Investigation Method of Transient Modes in Induction Machines with Rotor Cage Asymmetry

Abstract

This chapter presents investigation methods of transient modes in induction machines with asymmetrical rotor squirrel cage. This asymmetry can arise in the operation (for example, in case of bar breakage, especially at overloads), or in manufacturing (for example, due to low-quality soldering in bar-end ring joint).

Brief Conclusions

• 1. To study transients in induction machines with asymmetry in rotor cage, a number of machine parameters should be preliminarily determined: Mutual induction factors between rotor and stator loops and also equivalent parameters of secondary loops (of rotor) for both fields (direct and additional).

• 2. Calculation of these parameters requires:

  • preliminary calculation of current distribution in asymmetrical cage elements;

  • expansion of the current distribution step curve in bars of rotor periphery of this cage into harmonic series in complex plane;

  • determination of amplitude complex value of MMF direct and additional mutual induction fields (fields in gap).

• 3. Investigation of transients in induction machine with rotor cage asymmetry is reduced to that of transients in two equivalent symmetrical magnetically coupled loops (of rotor and stator); these two loops are in direct field of mutual induction, and two others—in additional field. Mutual induction factors between rotor and stator loops and also equivalent parameters of rotor and stator secondary loops corresponding to both fields (direct and additional) differ. Therefore, attenuation factors of transient currents are not equal for each of this field.

• 4. For the investigation of transients in each of these two symmetrical magnetically coupled loops it is expedient to use the “Theory of rotating field” [1, 3]; expediency of its choice is based on the fact that machines are characterized by constant air gap: $\updelta \ne {\text{f(x)}}$, where $\text{0} \ne {\text{x}} \le {\text{T}}$.

List of symbols

${\text{B}}_{\text{RES,DIR}} ,{\text{B}}_{\text{RES,ADD}}$ :

FLUX density amplitude in air gap of the main harmonic respectively of direct and additional fields

${\text{C}}_{1} ;{\text{C}}_{2} ;{\text{C}}_{3} ;{\text{C}}_{4}$ :

Constants determined by initial conditions

${\text{D}}_{\text{INN}}$ :

Stator core diameter

${\text{E}}_{\text{STAT,DIR}}$ :

Stator winding phase EMF amplitude induced by direct field of mutual flux density with amplitude ${\text{B}}_{\text{RES,DIR}}$ and frequency ${\text{f}}_{1}$

${\text{E}}_{\text{STAT,ADD}}$ :

The same, for additional field with amplitude B_RES,ADD and frequency f_ADD

${\text{F}}_{\text{STAT,DIR}} , {\text{F}}_{\text{STAT,ADD}}$ :

Amplitudes of MMF harmonics respectively for stator direct and additional fields

${\text{F}}_{\text{SEC,DIR}} ,{\text{F}}_{\text{SEC,DIR}}$ :

Amplitudes of MMF harmonics respectively for rotor direct and additional fields

$\Delta {\text{F}}_{\text{SEC,DIR}} ,\Delta {\text{F}}_{\text{SEC,ADD}}$ :

Amplitudes of MMF harmonics of rotor additional currents respectively for direct and additional fields

${\text{F}}_{{{\text{NO}} - {\text{LOAD}},{\text{DIR}}}} ,{\text{F}}_{{{\text{NO}} - {\text{LOAD,ADD}}}}$ :

Amplitudes of no-load MMF respectively for direct and additional fields

$\text{G}_{1} ,\text{G}_{2} ,\text{G}_{3} \text{,G}_{4}$ :

Constants determined by initial transient conditions

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

Resulting currents in short-circuited ring portions and in bars (after breakage)

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

Current amplitudes in short-circuited ring portions and in bars (before breakage)

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

Amplitudes of additional currents in short-circuited ring portions and bars caused by cage asymmetry

${\text{I}}_{\text{STAT,DIR}}$ :

Stator current amplitude with network frequency $\upomega_{1}$

${\text{I}}_{\text{STAT,ADD}}$ :

Stator current amplitude with frequency ${\text{f}}_{\text{ADD}}$

${\text{i}}_{\text{STAT,DIR}} \text{,}{\text{i}}_{\text{SEC,DIR}}$ :

Instantaneous values of transient currents for direct field respectively in stator and rotor

${\text{i}}_{\text{STAT,ADD}} \text{,}{\text{i}}_{\text{SEC,ADD}}$ :

The same for additional field

${\text{J}}_{{ ( {\text{NP)}}}}$ :

Current amplitude in bar with number ${\text{N}}_{\text{P}}$ before its breakage

K:

Iteration number

${\text{k}}_{\text{CAR}}$ :

Carter’s factor

${\text{k}}_{\text{SAT}}$ :

Magnetic circuit saturation factor

${\text{k}}_{\text{W}}$ :

Winding factor

${\text{L}}_{\text{SEC,DIR}}^{{\prime }}$ :

Leakage inductance of secondary loop for direct field

${\text{L}}_{\text{SEC,ADD}}^{{\prime }}$ :

Leakage inductance of secondary loop for additional field

${\text{L}}_{\text{STAT,DIR}}$ :

Total inductance of stator winding phase for direct field

${\text{L}}_{\text{STAT,ADD}}$ :

Total inductance of stator winding for additional field

${\text{L}}_{\text{STAT,PH}}$ :

Leakage inductance of stator winding phase

${\text{L}}_{\text{SEC,DIR}}$ :

Total inductance of rotor loop for direct field

${\text{L}}_{\text{SEC,ADD}}$ :

Total inductance of rotor loop phase for additional field

P:

Number of broken cage elements

n:

Order of spatial harmonic

${\text{M}}_{\text{DIR}}$ :

Mutual induction factor between stator and rotor loops for direct field

${\text{M}}_{\text{ADD}}$ :

Mutual induction factor between stator and rotor loops for additional field

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

Number of bars in cage

N:

Number of cage element (bar or ring portion)

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

Asymmetrical (broken) bar number

p:

Number of pole pairs in machine

${\text{R}}_{{{\text{SEC}},{\text{DIR}}}}$ :

A.C. resistance of secondary loop for direct field

${\text{R}}_{{{\text{SEC}},{\text{ADD}}}}$ :

A.C. resistance of secondary loop for additional field

${\text{R}}_{{{\text{STAT}},{\text{PH}}}}$ :

A.C. resistance of stator winding phase

${\text{S}}_{\text{SL}}$ :

Slip

T:

EMF expansion period

t:

Own time

${\text{t}}^{{\prime }}$ :

Time counted in one loop relative to the second

${\text{U}}_{{{\text{STAT}} . {\text{PRESET}}}}$ :

Amplitude of motor preset phase voltage

${\text{W}}_{\text{STAT}}$ :

Number of stator winding turns

${\text{Y}} = {\text{Y}}_{1} ,{\text{Y}} = {\text{Y}}_{2}$ :

Attenuation factors of transient currents for the main field

${\text{Y}} = {\text{Y}}_{3} ,{\text{Y}} = {\text{Y}}_{4}$ :

Attenuation factors of transient currents for additional field

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

Impedance of ring portion

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

Impedance of bar

${\text{Z}}_{\text{STAT,DIR}}$ :

Leakage impedance of stator winding at frequency ${\text{f}}_{ 1}$

${\text{Z}}_{\text{EXT,ADD}}$ :

Impedance of external network at frequency ${\text{f}}_{\text{ADD}}$

${\text{Z}}_{\text{STAT,ADD}}$ :

Leakage impedance of stator winding at frequency ${\text{f}}_{\text{ADD}}$

${\text{Z}}_{\text{SEC,DIR}}$ :

Impedance of secondary loop at frequency ${\text{f}}_{\text{SEC}}$ for the main field

${\text{Z}}_{\text{SEC,ADD}}$ :

Impedance of secondary loop at frequency ${\text{f}}_{\text{SEC}}$ for additional field

$\Delta \text{Z}$ :

Increase of bar impedance at its breakage

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

Impedance of unbroken bar

Z_R :

Impedance of ring portion

$\upgamma_{\text{SEC,DIR}} ,\upgamma_{\text{SEC,ADD}}$ :

Phase angles

$\upomega_{\text{SEC}}$ :

Circular frequency of EMF and currents in cage elements

$\upomega_{ 1}$ :

Network circular frequency

$\upomega_{\text{REV}}$ :

Angular speed of rotor rotation

$\upomega_{\text{MAIN}}^{{\prime }}$ :

Rotation speed of main field harmonics relative to stator (in direction of rotor rotation)

$\updelta$ :

Air gap

$\upmu_{0}$ :

Magnetic permeability of air

$\Phi _{\text{RES,DIR}} {,\varPhi }_{{{\text{RES}},{\text{ADD}}}}$ :

Main harmonic of mutual induction flux (flux in air gap) respectively for direct and additional fields,

${\upvarphi }_{\text{MAIN}} ,{\uppsi }_{\text{DIR}}$ :

Initial phase angles

$\upalpha,\upbeta$ :

Frequencies of transient currents respectively for stator and rotor direct field

$\uplambda,\upsigma$ :

Frequencies of transient currents respectively for stator and rotor additional field