On excitation forcing of a surge generator
Abstract Generated abstract
This paper analyzes forced excitation in impulse generators operated with pauses long enough for rotor transients to decay, where intermittent excitation can raise the magnetic flux before each pulse. Using a two-winding rotor model with excitation and damper circuits, the authors derive transient current and flux relations, then estimate how excitation forcing changes energy losses in stator steel and in the copper of the excitation and damper windings. The analysis suggests that moderate forcing can both increase maximum impulse power and reduce no-load losses, with steel losses falling markedly as the forcing coefficient rises from 1 to 2. For generators with damper-to-excitation time-constant ratios of about 0.5 to 0.7, the loss-minimizing forcing coefficient is estimated to be in the range 4 to 6.
Full Text
On Forcing the Excitation of an Impulse Generator
G. A. Sipailov, A. I. Verkhoturov
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
Forced excitation is widely used both in standard-design synchronous machines, to increase the stability of parallel operation, especially under emergency conditions, and in special electrical machines, in particular in burst-power generators, to maintain the voltage during a short circuit whose duration is approximately 10–20 periods of the e.m.f. In burst-power generators, whose operating process lasts about one period of the e.m.f., it is not meaningful to carry out forcing of this kind, since, in the presence of a powerful damping winding on the rotor, it is practically impossible to affect the voltage magnitude over so short a time interval.
When a burst-power generator operates in a mode in which the duration of the pauses between individual current pulses exceeds the total time constant of the rotor windings, intermittent excitation with a forced increase of the magnetic flux to the required value before each switching-on of the generator becomes expedient. In this case, on the basis of the condition of permissible thermal loading of the rotor, the generator e.m.f. can reach considerably greater values in comparison with the case in which excitation is carried out continuously.
The power of a burst-power generator in maximum kilovolt-amperes \(P_m\) can be expressed through the nominal rated apparent power \(P_n\) [1]:
\[ P_m = \frac{4 k_\phi^2 \cdot k_3^2}{x_d^{\prime\prime}} P_n, \]
where \(k_\phi\) is the voltage-forcing coefficient,
\(k_3\) is the stator-current attenuation coefficient,
\(x_d^{\prime\prime}\) is the subtransient inductive reactance in relative units.
With intermittent forced excitation it is possible to obtain a coefficient \(k_\phi\) equal to 1.2 and higher, which makes it possible to increase the maximum power of the burst-power generator by approximately a factor of 1.5. In this case, the increase in power occurs both as a result of the voltage increase and, to some extent, as a result of the decrease in subtransient inductive reactance due to saturation of the magnetic circuit.
Burst-power generators have higher no-load losses in comparison with turbogenerators of the same rated apparent power. This is due to the greater saturation of sections of the magnetic circuit.
impact generators, and also by the fact that, because of the reduced air gap, additional losses acquire a large specific weight. The main part of the no-load losses of an impact generator consists of losses in the steel, which, under continuous as well as intermittent excitation without forcing, exceed by 5–8 times or more the losses in the copper of the excitation winding.
The use of forcing under intermittent excitation has a substantial influence both on the magnitude and on the distribution of losses in copper and steel. In order to determine the excitation forcing coefficient \(k\) at which, for a given pulse repetition frequency, the losses will be minimal, it is necessary to establish the relation between them and to investigate the effect of forcing on the magnitude of losses in copper and steel.
Fig. 1. Diagram of the rotor windings of an impact generator.
To establish the indicated dependencies, let us consider the transient process of switching the excitation winding of an impact generator (Fig. 1) onto the constant voltage \(u_2\)
\[ \left. \begin{aligned} u_2 &= i_2 \cdot r_2 + L_2 \frac{di_2}{dt} + M_{23}\frac{di_3}{dt},\\ 0 &= i_3 \cdot r_3 + L_3 \frac{di_3}{dt} + M_{23}\frac{di_2}{dt}. \end{aligned} \right\} \tag{2} \]
where \(r_2, L_2\) 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.
Solving the system of equations (2), we find expressions for the currents:
\[ i_2=\frac{u_2}{r_2}+\frac{u_2}{bL_2} \left[ \left(1+\frac{\sigma p_2}{\delta_2}\right)e^{p_1t} - \left(1+\frac{\sigma p_1}{\delta_2}\right)e^{p_2t} \right], \tag{3} \]
\[ i_3=-\frac{u_2M_{23}}{bL_2L_3}\left(e^{p_1t}-e^{p_2t}\right), \tag{4} \]
where
\[ \delta_2=\frac{r_2}{L_2} \]
is the damping coefficient of the excitation winding,
\[ \delta_3=\frac{r_3}{L_3} \]
is the damping coefficient of the damper circuit,
\[ \sigma=1-k_c^2 \]
is the leakage coefficient of the circuits,
\[ k_c=\frac{M_{23}}{\sqrt{L_2\cdot L_3}} \]
is the coupling coefficient,
\[ p_{1,2}=\frac{1}{2\sigma}\left[-(\delta_2+\delta_3)\pm b\right] \]
are the roots of the characteristic equation,
\[ b=\sqrt{(\delta_2-\delta_3)^2+4k_c^2\delta_2\delta_3}. \]
The flux creating the voltage in the stator winding of the generator is proportional to the sum of the currents \(i_2\) and \(i_3\) and to the mutual-inductance coefficient between the rotor and stator windings \(M_{12}\):
\[ \varphi=\varphi_0 \left\{ 1+\frac{\delta_2}{b} \left[ \left(1+\frac{\sigma p_2}{\delta_2}-\frac{M_{23}}{L_3}\right)e^{p_1t} - \left(1+\frac{\sigma p_1}{\delta_2}-\frac{M_{23}}{L_3}\right)e^{p_2t} \right] \right\}. \tag{5} \]
In the general case, equation (5) cannot be solved with respect to time; however, it is substantially simplified if it is assumed that the leakage coefficient of the windings is equal to zero:
\[ \psi=\psi_0\left(1-Ae^{-\frac{t}{T}}\right), \tag{6} \]
where
\[ A=\frac{1}{\delta_2+\delta_3}\left(\delta_3+\frac{r_2}{\sqrt{L_2L_3}}\right), \]
\[ T=T_2+T_3 \]
is the total time constant of the rotor circuits.
If, without forcing the excitation, the flux increases to its nominal value in time \(t\), then with forcing by a factor \(k\) the flux increases to the same value in a shorter time \(t_1\). Hence the ratio of the fluxes is equal to:
\[ \frac{k\left(1-Ae^{-\frac{t_1}{T}}\right)} {1-Ae^{-\frac{t}{T}}}=1. \tag{7} \]
It may be assumed that, without forcing the excitation, the flux reaches its steady-state value in time \(t=\infty\). Then, using (7), we obtain
\[ t_1=T\ln\frac{kA}{k-1}. \tag{8} \]
Expression (8) has meaning only for \(k>1\). The losses in the steel of an impulse generator at constant frequency are proportional to the square of the flux. Then the energy losses in the generator steel during time \(t_1\) with forcing and, correspondingly, during time \(t\) without forcing may be represented as:
\[ Q_{c1}=Nk^2\psi_0^2\int_0^{t_1} \left(1-2Ae^{-\frac{t}{T}}+A^2e^{-\frac{2t}{T}}\right)\,dt, \tag{9} \]
\[ Q_c=N\psi_0^2\int_0^t \left(1-2Ae^{-\frac{t}{T}}+A^2e^{-\frac{2t}{T}}\right)\,dt, \tag{10} \]
where \(N\) is a quantity depending on the material, the degree of saturation, and the frequency.
Let
\[ k_{\phi c}=\frac{Q_{c1}}{Q_c}, \]
then, after integrating (9) and (10),
\[ k_{\phi c}= \frac{ k^2\left\{t_1+AT\left[2e^{-\frac{t_1}{T}}-0.5A\left(e^{-\frac{2t_1}{T}}-1\right)-2\right]\right\} }{ t+AT\left[2e^{-\frac{t}{T}}-0.5A\left(e^{-\frac{t}{T}}-1\right)-2\right] }. \tag{11} \]
Assuming that the transient process ends in the time \(t=3T\) and substituting in (11) the value of \(t\), from (8) we obtain
\[ k_{fs}= \frac{ k^2\left\{ -\ln\left(\frac{k-1+0,0498A}{kA}\right) + A\left[ 2e^{\ln\left(\frac{k-1+0,0498A}{kA}\right)} - \frac{ 0,5A\left(e^{2\ln\left(\frac{k-1+0,0498A}{kA}\right)}-1\right)-2 }{ 3+A(0,5A-1,9) } \right]\right\} }{ 3+A(0,5A-1,9) } \tag{12} \]
Denoting, by analogy with the preceding, by \(k_{fmv}\) the ratio of the energy of losses in the copper of the excitation winding under forcing, \(Q_{mv1}\), to the energy without forcing, \(Q_{mv}\), we obtain
\[ k_{fmv}= \frac{ k^2\left\{ t_1+BT\left[ 2\left(e^{-\frac{t_1}{T}}-1\right) - 0,5B\left(e^{-\frac{2t_1}{T}}-1\right) \right]\right\} }{ t+BT\left[ 2\left(e^{-\frac{t}{T}}-1\right) - 0,5B\left(e^{-\frac{2t_1}{T}}-1\right) \right] }, \tag{13} \]
\[ B=\frac{T_2}{T}. \]
Similarly, for the damper winding,
\[ k_{fmd}= \frac{ k^2\left(e^{-\frac{2t_1}{T}}-1\right) }{ e^{-\frac{2t}{T}}-1 } \tag{14} \]
The value of \(A\) from expression (6) may be taken as equal to unity, since
\[ \frac{r^2}{\sqrt{L_2L_3}}=\delta_2 \]
and, taking into account that \(e^{\ln x}=x\), the relations for the energy of losses in the steel (12), in the copper of the excitation winding (13), and in the copper of the damper winding (14) are written in the following form:
\[ k_{fs}=-0,625\left(k^2\ln\frac{k-0,95}{k}+0,95k+0,45\right) \tag{15} \]
\[ k_{fme}= \frac{ B\left[B(0,95k-0,45)-1,9k\right]-k^2\ln\frac{k-0,95}{k} }{ 3+B(0,5B-1,9) } \tag{16} \]
\[ k_{fmd}-1,9k=0,9 \tag{17} \]
The results of calculating the influence of excitation forcing on the magnitude of the energy of losses in steel and copper are presented in Fig. 2 and Fig. 3. In addition, Fig. 3 shows the influence of the parameters of the damper winding on the character of the distribution of copper losses as a function of the excitation-forcing coefficient.
As follows from Fig. 2, the losses in the steel, when the excitation-forcing coefficient is increased from \(k=1\) to \(k=2\), decrease by approximately 80 percent. With a further increase in \(k\), the losses decrease to a lesser degree.
The magnitude of the losses in copper, as follows from Fig. 3, changes differently with an increase in the excitation-forcing coefficient, depending on the ratio of the parameters of the damper winding and the excitation winding. However, in all cases, when the forcing coefficient is increased from 1 to 2, the losses in copper decrease. If \(0,5T_2 \leq T_3 \leq T_2\), then for \(k>2\) the losses in copper increase.
Fig. 2. Dependence of losses in steel on the excitation forcing coefficient.
Fig. 3. Dependence of losses in copper on the excitation forcing coefficient for different ratios of the rotor-winding parameters.
The optimal excitation forcing coefficient, from the standpoint of minimum energy losses, can be found from the condition of minimum losses in copper and steel. For impulse-power generators with the ratio
\[ \frac{T_3}{T_2}=0.5 \div 0.7, \]
this coefficient lies within the range 4–6.
Thus, the use of excitation forcing with intermittent excitation makes it possible not only to increase the impulse power,
but also to substantially reduce no-load losses, which is of great importance for impulse-power generators, since in them, because of the reduced air gap, the conditions for removing losses are worse than in turbogenerators.
References
- I. M. Postnikov. The problem of obtaining large values of short-duration electromagnetic power. Proceedings of LII, No. 2, 1937.