The Single-Cage Rotor: The Slot Leakage Circuit Loops

Abstract

With the simplifying provisions accepted in this work, the slot leakage impedance values of a single-cage rotor (impedance values of the slot bar of a single-cage rotor) are usually determined in consideration of the transverse leakage field taking place in the single rotor slot, which is caused by the currents induced in the conducting slot bar. The task of calculating the slot leakage field has been solved by many authors (A. Field, P. Emde, R. Richter, etc.). Below, on the basis of the results obtained in these studies, we consider the different approaches that can be used to determine the leakage impedance values of a single-cage rotor. We will proceed from the fact that the rectangular bar is located in the rectangular rotor slot, and we will use generally accepted assumptions.

Appendices A.9 Results of Calculations

9.1.1 A.9.1. The Relative Thickness of the Rotor Slot Bar Sub-Layers ε ₁₂ and ε ₂₂: Weak Skin Effect

The calculations of the relative thickness of the rotor slot bar sub-layers ε ₁₂ and ε ₂₂ were implemented using the condition obtained in (9.61). The results of the calculations are shown in Table A.9.1. For the known values of parameters ε ₁₂ and ε ₂₂ , the thickness of the rotor slot bar sub-layers h _Π1 and h _Π2 are determined from the condition h _Π1/h _Π2 = ε ₁₂/ε ₂₂. Then, taking into account that h _Π = h _Π1 + h _Π2, we have for h _Π1 and h _Π2 that

$$ \frac{h_{\Pi 2}}{h_{\Pi}}=\frac{1}{1+{\varepsilon}_{12}/{\varepsilon}_{22}}\kern0.5em \mathrm{and}\kern0.5em \frac{h_{\Pi 1}}{h_{\Pi}}=1-\frac{h_{\Pi 2}}{h_{\Pi}} $$

(A.9.1)

Table A.9.1 Values of the relative thickness of the rotor slot bar sub-layers depending on the relative depth of field penetration in the rotor slot bar at weak skin effect

9.1.2 A.9.2. The Relative Thickness of the Rotor Slot Bar Sub-Layers ε ₁₂ and ε ₂₂: Strong Skin Effect

In this case, the calculations of the relative thickness of the rotor slot bar sub-layers ε ₁₂ and ε ₂₂ are implemented using the expression given in (9.67). The results of the calculations are presented in Table A.9.2. When h _fcd  > h _Π, the values of h _Π1 and h _Π2 are determined by the expressions (A.9.1), with consideration for the condition h _Π1/h _Π2 = ε _1cz /ε _2cz . Here, the values of ε ₁₂ and ε ₂₂ arise from Table A.9.2. When h _fcd  ≤ h _Π, the values of h _Π1 and h _2E are calculated by the expressions (9.70).

Table A.9.2 Values of the relative thickness of the rotor slot bar sub-layers depending on the relative depth of field penetration in the rotor slot bar at strong skin effect

9.1.3 A.9.3. The Current Displacement Factors: An Analysis

Weak Skin Effect

At a weak skin effect, the current displacement factors are calculated by the expressions (9.6), (9.76), and (9.77). The results of the calculations are presented in Table A.9.3. It follows from this table that the expressions (9.6) and formulas (9.76) and (9.77) arising from the equivalent circuit in Fig. 9.6 provide the same results for the current displacement factors (columns 2, 3, 4, and 5, Table A.9.3). The data in columns 6 and 7 of Table A.9.3 were obtained by expressions (9.76) and (9.77) under the conditions that k _r1 = k _x1 = 1.0 and k ^′ _r2  = k ^′ _x2  = 1.0. In columns 8 and 9 of Table A.9.3, the values of the discrepancies (in %) among the calculated data presented in columns 2 and 3 and columns 6 and 7 are shown. The greatest discrepancy is observed in the area 1.3 < ε ₂ < 1.5, and it is at the level of (4–5)% for the resistance and (3–3.5)% for the leakage reactance of the rotor slot bar. Therefore, the calculated results obtained for current displacement factors k _r and k _x by expressions (9.76) and (9.77) (under the conditions that k _r1 = k _x1 = 1.0 and k ^′ _r2  = k ^′ _x2  = 1.0, columns 6 and 7), and with the use of expressions (9.6) (columns 2 and 3), are satisfactorily consistent when ε ₂ < 1.5. This means that the elements of the equivalent circuit in Fig. 9.6 can be determined by the simplified expressions (9.62) and (9.63) when ε ₂ < 1.5. The accuracy provided in this case is sufficient for practical calculations.

Table A.9.3 Values obtained by the analytical and proposed methods for the current displacement factors depending on the relative depth of field penetration in the rotor slot bar (weak skin effect)

Strong Skin Effect

At a strong skin effect (ε ₂ > 2.0), the current displacement factors arise on the basis of the equivalent circuit in Fig. 9.10, and they are determined by the expressions given in (9.78). The results of the calculations obtained by the expressions (9.6) and (9.78) are presented in Table A.9.4. It follows then that expressions (9.6) and (9.78) provide the same results for the current displacement factors, which is evident from the data in columns 2 and 3 and columns 4 and 5 (Table A.9.4). The data in columns 6 and 7 of Table A.9.4 were obtained by expression (9.78) under the conditions that k _r1 = k _x1 = 1.0 and k ^′′ _x2  = 1.0. In columns 8 and 9 of Table A.9.4, the values of the discrepancies (in %) between the calculated data presented in columns 2 and 3 and columns 6 and 7 are shown. The greatest discrepancy is observed for the resistance in the area 2.0 < ε ₂ < 2.5, and it is at a level of (5–6)%. The accuracy of determining the leakage reactance is at a level of (0.5–0.6)%. This means that the elements of the equivalent circuit in Fig. 9.10 can be determined by simplified expressions (9.62) and (9.69) when ε ₂ > 2.0. In this case, the accuracy provided is sufficient for practical calculations.

Table A.9.4 Values obtained by the analytical and proposed methods for the current displacement factors depending on the relative depth of field penetration in the rotor slot bar (strong skin effect)

From the analysis of factors k _r and k _x it follows that the area of preferred application of the equivalent circuit in Fig. 9.6 is ε ₂ < 1.5. The equivalent circuit in Fig. 9.10 is applicable when ε ₂ > 2.0. It follows from the data in Tables A.9.3 and A.9.4 that the equivalent circuits shown in Figs. 9.6 and 9.10 can be used in the area 1.5 < ε ₂ < 2.0. However, use of these equivalent circuits in this area of change in the parameter ε ₂ is associated with some deterioration of accuracy in the calculations. In accordance with the data in Table A.9.3, the use of the equivalent circuit in Fig. 9.6 is associated with the greatest discrepancy at the level of (5–6)% for the resistance and at the level of (3–3.5)% for the leakage reactance of the rotor slot bar. The greatest discrepancy obtained as a result of using the equivalent circuit in Fig. 9.10 in this area of change of parameter ε ₂ is at the level of (6–7)% for the resistance and at the level of (0.8–0.95)% for the leakage reactance of the rotor slot bar (Table A.9.4, columns 6 and 7). Thus we can assume that the equivalent circuits in Figs. 9.6 and 9.10 provide results acceptable for engineering practice when 1.5 < ε ₂ < 2.0.