On Taking Load into Account in the Design of the Stator Tooth Zone of a Shock Generator
Abstract Generated abstract
The paper examines how inductive load parameters affect the design of the stator tooth zone in a shock generator, where rotor kinetic energy is converted into electromagnetic energy during a current pulse. It derives a load coefficient relating delivered load energy to sudden short circuit energy, showing a maximum for a matched inductive load, then reports design calculations varying tooth width, slot proportions, slot number, pole number, frequency, and load reactance. The results indicate that optimal tooth zone geometry can shift when load reactance exceeds the matched value, especially under current density constraints in higher frequency multipole machines, while at lower frequency the same extremal geometry may remain optimal and reduced current density may permit more frequent operating cycles.
Full Text
ON ACCOUNTING FOR LOAD IN THE DESIGN OF THE TOOTH ZONE OF THE STATOR OF A SHOCK GENERATOR
K. A. KHOR'KOV, L. O. OGANYAN
(Presented by the joint seminar of the departments of electrical machines and apparatus and general electrical engineering)
In the event of a sudden short circuit of a shock generator, part of the kinetic energy of the rotor is converted into the electromagnetic energy of the leakage fields of the stator winding. If the short circuit occurred at zero emf, then the electromagnetic energy will be equal to
\[ Q_{\text{kz}}=\frac{x_{\text{ud}} I_{\text{ud}}^{2}}{2\omega}, \tag{1} \]
where
\(X_{\text{ud}}\) — inductive reactance of the stator winding under sudden short circuit,
\(\omega\) — angular frequency,
\(I_{\text{ud}}\) — shock current (allowing for the aperiodic component).
\[ I_{\text{ud}}=\frac{2U_{m0}k_z}{x_{\text{ud}}}, \tag{2} \]
where
\(U_{m0}\) — amplitude of the line voltage,
\(k_z\) — current attenuation coefficient.
When the stator winding of a shock generator is closed onto an inductive load, the fraction of kinetic energy converted into electromagnetic energy decreases. The amount of energy that the shock generator can deliver in one current pulse to an inductive load depends strongly on the load parameters.
\[ Q_{\text{n}}=\frac{x_{\text{n}} I_{\text{n}}^{\prime\prime 2}}{2\omega}, \tag{3} \]
where
\(x_{\text{n}}\) — inductive reactance of the load,
\(I_{\text{n}}^{\prime\prime}\) — amplitude of the supertransient current when the stator winding is closed onto the load.
\[ I_{\text{n}}^{\prime\prime}=\frac{2U_{m0}k_z}{x_{\text{ud}}+x_{\text{n}}}. \tag{4} \]
The magnitude of the current damping coefficient remains unchanged if the ratio of the inductive reactance of the load to its active resistance is equal to the ratio of \(X_y\) to the active resistance of the stator winding.
This assumption can be adopted without any substantial errors. Then the energy delivered by the shock generator to the load can be expressed in fractions of the energy of a sudden short circuit of the generator. The ratio
\[ k_n=\frac{Q_n}{Q_{kz}}=\frac{x_n x_{yd}}{(x_n+x_{yd})^2} \tag{5} \]
will be called the load coefficient. The load coefficient assumes its greatest value when the inductive reactance of the load is equal to the resistance \(X_{yd}\), that is, when the shock generator operates into a matched inductive load. In this case the energy delivered to the load is equal to \(0.25\) of the energy of a sudden short circuit. As the inductive reactance of the load is decreased or increased, the magnitude of the electromagnetic energy delivered to the load decreases. The dependence of the load coefficient on the ratio \(k_x=\dfrac{x_n}{x_{yd}}\), constructed according to expression (5), is shown in Fig. 1.
Fig. 1. Dependence of \(k_n\) as a function of \(k_x\)
It is assumed here that the parameters of the shock generator, including the geometry of the tooth zone, remain unchanged for different loads.
Studies of the optimal geometry of the stator tooth zone of a shock generator from the condition of obtaining maximum energy in a matched inductive load have been carried out quite fully [1, 2]. It is of interest to consider how the optimal geometry of the generator tooth zone changes when the load deviates from the matched value.
For this purpose, calculations were performed for a number of design variants of the stator tooth zone of a shock generator while varying the ratio of tooth width to tooth pitch \(\xi=0.5\div0.8\), the ratio of slot height to its width \(k_1=2\div3\), and the number of slots per pole and phase \(q=4\div10\). The magnitude of the inductive reactance of the load was varied from the matched value to \(2.5x_{yd}\). The geometry of the active part of the generator was determined by the following dimensions: stator bore diameter \(3.9\ \text{m}\), active-part length \(6\ \text{m}\). The rotor speed was taken as constant—\(750\ \text{rpm}\); the number of pole pairs was varied from \(2p=8\) to \(2p=2\), and correspondingly the generator voltage frequency from \(50\ \text{Hz}\) to \(12.5\ \text{Hz}\). As an example, Fig. 2 presents the dependences of \(Q\) and of the current density in the stator winding on \(\xi\) at \(k_1=2\) and \(q=6\) for 8- and 2-pole shock generators.
The asterisk (*) on the energy curves marks the points at which the current density in the stator winding reaches \(400\ \text{A/mm}^2\), which we have taken as the maximum permissible value. It is evident that the curves \(Q=f(\xi)\) have clearly expressed extrema. For an 8-pole generator, the extreme
the points lie on the line \(\xi=0.69\), and for the 2-pole case on the line \(\xi=0.83\). If the extremal value of the energy for the matched load \((k_x=1)\) is taken as unity in each case, then the extremal values of the energy at other \(k_x\), as fractions of the energy taken as unity, fall on curve 1 (Fig. 3). The character of this curve fully corresponds to the dependence \(k_{\mathrm{n}}=f(k_x)\) in Fig. 1.
Fig. 2. Dependence of the energy and current density on the function for \(k_1=2\) and \(q=6\):
a) for an 8-pole machine with \(f=50\ \text{Hz}\); b) for a 2-pole machine with \(f=12.5\ \text{Hz}\).
With the adopted limitation on the permissible current density in the stator winding for the 8-pole shock generator, attaining the maximum energy is impossible, and the optimum geometry of the generator stator will be determined by values significantly smaller than the extremal one,
Fig. 3. Dependence of the coefficient
\[ k_{\mathrm{n}1}=\frac{Q_{\mathrm{n}\max}}{Q_{\mathrm{n}\max}\big|_{k_x=1}} \]
(curves 1, 2) and of the current density (curves \(1'\), \(2'\)) on the value of \(k_x\): \(1, 1'\)—for a 2-pole machine with \(f=12.5\ \text{Hz}\); \(2, 2'\)—for an 8-pole machine with \(f=50\ \text{Hz}\).
especially for \(k_x = 1\). The decrease in \(Q_{\mathrm{n}}\) is different for different values of \(x_{\mathrm{n}}\): for a matched load it is large, while with increasing \(x_{\mathrm{n}}\) it is insignificant. This leads to the fact that the ratios of the energy \(Q_{\mathrm{n}}\) at \(x_{\mathrm{n}} > x_{\mathrm{ud}}\) to the energy under matched load will lie above curve 2, Fig. 3. Consequently, the optimal geometry of the stator tooth zone will differ for different loads. For a matched load, the value \(\xi = 0.61\) will be the smallest. As \(x_{\mathrm{n}}\) increases, the optimal \(\xi\) increases, reaching, at \(x_{\mathrm{n}} = 2.5 x_{\mathrm{ud}}\), the value \(\xi = 0.67\). Accordingly, for example, for \(x_{\mathrm{n}} = 2.5 x_{\mathrm{ud}}\), the energy increases from \(70\) MJ at \(\xi = 0.61\) to \(79\) MJ at \(\xi = 0.67\), i.e., by 13%.
When the generator frequency is decreased, the distribution pattern of \(Q_{\mathrm{n}} = f(\xi)\) and \(j = f(\xi)\) gradually changes. At the lowest frequency \(f = 12.5\) Hz \((2p = 2)\), when the generator is made within the dimensions of an eight-pole generator, the points \(Q_{\mathrm{n}(*)}\) with permissible current density move beyond the extremal \(\xi\). Therefore, the optimal geometry of the stator tooth zone will be determined for all \(k_x\) by one and the same value of the extremal \(\xi\), equal in this case to 0.83. The energy ratio, as indicated above, lies on curve 1 (Fig. 3). However, the current density in the stator winding at \(x > x_{\mathrm{ud}}\) decreases significantly, curve \(1'\), Fig. 3. For example, at \(k_x = 2.5\), the current density decreases by a factor of two relative to the permissible value (reaching only \(200\ \mathrm{A/mm^2}\)). This makes it possible to increase the number of operating cycles of the shock generator per unit time.
REFERENCES
- G. A. Sipailov, K. A. Khor'kov, V. Z. Khor'kova, V. S. Baklin, V. I. Andreev. On the choice of the geometry of the stator tooth zone of a shock generator. Izv. TPI, vol. 211, 1970.
G. A. Sipailov, K. A. Khor'kov, V. Z. Khor'kova, V. S. Baklin, V. I. Andreev. On the choice of the geometry of the stator tooth zone of a multipole shock generator. Izv. TPI, vol. 229, 1972.