Calculation of Optimal Reverse Voltage Values During Field Suppression of an Impact Generator
Abstract Generated abstract
This paper examines forced field suppression in impact generators equipped with damper windings, where rapid de-excitation is needed to reduce heating and damage risk after pulsed operation or internal faults. Using an unsaturated machine model with coupled excitation and damper circuits, it derives expressions for transient currents, flux decay time, and electromagnetic energy losses in the field winding, damper winding, and machine steel as functions of the applied reverse-voltage coefficient. The analysis shows that an optimal reverse-voltage coefficient exists for reducing terminal voltage to zero or to a specified residual value, minimizing total loss energy under the assumed conditions. Calculations indicate that operation at this optimum can reduce losses by about half relative to constant excitation over the same interval and by about one third relative to suppression with a reverse-voltage coefficient of one, provided winding and rotor heating limits are not exceeded.
Full Text
CALCULATION OF OPTIMAL REVERSE VOLTAGE VALUES DURING FIELD SUPPRESSION OF A SHOCK GENERATOR
G. A. Sipailov, A. I. Verkhoturov
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
Suppression of the magnetic field is the only method that makes it possible to limit the extent of damage to electrical machines during internal short circuits.
For a shock generator, problems of field suppression acquire particular urgency in connection with the specific conditions of its operation. To increase pulsed power, shock generators are often excited to a voltage higher than the rated value; 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, and the machine must be de-excited.
Shock generators have a powerful damping winding whose time constant is commensurate with the time constant of the field winding; therefore, even with complete disconnection of the field winding, no overvoltages arise on it, since the energy is transferred to the damper winding. On the other hand, the presence of the damper winding increases the time constant of the entire excitation system.
Since all damping circuits are short-circuited, the only possibility for carrying out rapid suppression is to act on the magnetic field of the damping circuits by some other magnetic field of opposite direction. Such a field may be the field of the field winding, if a voltage of opposite sign is applied to it. Reversal of the polarity at the rotor rings is easily accomplished by inverting an ionic or electric-machine exciter, as well as by using a circuit with an arc-suppression grid [1, 2, 3].
With the selected method of suppression, reducing the voltage from a specified value to zero can be accomplished at different values of the reverse voltage applied to the field winding. Therefore the question arises of choosing such a value of reverse voltage at which the total no-load electromagnetic losses during reduction of the voltage from the specified value to a given value would be minimal.
With a machine excitation system, it is considerably more convenient to change the polarity of the voltage applied to the field winding on the excitation side of the exciter. In the calculations, however, we shall assume that the polarity reversal is carried out directly in the rotor circuit of the shock generator. As calculations show, when the time constant of the exciter field circuit \(T_{\mathrm{v}}\) is small relative to ...
with respect to the total time constant of the rotor circuits \(T\) \(\left(\dfrac{T_{\mathrm{в}}}{T} \ll 0.05\right)\), the character of the transient process and the time of voltage decay from the specified value to zero, or to any other value, remain practically the same as during suppression directly in the excitation-winding circuit of the shock generator. This assumption is even more justified for a semiconductor or ion excitation system.
To solve the stated problem, let us consider field suppression of an unsaturated machine by changing the polarity of the voltage applied to the excitation winding of the shock generator:
\[ \left. \begin{aligned} r_2 i_2 + L_2 \frac{d i_2}{dt} + M_{23}\frac{d i_3}{dt} &= -k u_2,\\ r_3 i_3 + L_3 \frac{d i_3}{dt} + M_{32}\frac{d i_2}{dt} &= 0 \end{aligned} \right\}, \tag{1} \]
\(r_2, L_2\)—the active resistance and inductance of the excitation winding, taking into account the parameters of the exciter;
\(M_{23}, M_{32}\)—mutual inductances of the rotor windings;
\(k\)—the ratio of the reverse voltage applied to the excitation winding to the nominal voltage;
\(r_3, L_3\)—equivalent parameters of the damper system, taking into account the parameters of the rotor body and the wedge.
The error arising in the analytical determination of the equivalent parameters of the rotor body has a smaller effect on the nature of the transients during suppression than in turbogenerators, since the time constant of one body is approximately 10–20% of the time constant of the damper winding proper.
Since in shock generators special measures are taken to reduce the leakage fluxes of the rotor windings (shielding of the end parts, increasing the width of the slot for the damper winding in comparison with the width of the slot for the excitation winding, etc.), the total leakage coefficient of the rotor circuits lies within the range \(\delta \le 0.03 \div 0.05\), and in calculations of transients during suppression the leakage of the rotor circuits may be neglected. Taking this last assumption into account, the solution of system (1) has the form
\[ i_2 = \frac{u_2}{r_2}\left[(1+k)B e^{-\frac{t_{\mathrm{г}}}{T}} - k\right], \tag{2} \]
\[ i_3 = \frac{u_2}{r_2}(k+1)(1-B)e^{-\frac{t_{\mathrm{г}}}{T}}, \tag{3} \]
where \(T = T_2 + T_3\) is the total time constant of the rotor circuits;
\[ B = \frac{1}{1+\dfrac{T_3}{T_2}} \]
is a quantity depending on the ratio of the time constants of the damper system and the excitation winding.
The main flux in the machine is determined by the total current of both windings and by the coefficient of mutual inductance between the rotor and the stator
\[ \Phi_{\mathrm{г}} = \frac{u_2}{r_2} M_{12} \left[(1+k)e^{-\frac{t_{\mathrm{г}}}{T}} - k\right]. \tag{4} \]
The suppression time can be determined from the condition that the total current of the windings is equal to zero,
\[ t_r = T \ln \frac{k+1}{k}. \tag{5} \]
Let us express the loss energy released in the rotor windings when the voltage is reduced from a specified value to zero as a function of the coefficient of forced field suppression \(k\).
For the excitation winding,
\[ Q_{2r}=\int\limits_0^{t_r} i_2^2 \cdot r_2 \cdot dt = \frac{u_2^2}{r_2}\,T \left( k^2 \ln \frac{k+1}{k} - 2kB + B^2k + 0.5B^2 \right). \tag{6} \]
For the damper winding,
\[ Q_{3r}=\int\limits_0^{t_r} i_3^2 \cdot r_3 \cdot dt = 0.5\,\frac{u_2^2}{r_2}\,T(2k+1)B(1-B). \tag{7} \]
When the voltage is reduced from the specified value to zero, in addition to the losses in the rotor windings, losses are released in the steel of the machine, which are proportional to the square of the flux,
\[ Q_{\mathrm{st}r} = \frac{P_0'}{\Phi_0^2} \int\limits_0^{t_r} \Phi_r^2 \cdot dt = P_0'T \left( k^2 \ln \frac{k+1}{k} - k + 0.5 \right), \tag{8} \]
where \(P_0'\) is the no-load loss in the steel of the machine,
\(\Phi_0\) is the steady-state value of the flux,
\(w_1\) is the number of turns of the stator winding.
Adding the energy losses in the steel of the machine and in the rotor windings, we obtain the dependence of the total losses on the coefficient of forced field suppression,
\[ Q_{\Sigma r} = \frac{u_2^2}{r_2}\,T \left[ k^2(k_0+1)\ln \frac{k+1}{k} - (k_0+B)(k-0.5) \right], \tag{9} \]
where
\[ k_0=\frac{P_0'r_2}{u_2^2} \]
is the ratio of the electromagnetic no-load losses in the steel to the losses in the excitation winding.
Let us denote
\[ k_{\Sigma r}=\frac{Q_{\Sigma r}}{Q'_{\Sigma r}}, \]
where \(Q'_{\Sigma r}\) is the total energy loss during field suppression for \(k=1\); then
\[ k_{\Sigma r} = \frac{ k^2(k_0+1)\ln \dfrac{k+1}{k} - (B+k_0)(k-0.5) }{ 0.1931k_0 - 0.5B + 0.6931 }. \tag{10} \]
The results of the calculation by (10) are presented in Fig. 1 for \(k_0=10\) and
\[ \frac{T_3}{T_2}=0.4 \div 1. \]
It is seen from Fig. 1 that there is such a coefficient of forced field suppression at which the loss energy during the time of reducing the voltage from the specified value to zero is minimal.
Fig. 1. Comparison of total losses under forced field suppression with the losses at \(k=1\)
The value of the optimal forced field-suppression coefficient can be found by setting the derivative of \(Q_{\Sigma r}\) with respect to \(k\) equal to zero:
\[ 2k \ln \frac{k+1}{k+n} - \frac{k}{k+1} - \frac{k_0+B}{k_0+1} = 0 . \tag{11} \]
The transcendental equation (11) can be solved approximately with any desired degree of accuracy with respect to \(k\). The values of the forced field-suppression coefficients at which the losses will be minimal are denoted by \(k_{\mathrm{or}}\) and are called optimal. The values of the optimal forced field-suppression coefficients, determined from the solution of equation (11) for \(k=4 \div 10\) with variation of
\[ \frac{T_3}{T_2}=0.4 \div 1, \]
are presented in Fig. 2.
Fig. 2. Optimal coefficients of forced field suppression
Let us compare the energy of the losses released in the steel of the machine and in the rotor windings when operating with the optimal coefficient of forced de-excitation with the energy of the losses that would be released in the machine if the voltage at the generator terminals were not reduced, but the duration of action of the constant voltage were left equal to the time interval of field suppression with the optimal coefficient of forced de-excitation:
\[ k_{\mathrm{eg}1}=k_{\mathrm{og}}^{2}- \frac{(B+k_{0})(k_{\mathrm{og}}-0.5)} {(k_{0}+1)\ln \dfrac{k_{\mathrm{og}}+1}{k_{\mathrm{og}}}} . \tag{12} \]
The results of calculations by (12) are presented in Table 1 for \(\dfrac{T_3}{T_2}=0.4 \div 1\) and \(k_0=6 \div 10\).
Table 1
Values of the coefficients \(k_{\mathrm{eg}1}\)
| \(\dfrac{T_3}{T_2}\) | \(k_0=6\) | \(k_0=8\) | \(k_0=10\) |
|---|---|---|---|
| 1 | 0.42 | 0.45 | 0.47 |
| 0.8 | 0.43 | 0.41 | 0.48 |
| 0.6 | 0.45 | 0.47 | 0.49 |
| 0.4 | 0.59 | 0.4 | 0.51 |
As is evident from Table 1, the energy losses under constant excitation are approximately twice as large as the energy losses during suppression with the optimal coefficient.
If in equation (10) \(k_{\mathrm{og}}\) is substituted instead of \(k\), then this equation shows the ratio of the energy losses at the optimal coefficient of forced de-excitation to the energy losses at \(k=1\).
The calculation results show that, when the voltage is reduced with the optimal coefficient of forced de-excitation, the losses released are approximately 1.5 times smaller in comparison with the losses at \(k=1\).
In an analogous way one can find the coefficient of forced de-excitation at which a reduction of the voltage from a certain value to zero is accompanied by the release in the rotor of the minimum amount of electromagnetic-loss energy.
To find the optimal coefficients of forced de-excitation from the condition of a minimum of electromagnetic energy losses in the rotor, one can use equation (11), denoting in it by \(k_0\) the ratio of the no-load losses in the rotor steel to the losses in the field winding.
In the total energy losses released in the machine when the voltage is reduced from a specified 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 coefficient of forced de-excitation can be applied only in the case where the losses in the field winding, and also in the rotor as a whole during the reduction time, do not exceed the losses permissible under the heating conditions.
For machines with indirect cooling of the windings, where the limits of varia-
of permissible values of specific loads are small; with sufficient accuracy for practice the thermal state of the machine can be judged from the value of the specific thermal load.
The permissible specific thermal load \(A_p\) for rotors with indirect air cooling lies within the range \(0.9 \div 1.1\ \text{W}/\text{cm}^2\), and for the field winding, \(0.25 \div 0.3\ \text{W}/\text{cm}^2\) [4].
If the voltage is reduced not to zero, but to some arbitrary value, then the optimal coefficient of forced field suppression can be found from the equation
\[ 2k\ln \frac{k+1}{k+n} - (1-n)\left[\frac{k^2}{(k+1)(k+n)} + \frac{B+k_0}{k_0+1}\right]=0, \tag{13} \]
where \(n\) is the coefficient indicating to what value the voltage is reduced. The results of solving equation (13) are presented in Fig. 2 for \(n=0.5\), \(k_0=6\), and \(k_0=8\) when \(\dfrac{T_3}{T_2}=0.4 \div 1\).
Thus, in machines with a damper winding there exists such a coefficient of forced field suppression at which the total electromagnetic energy losses when the voltage at the generator terminals is reduced from a specified value to a definite value are minimal.
Carrying out field suppression with the optimal coefficient of forced field suppression makes it possible to reduce energy losses, in comparison with the losses under constant excitation over the same time, by approximately a factor of 2, and by almost a factor of 1.5 in comparison with the losses released in the machine when the voltage is reduced at \(k=1\).
Voltage reduction with the optimal coefficient of forced field suppression is expedient only in the case where the losses released during the reduction time in the field winding and in the rotor as a whole do not exceed the losses permissible under heating conditions.
REFERENCES
- N. N. Nikiforskii. On the question of forced field suppression of a synchronous machine. Elektrichestvo, 1963, No. 9.
- Fischer Rolf. On the selection of optimal values of reverse voltage and additional resistance for field suppression of a synchronous machine by express information. Elektricheskie mashiny i apparaty, No. 32, ref. 130, 1967.
- O. B. Bron. Automatic devices for magnetic-field suppression. Gosenergoizdat, 1961.
- V. V. Titov, G. M. Khutoretskii. Turbogenerators. Calculation and Design. “Energiya,” 1967.