Experimental Determination of the Inductive Reactances of Synchronous Machines

Yu. N. Kroneberg, V. G. Mosin

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

Much attention has long been devoted to the experimental determination of inductive reactances along the direct and quadrature axes—the parameters \(x_d\) and \(x_q\)—but even now this problem remains among the urgent ones. It is substantially complicated in the investigation of saturated machines, since the values of \(x_d\) and \(x_q\) become dependent on the excitation current and on the magnitude and character of the load. In the mathematical theory of electrical machines, and especially of large electric-power systems, in the overwhelming majority of cases one is forced to use equations in which the parameters \(x_d\) and \(x_q\) enter as constants; however, it is necessary that the adopted values provide the best approximation for real operating conditions.

Consequently, it is first necessary to determine the values of \(x_d\) and \(x_q\) as functions of external parameters, and then, using mathematical methods of processing, to pass to averaged values. Unfortunately, most of the known experimental methods do not make it possible to determine the parameters in arbitrary operating conditions [1, 2, 3]. Recently, methods free of this shortcoming have appeared [4, 5], but they involve increased complexity in instrumental implementation.

Below a method is proposed which, in its ideas, is close to that described in [4], but is less labor-intensive and requires a smaller amount of measuring apparatus, especially nonstandard apparatus.

The basis of the measuring system is a four-phase reference-voltage generator (RVG), rigidly coupled to the shaft of the synchronous generator (SG) under investigation and having the same polarity as it. To simplify the subsequent operations it is necessary that the EMF vectors of one of the phases of the RVG and of the SG coincide in direction (Fig. 1). In practice this is not difficult to accomplish, for example, by means of a coupling (M) that allows adjustment of the angular position of the shafts.

In the measurement circuit (Fig. 2), phase-sensitive voltmeters manufactured in series by industry are used, for example of type V5-2. Such instruments are intended for measuring the complex components of the vector of an applied voltage relative to a system of reference voltages shifted by \(90^\circ\). In other words, each of the two voltmeters with which the V5-2 instrument is equipped measures the projections of the applied voltage on the time axes of the reference voltages. In accordance with Figs. 1 and 2, the voltmeter \(V(0^\circ, 180^\circ)\) indicates the value \(U_d\), and \(V(90^\circ, 270^\circ)\)—\(U_q\). The same instrument (or even the very same one with corresponding switching of the input), connected through a noninductive-

Fig. 1. Vector diagram of a synchronous generator

… shunt (Sh), as a millivoltmeter, is used to measure the components \(I_q\)—mV \((0^\circ, 180^\circ)\) and \(I_d\)—mV \((90^\circ, 270^\circ)\).

In accordance with the vector diagram (Fig. 1), the sought values of the synchronous reactances are

\[ x_d=\frac{E_0-U_d-I_q r}{I_d}, \]

\[ x_q=\frac{U_q+I_d r}{I_q}, \]

where

\(r\) is the active resistance of the phase,

\(E_0\) is the no-load emf (with the load disconnected, its value is indicated by voltmeter \(V\) \((0^\circ, 180^\circ)\)).

An experimental check of the proposed method was carried out on a low-power generator \((P=1\ \text{kVA},\ U=150\ \text{V},\ f=200\ \text{Hz})\). In addition to the proposed method, the methods of no-load and short-circuit tests, slip, inductive and capacitive loads were also used to determine the parameters \(x_d\) and \(x_q\). All methods gave practically coincident values of the unsaturated reactance \(x_q\); moreover, it was established that in the machine under study saturation has no appreciable effect on it. The data for the unsaturated value of \(x_d\) also coincide (Fig. 3), but with increasing saturation, as …

Fig. 2. Circuit for measuring \(x_d\) and \(x_q\).

Fig. 3. Characteristics of a synchronous generator: 1 — no-load characteristic; 2 — rectified no-load characteristic; 3 — \(x_d\), obtained from the rectified no-load characteristic and the short-circuit characteristic and by the slip method; 4 — \(x_d\), determined from the no-load and short-circuit characteristics; 5 — \(x_d\), determined by the proposed method.

as can be judged from the no-load characteristic, \(x_d\) decreases noticeably, and, in addition, discrepancies are observed in the data obtained by different methods. Since saturation has a fundamental effect on the accuracy of all the methods used, except for the proposed one, its data should be considered the most reliable.

Thus, the proposed method for determining the parameters \(x_d\) and \(x_q\), based on the use of a four-phase power-frequency generator and phase-sensitive voltmeters, is sufficiently simple and makes it possible to carry out measurements under any operating conditions of a synchronous machine.

REFERENCES

1. A. I. Bertinov, V. G. Andreev. Determination of the parameters of magnetoelectric generators with a “star” type rotor. Proceedings of MAI, issue 133, Oborongiz, 1961.

2. M. N. Golodnov. Experimental determination of the synchronous reactance of three-phase salient-pole generators with permanent magnets. Elektrichestvo, 1958, No. 6.

3. I. E. Rodionov. Comparison of methods for determining the saturated parameters of synchronous salient-pole machines. Proceedings of UPI, collection 157, Sverdlovsk, 1967.

4. G. V. Karlov. A method for the experimental investigation of a synchronous machine in two axes of magnetic symmetry. Elektrichestvo, 1970, No. 9.

5. Yu. G. Buchholz, V. V. Zhulan, Yu. P. Rybkin, G. A. Shanshurov. Determination of parameters by estimating magnetic fields in the air gap of a synchronous machine. Izvestiya TPI, vol. 212, Tomsk University Press, 1971.