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

Vol. 212 (1971)

Commutation Armature Reaction in DC Machines with Deep Rotational Speed Control

A. I. Skorospeshkin, E. G. Chebotkov

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

In DC machines with deep control of rotational speed by field weakening, the m.m.f. of the commutation armature reaction \(F_k\) constitutes a considerable fraction of the m.m.f. of the main pole. For example, for a P50 machine at \(n = 3000\) rpm and \(I_a = 20\) A, the m.m.f. \(F_k\) is approximately \(22\%\) of the m.m.f. of the main pole. Therefore, in machines with deep speed control by field weakening, determination and accounting of \(F_k\) are necessary.

In [1] a method was proposed for calculating the m.m.f. \(F_k\) as applied to DC machines when controlling rotational speed from 600 to 2000 rpm. In the general case, machines with field weakening have a wider speed-control range. Therefore, the aim of the present work is to obtain an analytical method for calculating \(F_k\) when controlling rotational speed from 400 to 3200 rpm.

The m.m.f. of the commutation armature reaction is equal to

\[ F_k = i_{\mathrm{dcr}} w_{\mathrm{sa}}, \tag{1} \]

where

\[ \begin{aligned} i_{\mathrm{dcr}} &\text{ — average additional commutation current,}\\ w_{\mathrm{sa}} &\text{ — average number of turns of an armature section.} \end{aligned} \]

Here the average value of the additional commutation current, calculated per one pole of the machine, according to [1], is determined as follows:

for a simple lap winding,

\[ i_{\mathrm{dcr}} = 2 \frac{b_{\mathrm{sh}} - b_{\mathrm{m}}}{\pi b_k} i_{\mathrm{dm}} = m_1 i_{\mathrm{dm}}; \tag{2} \]

for a simple wave winding,

\[ i_{\mathrm{dcr}} = 2 \frac{b_{\mathrm{sh}} - b_{\mathrm{m}} + (b_k - \tau + y_k b_k)}{\pi \cdot b_k} \, i_{\mathrm{dm}} = m_2 i_{\mathrm{dm}}, \tag{3} \]

where

\[ \begin{aligned} b_{\mathrm{sh}} &\text{ — brush width,}\\ b_k &\text{ — commutator pitch,}\\ b_{\mathrm{m}} &\text{ — interlamellar distance,}\\ \tau = \frac{\pi D_k}{p} &\text{ — pole pitch along the commutator circumference,}\\ y_k &\text{ — winding pitch on the commutator,}\\ i_{\mathrm{dm}} &\text{ — amplitude value of the additional current corresponding to optimal commutation.} \end{aligned} \]

Thus,

\[ F_k = i_{\mathrm{dcr}} \cdot w_{\mathrm{sa}} = m i_{\mathrm{dm}} \cdot w_{\mathrm{sa}}, \tag{4} \]

where the coefficient \(m\), depending on the type of winding, is determined from expression (2) or (3).

The experimental determination of \(F_k\) in [1] was carried out using the procedure of [2]. However, its application on the machine under test revealed a significant experimental error. This error consists in the following.

According to [2], the dependence of the e.m.f. of the main pole \(F_{\mathrm{gp}}\) on the excitation current is recorded alternately in the generator and motor modes. Because of residual magnetization, a displacement of the dependences \(F_{\mathrm{gp}}=f(I_p/I_{\mathrm{ya}})\) parallel to themselves is possible. Therefore the point of intersection of the straight lines \(F_{\mathrm{gp}}=f(I_p/I_{\mathrm{ya}})\), corresponding to straight-line commutation, may be shifted to one side or the other from the true one, which causes a considerable error in the determination of \(F_k\). This error is especially large when the machine, for some reason, has a considerable e.m.f. of the longitudinal armature reaction, for example because of asymmetry of the magnetic system. Thus, for machine P50 the e.m.f. \(F_k\), determined by the procedure of [2], is shown in Fig. 1, where the straight lines 1, \(1'\) and \(F_{k1}\) were obtained in the first test; the straight lines 2, \(2'\) and \(F_{k2}\), in the repeated test; the straight lines 1, 2 correspond to the motor mode of machine operation, and the straight lines \(1'\), \(2'\), to the generator mode.

The discrepancy between \(F_{k1}\) and \(F_{k2}\) is about 40%. The error can be eliminated if the following method is used to determine \(F_k\).

With the aid of a Hall e.m.f. sensor mounted at the center of the main pole, the dependence \(F_{\mathrm{gp}}=f(I_p/I_{\mathrm{ya}})\) is recorded for any mode of operation of the machine at \(n=\mathrm{const}\) and \(I_{\mathrm{ya}}=\mathrm{const}\). The dependence obtained is approximated, by the least-squares method, by an equation of a straight line of the form

\[ F_{\mathrm{gp}}=a+bx. \tag{5} \]

Fig. 1

Using (5), the e.m.f. of the commutation armature reaction (Fig. 2) is equal to

\[ F_k=b(P-M), \tag{6} \]

where \(P\) is the value of \(I_p/I_{\mathrm{ya}}\) corresponding to optimum commutation and determined from the sparkless operating zones of the machine.

\(M\) is the value of \(I_p/I_{\mathrm{ya}}\) corresponding to straight-line commutation.

The value of the excitation current corresponding to straight-line commutation can be determined by means of the method of potential brush diagrams in the form [3]. However, when the brush width is small, this method may produce a considerable error because of the unequal

Fig. 2

of the distance from the edges of the brush of the conductors embedded in it. Therefore, strips of copper foil 0.05 mm thick were glued onto the trailing and leading edges of the brush. For insulation from the brush-holder clip, the foil is covered with kapron film. The conductors soldered to the foil are led out to the measuring instrument.

The experimental determination of \(F_k\) was carried out using the method described above while the tested machine was operating in motor mode. The test object was a series P direct-current machine of the 5th frame size, with deep speed control by field weakening.

Rated data of the tested machine

Supply voltage — 220 V.
Output power 2.2 kW.
Rotational speed 750/3000 rpm.
Excitation current 1/0.1 A.

The measurement of \(F_k\) was performed while regulating the rotational speed from 400 to 3200 rpm and the armature current from 5 to 20 A. To eliminate random error in determining the point of rectilinear commutation, four prepared brushes were installed simultaneously.

In [1], the expression for the amplitude value of the additional commutation current

\[ i_{\mathrm{dm}} = c \left( \frac{I_a}{2a} \right)^{\eta} \cdot n^x \tag{7} \]

was obtained for machine operation in the speed range 600–2000 rpm. At speeds above 2000 rpm, \(i_{\mathrm{dm}}\) varies according to another law, and not according to expression [7]. A dependence of any complexity can be represented by a polynomial of the corresponding order. Proceeding from this, a second-order rotatable experimental design [4] was drawn up for two independent variables \(x_1\) and \(x_2\), where

\[ x_1 = \frac{n - 1814}{1000}, \tag{8} \]

\[ x_2 = \frac{i_a - 6.25}{2.655}, \tag{9} \]

\(n\) — rotational speed in rpm,
\(i_a\) — current of the parallel branch of the armature.

The design matrix and the experimental results are summarized in Table 1, where \(y_1\) and \(y_1'\) are the emf of the commutation armature reaction for EG-63 brushes; \(y_2\) and \(y_2'\) are for EG-74, respectively, for current densities through the brush of \(1.25\)—\(5\ \mathrm{A/cm^2}\): \(y_1\) and \(y_2\); \(2.5\)—\(10\ \mathrm{A/cm^2}\): \(y_1'\), \(y_2'\).

Table 1

The variation intervals of the independent variables are given in Table 2.

Table 2

From the results of the experiments, regression equations of the form are determined

\[ y=b_0+b_1x_1+b_2x_2+b_{11}x_1^2+b_{22}x_2^2+b_{12}x_1x_2, \tag{10} \]

where the regression coefficients are:

\[ b_0=0{,}25(0y)-0{,}125\sum_{i=1}^{k}(iiy), \]

\[ b_i=0{,}125(iy), \]

\[ b_{ii}=0{,}125(iiy)+0{,}03125\sum_{i=1}^{k}(iiy)-0{,}125(0y), \]

\[ b_{ij}=0{,}25(ijy), \]

\[ (0y)=\sum_u y_u, \]

\[ (iy)=\sum_u x_{iu}\cdot y_{iu}, \]

\[ (iiy)=\sum_u x_{iu}^2\cdot y_{iu}, \]

\[ (ijy)=\sum_u x_{iu}\cdot x_{ju}\cdot y_u. \]

We analyze the obtained equations by means of the multiple correlation coefficient

\[ R=\sqrt{1-\frac{S_R}{\sum y^2}}, \tag{11} \]

where \(S_R=\sum y^2-\sum_{i=0}^{k} b_i\cdot(i_y)\) is the residual sum of squares due to unaccounted factors and experimental error.

For EG-63, at \(j=1.25\text{–}5\ A/cm^2\) we have

\[ R=0.993, \]

at \(j=2.5\text{–}10\ A/cm^2\)

\[ R=0.994, \]

for EG-74, at \(j=1.25\text{–}5\ A/cm^2\)

\[ R=0.993 \]

at \(j=2.5\text{–}10\ A/cm^2\)

\[ R=0.991. \]

On the basis of these results one may conclude that the regression equations describe the experimental results with sufficient accuracy.

Substituting into (10) the values of the variables \(x_1\) and \(x_2\) from (8), (9) and taking (4) into account, we obtain:

\[ F_k=mw_{ca}\left[b_0' + b_1'n + b_2'i_a + b_{11}'n^2 + b_{22}'i_a^2 + b_{12}'n\cdot i_a\right], \tag{12} \]

which, after transformations, gives

\[ F_k=mw_{ca}\left[b_0'' + b_1''D_a n + b_2'' i_a + b_{11}''D_k^2 n^2 + b_{22}'' i_a^2 + b_{12}''D_a n i_a\right]. \tag{13} \]

Expression (13) can be applied for calculating \(F_k\) under any commutation setting. Indeed, suppose we have commutation corresponding to point \(P'\) in Fig. 2. Then the m.m.f. of the commutation armature reaction is determined by an expression of the form

\[ F_k'=\frac{P'-M}{P-M}F_k, \tag{14} \]

where \(F_k\) is determined by expression (13), the coefficients of whose polynomial are presented in Table 3.

From analysis of the polynomial coefficients, the following conclusions can be drawn.

At a constant armature current, the m.m.f. \(F_k\) increases nonlinearly as the rotational speed increases. This nonlinearity is due to the coefficient \(b_{11}''\). The negative sign and the presence of the coefficient \(b_{11}''\) can be explained by the fact that, as the rotational speed increases, the resistance of the brush contact increases. Indeed, as the rotational speed increases, the time for formation of contact conductance decreases, which leads to an increase in the specific resistance of the contact.

The m.m.f. of the commutation armature reaction \(F_k\) as a function of the armature current at constant rotational speed has only slight nonlinearity, since the coefficient \(b_{22}''\) is three orders of magnitude lower than \(b_2''\). Therefore the coefficient \(b_{22}''\) may be neglected in calculations of the m.m.f. \(F_k\).

In addition, it follows from the experimental data that the m.m.f. \(F_k\) can constitute a significant part of the m.m.f. of the main pole. Thus, in experiment No. 4 of Table 1, corresponding to operation of the machine at \(F_{mp}=150\ aw\), the m.m.f. \(F_k\) amounts to 18% of the m.m.f. of the main pole.

Table 3

On the whole, as a result of the work performed, an analytical method is proposed for determining the n.s. of the commutating armature reaction, which can be used in determining \(F_k\) of direct-current machines with deep control of rotational speed under any commutation setting.

References

1. E. G. Chebotkov, Yu. P. Galishnikov. Commutating armature reaction in direct-current machines of the P series with controlled rotational speed. Izvestiya TPI, vol. 172, 1967.

2. V. A. Lifanov, A. G. Dorm. Investigation of the commutating armature reaction in direct-current machines using Hall sensors. Elektromekhanika, 1961, No. 3.

3. M. F. Karasev et al. Optimal Commutation of Direct-Current Machines. Transport Publishing House, 1967.

4. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extreme Experiments. Nauka Publishing House, 1965.