Proceedings of the Tomsk Order of the Red Banner of Labor
Polytechnic Institute named after S. M. Kirov

Vol. 190　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1968

MULTIPOLE PULSED-POWER GENERATOR AS AN ENERGY STORAGE DEVICE AND ENERGY SOURCE

V. F. Kulakov, G. A. Sipailov

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

The need to create pulsed generators of high unit power is determined by the growing requirements of modern science and technology, on the one hand, and by the economic advantages of large machines, on the other.

Unlike ordinary machines, an increase in the power of pulsed generators cannot be achieved by increasing the linear load, but only by increasing the volume of the machine and by somewhat increasing the magnetic induction at the lowest possible transient inductive resistance.

The dimensions of the rotor of a two-pole pulsed generator, widely used at present, are limited by the conditions of mechanical strength and do not make it possible to create a machine of sufficiently high pulsed power. The real possibility of increasing the power of pulsed generators lies in changing over to a multipole design of the machines.

To demonstrate this possibility, let us express the pulsed power of a single-phase generator \(P_{\mathrm{ud}1}\), by which is meant the product of the maximum value of the pulsed current, with allowance for its aperiodic component \(I_{\mathrm{ud}}\), and the amplitude value of the emf \(E_m\), through the geometrical ratios and parameters [1]:

\[ P_{\mathrm{ud}1}=\frac{2E_m^2 k_z^2 k_f^2}{x_d''}, \tag{1} \]

where

\(k_z\) — coefficient of attenuation of the amplitude of the pulsed current;

\[ k_f=\frac{E_f}{E_m} \]
— forcing coefficient.

\[ E_m = 2\pi f W_1 k_{w1}\frac{2}{\pi} k_f B_\delta \tau l, \tag{2} \]

where

\(f\) — frequency;
\(B_\delta\) — induction in the air gap;
\(l\) — active length;
\(k_f \approx 1\) — field-form coefficient.

In single-phase machines, the number of stator slots filled with the winding is taken equal to \(2/3\) of the total number of slots \(z\). Then the number of slots

per pole and phase

\[ q=\frac{z}{3p}\ \text{and}\ W_1=pq. \]

Expressing the pole pitch \(\tau\) in terms of the peripheral speed of the rotor \(v_{\mathrm{okr}}\) and the frequency \(f\),

\[ \tau=\frac{v_{\mathrm{okr}}}{2f}, \]

substituting

\[ l=\frac{\tau}{\beta}=\frac{v_{\mathrm{okr}}}{2f\beta}, \]

we obtain, for the mean value of the winding factor \(k_{w1}=0.82\),

\[ E_m=0.82\,\frac{q B_{\delta} v_{\mathrm{okr}}^{2}\rho}{f\beta}. \tag{3} \]

The subtransient inductive reactance is

\[ x''_d=x_{s1}+\left(\frac{1}{x_{ad}}+\frac{1}{x_{sv}}+\frac{1}{x_d}\right)^{-1}, \tag{4} \]

where \(x_{s1}, x_{sv}, x_{sd}\) are the leakage inductive reactances of the stator, field, and damper windings;

\(x_{ad}\) is the inductive reactance of the armature reaction.

The main portion of the subtransient reactance is the leakage reactance of the stator winding. As calculations and experiments carried out on a model of a salient-pole impulse generator show, the value of the subtransient reactance \(x''_d\) in a salient-pole machine with a damper winding may amount to \(1.28 \div 1.35\,x_{s1}\), which is somewhat higher than in a nonsalient-pole generator because of the large leakage of the salient-pole rotor circuits.

We shall find the stator leakage inductive reactance from the usual formula

\[ x_{s1}=4\pi f\mu_0 l\,\frac{W_1^2}{pq}\sum \lambda . \tag{5} \]

The leakage permeance is

\[ \sum \lambda=\lambda_p+\lambda_l+\lambda_\delta . \]

The slot-leakage permeance is

\[ \lambda_p=\left(\frac{h_1}{3b_p}+\frac{h_k+h_u}{b_p}\right)k_n, \tag{6} \]

where

\(k_n\) accounts for saturation of the tooth.

The total slot height \(h_p\) includes the wedge height \(h_k\), which is determined by the requirement of winding safety and is of the order of 2 cm; twice the insulation height \(h_u\), determined by the generator voltage; and the height of the stator-winding copper \(h_1\).

\[ h_p=h_1+h_k+2h_u. \tag{7} \]

Taking this into account, the slot permeance can be represented in the following form:

\[ \lambda_p=\left(\frac{1}{3}k_1+\frac{2h_k+h_u}{3\xi t_1}\right)k_n, \tag{8} \]

where \(t_1\) is the tooth pitch.

\[ \xi = \frac{b_n}{t_1} \quad \text{and} \quad k_1 = \frac{h_n}{b_n}. \]

The conductance of end leakage is

\[ \lambda_l = (0.47 l_{l\,\mathrm{av}} - 0.3\tau)\cdot \frac{q}{l}\cdot k_e, \tag{9} \]

where \(k_e\) is a coefficient allowing for the shielding effect of the bands. Since, with a single-phase concentric winding, the average length of the end portions is \(l_{l\,\mathrm{av}} \approx 1.2\tau\), then

\[ \lambda_l = 0.26\,\beta q k_e. \tag{10} \]

The conductance of differential leakage in salient-pole machines is

\[ \lambda_\delta = \frac{\delta'}{b_n + 0.8\delta'} = \frac{\delta'}{\xi t_1 + 0.8\delta'}, \tag{11} \]

where \(\delta'\) is the reduced air gap.

The damping coefficient of the surge-current amplitude \(k_3\) is determined mainly by two time constants—the damping constant of the supertransient process \(T''_d\), determined by the ratio of the leakage and active resistances of the rotor circuits, and the aperiodic component \(T_a\), whose damping depends on the active and individual leakage resistance of the stator winding,

\[ k_3 = \frac{1 + e^{-\frac{1}{2 f T''_d}} + 2e^{-\frac{1}{2 f T_a}}}{4}. \tag{12} \]

In order to reduce the damping of the supertransient component of the surge current, whose time constant is

\[ T''_d = \frac{\sigma_{vd} T_v T_d}{T_v + T_d}, \tag{13} \]

the values of the time constants of the excitation circuit \(T_v\) and the damper circuit \(T_d\) should be made, as far as possible, close to one another, i.e., the copper cross section of the damper winding should be increased to a value close to the copper cross section of the excitation winding.

The time constant of the aperiodic process is

\[ T_a = \frac{x''_d + x''_q}{2\omega r_1} \simeq \frac{1.3x_{s1}}{\omega r_1}, \tag{14} \]

where the active resistance of the stator winding is

\[ r_1 = \rho \frac{l_w pq}{s} = \frac{2\rho \tau pq\left(\frac{1}{\beta} + 1.2\right)}{k_1 \xi^2 t_1^2 k_{\mathrm{fill}}}, \tag{15} \]

where \(\rho\) is the specific resistance,

\[ l_w = 2(l + l_{l\,\mathrm{av}}) = 2\tau\left(\frac{1}{\beta} + 1.2\right) \]

is the average length of a turn.

The copper cross section is \(s = k_1 b_n^2 k_{\mathrm{fill}}\).

\(k_{\mathrm{fill}}\) is the coefficient of filling the slot with copper, which is a complex function of the slot dimensions and the generator voltage. It should be borne in mind that the induction in the stator teeth \(B_{z1}\) and, consequently, the induction in the air gap \(B_\delta = B_{z1}(1-\xi)\) can be increased when the generator frequency \(f\) is reduced. This circumstance is justified still

in that the tooth height in the pulse generator is small and the magnetizing current will increase only slightly. Saturation of the teeth lowers the leakage inductive reactance of the generator, thereby increasing its power.

The expression for the pulse power after substituting (3), (5) into (1) is

\[ P_{\mathrm{ud}1}=0.17\cdot 10^{6}\, \frac{B_{\delta}^{2}v_{\mathrm{okr}}^{3}qK_{3}^{2}k_{\phi}^{2}} {f^{2}\beta\Sigma\lambda\cdot \kappa}\,p; \tag{16} \]

here

\[ \kappa=\frac{x''_{d}}{x_{s1}}=1.28\div 1.35 \]

depending on the leakage of the rotor circuits;

\[ B_{\delta}=B_{z1}(1-\xi) \]

\(K_{3}\) and \(\Sigma\lambda\) are found from (6), (7), (8), (9).

From equation (12) it is evident that, all other parameters being equal, the power of the pulse generator increases linearly with an increase in the number of pole pairs \(p\).

It can be shown that the kinetic-energy density of the rotating rotor \(q_{\omega}\) is likewise determined by the peripheral velocity

\[ q_{\omega}=\frac{\frac{1}{2}J\omega^{2}}{V_{p}} =\frac{\gamma}{4}v_{\mathrm{okr}}^{2}; \tag{17} \]

here

\(V_{p}\) is the rotor volume,
\(\gamma\) is the mean density of the rotor material.

In order to improve the energy and weight characteristics of the pulse generator as a storage device, the peripheral velocity should, if possible, be increased. However, owing to the design features of multipole machines, the peripheral velocity of the rotor may be taken, from the conditions of mechanical strength, as no more than 160 m/sec in salient-pole generators with a disk rim and about 120 m/sec with a segmented rim.

The nature of the variation of pulse power per pole pair as a function of the peripheral velocity \(v_{\mathrm{okr}}\) and of the relative length of the machine

\[ \beta=\frac{\tau}{l} \]

is shown in Fig. 1.

The optimum dimensional ratios, in particular the optimum geometry of the toothed zone of the stator, are found from the condition of maximum pulse power (equation 16) as a function of the parameters entering into it.

The conclusions concerning the optimum geometry of the toothed zone of the stator of a salient-pole pulse generator are, in general, similar to the recommendations for the design of a nonsalient-pole pulse generator [2].

However, the relative depth of the slot

\[ k_{1}=\frac{h_{p}}{b_{p}} \]

has a somewhat larger value, of the order of 2.6–3.3.

In addition, if there are no special requirements on the generator voltage, it is recommended to choose a winding with a smaller number of slots per pole and phase, and a tooth pitch \(t_{1}\) of the order of 8–10 cm.

Under the condition of constant induction in the stator teeth, the pulse power depends comparatively little on the relative slot width

\[ \xi=\frac{b_{p}}{t_{1}} \]

and, within the range of its variation from 0.3 to 0.38, may be considered optimal. As an example, Fig. 2 gives curves illustrating

Fig. 1. Variation of the pulsed power per pair of poles as a function of the circumferential speed of the rotor

Fig. 2. Toward determining the optimum geometry of the stator tooth zone

curves giving the dependence of the pulsed power per pair of poles for an eight-pole, \(p = 4\), pulsed generator, whose rotor-surface circumferential speed is \(v_{\text{circ}} = 160\ \text{m/sec}\), on the parameters \(q\), \(K_1\), and \(\xi\). The generator emf frequency is \(f = 50\ \text{Hz}\), the induction in the stator teeth is \(B_{z1} = 1.8\ \text{T}\), \(\beta = 0.2\).

The curves are plotted for the optimum values

\[ K_1 = \frac{h_p}{b_p}, \]

in the case,

If this ratio deviates upward or downward, the power drops (dashed curves).

The pulse power of an eight-pole pulse generator, with successful parameter ratios, is therefore

\[ P_{\mathrm{ud}1} = p \cdot P_{\mathrm{udr}} = 4 \cdot 6.8 \cdot 10^3 = 27.2 \cdot 10^3 \ \mathrm{MVA}, \]

and the energy that the generator can transfer to an inductive matched load \((x_{\mathrm{n}} = x_d^{\prime\prime})\) in one period of emf is

\[ Q_{\mathrm{n}} = \frac{1}{4}\frac{P_{\mathrm{ud}1}}{\omega} = 21.6 \ \mathrm{MJ}. \]

In the case of the use of special circuits [3, 4], in particular when operating jointly with a capacitor bank, the amount of energy delivered by the generator can be increased by another factor of 2–3.

The use of multipole pulse generators as energy sources with an energy-transfer time on the order of 0.04–0.1 sec is very attractive. In this case the generator is designed for a reduced emf frequency, for example, 25, \(16^{2}/_{3}\), 12.5, etc. Hz; then, with the same stator-bore diameter \(D\) \((v_{\mathrm{okr}} = \mathrm{const})\) and length \(l\), and with a reduction in the number of pole pairs in comparison with a 50-Hz machine by factors of 2, 3, and more, the amount of transmitted energy correspondingly increases by factors of 2, 3, etc. In particular, the eight-pole generator shown in the example, if made with \(p = 2\), with a pulse-current duration of about 0.04 sec, will be able to transmit more than 40 MJ to an inductive load. This energy can be increased both by circuit solutions [3, 4] and by a further reduction in the frequency of multipole generators.

References

1. I. M. Postnikov. Obtaining maximum values of electrokinetic energy. Proceedings of LPI, No. 2, 1937.

2. G. A. Sipailov, K. A. Khorkov. On the choice of the principal dimensions of a pulse generator. Proceedings of TPI, 1964, vol. 132.

3. V. V. Ivashin, G. A. Sipailov. Method of accumulating energy from a pulse generator. Inventor’s Certificate No. 173823, Bulletin of Inventions, No. 16, 1965.

4. G. A. Sipailov, V. V. Ivashin, K. A. Khorkov, A. V. Loos. Obtaining large electromagnetic energies during joint operation of a pulse generator and a capacitor bank. Elektrichestvo, 1967, No. 8.