Accounting for the Influence of Slot Skew in Calculating Commutation in DC Machines

E. G. Chebotkov, A. I. Skorospelkin

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

In DC machines, slot skew is usually used to eliminate magnetic pulsations. Since in the commutation zone the induction is not constant over the width of the zone and varies according to a complex law, it should be expected that, because of slot skew, the waveform of the rotational emf of the commutated section due to external fields will differ from the form of the induction distribution in the air gap. Therefore, the aim of the present work is to take slot skew into account when calculating the rotational emf of the commutated section.

The emf of a conductor, taking slot skew into account, can be represented as the sum of the emfs induced in elementary conductors:

\[ E_{\mathrm{pr}}=\Delta e_1+\Delta e_2+\ldots+\Delta e_n = v\cdot \Delta l_1\cdot B_{\delta_1} +v\cdot \Delta l_2 B_{\delta_2} +\ldots+v\Delta l_n\cdot B_{\delta n}, \tag{1} \]

where

\[ \Delta e_1,\ \Delta e_2,\ldots \Delta e_n \]
are the emfs induced in the elementary conductors;

\[ \Delta l_1=\Delta l'_1\cos\varphi,\ldots \]
are the effective lengths of the elementary conductors;

\[ \Delta l' \]
is the length of an elementary conductor;

\[ \varphi=\operatorname{arc\,tg}\frac{b_c}{l_a}; \]

\[ b_c \]
is the skew magnitude,

\[ l_a \]
is the armature length,

\[ v \]
is the rotational speed of the armature,

\[ B_{\delta_1},\ B_{\delta_2},\ldots B_{\delta n} \]
are the induction values corresponding to the position of the elementary conductor under consideration.

Since in (1) the variables \(B_\delta\) and \(\Delta l\) vary in mutually perpendicular directions, it is difficult to represent this expression in the form of an integral sum. At the same time, the conductor emf, based on (1), can be represented in the form

\[ E_{\mathrm{pr}}=v\cdot l_a\cdot B_{\mathrm{sr}}, \tag{2} \]

where \(B_{\mathrm{sr}}\) is the value of the average induction for the conductor. The average value of the induction for a conductor, taking slot skew into account, is determined as

\[ B_{\mathrm{sr}}= \frac{ \displaystyle \int_{m_n}^{m_n+b_c} B_\delta\cdot dS }{b_c}, \tag{3} \]

where

\[ B_\delta=f(S) \]
is the distribution of induction along the circumference of the armature;

\[ m_n \]
is the quantity determining the position of the conductor relative to the reference axis.

Thus, for the e.m.f. of a conductor, taking account of the slot skew, we have the expression

\[ E_{\text{pr}} = v \cdot l_a \cdot \frac{\displaystyle \int_{m_n}^{m_n+b_c} B_\delta \cdot dS}{b_c}. \tag{4} \]

Consequently, with the full number of interpoles, the rotational e.m.f. of the section being commutated is equal to

\[ E_c = 2 \cdot E_{\text{pr}} \cdot w_c = 2 w_c \cdot v \cdot l_a \frac{\displaystyle \int_{m_n}^{m_n+b_c} B_\delta \cdot dS}{b_c}, \tag{5} \]

where \(w_c\) is the number of turns of the section.

Since the e.m.f. of a section is directly proportional to the mean induction of the conductor, in analyzing the influence of slot skew we shall confine ourselves to determining the induction \(B_{\text{sr}}\).

The induction in the air gap of the commutation zone of d.c. machines is the result of the action of several components, the most important of which are the fields of the interpoles and of the main poles and the cross reaction of the armature. Since the magnetic circuit in the commutation zone is unsaturated and, consequently, the superposition principle is applicable, we shall consider the influence of skew on the rotational e.m.f. from each field separately. We carry out the calculation of the distribution of the mean induction in the air gap of the commutation zone for the d.c. machine P51.

Mean induction of a conductor in the commutation zone from the field of the interpole

In [2] a method is given for calculating the field of the interpoles, on the basis of which the distribution of induction in the air gap of machine P51 was obtained—curve 1 in Fig. 1 for the interpole current \(I_w = 10a\).

To obtain the mean induction we shall use graphical integration—the trapezoidal method. With the choice of two quantities of intervals, \(n = 2n_1 = 12\), the latter provides sufficiently high accuracy of calculation.

As a result of the integration carried out and the calculation according to expression (3), the distribution of the mean induction of the interpole field in the neutral zone of the machine was obtained and is presented in Fig. 1 in the form of curve 2. The same figure shows the width of the interpole \(b_w\) and the width of the commutation zone \(b_k\).

From a comparison of the distribution of induction in the air gap with the distribution of the mean induction it follows that slot skew somewhat reduces the induction under the interpole and increases it outside the pole.

As can be seen from Fig. 1, the difference in inductions at certain points of the commutation zone reaches \(10 \div 15\%\), which is correspondingly significant in calculating commutation. In addition, the curve of the distribution of the mean induction differs noticeably in shape from the curve of the induction of the machine air gap—curve 2 (Fig. 1) has a more pronounced sinusoidal character than curve 1.

From the analysis carried out it is clear that taking account of slot skew in calculating the e.m.f. of the commutated section from the field of the interpoles is necessary.

Calculation of the armature-reaction field in the commutation zone and accounting for the influence of slot skew

The armature-reaction field has a significant influence on commutation, since its magnetomotive force usually amounts to \(0.75 \div 0.9\) of the magnetomotive force of the interpole. Therefore, when calculating commutation it is important to have the distribution of induction from the magnetomotive force \(F_{aq}\) in the commutation zone.

The calculation of the air-gap induction due to the transverse armature reaction is complicated by the fact that the magnetomotive force \(F_{aq}\) varies around the circumference of the armature. Since the currents of the sections undergoing commutation are smaller than the current of the parallel branch, the increase in the m.m.f. of the armature reaction \(F_{aq}\) in the commutation zone is slowed. Taking this into account, for somewhat accelerated commutation the m.m.f. of the transverse armature reaction under the center of the commutating pole, according to [4], is equal to:

\[ F_{aq}=\left(1-\frac{2}{3}\frac{b_k}{\tau}\right)\frac{A\cdot\tau}{2}. \tag{6} \]

At the end and at the beginning of the commutation zone, the m.m.f. of the armature reaction is determined in the form

\[ F'_{aq}=A\cdot\frac{\tau-b_k}{2}. \tag{7} \]

The difference between the magnetomotive forces \(F_{aq}\) and \(F'_{aq}\) is small and, for machines of series P sizes 1–6, amounts to \(5\div 10\%\) of the m.m.f. \(F_{aq}\).

Fig. 1.

The armature-reaction field in the commutation zone can be calculated using the method for calculating the field of commutating poles [2], under the assumption that the above-indicated variation of the magnetic potential around the armature circumference does not lead to a noticeable change in the parameters of the flux tubes. To verify the validity of this assumption, a series of experiments was carried out on modeling the armature-reaction field on conducting paper. As the experimental results showed, within the range of variation of the geometry and of the difference of the magnetomotive forces \(F_{aq}\) and \(F'_{aq}\) of machines of series P/1–6 sizes, no change occurs in the parameters of the flux tubes relative to the expressions for them in [2].

Thus, the calculation of the field in the commutation zone due to the transverse armature reaction can be carried out by the method of [2].

Proceeding from this, a calculation was made of the armature-reaction field in the commutation zone of machine P51 at \(I_a = 10a\). The results of the calculation are presented in the form of curve 3 in Fig. 1.

To verify the reliability of the calculation, an experiment was carried out to determine the induction of the armature-reaction field in the air gap of the machine. The distribution of induction was measured by means of a Hall emf sensor fastened to a tooth of the armature of machine P51, at armature current \(I_a = 10a\), \(i_v = 0\), \(I_w = 0\). The experimental data are presented in Fig. 1 as points. From a comparison of the calculated and experimental data it follows that the discrepancy between calculation and experiment does not exceed 5%. This once again confirms the possibility of determining the induction in the air gap of the commutation zone from the m.m.f. of the transverse armature reaction by means of the method [2].

As above, we estimate the influence of slot skew on the emf of a section by the distribution of the average induction along the armature circumference. The determination of the average

Fig. 2.

induction is carried out using expression (3) and graphical integration.

As a result of the calculations performed, we obtain the distribution of the average induction of the armature-reaction field, represented by curve 4 (Fig. 1).

From an analysis of curves 3 and 4 of Fig. 1 it follows that the slot skew somewhat reduces the induction in the commutation zone and increases it outside the zone. The reduction of induction at some points of the commutation zone reaches 10–15%. Consequently, slot skew can have a noticeable influence on the emf of the commutated section due to the field of the transverse armature reaction.

From the results obtained it follows that slot skew has an analogous influence on the emf of a section both from the field of the interpoles and from the field of the transverse armature reaction.

Distribution of the Average Induction from the Leakage Field of the Main Poles in the Commutation Zone

In [1], [3] methods are given for calculating the leakage field of the main poles in the commutation zone. There, too, the distribution of the field of the main poles in the neutral zone of machine P51 was obtained for saturated and unsaturated magnetic systems. Fig. 2 presents the distribution of induction in the air gap of the neutral zone of machine P51 for the field of the main poles, curve 1, at \(i_v = 0.6a\).

We determine the values of the average induction using expression (3) and the calculated curves of the distribution of induction in the air gap, obtained in [1,3] for machine excitation currents of \(0.6;\ 1.0;\ 1.4a\). The excitation current \(i_{\mathrm{v}} = 0.6a\) corresponds to an unsaturated pole shoe; at \(i_{\mathrm{v}} = 1a;\ 1.4a\), some sections of the pole shoe become saturated [3].

The calculation results are presented in the form of curves 2, 3, 4 in Fig. 2, respectively for excitation currents \(0.6;\ 1;\ 1.4a\).

Comparing the field distribution in the air gap (curve 1) with the distribution of the average induction (curve 2), corresponding to an excitation current \(i_{\mathrm{v}} = 0.6a\), the following conclusions may be drawn.

Curve 2 of the average induction corresponds to larger induction values than curve 1. The magnitude of the discrepancy between these curves increases toward the edge of the interpole and decreases somewhat beyond it.

Fig. 3.

The increase in the average induction relative to the induction in the air gap under the interpole is quite significant: in the present case, in some sections it reaches \(150 \div 400\) G, which has a substantial effect on commutation.

The increase in the average induction at the end of the commutation zone relative to the induction in the air gap is also of great importance. Thus, if for excitation currents of \(1\) and \(1.4a\) the induction at the end of the commutation zone was \(500 \div 600\) G [3], then for the same excitation currents, owing to slot skew, the average induction is already \(700 \div 900\) G.

Consequently, when calculating the e.m.f. of the commutated section due to the leakage field of the main poles, allowance for slot skew is necessary, since the latter has a substantial effect both on the magnitude and on the waveform of the e.m.f.

Since, in calculating commutation, the leakage field of the main poles is rarely taken into account, then, to illustrate its effect on the commutating e.m.f., let us consider the average induction due to the combined action of the field of the main and interpoles.

As was stated above, for the transverse circuit of a direct-current machine the principle of superposition is applicable. Therefore, we determine the combined average induction of the field of the main and interpoles as the algebraic sum of the components. The average induction of machine P15 obtained in this way in the commutation zone at \(I_w = 10a\) and excitation currents \(0.6\) and \(1.4a\) is shown in Fig. 3 in the form of curves.

Curve 1 represents the distribution of the average induction from the commutating-pole field at \(I_{\mathrm{v}} = 10\,a\).

Curve 2 is the combined average induction of the field of the main and commutating poles at an excitation current \(i_{\mathrm{v}} = 0.6a\), that is, with an unsaturated main-pole shoe.

Curve 3 is the combined average induction of the field of the main and commutating poles at \(i_{\mathrm{v}} = 1.4a\), corresponding to saturation of the pole shoe.

From a comparison of these curves it is evident that the leakage field of the main poles has a significant influence on the distribution of the average induction in the commutation zone. Thus, for example, even with an unsaturated magnetic circuit along the longitudinal axis and an unsaturated main-pole shoe—\(i_{\mathrm{v}} = 0.6a\)—the combined average induction, relative to the induction from the commutating pole, increases at the entering edge of the commutation zone by 28% and decreases at the leaving edge by the same amount. At an excitation current \(i_{\mathrm{v}} = 1.4a\), this change in induction at the leaving and entering edges of the commutation zone already reaches 50%.

In general, the following conclusions may be drawn from the work:

1. A method has been developed for taking slot skew into account when calculating the rotational emf of a section in the commutation zone due to external fields.

2. Slot skew has a significant influence on the distribution of the average induction in the commutation zone, changing it in magnitude and form. Slot skew has the greatest influence on the distribution of the average induction from the main-pole field, since beyond the commutating pole the leakage field of the main poles increases sharply.

3. With slot skew, the influence of the main-pole field on the resultant is intensified. For the excitation currents occurring in the present case, the average induction at the edges of the commutation zone is already 30–50% of the average induction of the commutating pole.

4. The possibility has been shown of calculating the armature-reaction field in the commutation zone using the procedure of [2]. The results of the calculation performed and of the experiment showed fairly good agreement—the discrepancy did not exceed 5%.

REFERENCES

1. A. I. Skorospeshkin, E. G. Chebotkov. Calculation of the main-pole field in the commutation zone of DC machines of the P series. Izvestiya TPI, vol. 190, 1968.

2. E. G. Chebotkov, A. I. Skorospeshkin. Calculation of the field of rectangular commutating poles of DC machines. Izvestiya TPI, vol. 212 (in press).

3. E. G. Chebotkov, A. I. Skorospeshkin. Calculation of the main-pole field in the commutation zone with allowance for saturation of the magnetic circuit. Izvestiya TPI, vol. 212 (in press).

4. I. N. Rabinovich, I. G. Shubov. Design of DC Electrical Machines. Energiya, 1967.