ANALYSIS AND EQUIVALENT CIRCUIT OF THE ROTOR–INDUCTOR CIRCUIT OF AN INSTALLATION FOR DETECTING CASTING DEFECTS IN SQUIRREL-CAGE ROTORS

A. I. ZAITSEV, M. P. TABINSKY

(Recommended by the scientific seminar of the Electromechanical Faculty)

Casting defects in a squirrel-cage rotor impair the operating characteristics of an induction motor. Despite strict operational control during manufacture of the motor, a rotor defect can often be detected only during the acceptance test of the assembled machine, when it can no longer be corrected and the rotor has to be scrapped. This leads to a considerable expenditure of labor, time, and materials, and disrupts the regularity of motor production. Meanwhile, a rejected rotor can be recast if the defect is found before mechanical machining of the rotor. According to data from the Kuzbass Elektromotor plant, casting defects in rotors cause the plant an annual loss of about 20,000 rubles. The question has not yet been studied of how the rotor asymmetry permitted by factory standards affects motor reliability. As is known, rotor asymmetry is permitted that leads to a five-percent fluctuation of the stator current during a short-circuit test.

According to [1], the breakage of one bar out of 46 is already impermissible; in this case, as experience shows, the fluctuation of the short-circuit current does not exceed 2%.

At present there is no effective quality control of rotor casting before mechanical machining. The control methods proposed at various times have not found application in production because of their complexity, low sensitivity, or low productivity [2, 3].

The authors have proposed, designed, and tested an installation for inspecting rotors before mechanical machining that is free of these shortcomings. The installation is universal, since it can be used to inspect rotors of various overall dimensions, with any slot configuration and number of slots.

The schematic diagram of the installation is shown in Fig. 1.

The rotor being inspected 1 is placed on two parallel steel cylindrical rollers 2, rotating synchronously in one direction. An inductor 3 with a Ш-shaped core is installed beneath the rotor. An excitation winding is wound on the middle magnetic core of the inductor ...

… winding, which is connected to an industrial-frequency alternating-current mains supply. Measuring windings, connected in opposition, are wound on the two outer magnetic cores. The inductor can be moved by means of a lead screw along vertical guides, which makes it possible to set the optimum gap between the rotor and the inductor for different rotor diameters. The gap may reach 2 mm or more.

Fig. 1.

The current \(\dot{i}_1\), flowing through the inductor winding \(\omega_1\), creates in the average magnetic core a magnetic flux which closes through the outer magnetic cores of the inductor via two sections of the rotor, I and II, and along leakage paths.

The magnetic fluxes penetrating sections I and II of the rotor induce emf’s in the bars of these sections; as a result, rotor currents \(I_{(u)1}\) and \(I_{(u)2}\) arise, which create their own magnetic fluxes directed opposite to the exciting fluxes. As a result, in the magnetic circuit the fluxes are distributed as shown in Fig. 1, where the following notation is used: \(\Phi_{\mathrm{ri}1}\), \(\Phi_{\mathrm{ri}2}\), \(\Phi_{\mathrm{ri}3}\) are the first, second, and third leakage fluxes of the inductor; \(\Phi_1\) and \(\Phi_2\) are the fluxes linked with the rotor bar and the inductor winding \(\omega_1\).

The fluxes \(\Phi_1\) and \(\Phi_{\mathrm{ri}2}\) induce a voltage \(\dot{U}_2\) in the measuring winding \(\omega_2\); the fluxes \(\Phi_2\) and \(\Phi_{\mathrm{ri}3}\) induce a voltage \(\dot{U}_3\) in the winding \(\omega_3\).

The measuring windings \(\omega_2\) and \(\omega_3\) are connected according to a differential scheme, and at the output terminals there arises the voltage difference

\[ \dot{U}_{\mathrm{out}} = \dot{U}_2 - \dot{U}_3 . \]

If bars with equal resistances are located in rotor circuits I and II, then the fluxes \(\Phi_1\) and \(\Phi_2\) will be identical. The fluxes \(\Phi_{\mathrm{ri}2}\) and \(\Phi_{\mathrm{ri}3}\) are identical by virtue of the symmetry of the inductor. In this case the output voltage is equal to zero.

Casting defects in the bars form cavities, constrictions, and breaks, which leads to a change in the active resistance of the bars. Let us consider the case in which the active resistance of the first rotor circuit is greater than the resistance of the second circuit. In this case the current of the second circuit creates a magnetic flux that displaces the exciting flux of the inductor into the first section, which leads to an increase in the flux \(\Phi_1\) (\(\Phi_1 > \Phi_2\)). As a result, the voltage \(\dot{U}_2\) becomes greater than the voltage \(\dot{U}_3\), and the voltage difference \(\dot{U}_2 - \dot{U}_3\) can reach a considerable value.

As the rotor rotates, a casting defect moves from the first section into the second; therefore, one casting defect corresponds to two maxima of the voltage \(\dot{U}_{\mathrm{out}}\).

To determine the sensitivity of the installation to various casting defects of the rotor bars, it is expedient to represent the actual inductor together with the rotor under investigation by an equivalent circuit. The principle of operation of the installation and the physical processes occurring in it will not change if the U-shaped magnetic core of the inductor is replaced by two…

U-shaped magnetic cores with two excitation windings \(w_I\) and \(w_{II}\), having \(w_1\) turns, connected in series. Thus, the W-shaped inductor together with the rotor is analogous, in its principle of operation, to two transformers in which the primary windings are connected in series and the secondary windings are closed through the resistances \(R_{p1}\) and \(R_{p2}\). The reduced equivalent circuit is shown in Fig. 2, where the following notation is used:

Fig. 2.                Fig. 3.

\[ r_1 \]
—the total active resistance of the excitation windings \(w_I\) and \(w_{II}\) of the inductor;

\[ L_{pн1};\ L_{pн2};\ L_{pн3} \]
—the first, second, and third leakage inductances of the excitation windings \(w_I\) and \(w_{II}\) of the inductor;

\[ M_1;\ M_2 \]
—the mutual inductances of the inductor excitation windings \(w_I\) and \(w_{II}\) with rotor circuits I and II;

\[ L'_{pp1};\ L'_{pp2} \]
—the reduced leakage inductances of rotor circuits I and II;

\[ R'_{p1};\ R'_{p2} \]
—the reduced active resistance of the rotor winding.

The equivalent circuit of Fig. 2 can be transformed into a series connection of equivalent impedances (Fig. 3):

\[ z_1 = r_1 + j\omega L_{pн1}. \tag{1} \]

\[ z_2 = j\omega L_{pн2} + \frac{(R'_{p2} + j\omega L'_{pp1})\,j\omega M_1} {R'_{p1} + j\omega (M_1 + L'_{pp1})}, \tag{2} \]

\[ z_3 = j\omega L_{pн3} + \frac{(R'_{p2} + j\omega L'_{pp2})\,j\omega M_2} {R'_{p2} + j\omega (M_2 + L'_{pp2})}. \tag{3} \]

The voltage on the measuring windings is equal to:

\[ \dot U_1 = \frac{w_2}{w_1}\, i_1 z_2, \tag{4} \]

\[ \dot U_2 = \frac{w_3}{w_{II}}\, i_1 z_3. \tag{5} \]

Since the measuring windings are connected in series opposition, the output voltage \(\dot U_{\mathrm{out}}\) is equal to the difference of the voltages \(\dot U_1\) and \(\dot U_2\).

Taking into account that $\omega_2=\omega_3=\omega_{\mathrm{i}}$, and that $\omega_{\mathrm{I}}=\omega_{\mathrm{II}}=\omega_1$, from formulas (4) and (5) it follows that:

\[ \dot U_{\mathrm{out}}=\dot U_1-\dot U_2= \frac{w_{\mathrm{i}}}{w_1} i_1(z_2-z_3)= \]

\[ = i_1\frac{w_{\mathrm{i}}}{w_1} \left[ \frac{\left(R'_{\mathrm{p}1}+j\omega L'_{\mathrm{pp}1}\right)j\omega M_1} {R_{\mathrm{p}1}+j\omega\left(M_1+L_{\mathrm{pp}1}\right)} - \frac{\left(R'_{\mathrm{p}2}+j\omega L'_{\mathrm{pp}2}\right)j\omega M_2} {R'_{\mathrm{p}2}+j\omega\left(M_2+L'_{\mathrm{pp}2}\right)} \right]. \tag{6} \]

Owing to the symmetry of the electromagnetic circuit of the installation, in the absence of casting defects in the rotor winding the following equalities are valid:

\[ M_1=M_2=M, \tag{7} \]

\[ L'_{\mathrm{pp}1}=L'_{\mathrm{pp}2}=L'_{\mathrm{pp}}. \tag{8} \]

\[ R'_{\mathrm{p}1}=R'_{\mathrm{p}2}=R'_{\mathrm{p}}. \tag{9} \]

In this case, according to formula (6),

\[ \dot U_{\mathrm{out}}=0. \]

Let us consider the case when casting defects lead to a change in the resistance of a bar in the first section of the rotor by an amount $\Delta R'$, and let us assume that equalities (7) and (8) are not changed in this case. A voltage will appear on the measuring windings:

\[ \dot U_{\mathrm{out}}= i_1\frac{w_{\mathrm{i}}}{w_1} \left[ \frac{\left(R'_{\mathrm{p}}+\Delta R' + j\omega L'_{\mathrm{pp}}\right)j\omega M} {R'_{\mathrm{p}}+\Delta R' + j\omega\left(M+L'_{\mathrm{pp}}\right)} - \frac{\left(R'_{\mathrm{p}}+j\omega L'_{\mathrm{pp}}\right)j\omega M} {R'_{\mathrm{p}}+j\omega\left(M+L'_{\mathrm{pp}}\right)} \right]. \tag{10} \]

The resistances $R'_{\mathrm{p}}$ and $\Delta R'$ in the denominators of the fractions may be neglected, since their magnitudes are much smaller than the inductive reactances of the equivalent circuit. After simple transformations we obtain

\[ \dot U_{\mathrm{out}}\approx j_1\frac{w_{\mathrm{i}}}{w_1}\, \frac{M\Delta R'}{M+L'_{\mathrm{pp}}}. \tag{11} \]

This formula agrees well with experiment if

\[ \Delta R'\ll 5R'_{\mathrm{p}} \quad \text{and} \quad \Delta R' + R'_{\mathrm{p}}\ll 0.1\,j\omega\left(M+L'_{\mathrm{pp}}\right). \]

A formula for calculating the output voltage when one bar is broken in the first section of the rotor is derived analogously $(\Delta R=\infty)$:

\[ \dot U_{\mathrm{out}}\approx i_1\frac{w_{\mathrm{i}}}{w_1}j\omega M \left( 1-\frac{1}{M+L'_{\mathrm{pp}}} \right). \tag{12} \]

It should be noted that in reality $M_1>M_2$, since the mutual inductance is affected by the displacement of the magnetic flux from the rotor section where the bar is intact into the section with the broken bar. Therefore, the experimentally measured value of $\dot U_{\mathrm{out}}$ is 20–30% greater than the calculated value when the equality $M_1=M_2$ is assumed.

Figure 4 shows an oscillogram of the output voltage of the inductor $\dot U_{\mathrm{out}}$ when testing the rotor of a KO 22-4 motor with one broken bar. The rotor was unmachined. For oscillography, the voltage was rectified and smoothed.

Figure 5 shows an oscillogram of the voltage $\dot U_{\mathrm{out}}$ when studying on the installation the rotor of a KO 22-4 motor rejected by the test station due to a 5% fluctuation of the short-circuit current. The oscillogram clearly shows the nature of the casting defect: two broken bars located next to one another (a), and two broken bars positioned around the circumference of the rotor (b, c).

Experimental verification of the proposed method for determining casting defects was carried out under production conditions at the Kemerovo Electrical-Machine-Building Plant “Kuzbasselektromotor.”

Fig. 4.

Fig. 5.

The principal technical data of the rotors tested are given in Table 1. The rotor slots were closed. The rotors were tested in the unmachined state, immediately after casting on a centrifugal machine and after cooling to a temperature of about \(100^\circ\text{C}\).

Table 1

As a result of the tests it was established that the installation clearly records linear defects of the bars that lead to a 2% fluctuation of current in the short-circuit test of the assembled motor

\[ \left(\frac{i_{\max}-i_{\min}}{i_{\max}}\cdot 100 \leq 2\%\right). \]

The output voltage when a section of the rotor with a broken bar passes over the inductor is 50% greater than with a normal bar (25 V and 5 V, respectively). In the presence of a casting defect that changes the active resistance of the bar by 40%, the output voltage becomes 100% greater than with a normal bar (10 and 5 V, respectively).

The process of inspection, setting up, and removing a rotor with a diameter of 200 mm and a weight of 50 kg takes no more than 30 sec.

The tests showed that roughness and local contamination of the rotor up to 2 mm thick do not have a noticeable effect on the operation of the installation.

Conclusions

The proposed installation for determining casting defects during rotor casting is simple to manufacture and operate and has high sensitivity. The simplicity of operation and measurement makes it possible easily to automate the operation of quality control of the casting of rotors of asynchronous motors.

References

1. Hiller, Ute-Maria: Einfluß fehlender Läuferstäbe auf die elektrischen Eigenschaften von Kurzschlußläufermotoren. ETZ—A, 1962, No. 4.

2. A. V. Safronov. Methods for Detecting Defects in the Rotors of Asynchronous Motors, TsBTI, Moscow, 1959.

3. Dan Teodorescu. A new method for determining casting defects in the bars of squirrel-cage rotors. VEP, 1963, No. 7.