i_{\mathrm{dm}}\) in expressions (2) and (3) may vary over very wide limits depending on the commutation adjustment of each particular machine. However, in the overwhelming majority of cases, commutation with one or another deviation corresponds to the middle of the sparkless-operation zone of the machine, so that in the subsequent discussion \(i_{\mathrm{dm}}\) is understood as the amplitude value of the additional current corresponding to optimal commutation.

The experimental determination of \(F_{\mathrm{k}}\) was carried out using the well-known method of M. A. Alyab’ev, improved by the use of Hall emf sensors [7]. As an analysis showed, the probable error in determining \(F_{\mathrm{k}}\) by this method, caused by displacement of the mean line of the current collector during machine reversal and by oscillations of the brush in the holder

brush holder, its value does not exceed 8–10%. All investigations of \(F_k\) were carried out on a series II DC machine of the 4th frame size.

Nominal data of the tested machine

Fig. 3. Dependence \(F_k = F_k(I_a)\):
a) rotational speed 1300 rpm,
b) rotational speed 2000 rpm.

Measurements of \(F_k\) were made under various operating conditions of the machine, with its rotational speed varied from 600 to 2000 rpm, this variation being effected by changing the magnetic flux of the main poles, and also at armature currents from 6 to 18 A. The experimental values of \(F_k\) are presented on the scale of the magnetomotive force of excitation, recalculated per 1 pole of the main magnetic circuit of the machine.

In Fig. 3 (a, b) the experimental dependences of \(F_k\) on the armature current are given, obtained for rotational speeds of 1300 and 2000 rpm at different currents in the auxiliary-pole winding. Processing of the experimental curves by the method of equal areas gives the following dependence:

\[ F_k = C_1 \left(\frac{I_a}{2a}\right)^\eta , \tag{5} \]

where \(I_a\) is the armature current,
\(a\) is the number of pairs of parallel branches of the armature winding,
\(C_1\) is a quantity that varies for different rotational speeds and different currents in the auxiliary-pole winding.

The exponent \(0<\eta<1\) in expression (5) is practically constant under the experimental conditions used and is equal in the present case to 0.97. Thus, the dependence of \(F_k\) on the armature current is by no means rectilinear, as follows from expression (1). This is apparently explained by the fact that the resistance of the brush contact does not remain constant—

constant when the armature current and, correspondingly, the current density in the sliding contact change. Convincing confirmation of the latter is provided by curves 1 and 2, given in [6] in Fig. 2 (a, b), which were obtained at nominal current densities in the contact of 6 and 12 A/cm², respectively. It should be noted that, in determining \(F_k\), the contact current density did not exceed 4 A/cm², which also explains the comparatively weak nonlinearity of the dependence \(F_k(I_a)\). It is clear that the greater the nominal current density in the brush contact, the more strongly \(\eta\) in expression (5) should differ from 1.

Taking into account the dependence on the rotational speed (the experimental curves in Figs. 3 and 4), the general empirical expression for \(F_k\) takes the following form:

\[ F_k = C_2 \left( \frac{I_a}{2a} \right)^{\eta} n^{\chi}. \tag{6} \]

Fig. 4. Dependence \(F_k = F_k(n)\).
a) armature current 12 A,
b) armature current 18 A.

The exponent \(\chi\), obtained in processing the experimental data, is equal to 0.6. Thus, despite the direct proportionality between \(e_{\text{res}} = e_k - e_r\) and the rotational speed of the machine, as the latter increases, \(F_k\), and consequently also the additional commutation current, increase nonlinearly. Hence it follows clearly that the total resistance of the brush contact increases with increasing commutator rotational speed. This once again confirms that the natural dynamic volt-ampere characteristic of the sliding contact is closely connected with the conditions under which it is formed. Indeed, as the rotational speed increases, the time for formation of the contact conductivity decreases proportionally, which as a result leads to an increase in the specific contact resistance.

We shall write expression (6) in the form:

\[ F_k = C_2 \left( \frac{I_a}{2a} \right)^{\eta} n \cdot n^{\chi - 1}, \tag{7} \]

which, after simple transformations, gives

\[ F_k=C_4\left(\frac{I_a}{2a}\right)^\eta \frac{D_a}{D_k^{\,1-\chi}} n^\chi . \tag{8} \]

From comparison of (4) and (8) we obtain

\[ i_{\mathrm{dm}}=C\left(\frac{I_a}{2a}\right)^\eta \frac{D_a}{D_k^{\,1-\chi}} n^\chi W_c, \tag{9} \]

where the numerical value \(C=0.208\cdot 10^3\).

Thus, the general expression for \(F_k\) has the form:

\[ F_k=mC\left(\frac{I_a}{2a}\right)^\eta \frac{D_a}{D_k^{\,1-\chi}} n^\chi W_c, \tag{10} \]

for a simple lap winding

\[ m_1=2\frac{b_{\mathrm{sh}}-b_m}{\pi b_k}, \tag{11} \]

and for a simple wave winding

\[ m_2=2\frac{b_{\mathrm{sh}}-b_m+(b_k-\tau+y_k b_k)}{\pi b_k}. \tag{12} \]

The coefficients \(\eta\) and \(\chi\) in expression (10) are determined by the characteristics of the brush contact. Determined experimentally for the ET-49 brush at a nominal current density of \(1.5\text{--}4\ \mathrm{A/cm^2}\), these coefficients are respectively equal to 0.97 and 0.6.

Thus, proceeding from optimal commutation, a calculation expression for \(F_k\) has been obtained which can be used in determining the m.m.f. of the commutation armature reaction of low-power DC machines operating over a wide range of rotational-speed variation at a nominal current density in the brush contact of \(1.5\text{--}4\ \mathrm{A/cm^2}\). The values of the coefficients \(C\), \(\eta\), \(\chi\) corresponding to higher current densities in the contact will be obtained in the course of further experimental studies of the commutation armature reaction.

REFERENCES

1. V. I. Radin, Reaction of commutation currents in an electric machine with transverse field. Elektrichestvo, No. 9, 1957.

2. A. I. Korostelev, On the question of the reaction of commutation currents in electrical-machine amplifiers with transverse field. Higher Educational Institutions, Electromechanics, No. 12, 1958.

3. V. T. Kas’yanov. Armature reaction of a DC machine. VEPI, No. 2, 1940.

4. N. M. Nemchenko. A new method for calculating the reaction of additional currents. Proceedings of the Kuban Agricultural Institute, issue 2, 1955.

5. M. S. Mikhailov-Mikulinsky. Reaction of the additional commutation current on the main field of a DC machine. VEPI, No. 7, 1949.

6. Yu. P. Galishnikov, A. I. Skorospeshkin. Natural volt-ampere characteristics of the sliding contact carbon brush–commutator. Izvestiya TPI, vol. 172, 1967.

7. V. A. Lifanov, A. G. Dorm. Investigation of the commutation armature reaction in DC machines using Hall sensors. Higher Educational Institutions, Electromechanics, No. 3, 1961.