On the Selection of the Geometry of the Stator Tooth Zone of an Impulse Generator
Abstract Generated abstract
This paper examines the choice of stator tooth zone geometry for high-power impulse generators, focusing on the number of slots per pole and phase, the tooth-width to pitch ratio, and the slot-height to slot-width ratio. Using a digital computer calculation based on turbogenerator design methods, the study evaluates variants under constraints on terminal voltage and current density, while accounting for insulation and wedge thickness, copper filling, leakage permeance, and saturation. The results indicate that the optimal tooth-width to pitch ratio generally falls within turbogenerator recommendations, but may increase to about 0.6 to 0.8 under strong excitation forcing, and that the optimal slot-height ratio depends on slot number and permissible current density rather than on maximum energy alone.
Full Text
ON SELECTING THE GEOMETRY OF THE STATOR TOOTH ZONE OF A SHOCK GENERATOR
(Presented by the scientific seminar of the Scientific Research Institute of Nuclear Physics)
G. A. SIPAILOV, K. A. KHORKOV, V. Z. KHORKOVA, V. S. BAKLIN,
V. I. ANDREEV
At present, increasing the power of an individual shock generator is an urgent research problem for electrical-machine designers. Shock generators of limiting power are needed not only for short-circuit laboratories, but also for conducting modern scientific research in the fields of solid-state physics, plasma, accelerator technology, etc.
Modern shock generators are made on the basis of existing turbogenerators. The design problems of turbogenerators have been developed quite fully, and this substantially facilitates the task of creating shock generators. However, a number of questions associated with the individual features of these machines, determined by their operating conditions, require further study and solution. For example, the question of designing the stator tooth zone has not been definitively resolved: clear recommendations for selecting the number of stator slots, the ratio between slot width and tooth pitch, and the ratio between slot height and its width are lacking.
The available recommendations [1, 3, 4] amount essentially to the following:
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The number of stator turns should be as small as possible, and the magnetic flux as large as possible.
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The depth of the slots should be as small as possible, and their width as large as possible.
At present, it is customary to reduce the height of the stator slots of a shock generator by a factor of 1.7–2.5 [2], and to take the ratio of slot height to its width equal to 2 (as compared with 10 for turbogenerators), [3, 4]. For example, such slot geometry is used in the Kapitsa–Kostenko generator [3], shock generators of the Elektrosila plant [4, 5, 6], generators of the firms Oerlikon [7], Westinghouse [8], Siemens—Schuchert-werke [9], and others. However, whether such slot geometry is optimal has not been definitively decided.
There are a number of statements warning against an excessive reduction of slot height. In [10] it is pointed out that the damping of the free components of the short-circuit current increases as a result of an increase in the active resistance of the stator winding when the volume of copper in the slot is reduced, while in [11] it is pointed out that the wedge thickness increases with increasing stator-slot width, i.e., that the proportion of leakage over the height of the wedge increases. Earlier we investigated the ratio of slot height to its width [12] and obtained results close to the recommendations of [3, 4]. However, in our calculations we did not impose strict limitations on the current density in the slot or on the maximum permissible voltage at the machine terminals.
This paper presents the results of a more complete study of the geometry of the tooth zone of a shock-power generator by means of an electronic digital computer. In the calculations, constraints were imposed on voltage and current density; the influence of the insulation thickness and wedge thickness on the filling of the slot with copper was taken into account; and the influence of saturation on the conductance of the slot-leakage fluxes was included. The calculations were carried out for a number of values of the induction in the machine air gap.
In deriving the equations on the basis of which the calculation program was compiled, the method for calculating turbogenerators [13, 14] was used.
As the efficiency criterion for the variants under consideration, the value of the energy delivered by the shock generator to an inductive load with matched resistance is adopted. Obviously, this energy is one fourth of the energy that can be stored by the generator stator winding under a sudden short circuit.
\[ W_{\mathrm{s}}=\frac{1}{4}\,\frac{x_{\mathrm{s}} I_{\mathrm{s}}^{2}}{2\omega} =\frac{1}{16}\,\frac{x_{\mathrm{s}} I_{\mathrm{s}}^{2}}{\pi f}, \tag{1} \]
where
\(x_{\mathrm{s}}\) is the inductive resistance of the stator winding of the shock generator under a sudden short circuit in the first instants of time;
\(I_{\mathrm{s}}\) is the current amplitude in the first half-period of a sudden short circuit of the generator onto the load;
\(f\) is the current frequency.
The current amplitude is determined as
\[ I_{\mathrm{s}}=\frac{2\sqrt{2}\,U k_{\mathrm{d}}}{Z_{\mathrm{s}}}, \tag{2} \]
where
\(Z_{\mathrm{s}}=r_{1}+j x_{\mathrm{s}}\) is the total impedance of the stator winding;
\(U\) is the rms voltage at the generator terminals, taking into account excitation forcing;
\(k_{\mathrm{d}}\) is the current-decay coefficient.
The value of the current-decay coefficient depends on the ratio between the inductive and active resistances of the stator winding. Obviously, the values of the decay coefficient decrease as the copper cross section of the winding is reduced.
\[ U=4.44 w_{1} k_{w1} f \Phi, \tag{3} \]
where
\(W_{1}\) is the number of series-connected turns of the stator winding.
For a single-phase, single-layer stator winding,
\[ w_{1}=\frac{p q}{a}, \tag{4} \]
\(a\) is the number of parallel branches of the stator winding;
\(p\) is the number of pole pairs of the machine;
\(q\) is the number of slots per pole and phase;
\(k_{w1}\) is the winding factor of the stator winding;
\(\Phi\) is the main flux of the machine.
For a machine with a nonsalient-pole rotor,
\[ \Phi=\frac{2}{\pi}\,B_{\delta}\tau l_{i}, \tag{5} \]
where
\(B_{\delta}\) is the induction in the air gap;
\(\tau\) is the stator pole pitch;
\(l_{i}\) is the calculated length of the active part of the machine.
The inductive resistance of the stator winding of a shock generator under a sudden short circuit depends on the design of the slot and end-winding ...
parts of the winding, on the size of the phase zone, and on the presence of a damper system on the rotor.
A single-phase stator winding occupying \(2/3\) of a pole pitch is analogous to two phases of a three-phase winding; therefore
\[ x_{\text{ud}}=\frac{x_d''+x_2}{2}, \tag{6} \]
where
\(x_d''\) is the subtransient inductive reactance of the stator winding along the direct axis;
\(x_2\) is the negative-sequence inductive reactance of the phases.
In the presence of a powerful damper system on the rotor, \(x_{\text{ud}}\approx 1.15x_s\) [12], where \(x_s\) is the leakage inductive reactance of the stator winding.
\[ x_s=4\pi f\mu_0\frac{w_1^2}{pq}l_i\Sigma\lambda, \tag{7} \]
\[ \Sigma\lambda=\lambda_{\text{p}}+\lambda_{\text{l}}+\lambda_{\text{k}}+\lambda_{\text{d}}. \tag{8} \]
The slot-leakage permeance of the stator winding is:
\[ \lambda_{\text{p}}=\left[\frac{h_{\text{M1}}}{3b_{\text{p}}}+\frac{h_{\text{k}}}{b_{\text{p}}}\right]k_{\text{n}}, \tag{9} \]
where \(h_{\text{M1}}\) is the height of the copper in the slot;
\(b_{\text{p}}\) is the slot width;
\(h_{\text{k}}\) is the height of the wedge and the insulation between the wedge and the winding copper;
\(k_{\text{n}}\) is the saturation coefficient, accounting for the decrease in slot-leakage permeance as the tooth saturation increases.
The value of the saturation coefficient depends on the number of slots per pole and phase, the tooth geometry, and also on the magnitude of the leakage flux.
We denote the ratio of the slot height to its width by
\[ k_1=\frac{h_{\text{p}}}{b_{\text{p}}}, \tag{10} \]
and the ratio of the tooth width to the tooth pitch by
\[ \xi=\frac{b_z}{t_z}. \tag{11} \]
The end-winding leakage permeance is
\[ \lambda_{\text{l}}=k_2\frac{\tau}{l_i}qk_{\text{e}}, \tag{12} \]
\(k_2\) is a coefficient accounting for the type of winding and the construction of the end parts. For a single-layer concentric subdivided winding, with \(2/3\) of the slots filled, the value is \(k_2=0.26\).
\(k_{\text{e}}\) is the shielding coefficient, accounting for the decrease in leakage of the end part when it is enclosed in damper cones.
The value of the shielding coefficient depends on the proximity of the damper rings to the winding copper and on their material.
The differential-leakage permeance is
\[ \lambda_{\text{d}}=\frac{k_{w1}^{2}t_z}{\pi^2 k_{\mu}\delta k_{\delta}}(\lambda'_z-\lambda'_b), \tag{13} \]
where \(k_{\mu}\) is the coefficient of the magnetic circuit, accounting for the influence of saturation;
\(k_{\delta}\) is the air-gap coefficient;
\(\delta\) is the air-gap size;
\(\lambda'_z\) is the leakage permeance for tooth harmonics;
\(\lambda'_b\) is the leakage permeance for spatial harmonics.
The permeance at the tooth tips is
\[ \lambda_{\text{k}}=0.2+\frac{\delta}{2t_z}. \tag{14} \]
In calculations on an electronic digital computer it was assumed that the following basic dimensions of the generator were known:
\(\tau\) — pole pitch of the stator;
\(l_i\) — calculated length of the active part of the machine;
\(\delta\) — magnitude of the air gap;
\(B_z\) — maximum permissible induction in the stator tooth.
Considering two-pole shock generators of limiting power, we proceed from the following values of the principal quantities:
\[ \tau = 2.06\ \text{m}, \qquad \delta = 0.03\ \text{m}, \qquad l_i = 6.0\ \text{m}. \]
The induction in the narrowest section of the stator tooth was taken equal to \(2.3\ \text{T}\), and, under forcing of the excitation, to \(2.65\) and \(3\ \text{T}\). In studying the geometry of the stator tooth zone, we assume that the rotor of the shock generator is ideal, that is, we do not impose strict limitations on the magnitude of the induction in the air gap.
We introduce limitations on \(B_\delta\) when making recommendations for selecting the number of slots per pole and phase of the stator in the corresponding section of the article. The ratio \(\xi\) has been adopted by us as one of the principal variable quantities. It relates the induction in the air gap of the machine to the induction in the narrowest section of the tooth,
\[ \xi = \frac{1}{k_c}\frac{B_\delta}{B_z}, \tag{15} \]
where \(k_c\) is the coefficient of filling of the stack with steel.
In addition to changing the ratio of the tooth width to the tooth pitch, the following parameters were varied: the ratio of the slot height to its width and the number of slots per pole and phase.
The ratio of the tooth width to the tooth pitch was varied within the limits \(0.4\)–\(0.8\); the ratio of the slot height to its width was varied within the limits \(1\)–\(3.5\); the number of slots per pole and phase was varied from \(6\) to \(24\).
The calculation results are presented in Tables 1–5.
Table 1
Calculation results for the shock generator with \(2p = 2\), \(f = 50\),
\(\tau = 2.06\ \text{m}\), \(l_i = 6\ \text{m}\), \(a = 2\), \(\xi = 0.4\)
| N | \(q\) | \(k_1\) | \(B_\delta\) (T) | \(B_z\) (T) | \(U\) (kV) | \(x_{\mathrm{ud}}\) (ohm) | \(I_{\mathrm{ud}}\) (kA) | \(j\) (A/mm²) | \(W_{\mathrm{n}}\) (MJ) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 8 | 2 | 0.985 | 2.65 | 5.74 | 0.0148 | 1058 | 31.80 | 6.60 |
| 2 | 10 | 2 | 0.985 | 2.65 | 7.17 | 0.0189 | 1037 | 50.70 | 8.00 |
| 3 | 12 | 2 | 0.985 | 2.65 | 8.60 | 0.0234 | 999 | 73.70 | 9.30 |
| 4 | 14 | 2 | 0.985 | 2.65 | 10.0 | 0.0284 | 958 | 101.9 | 10.3 |
| 5 | 16 | 2 | 0.985 | 2.65 | 11.5 | 0.0337 | 917 | 135.8 | 11.2 |
| 6 | 18 | 2 | 0.985 | 2.65 | 12.9 | 0.0394 | 877 | 177.2 | 12.0 |
| 7 | 20 | 2 | 0.985 | 2.65 | 14.3 | 0.0455 | 836 | 227.7 | 12.6 |
| 8 | 22 | 2 | 0.985 | 2.65 | 15.8 | 0.0504 | 822 | 299.2 | 13.5 |
| 9 | 24 | 2 | 0.985 | 2.65 | 17.2 | 0.0564 | 790 | 384.2 | 13.9 |
| 10 | 26 | 2 | 0.985 | 2.65 | 18.7 | 0.0629 | 753 | 491.4 | 14.1 |
| 11 | 28 | 2 | 0.985 | 2.65 | 20.0 | 0.0697 | 715 | 631.9 | 14.1 |
| 12 | 8 | 3 | 0.985 | 2.65 | 5.70 | 0.0189 | 933 | 15.60 | 5.20 |
| 13 | 10 | 3 | 0.985 | 2.65 | 7.20 | 0.0247 | 795 | 24.10 | 6.20 |
| 14 | 12 | 3 | 0.985 | 2.65 | 8.60 | 0.0302 | 779 | 35.40 | 7.30 |
| 15 | 14 | 3 | 0.985 | 2.65 | 10.0 | 0.0357 | 768 | 49.80 | 8.30 |
| 16 | 16 | 3 | 0.985 | 2.65 | 11.5 | 0.0417 | 748 | 67.00 | 9.30 |
| 17 | 18 | 3 | 0.985 | 2.65 | 12.9 | 0.0481 | 728 | 88.10 | 10.1 |
| 18 | 20 | 3 | 0.985 | 2.65 | 14.3 | 0.0549 | 706 | 113.6 | 10.8 |
| 19 | 22 | 3 | 0.985 | 2.65 | 15.8 | 0.0621 | 683 | 144.9 | 11.5 |
| 20 | 26 | 3 | 0.985 | 2.65 | 18.7 | 0.0777 | 636 | 233.8 | 12.5 |
| 21 | 28 | 3 | 0.985 | 2.65 | 20.1 | 0.0859 | 613 | 298.9 | 12.8 |
| 22 | 30 | 3 | 0.985 | 2.65 | 21.5 | 0.0946 | 588 | 385.9 | 12.9 |
| 23 | 32 | 3 | 0.985 | 2.65 | 22.9 | 0.1029 | 565 | 510.3 | 13.0 |
Table 2
Results of calculation of the shock generator with \(2P-2,\ f-50\) Hz,
\(\tau=2.06\) m, \(l_i=6\) m, \(a=2,\ \xi=0.5\)
| No. | \(q\) | \(k_1\) | \(B_\delta\) (T) | \(B_z\) (T) | \(U\) (kV) | \(x_{\mathrm{ud}}\) (ohm) | \(I_{\mathrm{ud}}\) (kA) | \(j\) (A/mm²) | \(W_{\mathrm{n}}\) (MJ) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 8 | 1 | 1.23 | 2.65 | 7.2 | 0.0086 | 1839 | 121 | 13.7 |
| 2 | 10 | 1 | 1.23 | 2.65 | 8.9 | 0.0113 | 2064 | 411 | 19.1 |
| 3 | 12 | 1 | 1.23 | 2.65 | 10.7 | 0.0144 | 1904 | 613 | 20.6 |
| 4 | 14 | 1 | 1.23 | 2.65 | 12.6 | 0.0178 | 1733 | 874 | 21.1 |
| 5 | 8 | 1.5 | 1.23 | 2.65 | 7.2 | 0.0113 | 1714 | 111 | 13.2 |
| 6 | 10 | 1.5 | 1.23 | 2.65 | 8.9 | 0.0144 | 1677 | 181 | 16.0 |
| 7 | 14 | 1.5 | 1.23 | 2.65 | 12.5 | 0.0210 | 1559 | 397 | 20.3 |
| 8 | 16 | 1.5 | 1.23 | 2.65 | 14.3 | 0.0250 | 1469 | 550 | 21.4 |
| 9 | 10 | 2 | 1.23 | 2.65 | 8.9 | 0.0133 | 1403 | 104 | 13.5 |
| 10 | 12 | 2 | 1.23 | 2.65 | 10.7 | 0.0216 | 1340 | 153 | 15.4 |
| 11 | 14 | 2 | 1.23 | 2.65 | 12.6 | 0.0249 | 1345 | 229 | 17.9 |
| 12 | 16 | 2 | 1.23 | 2.65 | 14.3 | 0.0290 | 1306 | 321 | 19.6 |
| 13 | 18 | 2 | 1.23 | 2.65 | 16.1 | 0.0336 | 1249 | 438 | 20.8 |
| 14 | 20 | 2 | 1.23 | 2.65 | 17.9 | 0.0384 | 1189 | 595 | 21.5 |
| 15 | 14 | 2.5 | 1.23 | 2.65 | 12.6 | 0.0294 | 1154 | 147 | 15.5 |
| 16 | 16 | 2.5 | 1.23 | 2.65 | 14.3 | 0.0348 | 1104 | 202 | 16.8 |
| 17 | 18 | 2.5 | 1.23 | 2.65 | 16.1 | 0.0384 | 1116 | 288 | 18.9 |
| 18 | 20 | 2.5 | 1.23 | 2.65 | 17.9 | 0.0433 | 1084 | 394 | 20.1 |
| 19 | 22 | 2.5 | 1.23 | 2.65 | 19.7 | 0.0486 | 1041 | 536 | 20.9 |
| 20 | 14 | 3 | 1.23 | 2.65 | 12.6 | 0.0326 | 1043 | 106 | 14.1 |
| 21 | 16 | 3 | 1.23 | 2.65 | 14.3 | 0.0383 | 1010 | 147 | 15.5 |
| 22 | 18 | 3 | 1.23 | 2.65 | 16.1 | 0.0445 | 973 | 199 | 16.7 |
| 23 | 20 | 3 | 1.23 | 2.65 | 17.9 | 0.0489 | 973 | 278 | 18.3 |
| 24 | 22 | 3 | 1.23 | 2.65 | 19.7 | 0.0543 | 952 | 382 | 19.5 |
| 25 | 24 | 3 | 1.23 | 2.65 | 21.5 | 0.0602 | 919 | 526 | 20.1 |
| 26 | 18 | 3.5 | 1.23 | 2.65 | 16.1 | 0.0483 | 901 | 153 | 15.5 |
| 27 | 20 | 3.5 | 1.23 | 2.65 | 17.9 | 0.0552 | 870 | 205 | 16.6 |
| 28 | 22 | 3.5 | 1.23 | 2.65 | 19.7 | 0.0624 | 840 | 276 | 17.4 |
| 29 | 24 | 3.5 | 1.23 | 2.65 | 21.5 | 0.0665 | 846 | 394 | 18.9 |
Table 3
Results of calculation of the shock generator with \(2P-2,\ f=50\) Hz,
\(\tau=2.06\) m, \(l_i=6\) m, \(a=2,\ \xi=0.6\)
| No. | \(q\) | \(k_1\) | \(B_\delta\) (T) | \(B_z\) (T) | \(U\) (kV) | \(x_{\mathrm{ud}}\) (ohm) | \(I_{\mathrm{ud}}\) (kA) | \(j\) (A/mm²) | \(W_{\mathrm{n}}\) (MJ) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 8 | 1 | 1.48 | 2.65 | 8.60 | 0.0069 | 3181 | 628 | 27.8 |
| 2 | 10 | 1 | 1.48 | 2.65 | 10.76 | 0.0095 | 2778 | 987 | 29.2 |
| 3 | 12 | 1 | 1.48 | 2.65 | 12.91 | 0.0120 | 2435 | 1496 | 28.9 |
| 4 | 14 | 1 | 1.48 | 2.65 | 15.06 | 0.0150 | 2079 | 2200 | 26.5 |
| 5 | 8 | 1.5 | 1.48 | 2.65 | 8.61 | 0.0090 | 2551 | 274 | 23.2 |
| 6 | 10 | 1.5 | 1.48 | 2.65 | 10.76 | 0.0110 | 2440 | 453 | 27.2 |
| 7 | 12 | 1.5 | 1.48 | 2.65 | 12.90 | 0.0140 | 2272 | 691 | 29.6 |
| 8 | 14 | 1.5 | 1.48 | 2.65 | 15.06 | 0.0170 | 2083 | 1011 | 30.6 |
| 9 | 6 | 2 | 1.48 | 2.65 | 6.40 | 0.0088 | 1996 | 77.4 | 13.9 |
| 10 | 8 | 2 | 1.48 | 2.65 | 8.60 | 0.0120 | 1920 | 141 | 17.7 |
| 11 | 10 | 2 | 1.48 | 2.65 | 10.76 | 0.0140 | 2049 | 257 | 23.3 |
| 12 | 12 | 2 | 1.48 | 2.65 | 12.90 | 0.0170 | 2000 | 404 | 26.9 |
| 13 | 14 | 2 | 1.48 | 2.65 | 15.00 | 0.0200 | 1900 | 599 | 29.3 |
| 14 | 10 | 2.5 | 1.48 | 2.65 | 10.76 | 0.0179 | 1625 | 154 | 18.7 |
| 15 | 12 | 2.5 | 1.48 | 2.65 | 12.91 | 0.0200 | 1725 | 261 | 23.6 |
| 16 | 14 | 2.5 | 1.48 | 2.65 | 15.0 | 0.0230 | 1699 | 396 | 26.7 |
| 17 | 16 | 2.5 | 1.48 | 2.65 | 17.2 | 0.0270 | 1632 | 578 | 28.7 |
| 18 | 10 | 3 | 1.48 | 2.65 | 10.8 | 0.0200 | 1460 | 112 | 16.9 |
| 19 | 12 | 3 | 1.48 | 2.65 | 12.9 | 0.0247 | 1411 | 171 | 19.5 |
| 20 | 14 | 3 | 1.48 | 2.65 | 15.1 | 0.0270 | 1489 | 276 | 23.7 |
| 21 | 16 | 3 | 1.48 | 2.65 | 17.2 | 0.0308 | 1469 | 410 | 26.4 |
| 22 | 12 | 3.5 | 1.48 | 2.65 | 12.9 | 0.0272 | 1289 | 130 | 17.9 |
| 23 | 14 | 3.5 | 1.48 | 2.65 | 15.1 | 0.0326 | 1246 | 191 | 20.0 |
| 24 | 16 | 3.5 | 1.48 | 2.65 | 17.2 | 0.0350 | 1310 | 301 | 23.8 |
| 25 | 18 | 3.5 | 1.48 | 2.65 | 19.4 | 0.0392 | 1295 | 448 | 26.0 |
| 26 | 20 | 3.5 | 1.48 | 2.65 | 21.5 | 0.0439 | 1255 | 668 | 27.4 |
Table 4
Results of the calculation of a shock generator with \(2P=2,\ f=50\) Hz,
\(\tau=2.06\) m, \(l_i=6\) m, \(a=2,\ \xi=0.7\)
| N | \(q\) | \(k_1\) | \(B_\sigma\) (T) | \(B_z\) (T) | \(U\) (kV) | \(x_{\mathrm{sh}}\) (ohm) | \(I_{\mathrm{sh}}\) (kA) | \(j\) (A/mm²) | \(W_{\mathrm{n}}\) (MJ) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 4 | 2 | 1.72 | 2.65 | 4.30 | 0.0059 | 1900 | 80.6 | 14.6 |
| 2 | 6 | 2 | 1.72 | 2.65 | 7.53 | 0.0065 | 3096 | 225 | 24.8 |
| 3 | 8 | 2 | 1.72 | 2.65 | 10.04 | 0.0083 | 3184 | 457 | 33.4 |
| 4 | 10 | 2 | 1.72 | 2.65 | 12.50 | 0.0106 | 3011 | 778 | 38.3 |
| 5 | 12 | 2 | 1.72 | 2.65 | 15.00 | 0.0130 | 2743 | 1228 | 40.1 |
| 6 | 4 | 3 | 1.72 | 2.65 | 5.02 | 0.0071 | 1934 | 36 | 10.5 |
| 7 | 6 | 3 | 1.72 | 2.65 | 7.53 | 0.0100 | 2000 | 89 | 16.3 |
| 8 | 8 | 3 | 1.72 | 2.65 | 10.04 | 0.0120 | 2164 | 188 | 23.6 |
| 9 | 10 | 3 | 1.72 | 2.65 | 12.55 | 0.0140 | 2316 | 356 | 30.6 |
| 10 | 12 | 3 | 1.72 | 2.65 | 15.06 | 0.0170 | 2286 | 594 | 35.5 |
| 11 | 14 | 3 | 1.72 | 2.65 | 17.50 | 0.0203 | 2175 | 947 | 38.1 |
Table 5
Results of the calculation of a shock generator with \(2P=2,\ f=50\) Hz,
\(\tau=2.06\) m, \(l_i=6\) m, \(a=2,\ \xi=0.8\)
| N | \(q\) | \(k_1\) | \(B_\sigma\) (T) | \(B_z\) (T) | \(U\) (kV) | \(x_{\mathrm{sh}}\) (ohm) | \(I_{\mathrm{sh}}\) (kA) | \(j\) (A/mm²) | \(W_{\mathrm{n}}\) (MJ) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 4 | 2 | 1.98 | 2.65 | 5.73 | 0.0028 | 5310 | 376 | 32.2 |
| 2 | 6 | 2 | 1.98 | 2.65 | 8.61 | 0.0041 | 5281 | 961 | 45.8 |
| 3 | 8 | 2 | 1.98 | 2.65 | 11.4 | 0.0059 | 4500 | 1805 | 49.4 |
| 4 | 10 | 2 | 1.98 | 2.65 | 14.3 | 0.0080 | 3760 | 3041 | 45.2 |
| 5 | 4 | 3 | 1.98 | 2.65 | 5.73 | 0.0057 | 2700 | 118 | 16.7 |
| 6 | 6 | 3 | 1.98 | 2.65 | 8.60 | 0.0058 | 3891 | 430 | 35.2 |
| 7 | 8 | 3 | 1.98 | 2.65 | 11.4 | 0.0075 | 3890 | 903 | 45.2 |
| 8 | 10 | 3 | 1.98 | 2.65 | 14.3 | 0.0097 | 3530 | 1624 | 48.2 |
Selection of the Ratio of Tooth Width to Tooth Pitch
For turbogenerators, the ratio of the tooth width to the tooth pitch is, as a rule, taken within the range 0.5–0.6. It is of interest to establish the optimal limits for selecting \(\xi\) for a shock generator as well. To this end, for a series of values of the induction in the air gap, the number of slots per pole and phase, and the slot geometry, the dependences of the energy delivered by the generator to the load and of the current density in the stator winding on \(\xi\) were calculated. The calculations show complete identity of the dependences for the variants considered. The characteristic curves are presented in Figs. 1 and 2. As a criterion in comparing the variants under consideration, in addition to the magnitude of the energy delivered to the load, we take the maximum permissible value of the current density in the stator winding. The current density in existing shock generators lies within the range from 150 A/mm² [2] to 270 A/mm² [3]. In [16] the dependence of the heating of the stator winding during one current pulse on the current density was obtained. The dependence is given below:
| \(j\), A/mm² | 100 | 200 | 300 | 400 | 500 | 600 | 700 |
|---|---|---|---|---|---|---|---|
| °C | 1.1 | 4.4 | 10.0 | 17.8 | 27.8 | 40.0 | 54.5 |
We consider it possible to allow a limiting value of the current density in the stator winding of the order of \(400\ \mathrm{A/mm^2}\).
It is evident from the tables and figures that, for any number of slots per pole and phase, the energy increases as \(\xi\) increases. However, the current density also increases. In Figs. 1 and 2, the values of the energy at current densities in the stator winding of the order of \(400\ \mathrm{A/mm^2}\) are marked with the symbol \((\times)\). Within the range of variation of the number of slots per pole and phase from \(q = 20\) to \(q = 6\), the energy increases as \(q\) decreases.
Fig. 1. Dependences of the energy \( /\!\!————/ \) and the current density \( /\!-\!-\!-\!/ \) on \(\xi\) and \(q\) for \(k_1 = 2\).
For 1, the energy is taken as \(10\ \mathrm{MJ}\).
Calculations show that, for each value of the number of slots per pole and phase, there is some limiting value of \(\xi\) (according to the permissible current density); moreover, as the ratio of slot height to slot width increases, the value of \(\xi\) increases somewhat. For a smaller number of slots per pole and phase, larger values of \(\xi\) should be adopted; for example, at \(k_1 = 2\) and \(q = 20\), the greatest permissible ratio of tooth width to tooth pitch is \(0.5\); for \(q = 16\), \(\xi = 0.55\), and for \(q = 8\), \(\xi = 0.8\).
Another criterion for the optimality of the choice of $\xi$ is the value of the induction in the machine air gap.
In [10] it is indicated that the rational limit of the induction in the air gap of the machine is $1.5\ \text{T}$. At present the firm CEM (France) has built and tested a model shock generator with a maximum value of the induction in the machine air gap of two tesla [15].
Fig. 2. Dependences of the energy $|—|$ and the current density $|---|$ on $\xi$ and $q$ for $k_1=3$.
Energy in $10\ \text{MJ}$ is taken as 1.
Taking the limiting value of the induction in the stator tooth as equal to $2.65\ \text{T}$, according to expression (15), we obtain the maximum ratios of the tooth width to the tooth pitch:
\[ \begin{aligned} \text{for } B_\delta &= 1.5\ \text{T} \qquad && \xi = 0.61,\\ \text{for } B_\delta &= 1.75\ \text{T} \qquad && \xi = 0.71,\\ \text{for } B_\delta &= 2.0\ \text{T} \qquad && \xi = 0.81. \end{aligned} \]
Thus, for shock generators the optimal ratio of the tooth width to the tooth pitch lies within the limits of the recommendations for turbogenerators. With considerable forcing of the excitation, the optimal value of the coefficient $\xi$ increases, in comparison with the recommendations for turbogenerators, and lies in the range $\xi = 0.6–0.8$, increasing as the number of slots per pole and phase decreases.
Selecting the Ratio of Slot Height to Slot Width
To reduce the leakage inductive reactance of a slot, it is recommended to reduce the slot height while increasing its width. In the preceding section we investigated the question of the expedient relationship between tooth width and tooth pitch, from which it follows that it is inadvisable to increase the slot width of the stator of a shock generator beyond \(0.5\,t_z\). In other words, the question of reducing the inductive reactance of the slot part of the stator winding is reduced entirely to selecting an expedient slot height. The criterion for the effectiveness of the selected ratio \(k_1\) is the amount of energy delivered by the generator at an allowable current density.
Obviously, for a selected slot width, reducing the slot height leads to an increase in the active resistance of the stator winding and, consequently, to an increase in the damping of the free components of the sudden-short-circuit current.
Fig. 3. Dependences of the energy /———/ and the current density /— — —/ on \(k_1\) and \(q\) at \(\xi=0.5\).
The energy in \(10\ \mathrm{MJ}\) and the current density in \(500\ \mathrm{A/mm^2}\) are taken as unity.
Figures 3 and 4 present the dependences of the energy delivered by the generator to the load on the ratio between the height and width of the stator slot. It is evident from the figures that, for any value of \(q\), reducing the slot height leads to an increase in energy only up to a certain limit.
At the same time, for each number of slots per pole and phase there will be its own critical values of \(k_1\). However, these values of \(k_1\) cannot be regarded as optimal, although the energy is maximal, since the current densities in the stator winding are considerably above the limit we have accepted as permissible. For example, at \(\xi=0.6\), for \(q=14\), the energy is maximal at \(k_1=1.5\), but in this case the current density in the stator winding is higher than that adopted by us as limiting. Reducing the current density to \(400\ \mathrm{A/mm^2}\) is accompanied by a decrease in energy of only \(2.5\%\). In this case the coefficient \(k_1\) increases to \(1.75\). For \(q=16\), the energy is maximal at \(k_1=1.75\), but the current density is 1.5 times greater than the permissible value. A current density of \(400\ \mathrm{A/mm^2}\) corresponds to \(k_1=2.2\), while the energy decreases by only \(5\%\). The energy values corresponding to a current density of \(400\ \mathrm{A/mm^2}\)
for different \(q\), are marked in Figs. 3 and 4 by (X). The coefficients \(k_1\) corresponding to these energies may be considered optimal. For each number of slots per pole and phase there is its own optimal \(k_1\), increasing with increasing \(q\).
It is evident from Figs. 3 and 4 that, for a given \(\xi\) and one and the same current density, such values of \(k_1\) can be selected for which the energy delivered by the generator to the load will be the same for different numbers of slots per pole and phase.
Fig. 4. Dependence of energy /————/ and current density /—·—·—/ on \(k_1\) and \(q\) at \(\xi=0.6\).
Energy of \(10\ \mathrm{MJ}\) is taken as 1.
For example, at \(\xi = 0.6\) and current density \(j = 400\ \mathrm{A/mm^2}\), the energy delivered by the generator to the load is equal to \(30\ \mathrm{MJ}\) for
\[ k_1 = 1.0 \quad \text{and} \quad q = 10,\qquad k_1 = 1.75 \quad \text{and} \]
\[ q = 14,\qquad k_1 = 2.5 \quad \text{and} \quad q = 18,\ \text{etc.} \]
Thus, we have shown the most expedient limits for varying the geometry of the tooth zone of a shock generator as compared with turbogenerators.
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