POWER, ENERGY, AND THE MACHINE CONSTANT OF AN IMPULSE GENERATOR

G. A. SIPAILOV, K. A. KHOR'KOV

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

In the modern literature, the term impulse generators is applied both to impulse-power generators and to breaking-power generators, although it is known that, despite a certain commonality in their design, there are essential differences between them [1]. Breaking-power generators are manufactured as three-phase machines and are used in breaking-power laboratories for testing high-voltage oil circuit breakers; the operating mode of such generators is the transient mode of a sudden short circuit, using 10–20 symmetrical periods of current.

Impulse-power generators are manufactured predominantly as single-phase machines; their operating mode is the supertransient mode of a sudden short circuit or of sudden connection to a load for the duration of one maximum half-wave of current, using the aperiodic component. Impulse-power generators are used to obtain superpowerful magnetic fields [2, 3], to create a magnetic field in installations for obtaining plasma [4, 5], and to test high-speed equipment [6]. It is also possible to use an impulse-power generator as a storage device and energy source for pulsed supply of the excitation windings of charged-particle accelerators [7, 8].

The questions of the theory and calculation of impulse-power generators have not yet received proper development. According to the existing calculation procedure for impulse- and breaking-power generators, when the principal dimensions are selected, the geometry of a standard generator intended for continuous operation is taken as the basis. The selection of many parameters is made mainly according to the intuition of the designer; the impulse power is determined at the end of the calculation. Therefore the question of creating a method for calculating an impulse-power generator, taking into account the principal features of its operation, has become urgent.

Since, when an impulse generator is used as a source for creating magnetic fields with large energy, the principal quantity characterizing the operation of the generator is the energy that the generator can deliver to the load, it will naturally be expedient to take the magnitude of this energy as the basis when selecting the principal dimensions of the generator.

In connection with this, in the present work, on the basis of the expressions obtained for the impact power, energy, and machine constant of an impulse generator, a relation is established between them which can serve as a basis in determining the principal dimensions of an impact-power generator.

Equations for Power and Energy

The maximum power of an impulse generator \(P_{\text{ud}}\) can be defined as the product of the maximum value of the current of the first half-wave \(I_{\text{ud}}\) (taking into account its aperiodic component) and the maximum value of the emf \(E_m\)

\[ P_{\text{ud}} = I_{\text{ud}} \cdot E_m . \tag{1} \]

The known expressions for the impact current [9, 10] are either very approximate,

\[ I_{\text{ud}} = \frac{1.8 \cdot 1.05 \cdot \sqrt{2}\, E}{X_d''}, \tag{2} \]

or too cumbersome:

\[ \begin{aligned} i_{\text{ud}}={}& - I_{dm}\cos(\omega t+\gamma_0) -(I'_{dm}-I_{dm})\cos(\omega t+\gamma_0)e^{-t/T'_d} \\ &-(I''_{dm}-I'_{dm})\cos(\omega t+\gamma_0)e^{-t/T''_d} + I_{am}\cos\gamma_0\, e^{-t/T_a}, \end{aligned} \tag{3} \]

where \(I_{dm}\), \(I'_{dm}\), \(I''_{dm}\), \(I_{am}\) are, respectively, the maximum values of the steady-state, transient, subtransient, and aperiodic components of the sudden short-circuit current, determined from the known equations, with \(I_{am}=I''_{dm}\);

\(T'_d\), \(T''_d\), \(T_a\) are the decay time constants, respectively, of the transient, subtransient, and aperiodic current components;

\(\gamma_0\) is the switching phase.

Since, in the case under consideration, only the first half-wave of the current is of interest, for its calculation equation (3) can be substantially simplified. For example, as a first approximation one may assume that the periodic component of the sudden short-circuit current is determined only by the subtransient current decaying with time constant \(T''_d\) (4,a), or is not decaying at all (4,b):

\[ i_n = I''_{dm}\cos(\omega t+\gamma_0)e^{-t/T''_d}, \tag{4,a} \]

\[ i_n = I''_{dm}\cos(\omega t+\gamma_0). \tag{4,b} \]

A comparative calculation of the periodic component of the current by equation (3) and by the simplified equations (4,a) and (4,b), carried out for a large number of generators, confirmed the possibility of such simpli-

tions. Let us present the results of calculating the periodic component of the current for a single-phase short circuit of the TI-25-2 generator (Table 1), having the following parameters:

\[ \begin{aligned} X_d'' &= 0.041, & T_d'' &= 0.286\ \text{s}, & I_d'' &= 15.2;\\ X_d' &= 0.177, & T_d' &= 2.29\ \text{s}, & I_d' &= 9.0;\\ X_d &= 1.290, & T_d &= 9.90\ \text{s}, & I_d &= 2.07;\\ X_2 &= 0.077, & T_a &= 0.10\ \text{s}, & I_a &= 15.2;\\ X_0 &= 0.080. \end{aligned} \]

Table 1

It is evident that calculation of the periodic component of the current by equation (4,a) gives underestimated values, while by equation (4,b) it gives overestimated values in comparison with the calculation data according to equation (3); in both cases the error is less than 2%. The calculation error can be substantially reduced if the values obtained from equations (4,a) and (4,b) are averaged. In this case the simplified

the equations for calculating the periodic component (5) and the total surge current (6) will have the form:

\[ i_n = I''_{dm}\cdot \cos(\omega t+\gamma_0)\, \frac{1+e^{-\frac{t}{T''_d}}}{2}; \tag{5} \]

\[ \begin{aligned} i_{\text{sur}}={}&-I''_{dm}\cdot \cos(\omega t+\gamma_0)\, \frac{1+e^{-\frac{t}{T''_d}}}{2} \; + \\ &+ I''_{dm}\cdot \cos\gamma_0 \cdot e^{-\frac{t}{T_a}} . \end{aligned} \tag{6} \]

The surge current reaches its maximum value at \(\omega t=180^\circ\) and \(\gamma_0=0\)

\[ I_{\text{sur}}= I''_{dm}\, \frac{1+e^{-\frac{\pi}{\omega T''_d}}+2e^{-\frac{\pi}{\omega T_a}}}{2} = 2I''_{dm}k_{\text{st}}, \tag{7} \]

where \(k_{\text{st}}\) is the current attenuation coefficient, with

\[ k_{\text{st}}= \frac{1+e^{-\frac{\pi}{\omega T''_d}}+2e^{-\frac{\pi}{\omega T_a}}}{4}. \tag{8} \]

To a first approximation

\[ e^{-\frac{\pi}{\omega T}} \approx 1-\frac{\pi}{\omega T}, \]

then

\[ k_{\text{st}}=1-\frac{0.005}{T_a}-\frac{0.0025}{T''_d}. \tag{8a} \]

Power of the surge generator

\[ P_{\text{sur}}=I_{\text{sur}}E_m=2I''_{dm}E_m k_{\text{st}}. \tag{9} \]

In the ideal case, in the absence of attenuation, \(k_{\text{st}}=1\)

\[ P_{\text{sur}}=2I''_{dm}E_m. \tag{9a} \]

In this case, the electromagnetic energy of the surge generator at \(\gamma_0=0\) will be

\[ \begin{aligned} Q &=\int_0^{\pi/\omega}(i_n+i_a)\,e\,dt = -\int_0^{\pi/\omega} I''_{dm}\cos\omega t\cdot E_m\sin\omega t\cdot dt \; + \\ &\quad +\int_0^{\pi/\omega} I''_{dm}E_m\sin\omega t\cdot dt =\frac{2}{\omega} I''_{dm}E_m . \end{aligned} \tag{10} \]

In the real case, for a sudden short circuit, taking attenuation into account, this energy will be somewhat reduced:

\[ Q=-\int_0^{\pi/\omega} I''_{dm}E_m\cos\omega t\cdot \sin\omega t\, \frac{1+e^{-\frac{t}{T''_d}}}{2}\cdot dt \; + \]

\[ + \int_{0}^{\pi/\omega} I''_{dm} E_m \sin \omega t\, e^{-t/T_a}\, dt = \]
\[ = \frac{\frac{1}{2}\omega I''_{dm}E_m}{\left(\frac{1}{T''_d}\right)^2+(2\omega)^2} \left(e^{-\frac{\pi}{\omega T''_d}}-1\right) + \]
\[ + \frac{\omega I''_{dm}E_m}{\left(\frac{1}{T_a}\right)^2+\omega^2} \left(1+e^{-\frac{\pi}{\omega T_a}}\right). \tag{11} \]

Since \(\left(\dfrac{1}{T_a}\right)^2 \ll \omega^2\) and \(\left(\dfrac{1}{T''_d}\right)^2 \ll (2\omega)^2\), then, neglecting the terms \(\left(\dfrac{1}{T_a}\right)^2\) and \(\left(\dfrac{1}{T''_d}\right)^2\), we obtain:

\[ Q=\frac{I''_{dm}E_m}{\omega} \left[ \frac{7}{8} +e^{-\frac{\pi}{\omega T_a}} +\frac{1}{8}e^{-\frac{\pi}{\omega T''_d}} \right] = \frac{2}{\omega}I''_{dm}E_m \kappa_{\mathrm{en}}, \tag{11a} \]

where \(\kappa_{\mathrm{en}}\) is the energy damping coefficient

\[ \kappa_{\mathrm{en}}= \frac{ \frac{7}{8} +e^{-\frac{\pi}{\omega T_a}} +\frac{1}{8}e^{-\frac{\pi}{\omega T''_d}} }{2} \tag{12} \]

or, in a first approximation,

\[ \kappa_{\mathrm{en}}=1-\frac{0.005}{T_a}-\frac{0.000625}{T''_d}. \tag{12a} \]

It follows from equations (8a) and (12a) that the current damping coefficient is somewhat smaller than the energy damping coefficient; however, the difference between the coefficients \(\kappa_{\mathrm{cur}}\) and \(\kappa_{\mathrm{en}}\) is small. For example, for the generator considered above, the difference amounts to only \(0.69\%\) of the absolute value of the coefficient \(\kappa_{\mathrm{cur}}\). Therefore, in calculations one may use, both for current and for energy, a single damping coefficient \(\kappa_d\) according to (8a).

If excitation forcing is used, the latter must be taken into account by the coefficient \(\kappa_f^2\); in this case formula (11) takes the form:

\[ Q=\frac{2}{\omega}I''_{dm}E_m\kappa_d\kappa_f^2. \]

However, when a powerful damping system is present and when the repetition frequency of the pulses is high, excitation forcing of impulse generators used in power-supply circuits for charged-particle accelerators is inadvisable. Therefore, in the subsequent exposition we take \(\kappa_f=1\).

Effect of the load on the magnitude of the energy delivered by the generator

The magnitude of the energy that an impulse generator can deliver to the load depends on the load parameters. In the case under consideration,

the generator is intended for operation into an inductive load \((X_{\mathrm{n}} \gg r_{\mathrm{n}})\); in this case the magnitude of the load is taken into account by reducing the current amplitude

\[ I''_{dm\mathrm{n}}=\frac{2E_m}{X'_d+X_2+2X_{\mathrm{n}}} =\frac{E_m}{X_{\mathrm{ud}}+X_{\mathrm{n}}} \tag{13} \]

and by changing the value of the damping time constant

\[ T_{a\mathrm{n}}=\frac{X_2+X_{\mathrm{n}}}{\omega\left(r_1+r_{\mathrm{n}}\right)}, \tag{14} \]

where \(X_{\mathrm{ud}}\) and \(r_1\) are the inductive and active resistances of the impulse generator, respectively, with

\[ X_{\mathrm{ud}}=\frac{X''_d+X_2}{2}; \]

\(X'_d\) and \(X_2\) are the inductive supertransient reactance and the negative-sequence reactance.

The change in \(T_a\) has a substantial effect on the magnitude of the damping coefficient. Table 2 presents calculated values of the damping coefficient of current (and energy) as a function of the ratio \(\dfrac{X_{\mathrm{n}}}{X_{\mathrm{ud}}}\) at \(r_{\mathrm{n}}=0\), for various values of the damping time constant \(T\).

Table 2

When an impulse generator is suddenly switched onto a load, the portion of the kinetic energy of the rotating rotor that is converted into the energy of the magnetic field of the load and of the stator leakage fields, \(Q_{\mathrm{em}}\), will be reduced in comparison with the short-circuit energy \(Q\). This reduction in energy can be taken into account by the coefficient \(\kappa_{\mathrm{em}}\)

\[ \kappa_{\mathrm{em}}=\frac{Q_{\mathrm{em}}}{Q} =\frac{X_{\mathrm{ud}}}{\left(X_{\mathrm{ud}}+X_{\mathrm{n}}\right)} \cdot\frac{\kappa_{\mathrm{zn}}^{2}}{\kappa_{z}^{2}}, \tag{15} \]

where \(\kappa_{\mathrm{zn}}\) is the damping coefficient of the impulse current under load; then

\[ Q_{\mathrm{em}}=\kappa_{\mathrm{em}}Q =\frac{2}{\omega} I''_{dm}E_m\kappa_z\kappa_{\mathrm{em}}. \tag{16} \]

Table 3 presents the calculated values of the coefficient \(\kappa_{\text{em}}\) as a function of the ratio \(\dfrac{X_{\text{n}}}{X_{\text{ud}}}\) at \(r_{\text{n}}=0\) for various values of the impact-current damping coefficient.

Table 3

Figure 1 shows the dependence of \(\kappa_{\text{em}}\) on \(\dfrac{X_{\text{n}}}{X_{\text{ud}}}\) for various \(r_{\text{n}}\).

The part of the electromagnetic energy that is converted into the energy of the magnetic field of the load, \(Q_{\text{n}}\), can be determined from the proportion

\[ \frac{Q_{\text{n}}}{Q_{\text{em}}} = \frac{X_{\text{n}}}{X_{\text{n}}+X_{\text{ud}}} = \kappa . \]

Hence

\[ Q_{\text{n}}=\kappa Q_{\text{em}}=\kappa \kappa_{\text{em}} Q=\kappa_{\text{n}}Q . \]

Fig. 1. Dependence of the coefficient \(\kappa_{\text{em}}\) on the ratio \(\dfrac{X_{\text{n}}}{X_{\text{ud}}}\) for \(1—r_{\text{n}}=0;\ 2—r_{\text{n}}=r_{\text{g}};\ 3—r_{\text{n}}=3r_{\text{g}}\).

Fig. 2. Dependence of the coefficient \(\kappa_{\text{n}}\) on the ratio \(\dfrac{X_{\text{n}}}{X_{\text{ud}}}\) at \(r_{\text{n}}=0\).

where \(\kappa_{\mathrm{n}}\) is the load factor, indicating what part of the electromagnetic energy corresponding to a short circuit is converted into the energy of the magnetic field of the load, with

\[ \kappa_{\mathrm{n}}=\kappa\cdot\kappa_{\mathrm{em}} = \frac{X_{\mathrm{ud}}X_{\mathrm{n}}}{(X_{\mathrm{ud}}+X_{\mathrm{n}})^2}\cdot \frac{\kappa_{\mathrm{zn}}^2}{\kappa_{\mathrm{z}}^2}. \tag{17} \]

For convenience in calculating the coefficient \(\kappa_{\mathrm{n}}\), equation (17) may be written as

\[ \kappa_{\mathrm{n}}= \frac{y}{(1+y)^2}\cdot \frac{\kappa_{\mathrm{zn}}^2}{\kappa_{\mathrm{z}}^2}, \tag{17a} \]

where

\[ y=\frac{X_{\mathrm{n}}}{X_{\mathrm{ud}}}. \]

The values of the coefficient \(\kappa_{\mathrm{n}}\) as a function of load are presented in Fig. 2.

Machine Constant of an Impulse Generator

Existing methods for calculating normal-design alternating-current machines relate the principal dimensions to the rated, or apparent, power of a three-phase machine by means of the so-called machine constant. In the case of calculating a single-phase machine, the corresponding conversion of the apparent power is performed. Therefore, in order to establish the relation between the principal dimensions of an impulse generator and its power, it is necessary, first, to find the dependence between the impulse power and the rated power of a single-phase synchronous machine in whose dimensions the impulse generator is made.

The rated, or apparent, power of a single-phase synchronous machine is

\[ P_1=U_1\cdot I_1=\frac{U_{m1}\cdot I_{m1}}{2}. \]

The ratio of the powers of an impulse generator and a single-phase synchronous machine made in the same dimensions is

\[ \frac{P_{\mathrm{ud}}}{P_1} = \frac{2E_m I''_{dm}\kappa_{\mathrm{z}}}{\frac{1}{2}U_{m1}I_{m1}} = 4\kappa_{\mathrm{z}}\kappa_i\frac{E_m}{U_{m1}}, \]

where \(\kappa_i\) is the impulse-current coefficient, with

\[ \kappa_i=\frac{I''_{dm}}{I_{m1}}= \frac{E_m}{X_{\mathrm{ud}}}\cdot\frac{1}{I_{m1}}. \]

In relative units,

\[ x_{\mathrm{ud}}=X_{\mathrm{ud}}\;(\mathrm{ohm})\cdot\frac{I_{m1}}{U_{m1}}, \]

then, taking \(U_{m1}=E_m\), which is quite permissible, we find:

\[ \kappa_i=\frac{1}{x_{\mathrm{ud}}}, \tag{18} \]

\[ \frac{P_{\mathrm{ud}}}{P_1}=4\kappa_{\mathrm{z}}\kappa_i. \tag{19} \]

The ratio between the apparent powers of single-phase and three-phase synchronous machines, built in the same frame sizes with the same number of slots and with single-layer windings on the stator, can be determined by equation (20), which is obtained from the condition of equality of the stator copper losses of the two machines,

\[ \kappa_m=\frac{P_1}{P_3}=\frac{2}{3}\sqrt{\frac{z'_1}{z_1}\cdot \sin \frac{z_1}{z'_1}\frac{\pi}{2}}, \tag{20} \]

where \(z'_1\) is the number of stator slots,
\(z_1\) is the number of filled stator slots.

Usually

\[ \frac{z_1}{z'_1}=\frac{2}{3}, \]

and in this case

\[ \kappa_m=\frac{1}{\sqrt{2}}. \]

For three-phase machines, the calculated power is related to the principal dimensions by means of Arnold’s machine constant \(C_A\):

\[ P_3=\frac{D^2\cdot l\cdot n}{C_A}. \tag{21} \]

Expressing the impact power through the calculated power of the three-phase machine, we find

\[ P_{\text{ud}}=4\kappa_z\kappa_t\kappa_m P_3 =4\kappa_z\kappa_t\kappa_m\frac{D^2ln}{C_A} =\frac{D^2ln}{C_{\text{ud}}}, \tag{22} \]

where \(C_{\text{ud}}\) is the machine constant of the impact generator, with

\[ C_{\text{ud}}=\frac{C_A}{4\kappa_z\kappa_t\kappa_m}. \tag{23} \]

For the mean value of the damping coefficient \(\kappa_z=0.93\), the machine constant of a single-phase impact generator with

\[ \frac{z_1}{z'_1}=\frac{2}{3} \]

will be equal to

\[ C_{\text{ud}}=0.38\,x_{\text{ud}}C_A. \tag{24} \]

It follows from equation (24) that the machine constant of an impact generator is not a constant quantity; it changes with variation in the generator power and its reactance. For example, for impact generators built in the frames of turbogenerators:

\[ \text{at } 50\text{--}150\ \text{MVA}\quad C_{\text{ud}}=(0.7\div 0.9)x_{\text{ud}}\ \frac{\text{cm}^3}{\text{J}}; \]

\[ \text{at } 5\text{--}10\ \text{MVA}\quad C_{\text{ud}}=(1.1\div 1.4)x_{\text{ud}}\ \frac{\text{cm}^3}{\text{J}}. \]

For an impact generator built in the frame of a 30 MVA turbogenerator, the machine constant in \(\text{cm}^3/\text{J}\) is numerically equal to the reactance \(x_{\text{ud}}\), expressed in relative units.

The obtained relationships between energy and power and the principal dimensions make it possible to determine the power of an impact generator for producing the required energy in the load with specified parameters and to determine the volume of the active part of the generator for the adopted prelimi-

relative value of the reactance \(x_{ud}\), which is desirable and possible to obtain in the machine being designed.

The indicated relations are the basis for developing a method of calculating impulse-power generators, which will be presented in subsequent papers.

REFERENCES

1. I. M. Postnikov, On the design of an impulse generator, Trudy LPI, 209, 1960.

2. P. L. Kapitza, Proc. Roy. Soc. A, 115, No. 772, 1927; A, 42, No. 235, 1930.

3. I. D. Cockcroft, Trans. Roy. Soc. A, 227, 1928.

4. T. Miyamoto et al., Methods of obtaining stable plasma. Atomic Technology Abroad, No. 10, 1958.

5. A. P. Bezbatchenko et al., Physics of plasma and problems of controlled thermonuclear reactions. Vol. IV, Publishing House of the Academy of Sciences of the USSR, 1958.

6. O. B. Bron, L. B. Halperin, Experiments on interrupting very large short-circuit currents. Elektrichestvo, No. 4, 1940.

7. Ya. P. Terletsky, On the stability of electron motion in betatron-type induction accelerators. Scientific Notes of Moscow State University, 95, book 4, 1946.

8. V. V. Ivashin, G. A. Sipailov, Commutation of the current of an impulse generator. Problems in the theory and design of electrical machines. Interuniversity collection of papers, Novosibirsk, 1963.

9. L. M. Piotrovsky, M. P. Kostenko, Electrical Machines. Gosenergoizdat, 1959.

10. M. P. Kostenko, Electrical Machines. Special Part, Gosenergoizdat, 1949.