Study of the Operation of a Shock Generator with an Inductive Load in Energy Storage Mode
Abstract Generated abstract
This paper analyzes the operation of a pulse generator feeding an inductive load in an energy accumulation circuit, focusing on the effect of rotor speed reduction during energy transfer. The authors formulate differential equations for the generator, load, and switching intervals, incorporate energy balance relations for rotor kinetic energy and electrical losses, and implement the process on an MN-14 analog computer for varying load inductance and initial rotor energy. The calculations indicate that assuming constant rotor speed can substantially overestimate attainable load current and stored electromagnetic energy, especially for larger load inductance or smaller flywheel energy reserves. The study concludes that finite rotor kinetic energy must be included in design estimates, and that increasing flywheel energy can improve accumulation in the load.
Full Text
PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
named after S. M. Kirov
Vol. 301 1975
STUDY OF THE OPERATION OF A PULSE GENERATOR WITH AN INDUCTIVE LOAD IN THE ENERGY-ACCUMULATION MODE
V. I. Andreev, E. I. Benyukov, K. A. Khor'kov
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
When a pulse generator operates with an inductive load, at the instant of the maximum current in the load an energy is concentrated that reaches 10–15% of the energy stored in the flywheel masses of the rotor. The fraction of energy transferred by the pulse generator to the load can be increased if energy accumulation in the load is carried out. A number of circuits implementing this idea are described in [1, 2, 3, 5]. The investigation of circuits for energy accumulation in an inductive load was carried out in the cited works in a first approximation, under the assumption that the rotational speed of the generator rotor is constant. Meanwhile, the rotational speed of the generator rotor decreases considerably in the process of transferring energy to the load, and this has a substantial effect on the calculation results. We shall study the influence of the decrease in the rotor rotational speed on the process of energy transfer to the load using the example of the simplest of the energy-accumulation circuits, shown in Fig. 1.
Fig. 1. Circuit for accumulation of energy in an inductive load from a pulse generator
The analysis of the phenomenon of sudden short circuit of a synchronous machine presents no difficulties [4]. Taking into account the pulsed nature of the operation of the pulse generator in the energy-accumulation circuit, one may use the law of constancy of flux linkage for the rotor circuits, i.e., assume that the active resistances of the field winding and the damper winding are zero. In this case the equations of equili-
... the stator winding e.m.f. when an inductive load is suddenly connected can be represented in the form
\[ (L_{\mathrm{g}}+L_{\mathrm{н}})\frac{di_{\mathrm{г}}}{dt}+(r_{\mathrm{г}}+r_{\mathrm{н}})i_{\mathrm{г}} =E_m\sin(\omega t+\psi), \tag{1} \]
where \(L_{\mathrm{г}}=L''_{d}\) is the subtransient inductive reactance of the stator winding of the shock generator,
\[ \begin{aligned} L_{\mathrm{н}}&\text{ is the load inductance,}\\ r_{\mathrm{г}}=r_{\mathrm{н}}&\text{ are the active resistances of the generator winding and the load,}\\ E_m&\text{ is the amplitude of the generator e.m.f.,}\\ \omega&\text{ is the angular frequency,}\\ \psi&\text{ is the phase angle of the e.m.f. at the moment of closing onto the load.} \end{aligned} \]
The switching device K-2 is open.
Fig. 2. Block diagram of the mathematical model of the process of energy accumulation in an inductive load
The process of current variation in the generator—load circuit is modeled on integrators 1 and 2. The block diagram of the mathematical model is shown in Fig. 2. As a rule, it is assumed that switching on the generator occurs at zero e.m.f. \((\psi=0)\). When the “start” button is pressed, the positive half-wave of the generator e.m.f. is applied through the normally closed contacts of relay P1 to the input of amplifiers 1 and 2, and the analog machine works out the solution of equation (1). When the current in the generator circuit reaches its amplitude value, the load is shunted by a short-circuit. From this moment in time \((t_1)\), the processes in the load and generator circuits are described independently of one another.
\[ L_{\mathrm{г}}\frac{di_{\mathrm{г}}}{dt}+r_{\mathrm{г}}i_{\mathrm{г}}=E_m\sin\omega t, \tag{2} \]
\[ L_{\mathrm{n}}\frac{di_{\mathrm{n}}}{dt}+r_{\mathrm{n}}i_{\mathrm{n}}=0. \tag{3} \]
In the mathematical-model circuit, the operation of closing K-2 is implemented by means of relays P1, P2, P3. The current in the load decays with the time constant of the load circuit. The generator current, under the action of the negative emf in the winding, tends to zero. At the instant when the generator current passes through zero \((t_2)\), K-1 is opened. This operation is implemented by relay P3. The generator operates in no-load mode until the end of the negative half-wave of the emf. At the instant when the generator emf passes through zero into the positive half-wave \((t_3)\), K-1 is again closed and the generator is short-circuited. Switching in the mathematical-model circuit is carried out by relay P2. The generator current rises sharply. The value of the generator current can be found from the solution of equation (2). The current in the load continues to vary according to equation (3). The current in the short-circuit branch falls:
\[ i_{\mathrm{z}}=i_{\mathrm{n}}-i_{\mathrm{g}}. \tag{4} \]
At the instant \((t_4)\), when the generator current reaches the value of the load current, \(i_{\mathrm{z}}=0\); the short-circuit is opened (K-2 is opened). The generator is connected to the load and delivers to the load the next portion of energy. The process of current variation in the generator-load circuit is described by equation (1). Switching in the model circuit at \(i_{\mathrm{z}}=0\) is carried out by relay P1. From this instant, the processes are repeated in the order described above.
As electromagnetic energy is accumulated in the load, the reserve of kinetic energy of the rotor decreases and the rotational speed of the rotor falls. The amplitude and angular frequency of the generator emf change. Formation of the generator emf \(e_{\mathrm{g}}\) is carried out by multiplication units BP24 and BP25 and by amplifiers 11--12. Correction of the instantaneous value of the angular frequency is performed by adder 7 and nonlinear unit BN14. Signals proportional to the initial value of the kinetic energy stored by the rotor of the impact generator \(T_0\), to the energy losses in the active resistances of the generator windings \(Q_{\mathrm{rg}}\) and of the load \(Q_{\mathrm{rn}}\), to the amount of energy transferred by the generator to the load \(Q_{\mathrm{n}}\), and to the instantaneous value of the electromagnetic energy in the generator winding \(Q_{\mathrm{g}}\) are applied to the input of amplifier 7. In this case
\[ Q_{\mathrm{n}}=\int_{0}^{t_1} e_{\mathrm{r}} i_{\mathrm{n}}\,dt+\int_{t_4}^{t_5} e_{\mathrm{r}} i_{\mathrm{n}}\,dt+\ldots . \tag{5} \]
The limits of integration \(0\)--\(t_1\), \(t_4\)--\(t_5\), ... correspond to the periods of energy accumulation in the load. In these same periods, the energy losses in the active resistance of the load are taken into account.
\[ Q_{\mathrm{rn}}=\int_{0}^{t_1} r_{\mathrm{n}}i_{\mathrm{n}}^{2}\,dt+\int_{t_4}^{t_5} r_{\mathrm{n}}i_{\mathrm{n}}^{2}\,dt+\ldots . \tag{6} \]
The integration limits are worked out by means of relay P4.
The energy stored in the electromagnetic field of the stator winding is
\[ Q_{\mathrm{g}}=L_{\mathrm{g}}\frac{i_{\mathrm{g}}^{2}}{2}. \tag{7} \]
During the first half-wave of the current, this energy is taken from the reserve of kinetic energy of the rotor, and then, during the second half-wave, as the generator current decreases, it is returned to the rotor. Energy losses in the active resistance of the stator winding
\[ Q_{rr}=\int_{0}^{t_2} r_r I_r^2\,dt+\int_{t_3}^{t_6} r_r I_r^2\,dt+\ldots, \tag{8} \]
where \(0—t_2,\ t_3—t_6\ldots\) are the periods corresponding to the flow of generator current.
The nonlinearity block performs the operation of extracting the square root
\[ \omega=\sqrt{\frac{2}{J}\left(T_0-Q_n-Q_g-Q_{gn}-Q_{rr}\right)}, \tag{9} \]
where \(J\) is the moment of inertia of the rotor.
The operation of a pulse generator on an inductive load in the energy-storage mode was investigated on an MN-14 analog computer. In the calculations the following initial parameters were adopted:
\(L_g=5\cdot10^{-5}\ \text{H};\ r_g=5\cdot10^{-4}\ \Omega;\ \omega_0=314;\ E_{m0}=22000\ \text{V};\ r_n=kr_g;\ L_n=kL_g;\ k=1,\ 2,\ 3,\ldots,10.\) The initial reserve of kinetic energy of the rotor was varied from \(T_0=5\cdot10^8\ \text{J}\) to \(2\cdot10^8\ \text{J}\). As an example, Figs. 3 and 4 give the curves obtained on the MN-14 for the variation of the generator and load currents, the generator emf, the angular frequency, and the kinetic energy of the rotor for the limiting values of \(T_0\) at \(L_n=L_g\).
Fig. 3. Calculated curves for \(T_0=5\cdot10^8\) J and a conversion coefficient equal to 0.05
The variants considered for \(L_n=L_g\) are summarized in Table 1. As the criterion for evaluating the efficiency of energy storage, the value of the pulse current (\(I_{\text{ud}}\)) of a sudden short circuit at the terminals of the stator winding at \(\omega=\text{const}\) and the amount of electromagnetic energy stored in this case in the leakage fields of the winding (\(Q_{kz}\)) were adopted.
Calculations show that a change in the rotational speed has a substantial effect on the magnitude of the maximum attainable current in the load and on the stored electromagnetic energy. The amplitude of the load current reaches a value close to that of the pulse current in three—
Fig. 4. Calculated curves for \(T_0 = 2 \cdot 10^8\) J and a conversion coefficient equal to 0.125.
four periods of energy accumulation. In the subsequent periods there occurs only compensation for the energy losses in the active resistance of the load. For the same parameters of the generator winding and the load, the energy delivered to the load decreases as the magnitude of the initial reserve of kinetic energy of the rotor \((T_0)\) decreases. As \(T_0\) decreases, the oscillations of the rotor rotation speed increase. Calculations show that, with an excessive decrease in the reserve of kinetic energy, the oscillations of the rotation speed become so large that the accumulation process may be disrupted, characterized by stoppage of the rotor of the pulsed generator. The generator current becomes aperiodic and decreases with a time constant of the short-circuited stator winding. For example, at \(T_0 = 200\) MJ and \(L_n = L_g\), such a disruption occurs in the 8th period of the process, although the remaining value of the rotor kinetic energy after the preceding operating period is ten times greater than the share of energy drawn into the load in this period.
The influence of the change in the rotor rotation speed on the energy-accumulation process manifests itself to an increasing degree when \(L_n > L_g\). Tables 2 and 3 give the values of the parameters under study for \(L_n = 4L_g\) and \(8L_g\), for \(T_0 = 200\) MJ. Contrary to the expected fourfold increase in accumulated energy at \(L_n = L_g\) (without taking into account the influence of speed) and the transfer of the full energy reserve \(T_0\) at \(L_n = 8L_g\), in the load we obtain only an amount of energy exceeding the energy of a sudden short circuit of the generator by 20–25%. The energy-accumulation process lasts 7–8 periods.
Thus, the studies have shown that the estimate of the energy delivered to the load from a pulsed generator for a finite value of the expended kinetic energy of the rotor must be made taking into account the influence of the rotor rotation speed on the accumulation process. The amount of energy accumulated in the load can be increased by increasing the reserve of kinetic energy of the rotor with the aid of a flywheel.
Table 1
| Parameter | Dimension | Energy-accumulation periods: 1 | 2 | 3 | 4 | 5 | 6 | Note |
|---|---|---|---|---|---|---|---|---|
| \(I_{\mathrm{n}}\) | rel. units | 0.50 | 0.725 | 0.83 | 0.873 | 0.892 | 0.892 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 500\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | MJ | 10.6 | 22.2 | 29.2 | 32.3 | 33.6 | 33.6 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 500\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | rel. units | 0.250 | 0.523 | 0.687 | 0.76 | 0.79 | 0.79 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 500\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\Delta Q\) | % | 31.9 | 34.1 | 19.7 | 9.9 | 4.4 | 0 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 500\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(T\) | % | 97.0 | 92 | 88 | 84.3 | 81 | 78 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 500\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\omega\) | % | 99 | 98 | 95 | 93.8 | 92 | 90.4 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 500\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(I_{\mathrm{n}}\) | kA | 643 | 917 | 1040 | 1100 | 1120 | 1130 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 333\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(I_{\mathrm{n}}\) | rel. units | 0.494 | 0.705 | 0.8 | 0.845 | 0.861 | 0.868 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 333\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | MJ | 10.3 | 21 | 27 | 30.3 | 31.5 | 32 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 333\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | rel. units | 0.243 | 0.494 | 0.635 | 0.713 | 0.741 | 0.753 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 333\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\Delta Q\) | % | 32.2 | 33.4 | 18.8 | 10.3 | 3.8 | 1.5 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 333\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(T\) | % | 96.0 | 88.3 | 81.8 | 75.8 | 70.7 | 65.6 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 333\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\omega\) | % | 98 | 94.7 | 91.5 | 88 | 85.3 | 81.8 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 333\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(I_{\mathrm{n}}\) | kA | 627 | 910 | 1030 | 1080 | 1110 | 1110 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 250\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(I_{\mathrm{n}}\) | rel. units | 0.483 | 0.70 | 0.793 | 0.83 | 0.854 | 0.854 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 250\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | MJ | 9.8 | 20.7 | 26.5 | 29.8 | 30.8 | 30.8 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 250\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | rel. units | 0.231 | 0.487 | 0.623 | 0.701 | 0.725 | 0.725 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 250\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\Delta Q\) | % | 31.8 | 35 | 19.0 | 10.9 | 3.3 | 0 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 250\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(T\) | % | 95 | 84 | 74.5 | 67 | 59.8 | 52.5 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 250\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\omega\) | % | 97.5 | 91.8 | 86.9 | 82 | 78 | 72.5 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 250\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(I_{\mathrm{n}}\) | kA | 605 | 882 | 1000 | 1040 | 1060 | 1060 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 200\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(I_{\mathrm{n}}\) | rel. units | 0.465 | 0.678 | 0.77 | 0.80 | 0.815 | 0.815 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 200\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | MJ | 9.14 | 19.4 | 25 | 27 | 28.4 | 28.4 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 200\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(Q_{\mathrm{n}}\) | rel. units | 0.215 | 0.457 | 0.588 | 0.635 | 0.67 | 0.67 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 200\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\Delta Q\) | % | 32.2 | 36 | 19.8 | 7.0 | 6.0 | 0 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 200\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(T\) | % | 94 | 81.2 | 69.3 | 60 | 50.6 | 40.6 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 200\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
| \(\omega\) | % | 97 | 88 | 82 | 77 | 70.7 | 63.7 | \(I_{\mathrm{ud}} = 1300\ \mathrm{kA}\) \(Q_{\mathrm{sc}}^{*} = 42.5\ \mathrm{MJ}\) \(T_0 = 200\ \mathrm{MJ}\) \(\omega_0 = 314\ \mathrm{el.\ rad/s}\) \(L_{\mathrm{n}} = L_{\mathrm{g}}\) |
Table 2
| Parameter | Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Note |
|---|---|---|---|---|---|---|---|---|---|
| $I_n$ | kA | 226 | 408 | 532 | 612 | 670 | 707 | 720 | $I_{yn}=1300$ kA |
| $I_n$ | rel. units | 0.174 | 0.314 | 0.408 | 0.47 | 0.515 | 0.544 | 0.554 | $I_{yn}=1300$ kA |
| $Q_n$ | mJ | 5.11 | 16.7 | 28.3 | 37.4 | 44.8 | 50.0 | 51.8 | $Q_{kz}=42.5$ mJ |
| $Q_n$ | rel. units | 0.120 | 0.393 | 0.666 | 0.88 | 1.05 | 1.18 | 1.22 | $Q_{kz}=42.5$ mJ |
| $\Delta Q$ | % | 9.8 | 22.4 | 22.4 | 17.5 | 14.3 | 10.0 | 3.6 | $T_0=200$ mJ |
| $\tau$ | % | 96 | 90.7 | 80.8 | 69.3 | 58.1 | 47.5 | 36.9 | $\omega_0=314$ el. rad/sec |
| $\omega$ | % | 98 | 95 | 90 | 83.8 | 76.8 | 69 | 60.6 | $L_n=4L_{\mathrm{g}}$ |
Table 3
| Parameter | Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Note |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \multicolumn{9}{c}{Energy accumulation periods} | |||||||||||
| \(I_{\mathrm{n}}\) | kA | 124 | 234 | 328 | 394 | 445 | 481 | 503 | 518 | 518 | \(I_{\mathrm{y}} = 1300\) kA |
| \(I_{\mathrm{n}}\) | p.u. | 0.095 | 0.18 | 0.252 | 0.303 | 0.343 | 0.370 | 0.387 | 0.40 | 0.40 | \(I_{\mathrm{y}} = 1300\) kA |
| \(Q_{\mathrm{n}}\) | MJ | 3.07 | 11.0 | 21.7 | 31.0 | 39.6 | 46.3 | 50.6 | 53.7 | 53.7 | \(Q_{\mathrm{n}} = 42.5\) MJ |
| \(Q_{\mathrm{n}}\) | p.u. | 0.072 | 0.259 | 0.511 | 0.73 | 0.934 | 1.09 | 1.19 | 1.26 | 1.26 | \(Q_{\mathrm{n}} = 42.5\) MJ |
| \(\Delta Q\) | % | 5.3 | 15.1 | 19.7 | 17.7 | 16.2 | 12.6 | 7.9 | 5.0 | 0 | \(T_{\mathrm{e}} = 200\) MJ |
| \(T\) | % | 96 | 94 | 88 | 78 | 68 | 57 | 47 | 36 | 26 | \(\omega_0 = 314\) electrical rad/s |
| \(\omega\) | % | 98 | 96 | 94 | 88 | 82 | 75.5 | 68 | 60 | 51 | \(L_{\mathrm{n}} = 8\) H |
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