PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 98 1960

ON THE DETERMINATION OF TURBOGENERATOR RESISTANCE IN CALCULATING INTERNAL SHORT-CIRCUIT CURRENTS

K. K. RYAZANTSEV

(Presented by Prof., Doctor of Technical Sciences I. D. Kutyavin)

The existing method for calculating currents in internal short circuits (s.c.) of generators is based on the following [1, 2, 3].

1. The position of the s.c. point on the stator winding is characterized by the fraction \(a\) of the short-circuited turns, counted from the neutral.

2. The electromotive force \(E_a\), induced in the short-circuited part of each phase, is approximately proportional to \(a\), i.e.

\[ E_{\phi a}=aE_{\phi}, \tag{1} \]

where \(E_{\phi a}\) is the phase emf corresponding to \(a=1\).

1. The magnetic flux of a two-pole generator, linking with each of the turns of a phase, consists of two parts: the first of them links only with the given turn (“self flux” — \(\Phi_c\)), and the second passes through all turns of the phase winding (“common flux” — \(\Phi_{ob}\)). The leakage flux of the slot part of the winding is a self flux. The leakage flux of the end connections and the resultant flux produced by the joint action of the stator and rotor are common fluxes.

2. The inductive resistance of a stator phase consists of two components \(x_c\) and \(x_{ob}\), determined by the fluxes \(\Phi_c\) and \(\Phi_{ob}\).

The inductive resistance of the short-circuited part, in the first approximation, is equal to

\[ x_a=a x_c+a^2 x_{ob}. \tag{2} \]

The active resistance of the same part of the turns is determined by the relation

\[ r_a=ar, \]

where \(r\) is the resistance of the entire phase.

Let us consider these assumptions in more detail, with the aim of introducing certain refinements into the determination of the resistances of the individual parts of the generator. In doing so we shall have in mind the case of a symmetrical three-phase s.c. inside an unloaded two-pole turbogenerator disconnected from the station busbars.

Estimating the Distance to the Short-Circuit Point

Estimating the distance from the neutral to the fault location by the quantity \(\alpha\) does not always make it possible to calculate the short-circuit current correctly. This is hindered by three circumstances. The first of them is that each stator turn has only two active sides (Fig. 1), the length of which, according to [4], in generators rated 12–50 MW is only 46–68% of the turn length; therefore the length of the end portions cannot be neglected. It follows from this that, when calculated by expression (1), the e.m.f. and the current corresponding to the short-circuit point will be found correctly if the fault location cuts off a part of a turn in which the same part of the turn e.m.f. is induced. This occurs in cases where an integral number of turns is short-circuited, or a fractional number that is a multiple of 0.25 (Fig. 1).

Fig. 1.

Fig. 2.

The second circumstance, because of which expression (1) is inaccurate even when an integral number of turns is short-circuited, is that the e.m.f.’s of the conductors \(E_{cm}\) of two adjacent slots are shifted by the angle [5]

\[ \gamma^\circ = \frac{360p}{Z}, \]

where \(p\) is the number of pole pairs, and \(Z\) is the number of stator slots.

With the series connection of \(q\) conductors lying under one pole, the resulting e.m.f. \(E_q\) is obtained by geometrical addition of the vectors \(E_{cm}\) (Fig. 2), and therefore it is smaller than the product \(qE_{cm}\). This is taken into account by the winding distribution factor \(k_p\), which (for the fundamental harmonic) is equal to

\[ k_p = \frac{E_q}{qE_{cm}} = \frac{OB}{q\,OA} = \frac{\sin \dfrac{q\gamma}{2}}{q \sin \dfrac{\gamma}{2}} < 1. \]

Since in a turbogenerator there is an integral number of slots per pole and phase, and, moreover, the arrangement of the phase slots under both poles is the same, the distribution factor found above will also have this value for the entire winding as a whole. Therefore

\[ E_\phi = w E_\sigma k_p, \]

where \(E_v\) is the e.m.f. of one turn, and \(w\) is the number of turns per phase.

If the number of turns is reduced to the value \(\alpha w\), then the new value of the phase e.m.f. will be

\[ E_{\phi\alpha}=\alpha w E_v k_{p\alpha}>\alpha E_\phi, \]

since \(k_{p\alpha}>k_p\).

In determining \(k_{p\alpha}\), it must be borne in mind that for \(\alpha<1\) the symmetry of the half-groups of turns lying under both poles is disturbed, and each half-group of turns will have its own \(k_{p\alpha}\).

However, the change in \(k_{p\alpha}\) need be taken into account only for short circuits in turns that are first from the neutral, since here there will be the greatest error in calculating \(E_{\phi\alpha}\) from expression (1).

The third circumstance is that, when the location of the short circuit is shifted, for example from point H to point P (Fig. 1), the value of \(\alpha\) increases, and with it increases the first term on the right-hand side of the expression

\[ x_\alpha=\alpha x_c+\alpha^2 x_{ob}, \]

which should not occur, since \(x_c\) represents the resistance of the slot portion. This introduces an additional inaccuracy in determining the short-circuit current, again if the condition that the number \(\alpha w\) be a multiple of a quarter-turn is not fulfilled. To avoid error for any value of \(\alpha\), the e.m.f. and resistances should be found for individual sections of the short-circuit circuit, having before one the diagram of the stator winding.

Dependence of the generator resistances on the value of \(\alpha\)

Active resistance \(r\)

The active resistance of the generator phase has a small value. For example, for a T2-100-2 turbogenerator it is equal to \(0.00216\ \Omega\) at \(75^\circ\). The active resistance should be taken into account when determining the short-circuit current for a fault near the neutral. It may be determined from the expression

\[ r_\alpha=\alpha r. \]

Inductive resistance \(x_{ob}\), corresponding to the stator reaction flux

The dependence of the inductive resistance of the short-circuited part of the phase on \(\alpha\) is complex, and, as will be seen below, it can be taken into account only approximately. We shall clarify this by considering a simplified picture of the magnetic fluxes corresponding to the initial instant of the short circuit.

Before the occurrence of the short circuit there exists a flux in the air gap \(\Phi_0\) and a leakage flux of the field winding \(\Phi_{vs}\) (Fig. 3). The short-circuit current that appears creates a magnetic flux which may be divided into the stator reaction flux \(\Phi'_{ad}\) and the leakage fluxes \(\Phi_s\). The flux \(\Phi''_{ad}\) passes through the stator, the air gap, and also along the leakage paths of the damper system and the rotor field winding. The leakage fluxes \(\Phi_s\) embrace the slot and end-winding portions of the stator winding.

Fig. 3.

According to [1], it may be taken approximately that the flux of longitudinal reaction is the total flux linking with the entire winding of each phase. Bearing in mind that the principal reluctances for it are the air gap and the leakage paths, one may write

\[ x''_{\mathrm{ob}}=\omega L''_{\mathrm{ob}}=\omega w^2 \lambda''_{\mathrm{ob}}, \tag{3} \]

where \(\lambda''_{\mathrm{ob}}\) is the magnetic permeance for the flux of longitudinal reaction of the stator.

On the basis of expression (3), one may assume that the reactance \(x''_{\mathrm{ob}}\) has a quadratic dependence on the number of short-circuited turns \(\alpha w\). In general, an analogy is obtained with the variation of the reactance of series-connected coils embracing a common magnetic core and having one turn each, i.e.,

\[ x''_{\mathrm{ob}\,\alpha}=\omega(\alpha w)^2\lambda''_{\mathrm{ob}}=\alpha^2 x''_{\mathrm{ob}}=\alpha^2(x''_d-x_s). \tag{4} \]

Accordingly, for the transient and steady-state regimes we have

\[ x'_{\mathrm{ob}\,\alpha}=\alpha^2(x'_d-x_s), \tag{5} \]

\[ x_{\mathrm{ob}\,\alpha}=\alpha^2(x_d-x_s)=\alpha^2 x_{ad}. \tag{6} \]

Inductive reactance \(x_p\), corresponding to the slot-leakage flux.

The leakage flux of the stator winding \(\Phi_s\) is divided into the following components:

1) slot-leakage flux; 2) leakage flux of the end portions;
3) tooth-tip leakage flux; 4) leakage flux in the phase zones, or belt-leakage flux [7].

According to [8], the proportion of the total leakage reactance \(x_s\) due to the last two components is only \(3\%\), and therefore they are not considered below.

The leakage flux across each stator slot may be regarded as existing separately from analogous fluxes of adjacent slots [8]. Therefore, with a single-layer winding, a change in the number of short-circuited stator turns is similar to a change in the number of series-connected single-turn coils having no common magnetic core [1]. This makes it possible to assume that, in this case, the inductive reactance of the winding lying in the slots depends linearly on \(\alpha\), i.e.,

\[ x_{p\alpha}=\alpha x_p. \tag{7} \]

Here \(x_p\) is the reactance of the entire slot portion of the winding of one phase. But such an expression, as was indicated above, correctly reflects the actual dependence of \(x_\alpha\) on \(\alpha\) only when the number of turns is an integer, or a fraction that is a multiple of 0.25.

In modern turbogenerators, exclusively two-layer lap windings with a shortened pitch and with an integral number of slots per pole and phase are used [6, 9]. Expressions for calculating the inductive reactance of the slot portion of such a winding may be found, for example, in [4, 7, 8, 9]. One of them, taken from [9], is given below:

\[ x_p=k_x\frac{2pl}{Z}\cdot\frac{3\beta+1}{4}\cdot\frac{h_1+3h_3}{3b}\,\omega_0. \tag{8} \]

where, for a star connection of the phases,

\[ k_x = 0.407 \left( \frac{w}{10} \right)^2 \frac{I}{pU}, \]

\(l\) — active length of the stator, in cm,
\(\beta\) — relative winding pitch,
\(I, U\) — stator current and voltage,
\(w\) — number of turns of one phase.
For the remaining notation, see Fig. 4.

Fig. 4.

Fig. 5.

Since there are two bars in a slot, the number of stator slots is \(Z = 3w\), if the winding has no parallel branches.

Substituting the expressions for \(Z\) and \(k_x\) into (8), we obtain

\[ x_n = 2.26 \cdot 10^{-4} \frac{I}{U}\, l(3\beta + 1)\, \frac{h_1 + 3h_3}{8}\, w = c \cdot w . \tag{9} \]

It follows from this that \(x_n\) is directly proportional to the number of turns of the phase \(w\). This would seem to provide grounds for determining the value of \(x_n\), as for a single-layer winding, from expression [7]. However, proceeding in this way, one may obtain an error, especially in the case of a short circuit in the turns nearest the neutral. The point is that two bars lie in the slot, being under different conditions and belonging to different turns.

The coefficients of self- and mutual inductance of the bars can be found from the expressions (see Fig. 4):

\[ L_1 = \mu_0 \frac{4h + 3h_2 + 3h_3}{3b}\, l, \tag{10} \]

\[ L_2 = \mu_0 \frac{h + 3h_3}{3b}\, l, \tag{11} \]

\[ M = \mu_0 \frac{h + 2h_3}{2b}\, l. \tag{12} \]

For example, for one turbogenerator the ratios between the coefficients calculated by (10–12) were as follows:

\[ L_1 : L_2 : M = 100 : 23.9 : 48.4 . \]

The e.m.f.’s of self- and mutual induction, \(E_L\) and \(E_M\), arising in each of the bars, are shifted in phase by the angle \(\Theta\), just as are the currents in the bars (Fig. 5). To determine the equivalent inductance, for example of the first bar, let us decompose the e.m.f. of mutual induction

\[ E_{1M}=-\omega M I_2 \]

into components

\[ E'_{1M}=-\omega M I_2 \cos \Theta;\qquad E''_{1M}=-\omega M I_2 \sin \Theta. \]

The voltage applied to the end of the bar must have the components

\[ U_L=-E_{1L};\qquad U'_{1M}=-E'_{1M};\qquad U''_{1M}=-E''_{1M}. \]

The first two components may be taken into account by simple summation, namely:

\[ U_L+U'_{1M}=\omega (L+M\cos\Theta) I_1, \]

since \(I_1\) and \(I_2\) are equal in absolute value. The third component, by its nature, corresponds to an active voltage drop, coinciding in phase with the current \(I_1\). It may be taken into account by introducing into the generator equivalent circuit a fictitious resistance

\[ r_{\phi}=\omega M\sin\Theta. \]

On this basis one may write:

\[ U''_{1M}=\omega M I_1\sin\Theta=r_{\phi}I_1. \]

The phase-displacement angles \(\Theta\) between the currents in the upper and lower bars of a slot depend on the shortening of the pitch [8]. They are the same in all slots only for three pitch values: \(\tau\), \(2/3\,\tau\), and \(1/3\,\tau\). In this case the displacement angles are respectively equal to \(0^\circ\), \(60^\circ\), and \(120^\circ\).

For intermediate pitch values there is no longer equality of the angles. To clarify this, the directions of the currents should be shown on the stator-winding diagram, starting from the neutral. For example, for one generator it was found in this way that, for each group of turns in eight slots, \(\Theta=0\); in four slots \(\Theta=+60\); and in the last four slots it is equal to \(-60\). In an internal short circuit, currents flow in both layers of each slot not for every location of the fault, but only when it is at a definite distance from the neutral, as can be traced from the winding diagram. Therefore the resistance found from expression [7] will differ substantially from the actual one owing to the absence of the e.m.f. of mutual induction. It follows that, for small values of \(a\), the resistance must be found for certain slots and turns separately. Let us also note that, in order to obtain the largest possible short-circuit currents, the upper bar with the smaller inductance of the slot part must be connected to the neutral last.

The inductive reactance \(x_l\), corresponding to the leakage flux of the end portions.

Usually the stator of a turbogenerator has basket-type end connections located on conical surfaces [6, 7]. A part of them is shown in Fig. 6, taken from [4]. The length of the end portions of the gene-

generators rated 12–50 MW is 54–32% of the length of a stator-winding turn. The portion of the resistance due to leakage fluxes around these parts is approximately 65% of the total leakage resistance. Therefore, the importance of clarifying the dependence of \(x_l\) on \(\alpha\) is evident.

The currents in the conductors (Fig. 6) can be resolved into axial \(I_a\) and tangential components \(I_t\). The former create a rotating flux, similar to what occurs in the air gap. For tangential currents, the end portions constitute a system of ring conductors encircled by their own fields [7]. To this it must be added that, in the end connections, conductors carrying currents shifted in phase by unequal angles are located in close proximity to one another.

On the leakage fields and, consequently, on \(x_l\) in particular, an influence is exerted by the shortening of the winding pitch and by the material of the rotor retaining ring steel.

The pattern of the magnetic field around the end connections is complex, as a result of which the accounting for leakage in this part of the generator proves more difficult than in the slot portions. This, apparently, explains the discrepancy in the results given by different formulas when calculating \(x_l\) even for the entire phase as a whole.

Fig. 6.

As \(\alpha\) decreases, an increasing number of conductors carry no current, and the initial pattern of the leakage fields—on which the formulas for the resistance of the phase as a whole were derived—is increasingly distorted. Therefore, the expression for \(x_{l\alpha}\) will give values with an ever-increasing error as one approaches the most important region near the neutral. For small values of \(\alpha\), it appears advisable to use another method for calculating the resistance, for example the method of sections described in [10], taking into account the special features of the end portions (configuration, the presence of mutual inductance between adjacent conductors, etc.).

Conclusions

1. As the authors of method [1] themselves indicate, the expressions they propose for determining the e.m.f. and inductive reactance are valid only as a first approximation.

2. In the present article the following has been established: 1) the e.m.f. and inductive reactances are determined by expressions (1, 2) with the least accuracy for a short circuit in turns located near the neutral; 2) in order to avoid error, for any value of \(\alpha\) the e.m.f. and resistances should be found for individual sections of the short-circuit circuit; 3) the resistance of the slot portion of the stator winding may be found, for the case of a three-phase short circuit, with sufficient accuracy if the current distribution in the winding scheme is taken into account; 4) further clarification is necessary regarding the calculation, for \(\alpha < 1\), of the resistance associated with the existence of leakage fluxes of the stator end portions.

References

1. N. F. Margolin and A. B. Tsentsin. A Method for Calculating Currents in Internal Faults in Synchronous Generators, 1937.

2. M. F. Kostrov, I. I. Solov′ev, and A. M. Fedoseev. Fundamentals of Relay-Protection Engineering. GEI, 1944.

3. A. M. Fedoseev. Relay Protection of Electric Power Systems. GEI, 1952.

4. A. Ya. Berger. Turbogenerators. Part II, ONTI, 1937.

5. L. M. Piotrovskii. Electrical Machines. GEI, 1940.

6. I. M. Postnikov. Design of Electrical Machines. GITL, Ukrainian SSR, 1952.

7. A. E. Alekseev and M. P. Kostenko. Turbogenerators. GEI, 1939.

8. A. Ya. Berger, V. I. Sysoev, and V. A. Vasil′ev. Basic Elements of the Newest Theory of the Synchronous Machine. ONTI, 1936.

9. E. G. Komar. Problems in the Design of Turbogenerators. GEI, 1955.

10. P. L. Kalantarov and L. R. Neiman. Theoretical Foundations of Electrical Engineering. GEI, 1951.