• most commonly used machines nowadays• designed in small and medium power range

Advantages: Disadvantages: Low cost Construction simplicity

Possibility of their starting Speed control in wide p y

Lower moment of inertia Reliability

pranges

y Security in work Easy maintenance

Solution:use of power electronics andEasy maintenancecomputer‐oriented control

Static models: allow obtaining static 

Dynamic models: parameters represented  allow obtaining static 

characteristics of the machine;

parameters represented as time‐dependent variables;machine;

Models:

variables;

Models:Models: Rotating  fields  theoryS       h   

Models: dq – model

l d i i   h Space  vector  theory   coupled‐ circuits theory multiple coupled circuits 

htheory

Based on following assumptions:

permeability of stator and rotor core is infinitely high; iron losses are neglected; iron losses are neglected; all of the parameters are constant and concentrated; rotor bars are isolated from the rotor core; rotor bars are isolated from the rotor core;

Electrical part equations: Mechanical part equations:p q p q

Inductances: calculated using winding function approach; turns function:turns function:

winding function: winding function:

i d   l l i inductances calculation:

Self inductances of stator turns and rotor loops are constant  values;;

Mutual inductances are angle‐dependent;

Fig.1 Mutual inductances between stator phase and one rotor loop

Stator resistances: three phases with equal 

Rotor resistances: 28 bars connected with p q

parameters in each of them;

small resistances between them;

previous relations could be solved using iterative procedure;

one of the most common techniques is Euler’s method qof integration;

Fig.2 Angular speed of the healthy machineg g p y

Fig.3 Electromagnetic torque of the healthy machine

Fig.4 Line current, phase A, of the healthy machine

Lrr

konstanta

zeros(28,1)

Ur

LrrLssmod

Rr

Constant1

Lar

Lssmod

Lsr

PSIs

PSIr

Lrs

Irfcn

MatrixMultiply

Product1Lar

LbrLsr

Lssmod

Lssmod46

1/J

Lbr

konstanta2

konstanta1

U_statora

Ir

D:[3,1]

Signal Specification4

D:[28,1]

Signal Specification2

Rotor current

Product2

Lcr

DLar

DLbr

DLcr

Lsrprimmedj

Is

Ir

Lsrprim

Tefcn

Generated torque

DLar

Lcr

konstanta3

Ir

Lssmod

PSIs

Isfcn

[3x3]Matrix

Multiply1s

DLcr

tetain

Lrs

Mutual inductances

-20

Load torque

1s

1s

DLbr

konstanta5

konstanta4 Lsr

Stator currentRs ProductPSI_s

1s

PSI_r

Angular velocity

wteta

DLcr

konstanta6

D:[3 28]D:[3,28]

Signal Specification1

Mechanical angle

Machine is exposed to various impacts, that sooner or later, inevitably lead to faults;

fault‐detection techniques are the issues of growing importance in technical practice;

most common faults:

rotor bar failure; stator  winding faults;g missing slot wedges; rotor  eccentricity;

one of the most common failures in the exploitation of induction machines;

occurs due to mechanical, thermal, dynamic or magnetic strains;

doesn’t provoke significant problems in industrial environment, as machine continues working with negligible distortion of its performance;

additional problem:leads to the redistribution and increase of the rotor currents of adjacent bars, that can lead to failure jmultiplication; 

:

rotor bar defect can be taken into account by increasing the amount of several rotor bars resistance, g ,as it’s case in the real machine;

HARMONIC SPECTRUM OF THE MACHINE

Upper RSH:

Lower RSH:

Harmonics due to rotor bar defect:

slip‐dependent components of the stator line current spectrum, due to the rotor bar defect:

speed pulsation component: speed pulsation component:

Fig.5 Angular speed of the healthy machine and machine with the rotor bar defect respectivelyrotor bar defect, respectively

Fig.6 Electromagnetic torque of the healthy machine and machine with the rotor bar defect respectivelywith the rotor bar defect, respectively

Fig.7 Line current, phase A, of the healthy machine and machine with the rotor bar defect, respectivelyy

Basic harmonic and the  RSHupper RSH;

Basic harmonic, upper ppRSH, slip‐dependent components and speed pulsation component;

Fig.8 Stator line current spectrum of the healthy machine and machine with the

rotor bar defect, respectively

very perfidious fault, due to the it’s concealment;

conventional protection systems doesn’t react to them;

fault evolvs into phase to phase or phase to ground fault, provoking complete breakdown of the machine;, p g p ;

Reasons that lead to the inter‐turn short circuit:

insulation aging;insulation aging; high dynamic stresses; intensified vibrations; intensified vibrations; moisture;hi h  high temperature;

overvoltages; transient operating mode; chemical factors;

this fault can be modeled by adding additional, short circuited phase;

number of turns of the faulted phase is reduced and added to the short circuited phase;p ;

Fig.9 Winding of healthy and faulty phase, and faulty and short circuited phase, respectively

Fig.10 Mutual inductance of healthy phase and one rotor loop and faulted phase and one rotor loop, respectively

Fi 11 M t l i d t f f lt d h d t lFig.11 Mutual inductance of faulted phase and one rotor loop and short circuited phase and one rotor loop, respectively

Fig. 12 Angular speed and developed electromagnetic torque of the machine with inter-turn short circuit

Fig.13 Line current of injured stator phase A (line) and current in shorted phase D (dot)

Common assumption in induction motor analysis is l tl     th  t t   d  tslotless or smooth stator and rotor;

More detailed analysis needs exact double slotted structure modeling;

Air gap is divided into two parts; Air gap is divided into two parts;

Once, smooth rotor and slotted stator is observed;Once, smooth rotor and slotted stator is observed;

After that, conversely, smooth stator and slotted rotor is , y,analysed;

As iron permeance is assumed infinitely high, superposition could be applied;

Two air gap length could be summed for every different rotor position;o o pos o ;

Fig. 14 Upside down: resultant air-gap length and air-gap permeance function:=0, S=36, R=32, bss=0.5s, brs=0.5r, g0=0.5mm.