SETUP FOR EXPERIMENTAL INVESTIGATION OF ELECTRODYNAMIC FORCES IN THE END WINDINGS OF A SHOCK GENERATOR

G. A. SIPAILOV, V. I. POPOV

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

The ability to determine correctly the electrodynamic forces arising in individual structural elements of electrical machines is one of the principal tasks in their design. This applies especially to shock generators, in which the electrodynamic forces substantially exceed the forces arising in elements of electrical machines of conventional design under normal operating conditions.

The greatest difficulties, both from the standpoint of calculating electrodynamic forces and of withstanding them, arise in designing the end windings of stator windings. This is due to the complex spatial distribution of the winding bars in the end-winding zone and to the influence of the surrounding ferromagnetic surfaces.

Usually, when calculating electrodynamic forces in end windings, simplifying assumptions are adopted in order to facilitate the calculation; these assumptions naturally introduce errors into the calculation results [1, 2, 3]. On the other hand, reducing the number of assumptions complicates the calculation procedure to such an extent that its application becomes possible only with the aid of digital computers [5, 4].

Thus, the problem of a sufficiently accurate calculation of electrodynamic forces in the end windings of electrical-machine windings requires further theoretical and experimental investigation. In this connection, it is proposed, on the basis of experimental studies of the electrodynamic forces arising in conductors that model the end windings of a single-layer concentric winding and are located in various media, to develop a calculation procedure that is sufficiently complete and acceptable from the standpoint of labor input. The present article gives a description of the setup and sets forth the procedure for experimental investigations. Subsequently, a method will be given for calculating electrodynamic forces in the end windings both without taking into account, and with taking into account, the influence of the surrounding surfaces.

In order to develop the experimental technique and subsequently determine the share of influence, on the magnitude of the forces, of ferromagnetic and other surfaces surrounding the end windings, at the first stage the experiments were carried out on models consisting of conductors arranged in air. One of these models, in which the arrangement of the conductors corresponds to the configuration of the end winding of a single-phase single-layer concentric winding of a two-pole shock generator, is shown in Fig. 1, a. In order to exclude from the experimental results the influence of twisting, the magnitude of which depends on

length of the curvilinear portion; all the conductors, by means of special fastening, were placed under identical conditions.

To determine the forces acting on the conductors and to estimate the strength of the conductors, it is necessary to know the mechanical stresses arising in them. However, mechanical stresses cannot be measured directly. They can be determined if the forces applied to the conductor and the geometrical dimensions of the conductor are known, or if the deformations of the conductor and their distribution are known. Since deformation of the conductor is associated with a change in its dimensions, it can be measured directly.

For measuring various mechanical quantities converted into deformations, wire transducers are used—resistance strain gauges bonded to the part under test. Their relatively small dimensions, simple shape, light weight, possibility of use in places that are difficult to access during operation, reliability, and small measurement error open up broad possibilities for their use in experimental investigations.

As deformation sensors in the installation under consideration, the diagram of which is shown in Fig. 1, wire strain gauges of type PKB-10-200, with a 10 mm gauge length and a resistance of 200 ohms, were used. When the gauges are connected according to the balanced-bridge circuit (Fig. 1, b), in the event of deformation of the part, an unbalance voltage \(\Delta U\) appears in the measuring diagonal of the bridge as a result of the change in the resistance of the strain gauges due to deformation

\[ \Delta U = \frac{1}{4} U k_{\mathrm{т}}\varepsilon, \]

where \(U\) is the bridge supply voltage,
\(K_{\mathrm{т}}\) is the strain-sensitivity coefficient of the strain gauge,
\(\varepsilon\) is the measured relative deformation.

One of the main characteristics determining the quality of a strain gauge is its strain-sensitivity coefficient \(k_{\mathrm{т}}\). It is necessary that all strain gauges selected for carrying out the experimental investigations have the same coefficient \(k_{\mathrm{т}}\), so that the measured signal taken from the strain gauges is proportional only to deformation. Therefore, before bonding the strain gauges to the model under investigation, for 10 percent of the batch of strain gauges selected for the experiment the dependence \(k_{\mathrm{т}} = f(n_{\mathrm{т}})\) was taken (Fig. 2, a), which confirmed that the strain-sensitivity coefficient may be taken as constant and, for this batch, equal to \(k_{\mathrm{т}} = 1.96\).

To determine the forces acting on the end portions of the model, active gauges were bonded in the middle part of each curvilinear conductor on two opposite sides, with their grids along the conductor, and were connected into the measuring circuit according to the bridge scheme. To eliminate pickups in the measured signal, all wiring was made with shielded wire with the shield grounded. For supplying the strain-gauge bridge and receiving the measured signal proportional to deformation, a four-channel strain-gauge amplifier UT4-1, equipped with an electrical calibration device, was used. Recording of the deformations was carried out simultaneously for three conductors together with recording of the current feeding the model. A shock generator was used as the power source for the end-part model under study. Since the time between current pulses of the shock generator can be set arbitrarily large, there is no need to provide temperature compensation of the strain gauges, since the heat released in one pulse is insignificant. This makes it possible to use a circuit with two active gauges. The connection circuit and the bonding locations of the active strain gauges are indicated in Fig. 1.

a)

Labels in the drawing: \(R\), \(\gamma\); 1, 2, 3, 4, 5, 6; strain gauge.

b)

Fig. 1. Diagram of the experimental setup: IM — measuring bridge, BU — balancing device, TU — calibration device, F — carrier-frequency filter, FD — phase-sensitive detector, U — amplifier, GNC — carrier-frequency generator, IU — measuring device, \(T_1, T_2\) — active strain gauges, \(T_3, T_4\) — reference strain gauges.

The characteristics of the measurement process and the properties of the elements of the measuring circuit predetermine the presence of quite definite errors—measurement inaccuracies. It is therefore necessary to make a realistic estimate in advance of the magnitude of the measurement errors, since neglecting them may lead to a large distortion of the measurement results.

The main sources of error may be:

a) instability of the strain-sensitivity coefficient,

b) nonlinearity of the amplitude characteristic of the measuring circuit [6].

As was shown (Fig. 2, a), for the selected batch of strain gauges \(k_t=\mathrm{const}\). As for the nonlinearity of the amplitude characteristic, in order to evaluate it the dependence of the output current of the measuring circuit on the relative strain was recorded (Fig. 2, b).

From the experimental data it follows that the nonlinearity of the amplitude characteristic is no more than 3 percent. The forces acting on the conductors when current flows through them were determined as follows. Before each measurement, a calibration oscillogram was recorded, the form of which is shown in Fig. 3, a. The calibration signal in each channel of the amplifier corresponds to \(\varepsilon_t=1\cdot10^{-4}\) rel. units of strain. Then, at unchanged amplification, an operating oscillogram was recorded while the shock current was passed through the conductors of the model. The form of the oscillogram is shown in Fig. 3, b. From the operating and calibration oscillograms the value of the measured relative strain \(\varepsilon_i\) was determined.

The bending moment acting on the conductor, measured by the scheme with two active gauges, is

\[ M_i=\frac{1}{2}\varepsilon_i EW, \]

where

\[ E \text{ — modulus of elasticity of the material; for copper } E=1.23\cdot10^6\ \frac{\mathrm{kg}}{\mathrm{cm}^2}, \]

\[ W=\frac{1}{6}ab^2 \text{ — section modulus in bending.} \]

Since the section on which the bending moment was measured is small, the specific load may be considered uniformly distributed. From the theory of strength of materials it is known that, for a beam with a uniform load, rigidly fixed at its ends, the bending moment at the middle is equal to

\[ M_i=\frac{1}{24}F_a l. \]

Since in our case \(\varphi=57^\circ=1\ \mathrm{rad}\), \(R=24\ \mathrm{cm}\), then \(l=R\varphi=24\ \mathrm{cm}\), and \(M_i\) is numerically equal to \(F_a\) in kg. Hence,

\[ F_a=\frac{1}{2}\varepsilon_i EW. \]

The axial forces measured by the method described, arising in the conductors when a current of 4000 A flows through them, are presented in Table 1; for comparison, calculated data are also given there.

Table 1

Fig. 5. Characteristics of the measurement-error assessment

a)

\(K_r\)

\(N\), strain-gauge point

b)

\(\xi \cdot 10^{-4}\) (relative units)

\(\bar I_{\mathrm{out}} \times 10\) (mA)

Fig. 3. Oscillograms: a — calibration, b — operating.

The satisfactory agreement between the results of calculation and experiment, obtained for a comparatively simple arrangement of conductors that is readily amenable to calculation, confirms the possibility of using the experimental setup to determine forces in all other cases, including those for which a sufficiently accurate calculation method does not yet exist. The reliability of the experimental data makes it possible to take sufficiently fully into account the influence of ferromagnetic and other surfaces surrounding the end portions of windings when calculating electrodynamic forces.

REFERENCES

1. G. B. Kholyavskii. Calculation of Electrodynamic Forces in Electrical Apparatus. GEI, 1962.

2. I. F. Calvet. Forces in Turbine Generator Stator Windings. Trans. of the A.I.E.E. 1931. 50, p. 173.

3. D. Harrington. Forces in Machine End Windings. Trans. of the A.I.E.E., 1952. Vol. 71, Part III, pp. 849–859.

4. A. I. Verkhoturov, G. A. Sipailov, K. A. Khorkov. Method for calculating forces in the end portions of a single-layer winding with the aid of an electronic digital computer. Izvestiya TPI, vol. 139, 1965.

5. A. I. Verkhoturov, G. A. Sipailov, K. A. Khorkov. Forces in the end portions of the stator winding of a shock generator. Izvestiya TPI, vol. 145, 1965.

6. Z. Ruzga. Electrical Resistance Strain Gauges. GEI, 1961.