SPECIFIC ENERGY OF AN IMPACT GENERATOR

G. A. SIPAILOV, K. A. KHOR’KOV

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

In many practical cases, in order to reduce the average power of the energy source, a pulsed mode of supply becomes necessary. For example, in domestic and foreign practice, experiments on controlled thermonuclear reactions are conducted mainly in a pulsed mode. High-energy charged-particle accelerators, arc-type aerodynamic tubes for testing models of guided projectiles and spacecraft, and certain other installations also operate in a pulsed mode [1, 2].

When operating in a pulsed mode, a very large amount of energy is usually required; as a rule, it is accumulated in a corresponding storage device over a long period of time, measured in seconds, and then, over a short interval of time, measured in microseconds or fractions of a second, is discharged into the load.

It is known that energy can be stored in the form of electric charge in capacitors, magnetic flux in inductances, mechanical energy in electrical machines, and chemical energy in storage batteries, etc. These various forms of energy storage, as a rule, do not exclude but complement one another. For example, energy storage in capacitors is most expedient in those cases where the discharge time is measured in microseconds and the magnitude of the stored energy does not exceed 1 MJ. Magnetic storage becomes justified at energies above 1 MJ and when energy is delivered over milliseconds. Electromechanical storage can be applied over a very wide range of energies, with discharge times from hundredths of a second to several seconds.

From the point of view of minimum losses, energy storage in capacitors is the most advantageous. However, on the other hand, the electrical method of storage is the most expensive and bulky, since the cost of capacitors is high and their energy density is only \(0.02\)–\(0.3\ \text{J}/\text{cm}^3\). The energy stored in the electric field per unit volume of a capacitor, in \(\text{J}/\text{cm}^3\), is equal to:

\[ q_E = 4.44 \cdot 10^{-14}\varepsilon E^2, \tag{1} \]

where \(\varepsilon\) is the dielectric constant,

\(E\) is the electric-field strength in \(\text{V}/\text{cm}\).

Table 1 gives data on the energy density in capacitors with various dielectrics.

Table 1

In ironless inductive storage devices, the energy density is determined by equation (2).

\[ q_L=\frac{k}{2}\frac{1}{\mu_0}B^2, \tag{2} \]

or

\[ q_L=0.398\,kB^2; \tag{2a} \]

where $q_L$ is the energy reserve per unit volume enclosed by the coil turns, in J/cm³;

$\mu_0$ is the magnetic permeability of air;

$B$ is the magnetic induction in T;

$k$ is a coefficient allowing for the coil geometry: $k=f\!\left(\dfrac{l}{D}\right)$,

\[ \text{for } \frac{l}{D} \gg 10 \quad k=1,\quad \text{for } \frac{l}{D}=1.5 \quad k=0.25; \]

$D$ is the mean diameter of the coil;

$l$ is the height of the coil.

Practically acceptable values of induction in ironless inductive storage devices may lie within the range $B=0.5 \div 10$ tesla; in this case the energy density in them, in accordance with equation (2a), will vary from 0.1 to 40 J/cm³. In known inductive storage devices the energy density varies within smaller limits; for example, in the inductance coil of the University of Michigan, having outside and inside diameters of 3 and 1.5 m respectively and a height of 0.9 m, made of 220 turns of aluminum 169-conductor cable and designed to store an energy of 1 MJ, the energy density is 0.3 J/cm³ [3]. In an inductance coil designed to store an energy of 100 MJ with a source power of 10 thousand kW, the energy density is 1.6 J/cm³ [2]. However, inductive storage devices have a number of disadvantages, of which the most important are low efficiency due to large energy losses, the complexity of switching the stored energy, and the need for powerful power supplies for the storage device.

Energy storage in rotating flywheel masses is the most effective; rotating rotors of electrical machines of limiting dimensions are capable of storing more than 1000 MJ of energy; however, in this case there are difficulties associated with delivering the energy to the load.

The conversion of the energy of rotating flywheel masses into electromagnetic energy is carried out both with the aid of standard and with the aid of special electrical machines, such as impact generators, unipolar generators, etc.; moreover, the capabilities of special machines, both from the standpoint of design features,

both from the standpoint of the methods of commutation have still not been completely exhausted.

Let us consider the possibilities of a surge-power generator—a single-phase synchronous machine of the turbogenerator type—with respect to the specific kinetic energy stored by the rotating rotor, and that part of it which is converted into the electromagnetic energy of the stator. In the event of a sudden short circuit of the impact generator for the period of one maximum half-wave of current, the electromagnetic energy obtained in the stator from the kinetic energy stored in the rotor will be [4]

\[ Q=\frac{P_{\mathrm{ud}}}{\omega}=\frac{D^{2}ln}{\omega C_{\mathrm{ud}}}; \tag{3} \]

where \(C_{\mathrm{ud}}\) is the machine constant of the impact generator.

For a single-phase generator the value \(C_{\mathrm{ud}}\), with a sufficient degree of accuracy, may be determined as

\[ C_{\mathrm{ud}}=0.38\,x_{\mathrm{ud}}C_A; \tag{4} \]

where \(C_A\) is the machine constant of a generator of normal design; \(x_{\mathrm{ud}}\) is the inductive reactance of the impact generator.

For a preliminary choice of the value \(x_{\mathrm{ud}}\), the following relation may be used:

\[ x_{\mathrm{ud}}\simeq 1.15\,x_s, \]

where \(x_s\) is the leakage reactance of the stator winding.

In existing impact and interrupting-power generators, the value \(x_{\mathrm{ud}}\), expressed in relative units, is usually in the range: \(x_{\mathrm{ud}}=0.02\div 0.05\).

The machine constant of an impact generator for machines with the dimensions of turbogenerators of powers from 20 to 200 MW, without taking account of forcing the excitation, varies within the limits

\[ C_{\mathrm{ud}}=(0.7\div 1.2)x_{\mathrm{ud}}\ \mathrm{cm}^{3}/\mathrm{J}, \]

the smaller values applying to machines of higher power. The electromagnetic energy of an impact generator per unit volume of the rotor \(q_p\), and correspondingly per unit volume of the entire active part of the machine \(q_a\), will be equal to:

\[ q_p=\frac{2}{\pi^2 C_{\mathrm{ud}}C_1}\simeq \frac{0.5}{x_{\mathrm{ud}}C_A C_1}; \tag{5} \]

\[ q_a=\frac{2}{\pi^2 C_{\mathrm{ud}}C_2}\simeq \frac{0.5}{x_{\mathrm{ud}}C_A C_2}; \tag{5a} \]

where

\[ C_1=\left(\frac{D_p}{D}\right)^2,\qquad C_2=\left(\frac{D_a}{D}\right)^2, \]

\(D_p\) is the rotor diameter,
\(D\) is the diameter of the stator bore,
\(D_a\) is the outside diameter of the stator.

In the event of sudden connection of the impact generator to an inductive load \(x_n\), a smaller part of the kinetic energy of the rotor will pass into the electromagnetic energy of the stator and the load than in the case of a short circuit. This reduction may be taken into account by the coefficient \(k_{\mathrm{em}}\).

\[ Q_{\mathrm{em}}=P_{\mathrm{ud}}\frac{k_{\mathrm{em}}}{\omega} =\frac{D^{2}ln}{C_{\mathrm{ud}}}\frac{k_{\mathrm{em}}}{\omega}, \tag{6} \]

where

\[ k_{\mathrm{em}}=\frac{x_{\mathrm{ud}}}{x_{\mathrm{ud}}+x_{\mathrm{n}}}\left(\frac{k_{\mathrm{zn}}}{k_{\mathrm{z}}}\right)^2, \]

\(k_{\mathrm{z}}\) is the current damping coefficient under short-circuit conditions,
\(k_{\mathrm{zn}}\) is the current damping coefficient under load.

Accordingly, the electromagnetic energy per unit volume of the rotor \(q_{\mathrm{rn}}\) and per unit volume of the entire active part of the machine \(q_{\mathrm{an}}\) under load will be

\[ q_{\mathrm{rn}}=\frac{2k_{\mathrm{em}}}{\pi^2 C_{\mathrm{ud}} C_1}, \qquad q_{\mathrm{an}}=\frac{2k_{\mathrm{em}}}{\pi^2 C_{\mathrm{ud}} C_2}. \tag{7} \]

It follows from equations (5) and (7) that the specific electromagnetic energy obtained in the stator from the kinetic energy of the rotating rotor depends mainly on the reactive resistances of the generator and the load, and on the quantities determining the machine constant \(C_A\) of the generator, within whose dimensions the impact generator is made.

Since, with a change in the number of poles, the coefficient \(C_1\) changes only slightly, while the coefficient \(C_2\) decreases appreciably as the number of poles increases, then, for identical values of \(x_{\mathrm{ud}}\) and \(C_A\), for high-speed (b) and low-speed (t) machines we obtain the following relations between their specific electromagnetic energies:

\[ \frac{q_{\mathrm{rt}}}{q_{\mathrm{rb}}} = \frac{(x_{\mathrm{ud}} C_A C_1)_{\mathrm{b}}}{(x_{\mathrm{ud}} C_A C_1)_{\mathrm{t}}} \simeq 1, \tag{8} \]

\[ \frac{q_{\mathrm{at}}}{q_{\mathrm{ab}}} = \frac{(x_{\mathrm{ud}} C_A C_2)_{\mathrm{b}}}{(x_{\mathrm{ud}} C_A C_2)_{\mathrm{t}}} >1. \tag{9} \]

Table 2 gives the values of the specific electromagnetic energies \(q_{\mathrm{r}}\) and \(q_{\mathrm{a}}\) as functions of the inductive reactance \(x_{\mathrm{ud}}\) for two-pole machines of limiting dimensions \((C_A=1.8\ \text{cm}^3/\text{J})\), with averaged values of the coefficients: \(C_1=0.94\) and \(C_2=3.2\).

Table 2

To judge the effectiveness of mechanical energy storage, let us determine what part of the stored kinetic energy of the rotating rotor is converted into electromagnetic energy during one pulse.

The kinetic energy of a rotor having moment of inertia \(I\) and rotating with angular velocity \(\omega\) is

\[ T=I\frac{\omega^2}{2}, \tag{10} \]

where

\[ I=\Sigma mr^2=\frac{1}{8}MD_{\mathrm{r}}^2, \qquad M=\gamma l\frac{\pi D_{\mathrm{r}}^2}{4}. \]

Kinetic energy per unit volume of the rotor

\[ \Delta T=\frac{T}{V_{\mathrm{r}}}=\frac{1}{16}\,\gamma\left(\frac{2\pi f}{p}\right)^2 D_{\mathrm{r}}^2 . \tag{11} \]

For the values \(\gamma=7800\ \mathrm{kg}/\mathrm{m}^3\) and \(f=50\ \mathrm{Hz}\), we obtain

\[ \Delta T=48.1\left(\frac{D_{\mathrm{r}}}{p}\right)^2\ \mathrm{J}/\mathrm{cm}^3, \]

where \(D_{\mathrm{r}}\) is the rotor diameter in meters,
\(p\) is the number of pole pairs.

The values of the kinetic-energy density for different diameters of two-pole rotors rotating at a speed of 50 rev/sec are given in Table 3.

Table 3

The ratio between \(q_{\mathrm{r}}\) and \(\Delta T\) indicates the degree of utilization of the machine volume. We shall call the ratio of \(q_{\mathrm{r}}\) to \(\Delta T\) the machine utilization factor

\[ \eta_{\mathrm{m}}=\frac{q_{\mathrm{r}}}{\Delta T}\cdot 100\%. \]

The values of the utilization factors for a number of generators of impact and breaking capacity are given in Table 4.

Table 4

It follows from Table 4 that, under a sudden short circuit, in the impact generators considered, from 9 to 25% of the kinetic energy of the rotating rotor is converted during one pulse into electromagnetic energy of the stator. In the case of a sudden connection to a load, the values of \(\eta_{\mathrm{m}}\) will be reduced in accordance with the value of the coefficient \(k_{\mathrm{sh}}\). The presence of a flywheel on the rotor shaft will also affect only the value of \(\eta_{\mathrm{m}}\). The greater the additional flywheel masses, the smaller the utilization factor will be.

It follows from equation (11) that, provided the peripheral speed is the same

\[ \left(\frac{D_{\mathrm{r}}}{p}=\mathrm{const}\right), \]

the specific kinetic energy of the rotating rotor does not depend on the number of poles of the machine. However, since the rotors of high-power multipole machines are made laminated-

then the permissible limiting peripheral speeds in them have to be reduced to values amounting to approximately 75% of the limiting peripheral speeds of solid rotors. Therefore the density of kinetic energy in multipole machines is, in general, less than in two-pole machines. But the total store of kinetic energy in multipole machines may be made greater than in two-pole machines. In this respect one should consider the possibilities of 4-, 6-, and 8-pole machines; here four-pole machines, whose rotors can be made solid, are of special interest.

At the present level of development of the metallurgical industry, on the basis of permissible mechanical stresses, the rotor of a two-pole generator can be made with a limiting diameter of about 1.2 m, and that of a four-pole generator with about 2 m. It should be noted that an impact generator is, as a rule, driven in rotation by an asynchronous motor; therefore its speed cannot exceed the synchronous speed, and the coefficient of runaway speed cannot be greater than unity.

Table 5 gives the values of the specific energy and the total store of kinetic energy for two-pole and four-pole impact-power generators made within the limiting dimensions.

Table 5

If it is assumed that about 10% of the kinetic energy stored in the rotor is converted into electromagnetic energy, then in this case an energy of more than 100 MJ can be obtained from a four-pole impact generator of limiting dimensions.

REFERENCES

1. R. Curruthers. Proc. Inst. Elect. Eng., A 106 S No. 2, 166–172, 1959.
2. G. M. Goncharenko, L. P. Poberezhnyi, I. P. Vereshchagin. Transactions of MEI, Electric Power Engineering, issue 45, 1963.
3. H. C. Early, R. C. Walker. Communication and Electronics, vol. 31, 1957.
4. G. A. Sipailov, K. A. Khorkov. Power, energy, and machine constant of an impact generator. Izvestiya TPI, vol. 132, 1964.