Experimental Study of Intermittent Excitation on a Model of a Pulse Power Generator

G. A. Sipailov, A. I. Verkhoturov

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

Recently, complex transient processes in large electrical machines have been successfully studied on physical models that reproduce the entire complex of phenomena characterizing the process under investigation. This complex may also include aspects of the phenomena that do not lend themselves to mathematical description and cannot be taken into account in the process equations. In addition, physical models make it possible to considerably shorten the duration of investigations and the material costs involved, which is especially important for pulse generators, experimental studies of which are associated with currents of hundreds of thousands of amperes and voltages of tens of kilovolts and, consequently, are distinguished by the extremely high cost of the installations.

In pulse generators, in order to increase the pulse power and reduce no-load losses, forcing of the excitation is applied before short-circuiting the stator winding [1]. In the intervals between pulses it is advisable to reduce the voltage to zero or to a certain value, since operation of the machine at a high no-load voltage may lead to impermissible overheating. Therefore it is of practical interest to study, on a model, the mode of intermittent excitation of a pulse generator and to choose the optimum forcing coefficient from the standpoint of minimum energy losses per one stator-current pulse.

To establish this dependence, let us consider the transient process of switching the excitation winding of a pulse generator onto a constant voltage:

\[ k u_2 = i_2 r_2 + L_2 \frac{d i_2}{dt} + M_{23}\frac{d i_3}{dt}, \]

\[ 0 = i_3 r_3 + L_3 \frac{d i_3}{dt} + M_{23}\frac{d i_2}{dt}, \tag{1} \]

where \(k\) is the forcing coefficient,

\(r_2, L_2\) are the parameters of the excitation winding,

\(r_3, L_3\) are the parameters of the damper winding,

\(M_{23}\) is the mutual inductance of the rotor windings.

Here and below we assume that \(M_{23}=\sqrt{L_2L_3}\), i.e., leakage is absent [2], and that \(L_2 \approx L_3\).

Solving the system of equations (1), we find expressions for the currents:

\[ i_2=\frac{\kappa u_2}{r_2}\left(1-\frac{\delta_3}{\delta_2+\delta_3}e^{-\frac{t}{T}}\right), \]

\[ i_3=-\frac{\kappa u_2}{\sqrt{L_2L_3}\,(\delta_2+\delta_3)}e^{-\frac{t}{T}}, \tag{2} \]

where

\[ \delta_2=\frac{r_2}{L_2} \]

is the damping coefficient of the excitation-winding current,

\[ \delta_3=\frac{r_3}{L_3} \]

is the damping coefficient of the current in the damper winding,

\[ T=T_2+T_3 \]

is the total time constant of the circuits.

The flux producing the voltage in the stator winding can be determined [3]:

\[ \Phi=\Phi_0\left(1-e^{-\frac{t}{T}}\right), \]

\[ \Phi_0=\frac{M_{12}u_2}{r_2 w_1}, \tag{3} \]

where \(w_1\) is the number of turns of the stator winding.

If, without forcing the excitation, the flux rises to its nominal value in a time \(t=3T\), then, with forcing by a factor \(\kappa\), the flux will reach the same value in a shorter time \(t_1\). Hence the ratio of the fluxes is

\[ \frac{\kappa\left(1-e^{-\frac{t_1}{T}}\right)} {1-e^{-\frac{t}{T}}}=1. \tag{4} \]

From this expression one can obtain the time \(t_1\):

\[ t_1=-T\ln\frac{\kappa-0.95}{\kappa}. \tag{5} \]

The process of field suppression by changing the polarity of the applied voltage is described by the system of differential equations (1), if \(\kappa u_2\) is replaced by \(-\kappa u_2\).

The solution of the resulting system of equations has the form:

\[ i_{2\text{г}}=\frac{u_2}{r_2}\left(\frac{1+\kappa}{1+m}e^{-\frac{t'_1}{T}}-\kappa\right), \tag{6} \]

\[ i_{3\text{г}}=\frac{u_2}{r_2}m\,\frac{1+\kappa}{1+m}e^{-\frac{t'_1}{T}}, \tag{7} \]

where

\[ m=\frac{r_2}{r_3}\sqrt{\frac{L_2}{L_3}}. \]

The flux that must be extinguished:

\[ \Phi_{\mathrm{g}}=\Phi_{0}\left[(1+\kappa)e^{-t'_1/T}-\kappa\right]. \tag{8} \]

The extinction time can be determined from the condition

\[ i_{2\mathrm{g}}+i_{3\mathrm{g}}=0. \tag{9} \]

After substituting into equation (9) the values of \(i_{2\mathrm{g}}\) and \(i_{3\mathrm{g}}\) from (8) and (7), we solve it with respect to \(t'_1\):

\[ t'_1=T\ln\frac{\kappa+1}{\kappa}. \tag{10} \]

The energy losses in the steel of the impulse generator under forced excitation and de-excitation can be represented as

\[ Q_{\mathrm{st}}=\frac{P'_0}{\Phi_0^2}\left(\int_{0}^{t_1}\Phi^2\,dt+\int_{0}^{t'_1}\Phi_{\mathrm{g}}^2\,dt\right), \tag{11} \]

where \(P'_0\) is the no-load loss in the generator steel at flux \(\Phi_0\). After substituting into (11) the values of \(\Phi\) from (3), \(\Phi_{\mathrm{g}}\) from (8), and integrating, we obtain:

\[ Q_{\mathrm{st}}=P'_0T\left[\kappa^2\ln\frac{\kappa+1}{\kappa-0.95}-1.95\kappa+0.05\right], \tag{12} \]

Let us denote \(\kappa_{\mathrm{st}}=\dfrac{Q_{\mathrm{st}}}{Q'_{\mathrm{st}}}\), where \(Q'_{\mathrm{st}}\) is the energy loss in the steel per cycle without forcing; then

\[ \kappa_{\mathrm{st}}=0.556\left(\kappa^2\ln\frac{\kappa+1}{\kappa-0.95}-1.95\kappa+0.05\right). \tag{13} \]

The energy of the losses in the copper of the excitation and damper windings per cycle can be represented in the following form:

\[ Q_{\mathrm{m}}=\int_{0}^{t_1}\left(i_2^2 r_2+i_3^2 r_3\right)dt+\int_{0}^{t_1}\left(i_{2\mathrm{g}}^2 r_2+i_{3\mathrm{g}}^2 r_3\right)dt. \tag{14} \]

Or finally:

\[ Q_{\mathrm{m}}=\frac{u_2^2T}{r_2}\left[\kappa^2\ln\frac{1+\kappa}{\kappa-0.95}+B(0.05-1.95\kappa)\right]. \tag{15} \]

By analogy with the preceding, let us denote by \(\kappa_{\mathrm{m}}\) the ratio of the losses in the copper of the excitation and damper windings under forcing to the loss energy at \(\kappa=1\); then:

\[ \kappa_{\mathrm{m}}= \frac{ \kappa^2\ln\frac{1+\kappa}{\kappa-0.95}+B(0.05-1.95\kappa) }{ 3.69-1.9B }. \tag{16} \]

The optimal forcing coefficient can be determined from the condition of equality of the energy losses in the copper of the rotor windings and in the machine steel over the operating cycle:

\[ \frac{ P'_0 T\left(\kappa^2 \ln \frac{\kappa+1}{\kappa-0.95}-1.95\kappa+0.05\right) }{ \frac{u_2^2 T}{r_2} \left[ \kappa^2 \ln \frac{1+\kappa}{\kappa-0.95} +B(0.05-1.95\kappa) \right] } =1. \tag{17} \]

Or, after simplification:

\[ (1-\kappa_0)\kappa^2 \ln \frac{\kappa+1}{\kappa-0.95} -1.95\kappa(\kappa_0+B)+0.05(\kappa_0-B)=0, \tag{18} \]

where \(B=\dfrac{T_2}{T}\); \(\kappa_0=\dfrac{P'_0 r_2}{u_2^2}\) is the ratio of the losses in the steel to the losses in the excitation winding at no load.

For known \(\kappa_0\) and \(B\), this transcendental equation can be solved approximately with any desired degree of accuracy with respect to \(\kappa\).

Circuit of intermittent excitation of a model pulse generator

Figure 1 shows the circuit of intermittent excitation of a model pulse generator (PG).

When button \(P\) is pressed, voltage is applied through the contacts of contactor \(B\) to the excitation winding \(OVV\) of the rotating exciter \(Ya\), which is directly connected to the excitation winding of the pulse generator \(OVG\). As soon as the voltage of the pulse generator reaches the specified value, voltage relay \(RN\) operates and closes its contact in the circuit of the induction sensor \(ID\). When the generator emf passes through zero, the sensor produces short pulses of positive and negative polarity. The positive pulse opens the thyratron \(TG\) in the discharge circuit of capacitor \(C\), whose discharge current closes apparatus \(K_1\). A stator-current pulse of duration about \(0.02\) sec passes through the load \(L_n\). The stator-current pulse has a small negative half-wave, during which apparatus \(K_2\) opens the stator circuit.

When apparatus \(K_1\) closes, its contact in the circuit of contactor \(I\) and time relay \(RV\) closes, which leads to the closing of contacts \(I\) and the opening of contacts \(B\) in the exciter excitation circuit. The current in the exciter excitation winding changes direction, and as soon as the voltage of the pulse generator becomes equal to zero, time relay \(RV\) opens its contact in the circuit of contactor \(I\). Before the circuit is started again, apparatuses \(K_1\) and \(K_2\) must be returned to their initial state.

The changes in voltage at the generator terminals and in the current in the excitation winding over the operating cycle are shown in Fig. 2. Figure 3 shows the dependence of the no-load losses in the steel as a function of the output voltage. The losses were determined by the mutual-load method (points denoted by \(\odot\)) and by the run-down method (points denoted by \(\times\)). The dependences of the losses as a function of output voltage are well approximated by the function

\[ \frac{P'_0 u_1^2}{u_{1n}^2}. \]

In Fig. 4 the dependence of \(k_c\) as a function of the forcing coefficient is presented; curve 1 was calculated from equation (13), and the dashed curve was obtained experimentally. Curve 2 shows the dependence of the ratio of the total energy losses in copper and steel under forcing, \(Q_{\Sigma \kappa}\), to the total losses without forcing, \(Q_{\Sigma 1}\), as a function of \(\kappa\).

Fig. 1. Circuit of intermittent excitation of a pulse-generator model.

The dependence of the energy losses in the copper of the excitation winding per cycle on the forcing coefficient is presented in Fig. 5 (curve 1). The experimental curve is shown by a dashed line. Curve 2 was calculated from equation (16) and represents the dependence of the ratio of the total energy losses in the copper of the rotor windings under forcing to the energy losses without forcing. The optimum forcing coefficient, calculated from equation (18) for the parameters of the model, is equal to 2.1.

All calculations were carried out for the following parameters of the pulse-generator model:

\[ u_2 = 40\ \mathrm{V}, \qquad I_2 = 30\ \mathrm{A}, \qquad \frac{T_3}{T_2} = 1, \qquad T = 1.6\ \mathrm{sec}, \qquad \kappa_0 = 16.5, \qquad B = 0.5. \]

Fig. 2. Oscillogram of the variation of voltage at the generator terminals and of the excitation current

Fig. 3. Dependence of losses in the steel on the output voltage:
(○) — mutual-load test; (×) — coast-down test

Fig. 4. Dependence of \(K_c\) and \(\dfrac{Q_{\Sigma k}}{Q_{\Sigma 1}}\) on the forcing coefficient

Fig. 5. Dependence of \(K_m\) on the forcing coefficient

The experimental curves were obtained as follows. The variation of the output voltage \(u_1\) and the excitation current \(i_2\) over an operating cycle was oscillographed for various values \(k = 1, 2, 3, 4, 6\). At \(k = 1\), the voltage rose to the steady-state value of \(40\ \mathrm{V}\) in 4.7 seconds. For all forcing coefficients the voltage rose to \(40\ \mathrm{V}\), after which the generator stator winding was connected to the load \(L_n\), and after 0.02 sec. de-excitation was carried out. The ratio of the energy losses in the steel with forcing to the energy losses without forcing was determined as the ratio of the squares of the areas described by the curves of voltage variation at the generator terminals; and for the energy losses in the copper of the excitation winding, as the ratio of the squares of the areas bounded by the curves of current variation in the excitation winding over the operating cycle.

Thus, intermittent excitation at the optimum forcing coefficient makes it possible to reduce the electromagnetic energy losses per stator current pulse by a factor of five in comparison with intermittent excitation without forcing at the same pulse repetition frequency.

REFERENCES

1. I. M. Postnikov. On the design of a shock generator. Proceedings of LPI, No. 209, Gosenergoizdat, 1960.

2. O. B. Bron. Automation of magnetic-field quenching. Gosenergoizdat, 1961.

3. G. A. Sipaylov, A. I. Verkhoturov. On forcing the excitation of a shock generator. Izvestiya TPI, vol. 160, 1966.