ON THE CHOICE OF THE STATOR WINDING OF A SHOCK GENERATOR

G. A. Sipailov, K. A. Khorkov

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

The stator windings of breaking-capacity and shock-capacity generators have a number of features that distinguish them from the windings of normal turbogenerators. These features are caused by the requirements of obtaining the maximum short-circuit current and a limited rate of its decay. To meet these requirements, it is necessary that breaking-capacity and shock-capacity generators have, as far as possible, a minimum number of turns on the stator and a maximum flux; that the leakage flux of the end portions be minimal; and, in addition, it is necessary to place the stator and rotor windings closer to the air gap and to provide an reinforced damper system [1, 2]. Therefore, designing the stator windings of breaking-capacity and shock-capacity generators is a complex and responsible task.

At present, breaking-capacity generators have found wide application in breaking-capacity laboratories in many countries of the world; therefore the issues of their design have been developed considerably more fully than for shock-capacity generators. This, in particular, applies to the choice of the type of stator winding. If, for breaking-capacity generators, which are three-phase machines, the choice of winding type is not a new problem, then for shock-capacity generators, which are predominantly single-phase machines, this problem requires further study and solution. In this case the problem is complicated by the variety of requirements imposed on the shape of the current curve of the shock generator, depending on its field of application.

In the present work, results are given of a theoretical and experimental investigation, carried out with the aid of the Minsk-1 calculating machine, on the choice of the type of stator winding and the number of stator slots completely and partially filled with winding for a shock generator intended to obtain magnetic fields with high energy, in particular for operation in power-supply circuits of charged-particle accelerators.

Choice of the Type of Stator Winding

Depending on the requirements imposed on the shape of the curve and the magnitude of the e.m.f. of the generator, as well as on the conditions for rational utiliza-

use of active materials, etc., one or another type of stator winding is selected. In principle, for a shock-power generator, both single-layer single-phase windings and three-phase double-layer windings with full or shortened pitch may be used; in operation, only two phases are used, while the third phase remains as a reserve. Eliminating the reserve phase, although it saves copper, is inexpedient, since it leads to considerable complications in fastening the end portions of the generator.

Single-layer windings, being more economical in the consumption of copper and insulation because of the absence of a reserve phase and interlayer insulation, become especially advantageous in those cases when, owing to the permissible voltage at the generator terminals, it is necessary to place only one effective conductor in each slot.

Since three-phase double-layer windings have been studied quite thoroughly, the present investigations will relate mainly to single-layer single-phase windings.

Many questions concerning the design and application of single-phase single-layer windings of electrical machines have until now given rise to disagreement. For example, different authors interpret differently the question of higher harmonics in the e.m.f. curve of a generator having a single-phase single-layer winding. Some unconditionally maintain that shortening the winding affects the magnitude of the higher harmonics to the same extent for both double-layer and single-layer windings, and on this basis draw the conclusion that the higher harmonics decrease as the winding pitch is shortened and that the operating properties of a single-phase single-layer winding are improved [3]. Others believe that reducing the pitch of a single-layer winding can lead only to an increase in the content of higher harmonics in the e.m.f. curve, and that therefore the rules existing for double-layer windings cannot be mechanically transferred to single-layer windings; in their opinion, shortening the pitch of single-layer windings is inexpedient [4].

We investigated the influence of shortening the pitch of a single-phase single-layer winding on the shape of the e.m.f. curve for windings of two types: involute, with equal section width, and concentrically subdivided. The experiment was carried out on the installation described earlier [5],

Table 1

In this case the generator under study had \(D = 104\) mm, \(Z'_1 = 24\), \(Z'_2 = 26\), \(\gamma = 0.7\), \(\delta = 4\) mm.

As a result of the experiment, the percentage content of each of the higher harmonics in the emf curve was determined,

\[ \frac{E_\nu}{E_1}\cdot 100\%, \]

for various shortenings of the winding pitch. The experimental data are presented in Table 1. Since for the 9th, 11th, 13th, 15th, and 17th harmonics the values \(E_\lambda = 0\), they are omitted from Table 1. It follows from Table 1 that most of the higher harmonics, namely all harmonics not multiples of 3, vary according to the same laws as for two-layer windings, both for the concentric and for the involute winding.

Figure 1 presents curves constructed from calculated data for the 5th and 7th harmonics of the emf; for comparison, points obtained as a result of the experiment are also plotted there:

Fig. 1. Variation of the magnitudes of the 5th and 7th harmonics of the emf as a function of the shortening of the winding pitch.

\[ \times \text{— for the equisection winding,} \]

\[ \circ \text{— for the concentric winding.} \]

It should be noted that pitch shortening has almost no effect on the tooth harmonics, which is also the case for a two-layer winding. Only harmonics that are multiples of three behave differently: for the involute winding they remain constant and practically equal to zero, while for the subdivided concentric winding they increase as the section shortening is increased (Fig. 2).

Figure 3 presents curves of the dependence of the distortion coefficient of the emf curve on the shortening of the winding pitch. As follows from the experiment, the distortion coefficient for a single-layer involute winding is determined mainly by the tooth harmonics and remains practically constant as the winding pitch changes. Shortening the pitch of a concentric winding leads to a sharp increase in the harmonics that are multiples of three, as a result of which the distortion of the emf curve increases; that is, with pitch shortening the overall shape of the emf curve does not improve, although individual harmonics do decrease. In addition, pitch shortening

Fig. 2. Variation of the magnitude of the 3rd and 27th harmonics of the e.m.f. as a function of the shortening of the winding pitch: 1 — theoretical curve; 2 — curve for an equal-section winding; 3 — curve for a concentric winding.

Fig. 3. Dependence of the distortion coefficient of the e.m.f. curve on the shortening of the winding pitch: 1 — theoretical curve; 2 — curve for an equal-section winding; 3 — curve for a concentric winding.

a single-layer winding is associated with a considerable reduction in the phase voltage, which leads to a decrease in the power and, consequently, in the energy delivered by the generator to the load. It follows from this that shortening the pitch of a single-layer winding is irrational.

From the point of view of obtaining maximum shock power, it is desirable to have the minimum shock resistance of the generator.

Table 2 gives the results of a comparative calculation of generator parameters, carried out within the dimensions of a 100 MW turbogenerator with a stator-bore diameter \(D = 115\) cm and length \(l = 420\) cm. The first row of the table corresponds to a generator having on the stator a single-phase, single-layer, distributed winding with full pitch, with the number of slots filled by the winding equal to \(2/3\) of the total number of slots. The second and third rows give data for two phases of a generator having on the stator a three-phase, double-layer winding with full and shortened pitch, respectively. Since the number of slots per phase of the single-phase single-layer winding is equal to the number of slots for two phases of the three-

Table 2

phase double-layer winding, it becomes correct to compare the external short circuit of the single-phase generator with the two-phase sudden short circuit of the three-phase generator and, correspondingly, the closing of the single-phase generator onto a load with the closing of two phases of the three-phase generator onto a single-phase load. The following notation is adopted in Table 2:

\(y\) — winding pitch in fractions of the pole pitch;
\(w_1\) — number of series-connected turns of a phase;
\(a_1\) — number of parallel branches of the stator winding;
\(k_{w1}\) — winding factor;
\(l_s\) — length of the end portion, taken as unity for the single-phase single-layer winding;
\(r_1\) — active resistance of a phase;
\(x_s, x''_d, x_2, x_{\mathrm{ud}}\) — inductive reactances, respectively, leakage, subtransient, negative-sequence, and shock; all are taken as unity for the single-phase single-layer winding;
\(G_{\mathrm{m}}, G_{\mathrm{iz}}\) — weight of the copper and insulation of the winding;
\(Q_{\mathrm{n}}\) — energy transferred by the generator to the load;
\(x_{\mathrm{n}}\) — load resistance in fractions of \(x_{\mathrm{ud}}\).

In calculating the different variants, only the stator winding was varied; in all other respects the generator remained the same. It follows from Table 2 that, both in terms of the amount of energy delivered to the load and in terms of copper and insulation consumption, the generator with a single-phase single-layer winding (row 1) has advantages over the generator with a double-layer winding with full pitch (row 2). This is explained by the fact that the single-layer winding has shorter end portions and less leakage in the end-winding region, as a result of which there is reduced-

the leakage reactance, and consequently the reactance \(x_{\mathrm{ud}}\), is reduced. In terms of the energy delivered to the load, the single-phase single-layer winding also has advantages over the double-layer winding with shortened pitch (line 3), which is explained by the decrease in voltage due to the shortening of the pitch of the double-layer winding.

Choice of the Number of Filled Slots

In ordinary single-phase synchronous machines, the number of stator slots filled by the winding is taken equal to \(2/3\) of the total number of slots. This is dictated by the desire to eliminate the third harmonic from the e.m.f. curve, as well as by economic considerations. Table 3 gives calculated and experimental data for the magnitudes of the higher harmonics as a function of the ratio

\[ \frac{q_1}{Q}=\frac{z_1}{z'_1}, \]

for \(z'_2=20,\ Q=12,\ \gamma=0.7,\ \delta=0.75\ \text{mm}\). The minimum distortion of the stator e.m.f. is obtained at

\[ \frac{q_1}{Q}=2/3 \]

(Fig. 4). This is due to the fact that, when deviating

Fig. 4. Dependence of the distortion coefficient of the e.m.f. curve on the ratio \(\dfrac{q_1}{Q}\): 1 — theoretical curve; 2 — experimental curve.

Table 3

At $\dfrac{q_1}{Q}$ of $2/3$ the third harmonics increase sharply, while all the others are at approximately the same level. An energy consideration of this question also speaks in favor of the ratio $\dfrac{q_1}{Q}=2/3$ (Table 4). Table 4 gives the results of calculating three variants:

1) $\dfrac{q_1}{Q}=5/6$, 2) $\dfrac{q_1}{Q}=2/3$ and $\dfrac{q_1}{Q}=1/2$ for $z'_1=24$.

The variant with a decrease in $\dfrac{q_1}{Q}$ is eliminated, since in this case the energy delivered by the generator to the load is significantly reduced. The variant with $\dfrac{q_1}{Q}=5/6$ may be accepted on an equal footing with the variant $\dfrac{q_1}{Q}=2/3$, since the increase in copper consumption is compensated by an increase in the generator energy, and at the same time both variants have identical distortions of the emf curve (Fig. 4).

Table 4

The choice of the type of stator winding (single-layer or double-layer) is inseparably connected with the choice of the number of stator slots $z'_1$.

Choice of the Number of Stator Slots

In order to determine the optimal number of stator slots of the shock generator, a number of variants were calculated on the Minsk-1 computer.

The calculation was carried out according to the method set forth in [6], for a number of values of the outer diameter $D_a$, the volume of the active part of the machine, the total number of stator slots $z'_1$, and the ratio of the slot height to its width $\kappa_1$. As the calculations showed, the character of the dependence of the shock power on the number of full stator slots and on the ratio of the slot height to its width is the same for different values of machine volumes and outer diameters $D_a$. Table 5 gives the dependences $P_{\mathrm{ud}}=f(z'_1 \kappa_1)$ for a number of values of $z'_1$ and $\kappa_1$ at a constant value $D^2l=7\ \mathrm{m}^3$ and a constant outer diameter $D_a=2\ \mathrm{m}$; in this case the power of the shock generator at $z'_1=36$ and $\kappa_1=2$ is taken as unity.

It follows from Table 5 that the optimal number of teeth, depending on the geometry of the stator slot, lies within the range $48 \div 60$, and that with an increase in slot height the maximum power is achieved with a larger number of slots.

Table 5

Since the type of stator winding, among other factors, is determined by the number of effective conductors in the slot $S_n$, the choice of the number of teeth can be related to the choice of the type of winding. The number of turns in a phase of the stator winding is equal to

\[ w_1=\frac{pE_1}{4.44\, f k_{w1} D l B_\delta}, \tag{1} \]

where $B_\delta$ is the induction in the air gap, in tesla;
$D$ and $l$ are the diameter of the stator bore and the length, in m.

If for a shock generator we take

\[ \frac{q_1}{Q}=2/3, \]

that is, $q_1=1/3\,z'_1$, or

\[ q_1=\frac{w_1 a_1}{S_n}, \]

and substitute the average value of the winding factor $k_{w1}=0.82$, then equation (1) takes the form

\[ z'_1=\frac{165 E_1 a_1\cdot 10^{-4}}{D l B_\delta S_n}. \tag{2} \]

Since in shock generators the number of parallel branches is usually $a_1=1$, it is easy to see from equation (2) that for machines of limiting dimensions the optimum number of slots is attained at $S_n=1$. This means that for shock generators built within the dimensions of machines of limiting power, it is preferable to choose a single-phase single-layer winding.

REFERENCES

1. I. M. Postnikov. Proceedings of LPI, No. 2, 1937; No. 209, 1960.
2. P. Kharmas. Electrical Machines for Special Purposes. GEI, 1960.
3. Müller. Deutsche Elektr., No. 4, 1956.
4. Kleinrat, Franke. Deutsche Elektr., No. 12, 27–30, 1956.
5. G. A. Sipailov, K. A. Khorkov. On higher harmonics in the e.m.f. curve of a shock generator. Problems in the Theory and Design of Electrical Machines. Interuniversity Collection of Papers. Novosibirsk, 1963.
6. G. A. Sipailov, K. A. Khorkov. On the choice of the principal dimensions of a shock generator. Proceedings of TPI, 1964, vol. 132.