FEATURES OF CALCULATING LEAKAGE REACTANCES OF INDUCTION MOTORS WITH AN ELECTROMAGNETIC INSERT ON THE ROTOR

E. A. Knipenberg, E. M. Guselnikov

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

Induction motors are the most frequently encountered element of a drive; therefore, on the basis of these electric motors the greatest number of modifications are developed, intended for various purposes. One of such modifications is a motor with a special insert.

Studies carried out at the Special Design Bureau of the “Sibelektromotor” plant showed that, using the leakage fields of short-circuited rings and part of the main flux of induction motors, it is possible to obtain considerable tractive forces for attracting, during starting, a special insert freely moving in the axial direction on the shaft, and also for subsequently holding the insert during operation in the rated mode.

The rotor of an electric motor with an insert (Fig. 1) consists of two parts: the special part B and the main part A, separated from each other by an insulating spacer 1 and connected by a common squirrel cage. On the shaft

Fig. 1. Induction motor with an electromagnetic insert:
a) longitudinal section of the rotor and stator, b) slot of the special part of the rotor, c) distribution of the magnetic flux along the length of the rotor.

a special insert 2 made of magnetically conducting material is mounted. The insert can move in the axial direction.

The main part of the rotor is assembled from ordinary laminations, while the laminations of the special part have the configuration shown in Fig. 1b. The insulating spacer has the same configuration as the laminations of the main part of the rotor. After the rotor stack has been cast with aluminum, the special part of the rotor is turned in such a way that the slots \(b_{\mathrm{sh}}\) are opened and the teeth are separated from one another.

When power is supplied to the motor, insert 2, under the action of the leakage fields of the short-circuiting ring \(\Phi_s\) and, in part, of the main flux \(\Phi_0\) passing through the special part of the rotor and the insert, overcomes the force of spring 3, is attracted to the end face of the rotor, and is held in the attracted position by the action of the motor’s main magnetic flux \(\Phi_0\) incident on the special part of the rotor; because of the presence of slots \(b_{\mathrm{sh}}\) and insulating spacer 1, this flux is forced to close through insert 2 as through the rotor yoke. The action of the leakage fields of the rotor short-circuiting ring at small slips is slight, since the currents in the short-circuiting ring are small.

The electromagnetic forces hold the insert in the attracted position as long as voltage is applied to the motor. After the voltage is switched off, the magnetic flux and the electromagnetic forces holding the insert in the attracted position disappear, and the insert, under the action of the spring, moves away from the rotor stack. If the insert is connected to some actuating mechanism, the indicated effect can be used for braking and releasing electric motors, as well as for other actions of various kinds, for example, switching in automation and control circuits, opening valves, etc.

The introduction into the rotor of a magnetically conducting sleeve 2, an insulating spacer 1, and special laminations with slots \(b_{\mathrm{sh}}\) leads to a substantial change in the leakage inductive reactance of the end portions of the stator and rotor, as well as in the slot leakage of the special part of the rotor.

a) Conductance of leakage fluxes between the slot walls

Figure 2 gives a diagram for calculating the slot conductance of the leakage fluxes of the special part of the rotor; the diagram is drawn with allowance for the above-described features of the motion of the magnetic leakage flux.

In Fig. 2 the following notation is introduced:

\(\lambda_1, \lambda_1'\) — the slot conductance and the conductance of the air gap between the rotor laminations, and also between the rotor end face and the insert, corresponding to the section of the slot with height \(h_0\);

\(\lambda_2, \lambda_2'\) — the slot conductance and the conductance of the air gap between the rotor laminations, and also between the rotor end face and the insert, corresponding to the section of the slot with height \(h\);

\(\lambda_3, \lambda_3'\) — the slot conductance and the conductance of the air gap between the rotor laminations, and also between the rotor end face and the insert, corresponding to the round part of the slot;

\(\lambda_{\mathrm{sh}}, \lambda_{\mathrm{sh}}'\) — the slot conductance and the conductance of the air gap between the rotor laminations, and also between the rotor end face and the insert, corresponding to the section of the slot with height \(h_{\mathrm{sh}}\).

According to the diagram presented, the total conductance of the leakage fluxes between the slot walls of the special part of the rotor is equal to

\[ \lambda_{\mathrm{p}} = \frac{ \left( \sum_{i=1}^{i=3} \lambda_i \times \sum_{i=1}^{i=3} \lambda_i' \right) \times [\lambda_{\mathrm{sh}} + \lambda_{\mathrm{sh}}'] }{ \left( \sum_{i=1}^{i=3} \lambda_i + \sum_{i=1}^{i=3} \lambda_i' \right) + (\lambda_{\mathrm{sh}} + \lambda_{\mathrm{sh}}') }. \tag{1} \]

Fig. 2. Conductance diagram of leakage-flux paths:
a) between the slot walls of the special part of the rotor;
b) around the short-circuiting rings of the rotor on the side of the electromagnetic insert

In this case, the conductance of the rotor leakage flux along the path from the slot to the insert is determined as the ratio of the total height of the corresponding section of the slot to the total value of the air gap \(\delta_1\) along the path of the flux to the insert and back, taking into account the different positions of the sections over the slot height,

\[ \lambda_1' = \frac{h_0}{\sum \delta_1}, \tag{2} \]

where

\[ \sum \delta_1 = 2 \left[ \delta_1 + \frac{l_c}{2}(1 - k_c) \right] \]

\(\delta_1\) — the magnitude of the air gap between the end face of the rotor and the insert at the height \(h_0\);
\(k_c\) — coefficient of steel filling of the rotor stack;
\(l_c\) — length of the special part of the rotor.

\[ \lambda_2' + \lambda_3' = \frac{ h + \dfrac{d_1 + d_2}{2} }{ 3 \sum \delta_{2,3} }, \tag{3} \]

where

\[ \sum \delta_{2,3}=2\left[\delta_{2,3}+\frac{l_c}{2}(1-\kappa_c)\right] \]

\(\delta_{2,3}\) is the magnitude of the air gap between the end face of the rotor and the insert at the height \(h+\dfrac{d_1+d_2}{2}\).

\[ \lambda_{\mathrm{sh}}'=\frac{h_{\mathrm{sh}}}{\sum \delta_{\mathrm{sh}}}, \tag{4} \]

where

\[ \sum \delta_{\mathrm{sh}}=2\left[\delta_{\mathrm{sh}}+\frac{l_c}{2}(1-\kappa_c)\right], \]

\(\delta_{\mathrm{sh}}\) is the magnitude of the air gap between the end face of the rotor and the insert at the height \(h_{\mathrm{sh}}\).

Since the magnitude of the air gap \(\delta_{2,3}\) is very large (not less than the height of the short-circuiting ring \(L\)—Fig. 3), the permeances \(\lambda_2' + \lambda_3'\) may be neglected.

Fig. 3. Short-circuiting ring of the rotor with an electromagnetic insert

If it is taken into account that the width \(b_{\mathrm{sh}}\) is usually 3–4 times smaller than the average slot width and that the height of the slit \(h_{\mathrm{sh}}\) is 10–15 times greater than the height \(h_0\), then, in deriving the formula for the permeances \(\sum_{i=1}^{i=3}\lambda_1\), it may be assumed that, in all sections of the slot, the leakage flux closes through the slit \(b_{\mathrm{sh}}\) and partly through the portion of the insert corresponding to the height of the slit.

This makes it possible to calculate the permeances \(\sum_{i=1}^{i=3}\lambda_1\) with an accuracy sufficient for practical purposes, as for an ordinary slot.

The permeance of the slit is calculated as

\[ \lambda_{\mathrm{sh}}=\frac{h_{\mathrm{sh}}}{b_{\mathrm{sh}}}. \tag{5} \]

b) Permeance of the leakage flux around the short-circuiting rings of the rotor on the side of the electromagnetic insert

Because of the presence of the insert, the short-circuiting ring of the rotor on the side of the special part is surrounded by ferromagnetic po-

surfaces (Fig. 3b), which leads to a significant increase in the reactive resistance of this ring. Therefore, when deriving the formula for calculating the leakage permeance around the end part of the rotor on the side of the electromagnetic insert, the short-circuiting ring is considered as a current-carrying conductor located in an open slot. This slot is completely open in the sections located opposite the rotor slots; in those sections that correspond to the teeth of the additional part of the rotor, in addition to the leakage flux between the walls of the electromagnetic insert, there also appears a leakage flux closing through the gaps \(\delta_n\) and \(\delta_b\) and the rotor teeth, which in this case play the role of a yoke.

In accordance with the indicated features of the distribution of leakage fluxes, Fig. 2b gives an equivalent circuit of the permeances of the leakage fluxes around the short-circuiting ring of the rotor on the side of the electromagnetic insert. In Fig. 2b the following notation is introduced:

\[ \lambda_{k1} \]
— the permeance of the short-circuiting ring in the sections corresponding to the rotor slots;

\[ \lambda_{k2} \]
— the permeance between the insert walls in the sections corresponding to the rotor teeth;

\[ \lambda'_{k2} \]
— the permeance of the leakage flux closing through the gaps \(\delta_n\) and \(\delta_b\) and the rotor teeth.

Taking Fig. 3 into account, the indicated permeances are determined as follows:

\[ \lambda_{k1}=\lambda_{k2}=\frac{h}{3b}, \tag{6} \]

\[ \lambda'_{k2}= \frac{ \dfrac{a_1}{\delta_n}\times\dfrac{a_2}{\delta_b} }{ \dfrac{a_1}{\delta_n}+\dfrac{a_2}{\delta_b} }. \tag{7} \]

The average length of the sections of the short-circuiting ring located opposite the rotor slots, \(l_1\), and of the ring sections located opposite the rotor teeth, \(l_2\), is respectively equal to:

\[ l_1=D_{\mathrm{cp}}\frac{b_{n\mathrm{cp}}}{b_{z\mathrm{cp}}+b_{n\mathrm{cp}}}, \tag{8} \]

\[ l_2=D_{\mathrm{cp}}\frac{b_{z\mathrm{cp}}}{b_{z\mathrm{cp}}+b_{n\mathrm{cp}}}, \tag{9} \]

where

\(b_{n\mathrm{cp}}\) and \(b_{z\mathrm{cp}}\) are the average widths of the slots and teeth of the special part of the rotor.

The total equivalent leakage permeance of the short-circuiting ring of the rotor on the side of the insert per unit rotor length, in accordance with the circuit of Fig. 2b and taking into account expressions (6, 7, 8, 9), is equal to

\[ \lambda_s=\lambda_{k1}+\lambda_{k2}+\lambda'_{k2} = \frac{\pi D_{\mathrm{cp}}}{l_\delta} \left[ \frac{h}{3b} + \left( \frac{ \dfrac{a_1}{\delta_n}\times\dfrac{a_2}{\delta_b} }{ \dfrac{a_1}{\delta_n}+\dfrac{a_2}{\delta_b} } \right) \frac{b_{z\mathrm{cp}}}{b_{z\mathrm{cp}}+b_{n\mathrm{cp}}} \right]. \tag{10} \]

When the gaps are equal, \(\delta_n=\delta_b\), expression (10) is written as

\[ \lambda_s= \frac{\pi D_{\mathrm{cp}}}{l_\delta} \left[ \frac{h}{3b} + \frac{a_1\times a_2\times b_{z\mathrm{cp}}} {\delta_n\times(a_1+a_2)\times(b_{z\mathrm{cp}}+b_{n\mathrm{cp}})} \right]. \tag{11} \]

c) Leakage permeance of the end parts of the stator winding on the side of the electromagnetic insert

Studies carried out on electric motors with a power of

0.2–7.5 kW have shown that, because the insert is brought close to the end portions of the stator winding, the leakage flux in the end portions increases by 20–25% (Fig. 1, c); therefore, when determining the leakage permeance of the end portions of the stator winding on the side of the electromagnetic insert, a correction factor \(k = 1.2 \div 1.25\) must be introduced into the generally accepted formulas. The larger values of the coefficient apply to windings with the number of pole pairs \(2p \geq 4\).