ON THE SELECTION OF THE DAMPER WINDING OF A SHOCK GENERATOR

G. A. SIPAILOV, L. V. LOOS

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

A shock generator is a single-phase synchronous generator of the turbogenerator type. The normal operating mode of a shock generator is sudden connection to a load at the moment when the generator emf passes through zero, and disconnection at the very first passage of the current through zero. A shock generator, like all single-phase synchronous generators, has a damper winding along the longitudinal and transverse axes.

However, the purpose of the damper system of a shock generator is not limited merely to damping the reverse field of armature reaction, as in single-phase synchronous generators. The damper winding of a shock generator also serves to reduce the supertransient inductive reactance of the stator winding. Since the damper winding of shock generators occupies up to 40% of the rotor-slot area [1], its correct selection is of great importance.

Many works [2], [3], and others are devoted to questions of the theory and calculation of damper windings of synchronous machines. In most cases, however, the damper winding is considered as applied to salient-pole synchronous machines.

The selection of the damper winding of a shock generator, in contrast to its selection for single-phase salient- and nonsalient-pole synchronous generators intended for operation in continuous duty, has the special feature that the normal operating mode of a shock generator, from the standpoint of established electrical-engineering concepts, is an emergency mode.

The nature of the electromagnetic processes occurring during a short circuit or sudden connection of a shock generator to a load whose inductance is, in magnitude, close to the leakage inductance of the generator stator winding is represented by differential equations composed for the stator and rotor circuits. Investigations of these processes by means of differential equations are in accord with modern ideas about transient processes in synchronous machines with damping circuits on the rotor, since in this case changes in leakage fluxes between the circuits are taken into account [4].

The use of modeling devices solves the problem of maximum correspondence to actual processes.

Studies of the influence of various parameters of the shock generator on the amplitude and on the shape of the shock-current curve were carried out on a mathematical model of the electromagnetic processes of a shock generator, created on the basis of the IPT-5 [5].

In the investigations, the following parameter values, in relative units, were taken as initial values:

\[ M = 1.0;\quad L_c = 1.05;\quad L_{\mathrm{B}} = 1.1;\quad L_{Dd}=L_{Dq}=1.03; \]

\[ r_c = 0.005;\quad r_{\mathrm{B}} = 0.005;\quad r_{D1}=r_{Dq}=0.005. \]

Where

\(M\) — mutual inductance between windings;
\(L\) — total inductances of the windings;
\(r\) — active resistances of the windings;
the subscripts for \(L\) and \(r\) denote: \(c\) — stator winding, \(b\) — field winding, Dd — damper winding along the longitudinal axis, Dq — damper winding along the transverse axis. In each case, the unit of current was taken to be that value which corresponded to parameter values equal to the initial ones.

GENERATOR WITHOUT A DAMPER WINDING

The investigation of a single-phase synchronous machine without a damper winding on the rotor is, essentially, of theoretical interest. These investigations help to assess more clearly the role of the active resistance of the stator winding, the inductance of the field winding, etc., in obtaining a current pulse of the required magnitude.

The differential equations of transient processes for the case of a synchronous machine without damper windings have the form:

\[ 0=i_1(r_1+r_{\mathrm{n}})+(L_1+L_{\mathrm{n}})\frac{di_1}{dt} +M\cos\omega t\,\frac{di_2}{dt} -i_2\omega M\sin\omega t; \]

\[ U_{\mathrm{B}}=i_2r_2+L_2\frac{di_2}{dt} +M\cos\omega t\,\frac{di_1}{dt} -i_1\omega M\sin\omega t. \tag{1} \]

The system of equations (1) cannot be integrated in general form, while numerical methods for solving it are rather laborious. The use of an analog computer makes it possible to solve this problem.

The method for solving differential equations describing the electromagnetic transient process of a shock generator having on the rotor, along with the field winding, a damper winding along the longitudinal and transverse axes is given in [5]. Using this method, one can solve the system of equations (1); for this it is necessary to disconnect from the solution circuit (Fig. 2) [5] amplifiers 9, 10, 11, 12, 13, 14, 15, 16.

Table 1 gives results showing the degree of influence of the active resistance of the stator winding on the magnitude of the currents in the stator winding and in the field winding; here the values of the currents at

\[ \lambda=\frac{r_c+r_{\mathrm{n}}}{L_c+L_{\mathrm{n}}}=0 \]

are taken as unity.

In shock generators, the reduction of the maximum stator current due to active resistance will be less than 5–8%.

Table 2 gives results showing the degree of influence of the inductance of the rotor circuit—or, in the case under consideration, the leakage inductance of the field winding—on the magnitude of the currents in the stator and rotor windings.

Table 2

With an increase in the inductance of the rotor circuit, the amplitudes of the currents in the stator and rotor windings decrease; in this case the current in the field winding approaches the constant value corresponding to no-load operation. In particular, this is evident from the oscillograms shown in Fig. 1, taken for two values of the inductances $L_{\mathrm{v}}$ with the other parameters unchanged and equal to the initial ones.

As a result of the investigations carried out, it may be concluded that the deviation of the stator-current curve from a sinusoid is greater the smaller the overall leakage coefficient, that is, the more perfect—

a

Fig. 1. Oscillograms of stator and rotor currents in the absence of a damper winding:
a) \(L_{\mathrm{в}} = 1.05\); b) \(L_{\mathrm{в}} = 4.0\).

the magnetic coupling between the windings. The relative weight of the higher harmonics in the stator- and rotor-current curves of a single-phase synchronous generator without damper windings is large.

There is a well-known means of eliminating the higher harmonics in the current curve by introducing a choke coil into the rotor circuit. However, such a method cannot be recommended for a shock generator, since in this case its supertransient reactance increases.

To eliminate the higher harmonics in the current curves, it is necessary to create on the rotor a multiphase winding capable of reacting to magnetic fields rotating relative to the rotor with fields rotating in the same way.

For a better clarification of the possibilities of damping, both the separate and the combined action of damper windings along the direct and quadrature axes are considered.

GENERATOR WITH A DAMPER WINDING ALONG THE QUADRATURE AXIS

In investigating a single-phase synchronous machine with a damper winding along the quadrature axis, all parameters remained constant; only the inductance of the damper winding along the quadrature axis was varied within the limits:

\[ L_{\mathrm{д}q} = 1.03 \div 1.2 . \]

Fig. 2. Oscillograms of the stator and rotor currents with a damper winding present only along the transverse axis. a) \(L_{Dq}=1.03\); b) \(L_{Dq}=1.2\); c) \(L_{Dq}=1.3\).

The results of these investigations are presented in Table 3.

Table 3

As a result of the investigations it was established that the inductance of the damper winding along the transverse axis affects both the amplitude value of the stator current and its waveform. At small values of the leakage of the damper winding, the surge current has a sharp-peaked crest (Fig. 2a). With an increase in the leakage of the damper winding, the crest of the surge-current curve becomes smooth (Figs. 2b, 2c).

GENERATOR WITH A DAMPER WINDING ALONG THE LONGITUDINAL AND TRANSVERSE AXES

With a complete damper winding present, the separate influence of changes in the inductance of the winding along the longitudinal and transverse axes on the magnitude and form of the short-circuit current was investigated. The degree of influence of the inductance \(L_{Dq}\) on the magnitude of the stator current in the presence of a damper winding along the longitudinal axis is shown in Table 3. In this case, the inductance of the damper winding along the transverse axis

Fig. 3. Oscillograms of stator and rotor currents in the presence of a damper winding along the longitudinal and transverse axes: a) \(L_{Dq}=1.05\); b) \(L_{Dq}=1.4\).

varied within the limits \(1.03 \div 1.4\), with all other parameters held constant.

Table 4

It follows from Table 4 that the damper winding along the transverse axis has the strongest influence on the magnitude of the stator current in the case when its inductance varies within the limits \(1 < L_{Dq} \leq L_{\mathrm{в}}\). If \(L_{Dq} \geq L_{\mathrm{в}}\), then a further increase in the inductance \(L_{Dq}\) affects the magnitude of the current only slightly.

An increase in the inductance \(L_{Dq}\), as in the case of a generator without a damper winding along the longitudinal axis, leads to smoothing of the peak of the shock-current curve.

Figure 3 presents an oscillogram for two values of the inductances \(L_{Dq}\), with the other parameters constant.

In studying the influence of the damper winding along the longitudinal axis on the magnitude of the shock current, its inductance was varied within the limits \(L_{Dd}=1.03 \div 1.4\), while the other parameters, equal to the initial ones, were held constant. The results of the investigation are presented in Table 5.

Table 5

It follows from Table 5 that, as the leakage of the damper winding along the longitudinal axis increases, the amplitude of the stator current decreases substantially. Therefore, from the standpoint of increasing the current amplitude, one should strive for the smallest possible values of the leakage inductance of the damper winding along the longitudinal axis. However, on the other hand, reducing the leakage of the damper winding means reducing the height of the copper occupied by this winding, which leads to an increase in its active resistance. At the same time, the time constant of the damper winding decreases, which leads to noticeable damping of the current already by the end of the first half-period of the e.m.f.

For a quantitative estimate of the influence of the active resistance of the damper winding on the penetration of the armature-reaction flux into the contour of the field winding, experiments were carried out in which the active resistance was varied over wide limits

\[ \delta=\frac{r_{Dd}}{L_{Dd}}=\frac{r_{Dq}}{L_{Dq}}=0 \div 0.1, \]

while the other parameters remained unchanged.

From the current oscillogram shown in Fig. 4a, taken at \(r_{Dd}=r_{Dq}=0\), it follows that the current in the field winding practically does not change during the pulse; that is, in the case when the damper circuit is superconducting, the armature-reaction flux does not penetrate into the field winding. But if the active resistance of the damper circuit is large, for example \(\delta=0.1\), then, as follows from the oscillogram shown in Fig. 4b, at the moment of short circuit the current in the field winding changes greatly.

Fig. 4. Oscillogram of the stator and rotor currents with a complete damper winding: a) $\vartheta=0$; b) $\vartheta=0.1$.

The maximum values of the currents in the field winding, obtained for different active resistances of the damper circuits, are presented in Table 6; here the rated field current is taken as the unit of current.

Table 6

In designing the damper system it is necessary to strive to make the ratio between the active resistance of the winding and its inductance not exceed $\delta < 0.01$. On the basis of this condition, one can choose the dimensions of the damper system and determine the fraction of the rotor-slot area occupied by the damper winding.

EFFECT OF THE LOAD

In determining the effect of the load on the amplitude value of the surge current, the load inductance, expressed in the same relative units as the generator inductances, was varied within the range $0 \div 0.6$ with the generator parameters unchanged.

Since the load inductance is added to the self-inductance of the stator winding, it may be considered that in the investigations the inductance of the stator winding varied within the range $1.05 \div 1.65$.

The results of the investigations are presented in Table 7; here the stator current at $L_{\mathrm{н}} = 0$ is taken as unity.

Table 7

Thus, as the load inductance increases, a sharp decrease in the stator current occurs.

CONCLUSION

As a result of investigations carried out with the aid of a mathematical model of the influence of individual parameters on the magnitude and character of the short-circuit current of a surge generator, it may be concluded that, in order to obtain the maximum value of the short-circuit current, the inductance of the damper winding along the longitudinal axis should be as small as possible; however, at the same time it is necessary to strive to ensure that the decrease in $L_{Dd}$ does not lead to such an increase in $r_{Dd}$ that the inequality $\delta < 0.01$ would be violated. From this point of view, in surge generators it is advisable to use such a rotor-slot shape in which the part of the slot occupied by the damper winding along the longitudinal axis has a greater width than the part of the slot occupied by the field winding.

The damper winding along the transverse axis has a lesser effect on the amplitude of the short-circuit current. A larger amplitude of the short-circuit current corresponds to a smaller leakage of the damp-

of the damper winding along the transverse axis. However, in order to obtain a smoother form of the current curve, one should strive for such values of the leakage of the damper winding along the transverse axis at which the system of rotor windings forms a multiphase system close to symmetrical.

REFERENCES

1. Technische Rundschau, No. 52, 1954. Powerful shock generators of the USA and Europe.
2. Ya. B. Danilevich, Yu. A. Kulik. Theory and calculation of damper windings of synchronous machines. Academy of Sciences of the USSR, 1962.
3. Yu. A. Kulik. Currents of the damping winding of a three-phase synchronous machine. VEP, No. 6, 1958.
4. L. G. Mamikonyants. On transient processes in synchronous machines with damping windings on the rotor. Elektrichestvo, No. 7, 1954.
5. A. V. Loos, G. A. Sipailov. Mathematical modeling of the electromagnetic processes of a shock generator. Izvestiya TPI, vol. 152, 1965.