Investigation Methods of Currents Distribution in Squirrel Cages with Asymmetry

Abstract

This chapter generalizes investigation methods of asymmetrical active U-shaped chain circuits (open and closed) developed in previous chapters. On their basis we obtained important results for practice regularities of currents distribution in asymmetrical squirrel cages (with breakages) induction machines.

Appendices

Appendix 10.1: Additional Currents in Eq. (10.2) for Squirrel Cage with Two (Not Adjacent) Asymmetrical Bars

The system of equations is given by:

$$ \sum\limits_{\text{i,j}} {{\text{a}}_{\text{i,j}} {\text{C}}_{{{\text{j}} + 2}} } = {\ddot {\text{I}}}_{\text{i}} \quad {\text{at}}\quad{\text{i}},{\text{j}} = 1,2,3,4. $$

Coefficients of system of equations (at ${\text{N}}_{{{\text{P}}_{1} }} = 0,\,{\text{N}}_{{{\text{P}}_{2} }} = {\text{N}}_{\text{P}}$):

$$ \begin{aligned} {\text{a}}_{1,1} & = - {\text{a}}_{2} - \frac{{\Delta {\text{Z}}_{0} }}{{{\text{Z}}_{\text{B}} }};\quad{\text{a}}_{1,3} = {\text{a}}_{1}^{{{\text{N}}_{0} - 1}} \left( {1 + \frac{{\Delta {\text{Z}}_{0} }}{{{\text{Z}}_{\text{B}} }}} \right);\quad{\text{a}}_{2,1} = {\text{a}}_{1}^{{{\text{N}}_{\text{P - 1}} }} \left( {1 + \frac{{\Delta {\text{Z}}_{{{\text{N}}_{\text{P}} }} }}{{{\text{Z}}_{\text{B}} }}} \right); \\ {\text{a}}_{2,3} & = - {\text{a}}_{1}^{{{\text{N}}_{\text{P}} - 1}} \left( {1 + {\text{a}}_{1} \frac{{\Delta {\text{Z}}_{{{\text{N}}_{\text{P}} }} }}{{{\text{Z}}_{\text{B}} }}} \right); \\ {\text{a}}_{3,1} & = - {\text{a}}_{1}^{{{\text{N}}_{\text{P}} }} \left( {1 + {\text{a}}_{2} \frac{{\Delta {\text{Z}}_{0} }}{{{\text{Z}}_{\text{B}} }}} \right);\quad{\text{a}}_{3,3} = {\text{a}}_{1}^{{{\text{N}}_{\text{P}} }} \left( {1 + \frac{{\Delta {\text{Z}}_{{{\text{N}}_{\text{P}} }} }}{{{\text{Z}}_{\text{B}} }}} \right);\quad{\text{a}}_{4,1} = 1 + \frac{{\Delta {\text{Z}}_{0} }}{{{\text{Z}}_{\text{B}} }}; \\ {\text{a}}_{4,3} & = - {\text{a}}_{1}^{{{\text{N}}_{ 0} - 1}} \left( {{\text{a}}_{1} + \frac{{\Delta {\text{Z}}_{ 0} }}{{{\text{Z}}_{\text{B}} }}} \right). \\ \end{aligned} $$

Right parts of the system of equations:

$$ \Pi _{1} = {\text{J}}_{0} \frac{{\Delta {\text{Z}}_{0} }}{{{\text{Z}}_{\text{B}} }};\quad\Pi _{2} = {\text{J}}_{{{\text{N}}_{\text{P}} }} \frac{{\Delta {\text{Z}}_{{{\text{N}}_{\text{P}} }} }}{{{\text{Z}}_{\text{B}} }};\quad\Pi _{3} = - {\text{J}}_{\text{NP}} \frac{{\Delta {\text{Z}}_{\text{NP}} }}{{{\text{Z}}_{\text{B}} }};\quad\Pi _{4} = - {\text{J}}_{ 0} \frac{{\Delta {\text{Z}}_{ 0} }}{{{\text{Z}}_{\text{B}} }}. $$

Note: Let us designate for generality the first index at system coefficients ${\text{a}}_{\text{i,j}}$ as i, and the second—as j. At values of i = 1, 2, 3, 4 and at even values j = 2k (for k = 1, 2,…) each coefficient ${\text{a}}_{\text{i,j}}$ can be obtained from the previous value ${\text{a}}_{\text{i,j}}$ (for j = 2k–1), if we replace ${\text{a}}_{1} \,{\text{by}}\,{\text{a}}_{2}$ and, respectively, ${\text{a}}_{2} \,{\text{by}}\,{\text{a}}_{1}$, for example ${\text{a}}_{ 1 , 2}$—from ${\text{a}}_{1,1} ,{\text{a}}_{4,4}$—from ${\text{a}}_{ 3 , 4}$, etc.

Appendix 10.2: Additional Currents in Eqs. (10.6) for Squirrel Cage with Three Adjacent Asymmetrical Bars

The system of equations is expressed by:

$$ \sum\limits_{\text{ij}} {{\text{a}}_{\text{i,j}} {\text{C}}_{\text{j}} =\Pi _{\text{i}} } \;{\text{at}}\;{\text{i, j}} = 1 , 2 , 3 , 4. $$

Coefficients in the system of equations (see Note to Appendix 10.1):

$$ \begin{aligned} {\text{a}}_{1,2} & = 1 + \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{1,3} = - {\text{a}}_{1} - {\text{a}}_{1}^{2} \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{1,4} = - {\text{a}}_{2} - {\text{a}}_{2}^{2} \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{2,1} = 1 + \frac{{\Delta \text{Z}_{1} }}{{\text{Z}_{\text{B}} }}; \\ {\text{a}}_{2,2} & = - {\text{a}}_{1} - {\text{a}}_{2} - \frac{{\Delta \text{Z}_{1} }}{{\text{Z}_{\text{B}} }} - \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{2,3} = {\text{a}}_{1}^{2} \left( {1 + \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{2,4} = {\text{a}}_{2}^{2} \left( {1 + \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }}} \right); \\ {\text{a}}_{3,1} & = - {\text{a}}_{1} - {\text{a}}_{2} - \frac{{\Delta \text{Z}_{1} }}{{\text{Z}_{\text{B}} }} - \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{3,2} = {\text{a}}_{2,1};\quad{\text{a}}_{3,3} = {\text{a}}_{1}^{{\text{N}_{0} - 1}} \left( {1 + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right); \\ {\text{a}}_{3,4} & = {\text{a}}_{2}^{{\text{N}_{0} - 1}} \left( {1 + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{4,1} = 1 + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{4,3} = - {\text{a}}_{1}^{{\text{N}_{0} - 1}} \left( {{\text{a}}_{1} + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right); \\ {\text{a}}_{4,4} & = - {\text{a}}_{2}^{{\text{N}_{0} - 1}} \left( {{\text{a}}_{2} + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{1,1} = {\text{a}}_{4,2} = 0. \\ \end{aligned} $$

Right parts of the system of equations:

$$ \Pi _{1} = \text{J}_{2} \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{2} = - \text{J}_{2} \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }} + \text{J}_{1} \frac{{\Delta \text{Z}_{1} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{3} = - \text{J}_{1} \frac{{\Delta \text{Z}_{1} }}{{\text{Z}_{\text{B}} }} + \text{J}_{0} \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{4} = - \text{J}_{0} \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}. $$

Appendix 10.3: Additional Currents in Eq. (10.7) for Squirrel Cage with Three Asymmetrical Bars: Two Bars Nearby, the Third—Next but One

The system of equations:

$$ \sum\limits_{{\text{i},\text{j}}} {{\text{a}}_{{\text{i},\text{j}}} } \text{C}_{\text{j}} =\Pi _{\text{i}} \quad{\text{at}}\quad\text{i},\text{j} = 1,2,3,4,5. $$

Coefficients in the system of equations (see Note to Appendix 10.1):

$$ \begin{aligned} {\text{a}}_{1,3} & = {\text{a}}_{2,4} = {\text{a}}_{2,5} = 1 + \frac{{\Delta \text{Z}_{3} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{1,4} = - {\text{a}}_{1}^{2} - {\text{a}}_{1}^{3} \frac{{\Delta \text{Z}_{3} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{1,5} = - {\text{a}}_{2}^{2} - {\text{a}}_{2}^{3} \frac{{\Delta \text{Z}_{3} }}{{\text{Z}_{\text{B}} }}; \\ {\text{a}}_{2,1} & = {\text{a}}_{2,2} = {\text{a}}_{3,3} = 1 + \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{2,3} = - {\text{a}}_{1} - {\text{a}}_{2} - \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }} - \frac{{\Delta \text{Z}_{3} }}{{\text{Z}_{\text{B}} }}; \\ {\text{a}}_{3,1} & = - {\text{a}}_{1} \left( {{\text{a}}_{1} + \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{3,2} = - {\text{a}}_{2} \left( {{\text{a}}_{2} + \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{4,1} = - {\text{a}}_{2} - \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}; \\ {\text{a}}_{4,2} & = - {\text{a}}_{1} - \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{4,4} = {\text{a}}_{5,1} {\text{a}}_{1}^{{\text{N}_{0} - 1}} ; \\ {\text{a}}_{4,5} & = {\text{a}}_{5,1} {\text{a}}_{2}^{{\text{N}_{0} - 1}} ;\quad{\text{a}}_{5,1} = 1 + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }} = {\text{a}}_{5,2} ;\quad{\text{a}}_{5,4} = - {\text{a}}_{1}^{{{\text{N}}_{0} - 1}} \left( {{\text{a}}_{1} + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right); \\ {\text{a}}_{5,5} & = - {\text{a}}_{2}^{{\text{N}_{0} - 1}} \left( {{\text{a}}_{2} + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{1,1} = {\text{a}}_{1,2} = {\text{a}}_{3,4} = {\text{a}}_{3,5} = {\text{a}}_{4,3} = {\text{a}}_{5,3} = 0. \\ \end{aligned} $$

Right parts of the system of equations:

$$ \Pi _{1} = \text{J}_{3} \frac{{\Delta \text{Z}_{3} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{2} = - \text{J}_{3} \frac{{\Delta \text{Z}_{3} }}{{\text{Z}_{\text{B}} }} + \text{J}_{2} \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{3} = - \text{J}_{2} \frac{{\Delta \text{Z}_{2} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{4} = \text{J}_{0} \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{5} = -\Pi _{4} . $$

Appendix 10.4: Additional Currents in Eq. (10.8) for Squirrel Cage with Three Asymmetrical Bars: Three Bars, Not Adjacent

The system of equations:

$$ \sum\limits_{\text{i,j}} {{\text{a}}_{\text{i,j}} {\text{C}}_{\text{j}} } =\Pi _{\text{i}} \;{\text{at}}\;{\text{i}},{\text{j}} = 1,2,3,4,5,6. $$

Coefficients in the system of equations (see Note to Appendix 10.1 ):

$$ \begin{aligned} {\text{a}}_{1,1} & = {\text{a}}_{1,2} = 1 + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{1,5} = - {\text{a}}_{1}^{{\text{N}_{0} }} \left( {{\text{a}}_{1} + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{1,6} = - {\text{a}}_{2}^{{\text{N}_{0} - 1}} \left( {{\text{a}}_{2} + \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }}} \right); \\ {\text{a}}_{2,1} & = - {\text{a}}_{2} - \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{2,2} = - {\text{a}}_{1} - \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad{\text{a}}_{2,5} = {\text{a}}_{1}^{{\text{N}_{0} - 1}} {\text{a}}_{1,1} ;\quad{\text{a}}_{2,6} = {\text{a}}_{2}^{{\text{N}_{0} - 1}} {\text{a}}_{1,1} ; \\ {\text{a}}_{3,1} & = {\text{a}}_{1}^{{\text{N}_{\text{P}} - 1}} \left( {1 + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{3,2} = {\text{a}}_{2}^{{\text{N}_{{\text{P}_{1} }} - 1}} \left( {1 + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right); \\ {\text{a}}_{3,3} & = - {\text{a}}_{1}^{{\text{N}_{{\text{P}_{1} }} - 1}} \left( {1 + {\text{a}}_{1} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{3,4} = - {\text{a}}_{2}^{{\text{N}_{{\text{P}_{1} }} - 1}} \left( {1 + {\text{a}}_{2} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right); \\ \end{aligned} $$

$$ \begin{aligned} & {\text{a}}_{4,1} = - {\text{a}}_{1}^{{\text{N}_{{\text{P}_{1} }} }} \left( {1 + {\text{a}}_{2} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{4,2} = - {\text{a}}_{2}^{{\text{N}_{{\text{P}_{1} }} }} \left( {1 + {\text{a}}_{1} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{4,3} = {\text{a}}_{1}^{{\text{N}_{{\text{P}_{1} }} }} \left( {1 + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right) \\ & {\text{a}}_{4,4} = - {\text{a}}_{2}^{{\text{N}_{{\text{P}_{1} }} }} \left( {1 + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{5,3} = - {\text{a}}_{1}^{{\text{N}_{{\text{P}_{2} }} }} \left( {1 + {\text{a}}_{2} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{5,4} = - {\text{a}}_{2}^{{\text{N}_{{\text{P}_{2} }} }} \left( {1 + {\text{a}}_{1} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }}} \right) \\ & {\text{a}}_{5,5} = {\text{a}}_{1}^{{\text{N}_{{\text{P}_{2} }} }} \left( {1 + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{5,6} = {\text{a}}_{1}^{{\text{N}_{{\text{P}_{2} }} }} \left( {1 + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{6,3} = {\text{a}}_{2} {\text{a}}_{5,5} ;\quad{\text{a}}_{6,4} = - {\text{a}}_{1} {\text{a}}_{5,6} \\ & {\text{a}}_{6,5} = - {\text{a}}_{1}^{{\text{N}_{{\text{P}_{2} }} }} \left( {{\text{a}}_{2} + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }}} \right);\quad{\text{a}}_{6,6} = - {\text{a}}_{2}^{{\text{N}_{{\text{P}_{2} }} }} \left( {a_{1} + \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }}} \right); \\ & {\text{a}}_{1,3} = {\text{a}}_{1,4} = {\text{a}}_{2,3} = {\text{a}}_{2,4} = {\text{a}}_{3,5} = {\text{a}}_{3,6} = {\text{a}}_{4,5} = {\text{a}}_{4,6} = {\text{a}}_{5,1} = {\text{a}}_{5,2} = {\text{a}}_{6,1} = {\text{a}}_{6,2} = 0. \\ \end{aligned} $$

Right parts of the system of equations:

$$ \begin{aligned} &\Pi _{1} = - \text{J}_{0} \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{2} = \text{J}_{0} \frac{{\Delta \text{Z}_{0} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{3} = \text{J}_{{\text{N}_{{\text{P}_{1} }} }} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }}; \\ &\Pi _{4} = - \text{J}_{{\text{N}_{{\text{P}_{1} }} }} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{1} }} }} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{5} = - \text{J}_{{\text{N}_{{\text{P}_{2} }} }} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }};\quad\Pi _{6} = \text{J}_{{\text{N}_{{\text{P}_{2} }} }} \frac{{\Delta \text{Z}_{{\text{N}_{{\text{P}_{2} }} }} }}{{\text{Z}_{\text{B}} }}. \\ \end{aligned} $$

Brief Conclusions

• 1. Investigation methods of symmetrical active open and closed chain circuits are generalized and used to study asymmetrical chain circuits class (open and closed). As a result of their generalization it has become possible to obtain regularities of currents distribution, important for practice, in asymmetrical squirrel cages (with damages). In particular, the following regularities of currents distribution in asymmetrical squirrel cage elements of induction machine are obtained:

  • besides currents for symmetrical squirrel cage elements there are additional currents with two components;

  • both components vary depending on the number of bar ring portion under the aperiodic law, one of which fades with increase of this number in amplitude, and another—increases.

• 2. The general problem to compute currents distribution in short-circuited rotor cage with P asymmetrical bars is reduced to the solution of algebraic system in general case of r linear equations (r $\le$ 2P + 2); its order does not depend on total number $\text{N}_{0}$ of bars in cage. This method has essential advantages in comparison with numerical methods based on the solution of a system of Kirchhoff’s equations for $\text{N}_{0}$ loops. In practical problems $\text{N}_{0} \le 70,\;{\text{and}}\;\text{P} \le \text{3} - \text{4}$. For small slips, real and imaginary parts of some system coefficients differ in several dozen times, therefore in practice, the solution of systems of high order $\text{N}_{0}$ with complex coefficients using numerical methods meets difficulties.

• 3. Besides the general problem, its special cases representing independent practical interest are solved:

  • asymmetrical squirrel cage with one and two damaged bars;

  • asymmetrical squirrel cage with three adjacent damaged bars;

  • asymmetrical squirrel cage with three damaged bars: two bars nearby, the third—next but one;

  • asymmetrical squirrel cage with three damaged bars: three bars, not adjacent.

List of Symbols

${\text{C}}_{1} , \ldots ,{\text{C}}_{{{\text{N}}_{\text{T}} }} , \ldots ,{\text{D}}_{ 0} , \ldots ,{\text{D}}_{{{\text{N}}_{\text{T}} }} , \ldots$ :

Constants for calculating currents determined from both Kirchhoff’s laws;

$\text{E}_{{\text{N}_{\text{P}} ,\text{N}_{\text{P}} - 1}}$ :

EMF in the loop $\left( {\text{N}_{\text{P}} ,\text{N}_{\text{P}} - 1} \right)$, induced by the resulting field in air gap;

$\underline{\underline{\text{I}}}_{\text{N}} ,\underline{\underline{\text{J}}}_{\text{N}}$ :

Currents in ring portions (segments) and in bars of squirrel cage at emergence of asymmetry (damage);

$\text{I}_{\text{N}} ,\text{J}_{\text{N}}$ :

Currents in ring portions (segments) and in bars of squirrel cage before asymmetry emergence;

$\Delta \text{I}_{{(\text{N})}} ,\Delta \text{J}_{{(\text{N})}}$ :

Additional currents in bars and ring portions of asymmetrical squirrel cage;

N:

Bar (ring portion) number of squirrel cage;

$\text{N}_{0}$ :

Number of squirrel cage bars;

$\text{N}_{{\text{P}_{1} }} ,\text{N}_{{\text{P}_{2} }}$ :

Numbers of asymmetrical bars of squirrel cage;

$\text{Z}_{\text{B}}$ :

Impedance of the symmetrical (undamaged) bar;

$\text{Z}_{\text{R}}$ :

Impedance of the ring portion between two adjacent bars;

$\text{Z}_{{\text{N}_{\text{P}} }} , \ldots ,\text{Z}_{{\text{N}_{\text{R}} }} , \ldots ,\text{Z}_{{\text{N}_{\text{S}} }}$ :

Impedances of asymmetrical (damaged) bars;

$\Delta \text{Z}_{{\text{N}_{\text{P}} }} , \ldots ,\Delta \text{Z}_{{\text{N}_{\text{R}} }} , \ldots ,\Delta \text{Z}_{{\text{N}_{\text{S}} }}$ :

Additional impedances of asymmetrical (damaged) bars