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Model of a Squirrel Cage Induction Machine with InterbarConductances

Christian Kral Anton Haumer Bernhard Kubicek Oliver WinterAustrian Institute of Technology

Giefinggasse 2, 1210 Vienna, Austria

Abstract

Conventional models of squirrel cage inductionmachines do not consider interbar currents. For theaccurate numerical investigation of electrical rotorasymmetries interbar currents have to be modeled. Amathematical model of such a machine with interbarconductances is presented in this paper. An approachfor the parametrization of the interbar conductances isintroduced and discussed.

Keywords: Squirrel cage induction machines, elec-trical rotor asymmetries, interbar currents, machinemodel

1 Introduction

The electric conductors in the squirrel cage of an in-duction machine the rotor bars and end rings areusually either made of copper or aluminum. For diecast rotors the conducting material is not electricallyinsulated from the sheet iron due to the casting pro-cess. Even if copper and aluminum have a muchgreater electric conductivity than the sheet iron, somefractions of the rotor currents may flow through therotor teeth, directly from bar to bar.In mains supplied induction machines the rotor barsare usually skewed, i.e., the bars are twisted in tangen-tial direction (Fig. 1 and 2) . Due to the skewing of therotor bars the following effects occur:

1. Torque saddle points due to harmonic magneticfield waves can be avoided [13]

2. The torque speed characteristic and performanceof the machine changes [4, 5]

3. Axial pull of the rotor caused by the tangentialcomponent of the bar currents

(a) unskewed rotor sheets (b) skewed rotor sheets

Figure 1: Rotor core of an induction machine

(a) unskewed cage (b) skewed cage

Figure 2: Squirrel cage of an induction machine, con-sisting of rotor core and a skewed or unskewed cage(bars and end rings)

4. Currents are flowing from bar to bar through thesheet iron the so called interbar currents

Interbar currents significantly arise in the case of bro-ken rotor bars or end ring segments. The intermittenceof the electric conductance of either a bar or end ringsegment causes the current to flow through alternativepaths in the sheet iron from bar to bar.For the numerical investigation of interbar currents insquirrel cage induction machines finite element meth-ods are usually applied [68]. Alternatively, modelsbased on equivalent circuits can be developed to inves-tigate the operational behavior of machines with inter-

Proceedings 7th Modelica Conference, Como, Italy, Sep. 20-22, 2009

The Modelica Association, 2009 861 DOI: 10.3384/ecp09430003

bar currents. In such models the skewing of rotor barscan be considered by means of skewing factors appliedto the calculation of the mutual inductances betweenstator and rotor [9, 10]. Interbar currents of electri-cally asymmetrical rotor cages are yet more complexto model since the rotor has to be discretized in axialdirection. A Modelica model considering such axialdiscretization is presented in this paper. To be moreprecise: the presented model takes interbar conduc-tances into account which in turn conduct the interbarcurrents.

2 Model Structure

The proposed induction machine model with interbarcurrents is derived from the ExtendedMachines librarywhich was presented in [11]. The ExtendedMachineslibrary includes a model of the stator and rotor wind-ing topology, core, friction and stray-load losses, anair gap model and thermal connectors for the couplingwith a thermal model or environment. From this li-brary the air gap, cage and winding function mod-els are extended and modified such way that an axialdiscretization including interbar conductances is mod-eled.A diagram of the proposed machine model is de-picted in Fig. 3. The stator network considersthe stator resistance (rs), the stator stray induc-tance (lssigma) and the air gap model (airgap),which can be seen as electromechanical power con-verter. Stray-load losses (strayLoad), core losses(coreStray and coreTerminal, respectively, foralternate use, depending on the model character) andfriction losses (friction) are also taken into ac-count. On the rotor side a multiphase squirrel cage(cage) is modeled. Each leg of the rotor phaseshas to be grounded (ground). Stator and rotorinertia (intertiaStator, inertiaRotor) areconnected with the air gap model and the shaft end(flange) and housing (support) connector, re-spectively.Any symmetric or asymmetric stator winding topology(windingStator) can be considered by specifyingthe location of the begin and end of each turn foreach phase. Details about the modeling of the statorwinding topology are explained in [11]. The topologyof the rotor cage is fully symmetrical and formed of theperiodic structure of bars and end rings as depicted inFig. 2. Electrical rotor asymmetries are model by mod-ified rotor bar and end ring resistances. The topologymodel of the squirrel cage is presented in section 3.

Figure 3: Modelica diagram of the proposed inductionmachine with interbar conductances

3 Cage Model

The presented cage model (cage) takes skewed rotorbars and interbar conductances into account. The in-terbar conductances are modeled in such a way, thateach rotor bar is axially discretized and each interme-diate node of two adjacent bars is connected by anan interbar conductance. A sketch of the rotor net-work model is presented in Fig. 4. This modeling ap-proach does not consider current paths through the ro-tor yoke [12], which seems to be a reasonable simpli-fication.The voltages induced in the rotor meshes are calcu-lated in the air gap model which is presented in sec-tion 4. These induced voltages are the terminal volt-ages v[m] of the cage model in Fig. 3. The currents as-sociated with the multiphase ports of the cage model,i[m], are rotor mesh currents.The cage consists of Nr rotor bars and thus Nr is alsothe number of end ring segments on both sides thedrive end (index a) and the non drive end (index b). Inaxial direction each rotor bar is subdivided into ni + 1segments. The rotor bar segments are tangentially con-nected by interbar conductances Gi[m]. In axial direc-tion the rotor is thus modeled by ni interbar conduc-tors. Each of these conductors represents the conduc-tance of the sheet iron including the contact conduc-tance. Due to this structure the rotor cage consists of(ni +1)Nr +1 elementary meshes, which are indicated


The Modelica Association, 2009 862

ieb non drive end

ib[ j]

Reb[ j] Leb[ j]

Lb[ j+1]

Rb[ j+1]

ib[ j+1]

Lb[ j+Nr+2]

Rb[ j+Nr+2]

ib[ j+Nr+2]

Lb[ j+Nr+1]

Rb[ j+Nr+1]

ib[ j+Nr+1]

Rb[ j+1]

Reb[ j+1] Leb[ j+1]

Lb[ j+2]

ib[ j+2]Rb[ j+2]

Reb[ j+2] Leb[ j+2]

Lb[ j+3]

Rb[ j+3]

ib[ j+3]

Lb[ j+Nr+3]

Rb[ j+Nr+3]

ib[ j+Nr+3]

Rb[ j]

Lb[ j]

Gi[ j+1] Gi[ j+2]

Lb[ j+Nr ]

Rb[ j+Nr ]

Gi[ j+Nr ] Gi[ j+Nr+1] Gi[ j+Nr+2]

ib[ j+Nr ]

Gi[ j] ii[ j] ii[ j+1] ii[ j+2]

i[i+Nr+1] i[i+Nr+2]

i[i+1] i[i+2]

ii[ j+Nr ] ii[ j+Nr+1] ii[ j+Nr+2]

ir[i]

ir[i+2Nr ]

ir[i+(ni1)Nr ]

ir[i+Nr ]

k(b)(a)

Rea[ j] Lea[ j]

Lb[ j+niNr+1]

Rb[ j+niNr+1]

ib[ j+niNr+1]

Rea[ j+2] Lea[ j+2]

Lb[ j+niNr+3]

Rb[ j+niNr+3]

ib[ j+niNr+3]

Rea[ j+1] Lea[ j+1]

Lb[ j+niNr+2]

ib[ j+niNr+2]

Gi[ j+(ni1)Nr] Gi[ j+(ni1)Nr+1] Gi[ j+(ni+2)Nr ]

Rb[ j+niNr ]

Lb[ j+niNr ]

Rb[ j+niNr+2]

ib[ j+niNr ]

drive end

i[i+niNr ] i[i+niNr+1] i[i+niNr+2]

ii[ j+(ni1)Nr+1]ii[ j+(ni1)Nr ] ii[ j+(ni1)Nr+2]

i[i]

i[i+Nr ]

Figure 4: (a) Equivalent circuit of rotor cage with interbar conductances; (b) location of size of skewed rotorloops, skewing angle k

in gray in Fig. 4. For these elementary meshes thevoltage equations have to be modeled.The rotor voltage equation of the elementary meshalong the end ring on the non drive end yields

0 =Nr

m=1

(

Reb[m](i[m] + ieb)+ Leb[m]d(i[m] + ieb)

dt

)

.

(1)

From the remaining rotor meshes two different typesof meshes have to be distinguished. An end ring mesh,is formed by one end ring segment, two bar segmentsand one interbar conductance. An intermediate meshis formed of two bar segments and two interbar con-ductances. The indices of the meshes and the currents,respectively, are numbered in ascending order fromleft to right and from the non drive end to the driveend.The rotor voltage equations for the end ring meshes of

the non drive end are

0 = v[m] + Rb[m]ib[m] + Lb[m]dib[m]

dt

Rb[m+1]ib[m+1] Lb[m+1]dib[m+1]

dt

+ Reb[m](i[m] + ieb)+ Leb[m]d(ir[m] + ieb)

dt

ii[m]Gi[m]

,

(2)

for m [1,2, . . . ,(ni + 1)Nr].

For the intermediate meshes the voltage equations are


dt


dt

ii[m]Gi[m]

+ii[mNr ]Gi[mNr ]

, (3)

for m [Nr + 1,Nr + 2, . . . ,niNr].



The remaining end ring meshes on the drive end are


dt


dt

+ Rea[mniNr ]i[m] + Lea[mniNr ]di[m]dt

+ii[mNr ]Gi[mNr ]

, (4)

for m [niNr + 1,niNr + 2, . . . ,(ni + 1)Nr].

For the interbar currents the following equations

ii[ j] = i[ j+Nr ] i[ j] (5)

apply for j [1,2, . . . ,niNr]. Additionally, the bar cur-rents and the mesh currents are related by

ib[ j] = i[ j+1] i[i+Nr ] (6)

ib[i+k] = i[i+k] i[i+k1] (7)

and j = Nr(n 1) with n [1,2, . . . ,ni + 1], and k [2,3, . . . ,Nr].

4 Air Gap Model

In the air gap model (airgap) the magnetic couplingof the ms phase stator winding and the Nr phase ro-tor cage (or winding) is modeled. Due to the axialsegmentation of the rotor bars, the air gap model andthe cage model are coupled through a multiphase con-nector with (ni + 1)Nr phases. The magnetic couplingbetween the windings of the stator and rotor is deter-mined by the respective number of turns and the actuallocation of the turns with respect to each other. Therelationship between the induced stator voltages, vs[l],and the induced rotor voltages, vr[m], and the respectivecurrents is[l] and ir[m] is expressed by

vs[ j] =3

l=1

Lss[ j,l]dis[l]dt

+(ni+1)Nir

m=1

ddt

(

Lsr[ j,m]ir[m])

, (8)

vr[m] =3

j=1

ddt

(

Lrs[m, j]is[ j])

+(ni+1)Nr

n=1

Lrr[m,n]dir[n]

dt, (9)

where Lss[ j,l] are the stator inductances,

Lsr[ j,m] = Lrs[m, j] (10)

are the mutual stator and rotor inductances, and Lrr[m,n]are the rotor inductances of the machine.For the stator it is assumed that the complex windingfactor, s[ j], and the number of turns, ws[ j], can be de-termined for each phase based on the exact topology

of each phase winding [13]. For the rotor cage thenumber of turns is equal to one,

wr[m] = 1 (11)

and the complex winding factor,

r[m] = sin(

pNr

)

ej2pm/Nr , (12)

is formed of the chording factor of two adjacent ro-tor bars, and the tangential location of the respectiverotor mesh assuming it were unskewed. In order toconsider the skewing of the rotor bars accordingly, thetangential displacement of the rotor meshes belongingto two particular rotor bars, is modeled by a complexdisplacement factor, r[m]. The inductances in (8) and(9) can thus be expressed as

Lss[ j,l] = Lws[ j]ws[l]Re(s[ j]s[l]), (13)

Lsr[ j,m] = Lws[ j]wr[m]r[m]Re(s[ j]r[m]r[m]ejm), (14)

Lrr[m,n] = Lwr[m]wr[n]r[m]r[n]Re(r[m]r[ j]r[m]

r[n]),

(15)

where L is the base inductance of one turn with a widthequal to one pole pair. In this equations the expression

r[m] =

{

12ni

if m refers to an end ring mesh1ni

if m refers to an intermediate mesh(16)

applies, which determines the axial length of each endring or intermediate mesh. The axial length of eachmesh is modeled such way, that two bar segments ofequal length e contribute to one interbar conductor(Fig. 5). Expression (16) is also used to determine theresistances and stray inductances of the bar segmentswith respect to the total bar resistance and stray induc-tance, respectively.In the induction machine model of Fig. 3 the wind-ing factors of the stator winding and the rotor cageare determined by the models windingStator andwindingRotor, respectively. These models com-pute the winding factors, number of turns, displace-ment factors and (16) and propagate them to the airgap model. The input parameters are the skewing an-gle k (Fig. 4), the number of phases and the remain-ing topology data of the stator winding.The electromagnetic torque of the induction machineis

Te =3

j=1

(ni+1)Nr

m=1

Lsr[ j,m]m

is[ j]ir[m], (17)

and applies to the stator and rotor inertia, with inversesigns, however. In this equation m indicates the elec-trical angle of the rotor with respect to the stator.



Gi[ j]

Gi[Nr+ j]

Gi[2Nr+ j]

e

e

e

e

e

e

bar segment

bar segment

bar segment

end ring mesh

intermediate mesh

intermediate mesh

bar segment

end ring mesh

non drive end

drive end

Figure 5: Sketch of two skewed rotor bars with ni = 3interbar conductances: two equal bar segments (lengthe) contribute to one interbar segment; this results indifferent bar segment lengths for end ring meshes andintermediate meshes

5 Practical Application

5.1 Model Development and Compilation

The model was developed with Dymola 6.0b and theC code was compiled in parallel on a 64 bit Linuxcluster. The compilation and execution times of themodels very much rely on the number of interbar seg-ments, ni. Two different simulations were carried outfor the cases ni = 4 and ni = 9. For the second casea newer version of the Gnu C Compiler (GCC) hadto be used in order to keep the compilation time ina reasonable range. The compilation of one modelrequired approximately 8 GB memory and took onehour. Hence, a 64 bit GCC version was use to com-pile the 32 bit files. The simulation model was usedin a project where 25 different machine configurationswere investigated. Since the different configurationscould be investigated independently the compilationsand simulations were performed in parallel on differ-ent nodes on the Linux cluster.

For each model a total (real time) simulation time spanof 12.5 s was performed. The cases ni = 4 and ni = 9required 10 and 72 hours of CPU time, respectively.For the computation of each simulation in the project,a total of approximately 7000 hours of CPU time accu-mulated. During the comparison of these two cases itturned out, that ni = 9 shows only a slight increase ofaccuracy and is not necessary for most investigations.

Figure 6: Squirrel cage rotor with respect to the con-figuration with one broken rotor bar (segment)

5.2 Parametrization of the Model

For the investigated project, the regular induction ma-chine parameters were determined from a electromag-netic design tool. This way the resistances and strayinductances of the bars and end ring were calculated.The design tool does not calculate the interbar conduc-tances, however, since these quantities very much relyon the manufacturing process and the contact conduc-tances between the conductor and the sheet iron. Inpractice, the interbar resistances between two bars canusually not be measured without destroying the rotorcage [1417].Significant interbar currents arise in the rotor if onebar or end ring of the machine is broken. There-fore the configuration of one broken bar (Fig. 6) andone broken end ring segment (Fig. 7) are investigated.In order to compare measurement and simulations amethod has to be applied which allows assessmentof the effects caused by a broken rotor bar or endring segment. Such a method is the Vienna Moni-toring Method (VMM), originally introduced in 1997[1820]. The VMM calculates a fault indicator whichis related with the fault extent. Interbar currents de-teriorate the fault indicator and thus the comparisonof measurement and simulation results can be used toparametrize the interbar conductances. In this contextit is assumed that the interbar conductances of the en-tire rotor topology are equal

Gi[m] =Giref

ni(18)

for m [1,2, . . . ,niNr].Measurement and simulations results for the two in-vestigated configurations and ni = 4 are depicted inFig. 8:



Figure 7: Squirrel cage rotor with respect to the config-uration with one broken (removed) end ring segment

0

0.01

0.02

0.03

0.04

0.05

1e3 1e6 1e9

faul

tind

icat

or

Giref [1/]

broken end ringbroken bar

Figure 8: Fault indicator of the VMM versus interbarconductance Giref

Broken end ring segment: The measured fault in-dicator is shown as horizontal dashed line. Thesimulation result to match decreases with increas-ing total interbar conductance Giref. The intersec-tion of the simulation and measurement resultsindicates well parametrized simulation model.

Broken rotor bar: The fault indicator obtained bymeasurements is depicted as horizontal solid line.The intersection of this line with the simulationresult leads to a solution close to the one obtainedfor one broken end ring segment.

From the two intersection points of Fig. 8 an averagevalue of Giref = 2.3105 1 is obtained, which satisfiesthe required accuracy of both fault configurations. Ifthe two intersection points were much further apart,this would indicate that the simulation model is notreflecting the real machine behavior.

6 Conclusions

This paper presents a mathematical model of a squir-rel cage induction machine with interbar conductancesand skewed rotor bars. For this purpose each rotor baris axially subdivided into bar sections which are tan-gentially connected by interbar conductors. This waya discretized rotor topology model is developed. TheModelica models including equations of the air gapand cage are presented in detail.The simulation model was used in a project wherethe impact of interbar currents, in combination withelectrical rotor asymmetries are studied. In order toparametrize the interbar conductances two differentfault configurations are investigated by measurementsand simulations. The obtained parameter of the totalinterbar conductance confirms that the proposed simu-lation model leads to reasonable results and accuratelyreflects real machine behavior under rotor fault condi-tions.

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