Calculation of Optimal Reverse Voltage Values with Allowance for Saturation during Field Suppression of a Shock Generator

CALCULATION OF OPTIMAL REVERSE VOLTAGE VALUES TAKING SATURATION INTO ACCOUNT DURING FIELD SUPPRESSION OF A SURGE GENERATOR

G. A. Sipaylov, A. I. Verkhoturov

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

The problem of field suppression for a surge generator is of particular importance because of the specific conditions of its operation. To increase pulse power, surge generators, before the stator-current pulse, are often excited to a voltage higher than the rated voltage; therefore, leaving the voltage after the stator-current pulse at such a high level even for a short time is dangerous from the standpoint of heating conditions.

In [1] it was shown that, when suppressing the field of a surge generator, there exists such a value of the reverse voltage at which the total electromagnetic no-load losses released in the generator during the suppression time are minimal. However, in calculating the optimal reverse voltage values, saturation of the magnetic circuit was not taken into account; and since surge generators, in order to make better use of active materials, are made with a saturated magnetic circuit, the assumption of constant inductance may lead to significant errors in the calculation.

This article considers the influence of changes in saturation of the magnetic circuit on the choice of optimal values of the reverse voltage applied to the field winding during field suppression, and gives an estimate of the error obtained when calculating the optimal coefficients without taking variable saturation into account.

When calculating transients in excitation circuits with allowance for

Fig. 1. On accounting for saturation by the method of piecewise-linear approximation

saturation, the method of piecewise-linear approximation [2] is most widely used; in this case the magnetization curve is, as a rule, replaced by two straight lines \(oa\) and \(ab\) (Fig. 1).

To solve the stated problem, let us consider field suppression of a saturated machine by changing the polarity of the voltage applied to the field winding of the surge generator. Neglecting the leakage fluxes between the damping and field windings, for the portion of characteristic \(ab\) (Fig. 1) we obtain the following system of differential equations:

\[ \left. \begin{aligned} r_2 i_2 + L_{2\beta}\frac{di_2}{dt} + \sqrt{L_{2\beta}L_{3\beta}}\frac{di_3}{dt} &= -\kappa u_2,\\ r_3 i_3 + L_{3\beta}\frac{di_3}{dt} + \sqrt{L_{2\beta}L_{3\beta}}\frac{di_2}{dt} &= 0 \end{aligned} \right\}, \tag{1} \]

\(r_2, r_3\) are the active resistances of the field and damping windings;

\(L_{2\beta}, L_{3\beta}\) are the inductances of the same windings in the saturated region;

\(\kappa\) is the multiple of the reverse voltage applied to the field winding relative to the nominal voltage.

The solution of this system of equations has the form:

\[ i_2 = -\frac{u_2}{r_2}\left[(1+\kappa)B e^{-\frac{t_1}{T_\beta}}-\kappa\right], \tag{2} \]

\[ i_3 = -\frac{u_2}{r_2}(1+\kappa)(1-B)e^{-\frac{t_1}{T_\beta}}, \tag{3} \]

where

\[ T_{2\beta}=\frac{L_{2\beta}}{r_2},\quad T_{3\beta}=\frac{L_{3\beta}}{r_3} \]

are the time constants of the field and damping windings in the saturated region;

\[ T_\beta = T_{2\beta}+T_{3\beta} \]

is the total rotor time constant in the saturated region;

\[ B=\frac{T_{2\beta}}{T_\beta} \]

is the ratio of the field-winding time constant to the total time constant in the saturated region.

The main flux in the machine is determined by the total current of both windings and by the coefficient of mutual induction between the rotor and the stator. On the portion \(ab\) (Fig. 1) the flux varies according to the relation:

\[ \Phi_\beta = \frac{u_2 M_\alpha}{r_2 w_1} \left\{ m(1-c)+c\left[(1+\kappa)e^{-\frac{t_1}{T_\beta}}-\kappa\right] \right\}, \tag{4} \]

where

\(M_\alpha\) is the coefficient of mutual induction between the rotor and the stator in the unsaturated region;

\[ m=\frac{I_k}{I_0} \]

is the ratio of the field current corresponding to the knee point \(a\) to the nominal current;

\(w_1\) is the number of turns of the stator winding;

\[ c=\frac{M_\beta}{M_\alpha} \]

is the ratio of the mutual inductances in the saturated and unsaturated portions.

The field-suppression time on the first segment can be determined from the condition that the total current of the rotor windings at the inflection point is equal to \(m \dfrac{u_2}{r_2}\):

\[ t_1 = T_\beta \ln \frac{\kappa + 1}{\kappa + m}. \tag{5} \]

Let us express the energy of the losses released in the steel of the machine during field suppression on segment \(ba\) as a function of the coefficient \(\kappa\)

\[ Q_{c\beta} = \frac{P_0' \cdot T_\beta}{[c + m(1-c)]^2} \left\{ \ln \frac{\kappa + 1}{\kappa + m} [\ln(1-c)-c\kappa]^2 + (1-m)\cdot c(2m - \right. \]

\[ \left. -\,1.5mc - \kappa c + 0.5) \right\}. \tag{6} \]

where \(P_0'\) is the no-load loss in the steel of the machine.

In the excitation winding

\[ Q_{v\beta} = \int_0^{t_1} i_2^2 r_2 \cdot dt = \frac{u_2^2 \cdot T_\beta}{r_2} \left\{ \kappa^2 \ln \frac{\kappa + 1}{\kappa + m} + (m-1)B \times [2\kappa - \right. \]

\[ \left. -\,B(\kappa + 0.5m + 0.5)] \right\}. \tag{7} \]

In the damper winding

\[ Q_{d\beta} = \int_0^{t_1} i_3^2 \cdot r_3 \cdot dt = -\frac{u_2^2}{r_2}(1-B)T_\beta(m-1)(\kappa + 0.5m + 0.5). \tag{8} \]

Adding the right-hand sides of equations (7) and 8), we obtain the energy of the losses released in the damper winding and the excitation winding:

\[ Q_{v\beta} + Q_{d\beta} = \frac{u_2^2}{r_2}T_\beta \left[ \kappa^2 \ln \frac{\kappa + 1}{\kappa + m} + B(m-1)(\kappa - 0.5m - 0.5) \right]. \tag{9} \]

After the transition of the magnetic system into the unsaturated state (segment \(oa\) in Fig. 1), the system of differential equations will have the previous form (1), but quantities with the index \(\alpha\) will enter them instead of the quantities with the index \(\beta\).

Taking into account the new initial conditions

\[ i_2(0) + i_3(0) = m\frac{u_2}{r_2}, \tag{10} \]

the expressions for the currents take the form:

\[ i_2 = \frac{u_2}{r_2} \left[ (\kappa + m)B e^{-\frac{t_2}{T_\alpha}} - \kappa \right], \tag{11} \]

\[ i_3 = \frac{u_2}{r_2}(\kappa + m)(1-B)e^{-\frac{t_2}{T_\alpha}}, \tag{12} \]

where

\[ T_{2\alpha} = \frac{L_{2\alpha}}{r_2}, \qquad T_{3\alpha} = \frac{L_{3\alpha}'}{r_3} \]

are the time constants of the excitation and damper windings in the unsaturated region,

\[ T_\alpha = T_{2\alpha} + T_{3\alpha} \]

is the total time constant of the rotor windings in the unsaturated region.

Since each of the sections of the rotor winding is under identical conditions with respect to the magnetic circuit, the ratio \(\dfrac{T_{2\alpha}}{T_\alpha}\) will be the same as \(-\dfrac{T_{2\beta}}{T_\beta}\), i.e., it can likewise be denoted by \(B\).

The suppression time is determined from the condition:

\[ i_2+i_3=0, \tag{13} \]

whence

\[ t_2=T_\alpha \ln \frac{\kappa+m}{\kappa}. \tag{14} \]

The flux in section \(oa\) is

\[ \Phi_\alpha=\frac{u_2 M_\alpha}{r_2 w_1} \left[(\kappa+m)e^{-\frac{t_2}{T_\alpha}}-\kappa\right]. \tag{15} \]

The dependence of the losses in the steel on \(\kappa\) for section \(oa\) is

\[ Q_{c\alpha}= \frac{P_0' T_\alpha}{[c+m(1-c)]^2} \left[ \kappa^2 \ln \frac{\kappa+m}{\kappa} +m(0.5m-\kappa) \right]. \tag{16} \]

The loss energy released in the field winding in section \(oa\) is equal to

\[ Q_{B\alpha}=\int_0^{t_2} i_2^2 \cdot r_2 \cdot dt = \frac{u_2^2}{r_2}T_\alpha \left[ \kappa^2 \ln \frac{\kappa+m}{\kappa} +mB(0.5mB+B\kappa-2\kappa) \right]. \tag{17} \]

In the damper winding,

\[ Q_{D\alpha}=\int_0^{t_2} i_3^2 \cdot r_3 \cdot dt = \frac{u_2^2}{r_2}T_\alpha mB(1-B)(0.5m-\kappa). \tag{18} \]

The total losses in the rotor windings during suppression for section \(oa\) are

\[ Q_{B\alpha}+Q_{D\alpha} = \frac{u_2^2}{r_2}T_\alpha \left[ \kappa^2 \ln \frac{\kappa+m}{\kappa} +0.5Bm(m-1) \right]. \tag{19} \]

Adding the energy losses in the machine steel (6), (16) and in the rotor windings (9) and (19), we obtain the dependence of the total losses on the coefficient of forced field suppression \(\kappa\):

\[ \begin{aligned} Q_{\Sigma n}=&\frac{u_2^2}{r_2}T_\alpha \Bigg\{ \kappa^2 \ln \frac{(\kappa+1)^c(\kappa+m)^{1-c}}{\kappa} +B n(0.5m-\kappa) \\ &+cB(m-1)(\kappa-0.5m-0.5) +\frac{p}{[c+m(1-c)]^2}\times \\ &\times \Big[ \kappa^2 \ln \frac{\kappa+m}{\kappa} +m(0.5m-\kappa) +c[m(1-c)-c\kappa]^2 \ln \frac{\kappa+1}{\kappa+m} \\ &\qquad +c^2(1-m)(2m-1.5mc-\kappa c+0.5c) \Big] \Bigg\}, \end{aligned} \tag{20} \]

where \(p=\dfrac{P_0' r_2}{u_2^2}\) is the ratio of the no-load electromagnetic losses in the steel to the losses in the field winding.

The value \(k\) at which the total electromagnetic no-load losses during field suppression of the machine will be minimal can be found by setting the derivative of \(Q_{\Sigma\mathrm{n}}\) with respect to \(k\) equal to zero,

\[ \frac{dQ_{\Sigma\mathrm{n}}}{dk}=0. \tag{21} \]

The transcendental equation (21) can be solved approximately with any desired degree of accuracy with respect to \(k\). The values of the field-forcing coefficients at which the losses will be minimal are denoted by \(k_{\mathrm{ogn}}\) and are called optimal. The values of the optimal field-forcing coefficients, determined from the solution of equation (21) for \(p=10\) with variation of

\[ \frac{T_3}{T_2}=0.8 \div 1,\quad m=0.5,\quad c=0.3, \]

are presented in Table 1. The same table gives the values of the optimal coefficients \(k_{\mathrm{og}}\), determined for these same parameters but at constant saturation.

Table 1

As is evident from Table 1, the optimal coefficients calculated with allowance for variable saturation and without allowance for saturation, for the same \(\dfrac{T_3}{T_2}\) and \(p\), practically coincide. The relative constancy of the optimal field-forcing coefficients with variation of saturation of the magnetic circuit can be explained by the fact that, for a prescribed mode, the optimal field-forcing coefficient is determined mainly by the ratio \(\dfrac{T_3}{T_2}\), which remains practically unchanged when saturation changes, since the rotor windings are in magnetically identical conditions.

In the total energy losses released in the machine when the voltage is reduced from a prescribed value to zero, the individual components of the losses behave differently as \(k\) changes. Thus, the energy losses in the steel decrease as \(k\) increases, whereas the losses in the field winding and in the damper winding increase. Therefore, voltage reduction with the optimal \(k\) can be applied only in the case where the losses in the field winding or in the rotor as a whole during the reduction time do not exceed the losses admissible under heating conditions.

Thus, when calculating the optimal values of the reverse voltage applied to the field winding of a surge generator, saturation of the magnetic circuit can be sufficiently accounted for by choosing saturated parameter values for the prescribed mode.

References

1. G. A. Sipailov, A. I. Verkhoturov. Calculation of optimal values of reverse voltage during field suppression of a surge generator. Proceedings of TPI, vol. 212, 1971.

2. O. B. Bron. Magnetic-Field Suppression Automata. Gosenergoizdat, 1961.