On the Selection of the Rotor Winding for a Pulsed Generator
Abstract Generated abstract
This paper examines the selection of rotor winding parameters for pulsed, or shock, generators where low electromotive-force pulsations and controlled harmonic content are required, especially for accelerator supply applications. Using theoretical harmonic analysis and experimental measurements on model generators with varied rotor slot numbers, filled-slot ratios, air gaps, and slot wedges, the authors evaluate how rotor tooth divisions, winding distribution, damper-slot geometry, and magnetic or composite wedges affect field pulsations, higher harmonics, and excitation-winding reactance. The study concludes that, for most practical shock generators, the rotor slot number should differ from the stator slot number by four tooth divisions, the filled rotor slots should be close to two thirds of the total, and composite wedges are useful for reducing higher and tooth harmonics in the stator phase e.m.f.
Full Text
ON THE SELECTION OF THE ROTOR WINDING FOR A SHOCK GENERATOR
G. A. SIPAILOV, K. A. KHORKOV.
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
The rotors of shock- and breaking-power generators, like the rotors of turbogenerators of normal design, are usually made as solid forgings with non-salient poles. The number of tooth divisions of the rotor, as well as the number of slots filled by the field winding, and the distribution of the winding among the slots depend on the purpose of the generator and on its connection scheme. For example, in the Kapitsa–Kostenko shock generator, intended for obtaining superstrong magnetic fields with a constant value of the maximum field magnitude over a short interval of time, the field winding consists of two parts. One of the parts of the winding has a pitch practically equal to the pole pitch, and the second has half the pole pitch. Each part of the winding has independent supply and regulation [1]. The rotor of the single-phase generator TO-12-2, produced by the Elektrosila plant also for the creation of superstrong magnetic fields, is constructed similarly [2].
Common to the rotors of shock- and breaking-power generators is a powerful damper system, which in some generators occupies up to 40% of the area of the rotor slot [3]. In this case the damper winding is distributed uniformly over the entire circumference of the rotor; in connection with this, in the zone of the large teeth of the rotor, special slots are milled for the damper winding.
The slots on the large tooth for the damper winding, like the main slots of the rotor, affect the magnitude of the pulsations of the magnetic flux. This influence in shock generators is considerably stronger than in turbogenerators of normal design, since the air gap in shock- and breaking-power generators is reduced by approximately a factor of two. Therefore, in those cases where the operating conditions of the generator impose the requirement that the pulsations of the e.m.f. not exceed a prescribed value, there arises the need for a detailed study of the influence of the rotor tooth structure on the form of the stator e.m.f. curve. In particular, such a requirement is of primary importance for shock-power generators intended for operation in supply circuits of charged-particle accelerators. In this case, in the rising part of the field or current of the generator that creates this field there must be no dips caused by higher, including tooth, harmonics. In particular ...
this applies to the initial portion of the current curve from zero to a value determined by the injection level of the accelerator.
In the present work, on the basis of theoretical and experimental investigations, a choice is made of the number of rotor slot divisions, the number of rotor slots filled by the field winding, and the number of turns of the field winding; the influence of the ratio \(\gamma = z_2/z_2'\) and of magnetic wedges in the rotor slots on the magnitude of the higher harmonics of the e.m.f. and on the magnitude of the supertransient inductive reactance is also considered.
Selection of the Number of Rotor Tooth Divisions
In turbogenerators of normal design, the solid rotor body provides a sufficient damping action, and therefore turbogenerators are usually made without a special damper winding or, in some cases, with a very weak damper winding. For this reason, all known methods for calculating damper systems for synchronous machines apply to machines with salient poles.
Depending on the operating mode and conditions of a synchronous machine, different requirements are imposed on the damper winding. In single-phase generators, for example, the principal task of the damper system consists in damping the reverse field. In this case the damper winding must be designed so that, as far as possible, it has minimum active resistance, minimum losses, good heat removal, etc.
In order to reduce additional losses in the damper system from currents of tooth frequency, it is recommended that the pitch of the damper winding \(t_2\) be chosen as close as possible to the stator tooth pitch \(t_1\) [4]:
\[ t_2 = (0.9 \div 1.2)t_1; \qquad n_c = \frac{b_p - t_2}{t_2} = \text{an integer}, \]
where \(n_c\) is the number of damper bars per pole,
\(b_p\) is the width of the pole arc.
In some works, when choosing the pitch of the damper winding, the condition is used of preventing the appearance of large amplitudes of tooth harmonics not only of the first, but also of higher orders, in order to obtain a high purity of the e.m.f. wave [5].
\[ t_2 = \frac{q_1}{q_1 \pm 1} t_1, \tag{1} \]
where \(q_1\) is the number of filled slots per pole and phase. In this case, in order to accommodate a sufficient number of damper bars per pole, it is recommended that the pitch of the damper winding always be taken smaller than the stator tooth pitch.
For single-phase machines, when the number of filled stator slots is equal to \(2/3\) of the total number of slots, \(q_1 = \frac{1}{3} z_1'\); then expression (1) will have the form
\[ z_2' = \left(1 - \frac{2\delta}{D}\right)(z_1' \pm 3). \]
Taking, for pulsed-power generators, the ratio of the gap \(\delta\) to the stator bore diameter \(D\) to be on average equal to 0.02, we obtain
\[ z_2' = 0.96(z_1' \pm 3). \tag{2} \]
Table 1 gives calculated values of the numbers of rotor tooth divisions \(z'_2\) for a given series of numbers of stator tooth divisions \(z'_1\). The calculation was made according to expression (2), with rounding to the nearest even number; here the “plus” sign in expression (2) corresponds to the second row of Table 1, and the “minus” sign to the third. The greatest deviation of the number of rotor tooth divisions in Table 1 from the values calculated by (2) is one tooth division.
Table 1
| 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| \(z'_1\) | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | |
| 1 | \(z'_2\) | 26 | 32 | 38 | 44 | 50 | 54 | 60 | 66 | 72 |
| 2 | \(z'_2\) | 20 | 26 | 32 | 38 | 44 | 50 | 54 | 60 | 66 |
Thus, taking rounding into account, condition (2) for determining the number of rotor tooth divisions of a shock-power generator may be represented in the form
\[ z'_2 = z'_1 + (2 \div 0)\quad \text{and}\quad z'_2 = z'_1 - (4 \div 6), \tag{2a} \]
where the first figures in parentheses in expression (2a) refer to the numbers of stator tooth divisions in the range \(z'_1 = 24 \div 54\), and the second to \(z'_1 = 54 \div 72\).
If one adheres to the ratio \(t_2 < t_1\) recommended for machines with salient poles, then for a shock-power generator one should take
\[ z'_2 = z'_1 + 2. \tag{2b} \]
However, with such a ratio of the numbers of stator and rotor slots there will be one common tooth harmonic from the fluxes \(\Phi_{z1}\) and \(\Phi_{z2}\), with frequency
\[ \nu = \frac{z'_1 + z'_2}{2}. \]
The amplitude of the common tooth flux will be equal to
\[ \Phi_z = \sqrt{\Phi_{z1}^{2} + \Phi_{z2}^{2} + 2\Phi_{z1}\Phi_{z2}\cos(\alpha - \beta)}, \]
where \(\alpha\) and \(\beta\) are the phase-shift angles of the fluxes relative to the coordinate origin.
When the shock generator is connected to the load at the instant when the emf passes through zero, \(\alpha - \beta = 0\), and then
\[ \Phi_z = \Phi_{z1} + \Phi_{z2}. \]
The tooth fluxes, adding algebraically, sharply worsen the shape of the coil emf curve (Fig. 1a) in comparison with the case when \(\Phi_{z1}\) and \(\Phi_{z2}\) have no common harmonic (Fig. 1b).
The theoretical and experimental investigations carried out in this connection to determine the optimal ratio of the numbers of stator and rotor slots showed that expression (2), obtained for salient-pole machines, requires some refinement when applied to machines with nonsalient poles.
It is known that increasing the number of rotor slots of a turbogenerator brings the magnetomotive-force curve of the field winding closer to a sinusoid and facilitates rotor cooling conditions. However, in turbogenerators, because of the condition of mechanical strength of the section at the base of the rotor tooth, the number of rotor slots is usually chosen somewhat smaller,
than the number of stator slots. But since shock generators, as a rule, have fewer slots on the stator than turbogenerators of the same dimensions, their rotor, without violating the condition of the necessary mechanical strength, may have more slots than the stator.
a
b
Fig. 1a. Curve of the emf of a stator phase:
a) the rotor and stator have a common tooth harmonic: \(z'_1 = 24,\ z'_2 = 26,\ \delta = 0.5\ \mathrm{mm},\ \gamma = 0.69\);
b) the rotor and stator do not have a common harmonic: \(z'_1 = 24,\ z'_2 = 20,\ \delta = 0.5\ \mathrm{mm},\ \gamma = 0.7\).
Studies have shown that instead of relation (2б), for shock generators it is expedient to adopt:
\[ z'_2 = z'_1 + 4. \tag{3} \]
If, for \(z'_2 = z'_1 + 2\), the stator tooth harmonics of frequency
\[ \nu = \frac{1}{2}(z'_1 + z'_2) \]
will have the same winding distribution factor as the fundamental harmonic, then for \(z'_2 = z'_1 + 4\) the distribution factor for the rotor tooth harmonics will be close to zero; that is, in the latter case the harmonics of the rotor tooth frequency will not exert a noticeable influence on the emf of the generator stator.
On the other hand, taking into account that the stator slots of a shock generator are usually made wider and that their number is correspondingly smaller than in turbogenerators, the relation \(z'_1 - z'_2 > 6\)
can be retained only for individual machines with special requirements regarding the shape of the current curve, and for cases when the number of slots on the stator is \(z'_1 > 54\).
Thus, the ratio between the numbers of slots on the stator and rotor of a shock generator that is acceptable for most practical cases will be
\[ z'_2 = z'_1 \pm 4. \tag{4} \]
Influence of the Depth of the Rotor Slots on the Field Pattern in the Air Gap
The slots of the damper winding on the large tooth of the rotor have a substantial influence on the field pattern in the air gap. In shock generators, the depth of the slots for the damper winding, at equal width, reaches 40% of the depth of the slots filled by the field winding. If it is assumed that the slots on the large tooth, like the main rotor slots, affect the magnitude of the pulsations of the magnetic field in the air gap to the same degree, then the calculation of the permeances of the rotor tooth layer is considerably simplified. In this case, when calculating the field pattern in the air gap of synchronous machines with nonsalient poles, it will be possible to apply the method developed in detail and investigated with reference to asynchronous machines [6].
To clarify the influence of the slots of the large rotor tooth on the field pattern in the air gap, a large number of experiments were carried out. In one of them, the emf of a full-pitch coil located on the stator of the generator under investigation was oscillographed; the generator had \(z_1 = 24\), \(z_2 = 20\), \(D = 104\) mm, \(\delta = 0.5\) mm. In the first case, the solid rotor with nonsalient poles had open slots on the large tooth, their depth being one third of the depth of the main slots (Fig. 2,a). In the second case, all the slots of the large tooth of the same rotor were filled throughout their entire volume with magnetic wedges (Fig. 2,b).
Fig. 2. Curve of the emf of a full-pitch coil.
a — with open slots on the large tooth of the rotor;
As follows from Fig. 2,a, in the presence of slots on the large tooth of the rotor whose depth is less than the depth of the slots occupied by the field winding, the emf pulsations over the entire pole arc are constant. Consequently, all the rotor slots of a shock generator, in terms of their influence on the magnitude of the pulsations of the magnetic field in the air gap, may be regarded as homogeneous. In the oscillogram (Fig. 2,b),
some excursions in the e.m.f. curve in the zone of the large tooth indicate an insufficiently tight fitting of the magnetic wedges.
In another experiment, by means of a frequency-spectrum analyzer, the content of higher harmonics in the e.m.f. curve of a coil with full pitch, located on the stator of the generator under investigation, was measured. In the first case all 22 slots of the solid rotor had a depth of 4.5 mm with an air gap \(\delta = 4\) mm. The results of measuring the e.m.f. of the higher harmonics for this slot depth, expressed as percentages of the e.m.f. of the first harmonic, are given in the row for \(E_1\) in Table 2. In the second case the depth of all rotor slots was doubled. The percentage content of the e.m.f. of the higher harmonics for a slot depth equal to 9 mm is presented in the row for \(E_2\) in Table 2.
Fig. 2. \(\sigma\)—with closed slots on the large rotor tooth,
Table 2
| \(\nu\) | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(E_1\) | 21.9 | 10.6 | 4.6 | 1.93 | 0.60 | 0.0 | 0.28 | 0.35 | 0.26 | 0.30 | 4.37 | 0.58 |
| \(E_2\) | 19.8 | 9.7 | 4.5 | 1.96 | 0.66 | 0.0 | 0.32 | 0.32 | 0.32 | 0.32 | 4.45 | 0.66 |
As follows from Table 2, when the slot depth is doubled, the content of higher-harmonic e.m.f. in the stator coil practically does not change.
Selection of the Number of Filled Rotor Slots
When selecting the ratio of the wound part of the rotor to its full circumference, \(\gamma = z_y/z_2\), for a shock generator one should proceed from the conditions of obtaining minimum pulsations of the excitation field and minimum inductive resistance of the excitation winding.
The magnetomotive force (m.m.f.) of the excitation winding of a nonsalient-pole rotor has a trapezoidal form. Expanding the m.m.f. curve into a Fourier series, one can obtain an expression for any \(\nu\)-th harmonic m.m.f.; in this case its amplitude will be equal to
\[ F_{b\nu}=\frac{8}{\pi^2}\frac{F_b}{\gamma}\frac{1}{\nu^2}\sin \nu\frac{\pi}{2}\cdot \sin \gamma\nu\frac{\pi}{2}. \tag{5} \]
Expression (5) can be represented in a somewhat different form:
\[ F_{b\nu}=\frac{8}{\pi^2}\frac{F_b}{\nu^2}\, \frac{\sin \gamma \nu \dfrac{\pi}{2}} {\gamma \sin \nu \dfrac{\pi}{2}} = \frac{8}{\pi^2\nu^2}k_{p\nu}F_b, \tag{6} \]
\[ k_{p\nu}= \frac{\sin \gamma \nu \dfrac{\pi}{2}} {\gamma \sin \nu \dfrac{\pi}{2}}, \tag{7} \]
where \(k_{p\nu}\) is the distribution factor of the rotor winding for the \(\nu\)-th harmonic.
The introduction of the distribution factor for rotor windings facilitates the problem of investigating the rotor field. In its physical meaning this factor is analogous to the distribution factor of the stator winding: the value of the factor \(k_{p\nu}\) depends on \(\gamma\) in the same way as the value of the stator-winding distribution factor depends on \(q_1\). For the rotor winding it is possible to choose such a value of \(\gamma\) for which \(k_{p\nu}=0\), i.e., by choosing the value of \(\gamma\) one can achieve the destruction of any specified harmonic in the field curve. The condition \(\Phi_\nu=0\), or correspondingly \(E_\nu=0\), will be satisfied for
\[ \gamma=\frac{2n}{\nu}, \]
where \(n=1,2,3\ldots\). For example, at \(\gamma=2/3\) the field curve will contain no third harmonic and no harmonics that are multiples of three [7].
An experimental investigation of the rotor-field curve was carried out on a setup in which the tested generator, with the number of stator slots \(z'_1=24\) and an air-gap value \(\delta=0.5\) mm, had three replaceable salient-pole rotors with the number of tooth divisions \(z'_2=20, 26\), and 28. To obtain the same m.m.f. with different numbers of filled rotor slots, the excitation current was regulated accordingly.
Table 3 presents the experimental results in the form of the sum of the higher harmonic e.m.f.’s of a full-pitch coil, expressed as a percentage of the e.m.f. of the fundamental harmonic. As follows from Table 3, the sum of the higher harmonic e.m.f.’s of the stator coil from the third harmonic up to the level determined by the lower limit of slot frequency,
\[ \sum_{\nu=3}^{15} E_\nu, \]
for all numbers of rotor teeth has the same form
Table 3
| \(z'_2=20\) | \(z'_2=20\) | \(z'_2=20\) | \(z'_2=26\) | \(z'_2=26\) | \(z'_2=26\) | \(z'_2=26\) | \(z'_2=26\) | \(z'_2=26\) | \(z'_2=28\) | \(z'_2=28\) | \(z'_2=28\) | \(z'_2=28\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\gamma\) | 0.3 | 0.5 | 0.7 | 0.23 | 0.38 | 0.54 | 0.69 | 0.95 | 1.0 | 0.28 | 0.43 | 0.57 | 0.71 |
| \(\displaystyle \sum_{\nu=3}^{15} E_\nu\) | 29.4 | 13.1 | 5.6 | 34.2 | 21.6 | 9.8 | 5.1 | 7.7 | 9.8 | 30 | 18.4 | 9.3 | 5.1 |
| \(\displaystyle \sum_{\nu=3}^{31} E_\nu\) | 56.4 | 51.4 | 48.6 | 57.5 | 53.5 | 52.0 | 51.5 | 51.5 | 52.5 | 59 | 53.5 | 52 | 51 |
and reaches a minimum value at \(\gamma = 0.67\). The sum of the higher harmonic e.m.f.’s of a coil from the third harmonic to the level determined by the upper limit of the slot frequency,
\[ \sum_{\nu=3}^{31} E_\nu, \]
for different numbers of rotor teeth has a less pronounced dependence on \(\gamma\); moreover, the values of \(\gamma\) at which this sum has a minimum lie in the range from 0.4 to 0.8.
For the higher harmonic e.m.f.’s of the stator phase, the dependence on the number of filled rotor slots is different from that for individual coils, since in this case the higher harmonics of the coils composing the phase are added geometrically.
Table 4
| \(z'_2 = 20\) | \(z'_2 = 20\) | \(z'_2 = 20\) | \(z'_2 = 26\) | \(z'_2 = 26\) | \(z'_2 = 26\) | \(z'_2 = 26\) | \(z'_2 = 26\) | \(z'_2 = 26\) | \(z'_2 = 28\) | \(z'_2 = 28\) | \(z'_2 = 28\) | \(z'_2 = 28\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\gamma\) | 0,3 | 0,5 | 0,7 | 0,23 | 0,38 | 0,54 | 0,69 | 0,85 | 1,0 | 0,28 | 0,43 | 0,57 | 0,71 |
| \(\displaystyle \sum_{\nu=3}^{15} E_\nu\) | 2,0 | 0,9 | 0,9 | 2,2 | 0,7 | 1,0 | 1,0 | 0,8 | 1,2 | 1,8 | 0,8 | 1,2 | 0,8 |
| \(\displaystyle \sum_{\nu=3}^{31} E_\nu\) | 9,6 | 8,2 | 7,8 | 42,8 | 41,6 | 41,5 | 41,2 | 41,2 | 41,3 | 14,2 | 10,1 | 7,6 | 6,9 |
Table 4 presents the experimental data in the form of the sum of the higher harmonic e.m.f.’s of the phase, expressed as a percentage of the e.m.f. of the fundamental harmonic, as a function of the number of filled rotor slots. It follows from Table 4 that the content of higher harmonics in the stator-phase e.m.f. curve depends only weakly on the number of filled rotor slots, but depends strongly on the ratio of the numbers of stator and rotor slots. In the case where the stator and rotor slots have a common harmonic, that is, when \(z'_2 = z'_1 \pm 2\), the content of higher harmonics in the stator-phase e.m.f. curve reaches a maximum. Thus, experiment confirms that the number of rotor slots of a shock-power generator should be selected according to expression (4), and not according to (1).
Although the value of \(\gamma\) has practically no effect on the magnitude of the pulsations of the stator-phase e.m.f., it nevertheless makes sense to find the optimum value of \(\gamma\) from the condition of obtaining minimum pulsation losses from the higher field harmonics in the air gap of the machine. As follows from Table 3, such a value of \(\gamma\) will be about 0.7. For example, at \(z'_2 = 28\) and \(\delta = 0.5\) mm, the magnitude of the pulsation losses due to the higher harmonics at \(\gamma = 0.71\) will be 3.4 times smaller than at \(\gamma = 0.57\), and 34 times smaller than at \(\gamma = 0.28\). A similar picture is obtained for other values of \(z'_2\).
The optimum value of \(\gamma\), from the condition of obtaining a minimum value of the subtransient reactance, depends on the relative area of the rotor slots \(s\) [8]. For the relative area
\[ s = \frac{z'_2 h_n b_n}{\frac{\pi}{4} D_p^2}, \]
varying within the limits \(s = 0.18\text{–}0.30\), with the relative slot depth \(\beta = h_n/D_p = 0.14\), the optimum values of \(\gamma\) are respectively equal to \(0.7\text{–}0.6\).
The choice of the number of turns of the field winding for selected values of $\gamma$ and of the number of rotor slots $z'_2$ is made in the same way as for turbine generators of normal design. The active and reactive resistances of the rotor winding, referred to the stator winding, depend neither on the number of turns of the rotor winding nor on the current density in it, but only on the volume of copper in the slot. In this connection, the dimensions of the copper of the field winding are determined by technological considerations, and the number of turns by an acceptable value of the field current.
Influence of magnetic wedges in the rotor slots on the field pattern in the air gap
The influence of magnetic wedges in the rotor slots, in particular in the slots of the large rotor tooth, on the magnitude of the pulsations of the magnetic flux in the air gap was investigated for two rotors with $z'_2 = 20$ and 22.
Table 5 presents the results of measuring the higher harmonics of the e.m.f. of a coil and of a stator phase of a generator whose rotor had 20 tooth divisions at $\gamma'_2 = 0.7$; in one case the slots of the large rotor tooth were open, and in the other they were filled with magnetic wedges throughout their entire volume.
Table 5
| $\displaystyle \sum_{\nu=3}^{15} E_\nu$ | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | $\displaystyle \sum_{\nu=3}^{31} E_\nu$ | Note | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Coil | 5,70 | 2,40 | 37,5 | 30,0 | 2,10 | 1,30 | 1,10 | 0,50 | 0,45 | 48,5 | without wedges |
| Coil | 15,80 | 8,25 | 15,4 | 8,0 | 7,15 | 3,30 | 0,74 | 2,05 | 1,98 | 26,3 | with wedges |
| Phase | 0,92 | 0,30 | 7,4 | 0,3 | 1,73 | 0,81 | 0,00 | 0,10 | 0,10 | 7,7 | without wedges |
| Phase | 0,83 | 1,00 | 3,0 | 0,3 | 6,40 | 3,30 | 0,00 | 0,20 | 0,10 | 7,9 | with wedges |
Table 6
| $\displaystyle \sum_{\nu=3}^{15} E_\nu$ | 17 | 19 | 21 | 23 | 25 | 27 | 29 | $\displaystyle \sum_{\nu=3}^{29} E_\nu$ | Note | |
|---|---|---|---|---|---|---|---|---|---|---|
| Coil | 29,5 | 1,18 | 1,58 | 2,77 | 41,0 | 6,86 | 6,34 | 5,17 | 51,6 | $\alpha = 0$ |
| Phase | 2,04 | 0,61 | 1,20 | 1,30 | 38,0 | 6,41 | 1,36 | 1,10 | 38,5 | $\alpha = 0$ |
| Coil | 17,7 | 2,77 | 7,16 | 11,8 | 17,2 | 6,80 | 6,65 | 4,00 | 30,2 | $\alpha = 0{,}35$ |
| Phase | 2,34 | 0,65 | 1,48 | 1,50 | 16,0 | 6,60 | 1,20 | 1,00 | 17,6 | $\alpha = 0{,}35$ |
| Coil | 15,2 | 4,92 | 2,10 | 11,6 | 11,5 | 6,90 | 3,50 | 5,50 | 24,8 | $\alpha = 0{,}52$ |
| Phase | 1,21 | 0,80 | 0,75 | 1,40 | 11,1 | 6,90 | 1,00 | 1,40 | 13,4 | $\alpha = 0{,}52$ |
| Coil | 38,3 | 6,30 | 5,30 | 2,60 | 0,70 | 2,94 | 1,17 | 0,97 | 39,5 | $\alpha = 0{,}87$ |
| Phase | 2,40 | 0,82 | 0,88 | 0,00 | 0,50 | 2,94 | 0,40 | 0,43 | 4,05 | $\alpha = 0{,}87$ |
As follows from the results of the experimental investigation, magnetic wedges in the slots of the large rotor teeth reduce the magnitude of the pulsations of the magnetic field in the air gap by approximately a factor of two (Table 5), which substantially reduces the surface and pulsation losses in the stator teeth. At the same time, however, the content of tooth harmonics of the e.m.f. in the stator phase remains practically constant, since, when the e.m.f. from the rotor tooth fluxes decreases, the e.m.f. from the stator tooth fluxes increases by approximately the same amount.
Table 6 presents the results of measuring the same quantities as in Table 5, but for a rotor having \(z'_2 = 22\), with an excitation winding placed in two large diametrical slots. In this case, the symmetrically arranged rotor slots were successively filled with magnetic wedges; the e.m.f. measurements were carried out for various values of \(\alpha\), equal to the ratio of the arc of the circumference having closed slots to the full circumference of the rotor.
The character of the variation of the third, fifth, and other harmonics of the e.m.f. is seen from Fig. 3, while the character of the variation of the tooth harmonics of the e.m.f. down to the level determined by the lower limit of the sub-tooth frequency as a function of \(\alpha\) is seen from Table 6.
Fig. 3. Character of the variation of the 3rd and 5th harmonics of the coil e.m.f. as a function of \(\alpha\).
As follows from Table 6, closing all the slots with magnetic wedges leads to a sharp reduction in the content of higher harmonics in the e.m.f. curve of the stator phase, and to a slight reduction of the higher harmonics in the field curve. The latter is explained by the fact that, on the one hand, the higher harmonics associated with the toothed structure of the rotor decrease, while, on the other hand, the higher harmonics from the third to the fifteenth, associated with the distribution of the rotor winding among the slots, increase because of the increased permeance of the air gap.
In addition, for a rotor with \(z'_2 = 20\), at \(\gamma = 0.7\) and \(\delta = 1.5\) mm, measurements were made of the higher harmonics of the coil e.m.f. and of the stator phase; in the first case all rotor slots were open, while in the second case all slots were filled with composite copper-steel wedges. Each of the component parts of a wedge occupied half the width of the slot. The character of the variation of the higher harmonics is presented in Table 7, from which it follows that the tooth harmonics in the rotor field curve are thereby reduced by a factor of 2–3, and in the e.m.f. curve of the stator phase by a factor of 1.5–2.
Table 7
| $\displaystyle \sum_{\nu=3}^{15} E_\nu$ | 17 | 19 | 21 | 23 | 25 | 27 | $\displaystyle \sum_{\nu=3}^{27} E_\nu$ | Note | |
|---|---|---|---|---|---|---|---|---|---|
| Coil | 3.60 | 1.00 | 20.0 | 15.5 | 0.80 | 0.40 | 0.25 | 25.5 | without wedges, |
| Phase | 0.83 | 0.10 | 3.95 | 0.10 | 0.10 | 0.10 | 0.00 | 4.05 | without wedges, |
| Coil | 3.95 | 0.40 | 10.5 | 5.10 | 0.50 | 0.35 | 0.25 | 12.5 | with wedges. |
| Phase | 0.61 | 0.05 | 2.50 | 0.05 | 0.15 | 0.10 | 0.00 | 2.55 | with wedges. |
Taking into account the fact that even with comparatively small gaps (in the case under consideration $\delta/D = 0.014$) the tooth harmonics considerably exceed the sum of all the other harmonics, such a reduction of the tooth harmonics may be considered acceptable, since it does not entail a substantial increase in the leakage reactance.
Conclusions
-
For shock generators it is advisable to take the number of rotor slots as differing from the number of stator slots by four tooth divisions.
-
The number of filled rotor slots, from the conditions of minimum flux pulsations in the gap and minimum inductive reactance of the field winding, should be taken as close to $2/3$ of the total number of rotor slots.
-
In order to reduce the content of higher harmonics, including tooth harmonics, in the e.m.f. curve of the stator phase, composite wedges should be used to close the rotor slots.
References
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