THE 8th INTERNATIONAL SYMPOSIUM ON ADVANCED TOPICS IN ELECTRICAL ENGINEERING May 23-25, 2013

Bucharest, Romania

Parameters Determination for the Induction Machine Mathematical Model,

Taking into Account Saturation and Skin Effect

Ion-Daniel Ilina Electrical Engineering Department, University Politehnica of Bucharest, 060042, Romania

ilinadaniel@yahoo.com

Abstract – The paper presents an experimental determination of the parameters of the mathematical model of the induction machine, taking into account magnetic saturation and skin effect. The magnetization inductance, the differential magnetization inductance, and the resistances and leakage inductances of the multi-shunt circuits which simulate the skin effect in the rotor, are determined. The induction machine d-q model is issued. The experimental tests are conducted with stand still rotor.

Keywords: induction machine parameters, experimental determination, saturation and skin effect.

I. INTRODUCTION

Magnetic saturation and skin effect represent two phenomena that can complicate enough the study of and determination of the parameters of the electric machines.

The relation between magnetic fluxes and currents underline the main inductivities and leakage inductivities of the induction machines, although the leakage field lines and those of the main field are partly overlapping. Hence, magnetic saturation influences both, the main inductivities (to a larger extent) and leakage inductivities (to a lesser extent).

The skin effect is noticeable mostly in massive conductors and in the jokes and main parts of the machine. The skin effect occurs mostly in squirrel cage induction machines. The higher the power of the machine is, the greater the skin effect gets.

Mathematical model of the skin effect and magnetic saturation, if considered simultaneously, can be achieved by using numerical methods for field calculation, or by estimating the resistances and inductivities (leakage and main) through experimental testing.

Due to the fact that resistances and rotor leakage inductivities depend on the skin effect, and the main inductivity depends on magnetic saturation, makes their determination more difficult, as well as designing the driving system. In other words, the machine has variable parameters, and the driving system implies increased computation power, increased memory requirement which means in the end bigger costs. These shortcomings can be eliminated if the rotor circuit with variable parameters that depend on frequency is replaced by few equivalent circuits that have constant parameters within a large frequency range. Such equivalence becomes better as the number of parallel circuits

gets higher, and is absolutely precise in case the number of circuits is infinity [2], [3], [14].

On the other hand, using mathematical models with orthogonal axis, with an optimal position of these axes for different types of machines and operational situation, can provide significant simplifications in determining the equivalent parameters of the machine.

Also, by using static methods it is possible to achieve results relatively easy, and with reduced costs. Such methods are very useful for large and very large electric machines. The static method can be used without affecting the construction of the driving system in which the induction machine is embedded.

This paper presents a static method for estimating the mathematical model parameters for an induction machine, taking into account the influence of skin effect and magnetic saturation. The method is implemented as an experimental test comprising two procedures:

- damping current test for determination of the magnetic curve, of the main inductivity and the differential magnetization inductivity;

- by measuring frequency response the resistances and leakage inductivities of the rotor circuits that simulate the skin effect, are determined [2], [3], [4], [17].

II. THE ORTHOGONAL MODEL OF INDUCTION MACHINE

The study of the skin effect and of the magnetic saturation upon the parameters of the three-phase induction machine was done by means of the orthogonal model of the machine. The real machine is made equivalent to a machine which has the windings placed on two orthogonal axes d and q, axes which are fixed against the rotating magnetic field. The equivalence between the real and orthogonal model machine is done using Park transformation [2], [3], [17].

The equations of the orthogonal model of the induction machine with two squirrel cages, that take into account the variation of the rotor parameters with frequency, for each of the two axes, are determined from the general equations of the orthogonal model [2-4]:

.2,1,,

,,

=Ψ

−=Ψ

−=

Ψ−Ψ

−=−Ψ+Ψ

−=−

jdt

dRi

dtd

Ri

dtd

URidt

dURi

QjQjQj

DjDjDj

drq

qsqqrd

dsd ωω (1)

978-1-4673-5980-1/13/$31.00 ©2013 IEEE

Although between the two rotor squirrel cages D1 and D2 and Q1, Q2 respectively, are magnetic coupling through the leakage paths, including those in the equations would complicate the model without gaining great advantages in parameters determination or for increasing accuracy of the model. Taking into account the general relation between the fluxes and currents with consideration of magnetic saturation and by separating the main fluxes from the leakage fluxes, the following relations are obtained [3], [4]:

.2,1

,,

,,

=

Ψ+=ΨΨ+=Ψ

Ψ+=ΨΨ+=Ψ

j

iLiL

iLiL

qmQjQjQjdmDjDjDj

qmqsqdmdsd

σσ

σσ

(2)

Also, it must be noticed that the model considers the general case when the parameters of the two squirrel cages, along the two axes, are not the same. As in most of the cases the squirrel cages are symmetrical, the relations are simplifying, and it can be considered that:

.2,1,, ===== jRRRLLL rjQjDjrjQjDj σσσ (3)

The main fluxes qmdm ΨΨ , depend on the total field

current and its components qmdm ii , along d and q axes [3],

[4]: .)(,)( qmmdmqmdmmdmdm iiLiiL =Ψ=Ψ (4)

where:

.

,,

22

2121

qmdmm

QQqqmDDddm

iii

iiiiiiii

+=

++=++= (5)

The equivalent circuit of the induction machine with constant parameters, along q axes is shown in Fig. 1. The Lqmt is machine’s differential magnetization inductance, and Rs and Lσs are the resistance and stator leakage inductance [3]. As the air-gap is constant, the d equivalent circuit along d axis is similar.

The machine differential main inductance Lqmt depends on

magnetic saturation and on field current, by a non-linear characteristic )( mqmt iL . This characteristic will be determined

experimentally by current damping testing. It can be said that the equivalent circuit of the induction machine takes into account simultaneously the non-linear phenomena inside machine: the skin effect and magnetic saturation. During experimental determination, the two influences are analyzed separately. The frequency response

tests are done at low current values (≈15% of the rated current).

III. CURRENT DAMPING TEST

These tests are made to determine the magnetization characteristics of the induction machine, and the main inductance and the differential magnetization inductance as a field current dependency.

Taking into account the symmetry of the induction machine, positioning the rotor is no needed. Following there are the calculations and experimental determinations made for q axis. The test is done with still rotor and stator windings v and w connected in series and passed by a direct current Iv0. Then the stator circuit is shortcut, and the damping process of the current is registered (Fig. 2) [10].

In this case only first equation corresponding to axis q from

all stator equations (1) is used:

.dt

dURi q

qsqΨ

−=− (6)

The test is made on the real machine, and the relation between the currents in the mathematical model and those in the real machine is obtained by using Park transformation [3]:

.02,0 =+== wvvqd iiiii (7)

If equation (6) is integrated, after the short-circuit, when 0=qU , it is obtained:

.)()(200

finalqinitialq

t

v

t

sqs dtiRdtiR Ψ−Ψ=⋅= ∫∫∞→∞→

(8)

Taking into account the flux relations (2) and that 0)( =Ψ finalq , it yields:

,2)(0∫

∞→

⋅⋅=Ψt

vsinitialq dtiR

.)()( initialqsinitialqqmi iL σ−Ψ=Ψ (9)

Since 2)( 0vinitialq Ii = , it results:

.22 00

vs

t

vsqmi ILdtiR σ−=Ψ ∫∞→

(10)

The stator resistance RS can be determined by DC measurement, while leakage inductance Lsσ can be determined from the short-circuit test of the induction motor [1], [8], [9]. Another way to determine the inductance is by approximating the leakage inductance with the homopolar inductance sL0 measured in AC, when stator windings are in

U1 U2

V1 V2

W1 W2

A =

Uuvw(t)

iu(t)

MAS

Uq

Iq

Rs jωLsσ Rr1 Rr2

jωLqmt

jωLr1σ jωLr2σ

Iqm IQ1 IQ2

Fig. 1. Equivalent circuit of the induction machine along q axis.

Fig. 2. Test circuit for damping current measurement.

series, by measuring the power, voltage and current [9]. Using the measured values the following can be determined:

.)3/(1,3/

20

200

10

2000

ss

s

RIUL

IPR

−=

=

ω (11)

As the magnetization of the machine iron core depends only on the current iq, results:

.2vqqmm iiii === (12)

At the same time: .**

qmqmm

qmqm i

iΨ=⋅

Ψ=Ψ

Therefore, from previous relation, the machine magnetization curve can be determined:

.2,)()( **vmmqmmdm iiii =Ψ=Ψ (13)

Hence:

.22 00

*vs

t

vsqm ILdtiR σ−=Ψ ∫∞→

(14)

and 20vm Ii = , respectively. Taking into account relation (14) the mathematical

expression can be used to determine the magnetization characteristic of the induction machine )(*

mqm iΨ , with all

terms known or possible to be determined. Moreover, the main inductance can be computed, as well as

differential magnetization inductance:

.,**

m

qmqmt

m

qmqm di

dL

iL

Ψ=

Ψ= (15)

Due to machine’s symmetry, the entire procedure described above for q axis, can be applied for d axis as well, obtaining a unique magnetization curve.

IV. FREQUENCY RESPONSE TESTS

The resistances and leakage inductances of the rotor circuits that simulate the skin effect can be determined from frequency response tests. In order to separate the magnetic saturation effect from the skin effect, these tests are done at low value of the currents (about 15% of the rated current).

The test circuit is similar to that used for the damping current test. However, in this case the stator windings are supplied from an AC voltage supply with variable frequency (Fig. 3) [10].

Taking into account the equivalent circuit (Fig. 1) it is

possible to determine the equivalent circuit impedance:

.)()(q

qqsq I

UjLjRjZ =+= ωωω (16)

During the tests there are measured voltage Uvw and current Iv. The relations between values measured on real machine and those of the orthogonal model are determined based on Park transformation [3]:

.2

,2,2/v

vw

q

qvqvwq I

UI

UIIUU

⋅=⋅== (17)

From relation (16) and (17) it results: [ ].2)(2 ωω jIjLRIU vqsvvw ⋅⋅⋅=⋅⋅− (18)

The equivalent circuit shown in Fig.1 contains two rotor circuits that simulate the skin effect. The more circuits are connected in parallel in the equivalent circuit, the better the skin effect is simulated. But for small size machines, using just one circuit or two circuits is sufficient.

In case only one circuit is used to simulate the skin effect, resolving the equivalent circuit yields the equivalent impedance:

2

111

)(TjTj

LLjRZ qmtssq ωωω σ +

+⋅++= and respectively:

.11

)()(2

1

TjTj

LLjL qmtsq ωω

ω σ ++

⋅+= (19)

In case of using two circuits to simulate the skin effect, the equivalent impedance will have the form:

( ) ( )( )( )( )43

21

1111

TjTjTjTjLLjRZ qmtssq ωω

ωωω σ ++++⋅++=

and respectively:

( ) ( ) ( )( ) ( ) .

1111

)(43

21

TjTjTjTj

LLjjL qmtsq ωωωω

ωω σ +⋅++⋅+

⋅+= (20)

Where: T1, T2, T3, T4 are time constants which can be written as analytical relations that depend on the parameters of equivalent circuit along q axis. These relations are:

- for one rotor circuit:

.

,)(

1

12

1

111

r

rqmt

qmtsr

rqmtsqmtsr

RLL

T

LLRLLLLLL

T

σ

σ

σσσσ

+=

+

++=

(21)

- for two rotor circuits:

( )

( )

( )[ ]

( )[ ].1

,1

,1

,1

21122121

43

212121

43

21122121

21

212121

21

rrqmtrrrrrr

rrqmtrrrr

rrqmts

qmtsrrrr

rr

rrqmts

qmtsrr

rr

RRLLRLRRR

TT

LLLLLRR

TT

RRLL

LLLRLR

RRTT

LLLL

LLLL

RRTT

+++=+

++=⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+++=+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

++=⋅

σσ

σσσσ

σ

σσσ

σσσ

σσσ

(22)

U1 U2

V1 V2

W1 W2

A

~ Uuvw(t)

iu(t)

MAT V

Fig. 3. Testing circuit for frequency response tests.

Taking into account relation (18) in the field of amplitude, a transfer function )( ωjLq to approximate the equality within a frequency interval as large as possible, must be found. From relations (19) and (20) it can be noticed that the transfer function is of first or second order, depending on the following unknowns:

21 ,, TTLL qmts +σ or .,,,, 4321 TTTTLL qmts +σ

V. EXPERIMENTAL RESULTS

Experimental tests were conducted on an induction machine with squirrel cage with the following rated data:

.][2820,86.0cos,][3.6/11,][380/220,][3

rpmnAIVUkWP

nn

nnn

=====

ϕ

The stator resistance Rs was measured and then the stator leakage inductance Lsσ was determined from a short-circuit test:

.][00752.0,][3.1 HLR ss =Ω= σ The voltage and current were determined with the help of

an oscilloscope with memory (Tektronix TDS 2000B) and a grid analyzer with memory (Chauvin-Arnoux CA8334). The calculations were done with Matlab/Simulink software [18]. A. The damping current tests

From the analysis of the computational method presented above, the following experimental stages can be described [10]:

- Phases v and w of the machine are connected in series and supplied with a direct current 0vI and the damping of the current is registered;

- The magnetic characteristics )(*mqm iΨ is determined

from relation (14); - The main inductance and differential magnetization

inductance )(,)( mqmtmqm iLiL are determined from relation (15).

These stages are repeated by increasing gradually the current in the stator, obtaining the magnetization characteristic and machine’s inductivities as a table or in a graph format.

The results obtained for all the stages are processed with Matlab/Simulink.

From testing circuit shown in Fig. 2 it can be observed that the stator windings must be supplied with a direct current for the current damping tests, which will permit the short-cut at windings terminals. The current will be produced by a DC shunt generator turned by a DC motor with separate excitation.

Fig. 4 show the damping of the stator current, for different values, for an induction machine with squirrel cage.

Fig. 4. The damping current test for Iv0 = 4A and Iv0 = 7.5A.

The following graphics are obtained with Matlab/Simulink software by processing data acquired for sequentially increased values of the stator current Iv0.

Fig. 5. The magnetic characteristic )(*

mqm iΨ

Fig. 6. The main inductance and differential magnetization inductance.

B. Frequency response tests Using the method presented above, there is possible to

establish the stages of mathematical estimation of the induction machine parameters, as follows [10]:

- The stator windings are fed from an AC voltage supply with variable frequency (Fig.3);

- The voltage and current Uvw(t) , iv(t) waveforms are acquired;

- The amplitude and phase shift of voltage and current are determined;

- Using relations (16) and (17), the amplitude and phase of the equivalent impedance and equivalent inductivity, can be determined;

- If only one rotor circuit is considered σ11 , rr LR transfer function with form (19) can be estimated which verifies the minimum error condition (18) for the entire range of measure frequencies;

- The unknown values result: ( 21 ,, TTLL qmts +σ );

- Determining the leakage uinductivity σsL , from equatino system (21), allow cmoputing the parameters of equivalent circuit σσ 11 ,,,, rrqmtss LRLLR ;

- In case of considering two rotor circuits σσ 2211 ,,, rrrr LRLR a transfer function can be

determined of form (20) that will approximate with minimum error the relation (18) for the entire range of measured frequencies;

- Then the following values result: 43432121 ,,,, TTTTTTTTLL qmts ⋅+⋅++σ ;

- Knowing the leakage inductivity and solving the system (22) the values of the equivalent circuit parameters can be determined:

σσσ 2211 ,,,,,, rrrrqmtss LRLRLLR . Fig. 7 shows two examples of the voltage and current

waveforms for different values of their amplitude and different supplied frequency. It can be noticed that the waveforms are not sinusoidal. Using Matlab/Simulink the fundamental harmonic was extracted.

Fig. 7. Voltage waveform and current waveform for: f = 10.48 [Hz]; Uvw = 7.403 [V]; Iv = 1.08 [A] f = 32.01 [Hz]; Uvw = 17.72 [V]; Iv = 2.103 [A]. After the acquisition of voltage (Uvw) and current (Iv), and

using relation (18), the dependence of the inductance amplitude against frequency is computed (Fig. 8).

Fig. 8. Inductance amplitude variation )( ωjLq versus frequency. In order to determine the transfer function, an output-error

model estimator from Matlab/Simulink software is used. The estimator can calculate the transfer function for a given input function for which the desired output functions is obtained with a minimum error. The estimator uses a recursive algorithm which verifies the following equation:

.)()()()( tentMzHtM kinputoutput +−⋅= (23)

Where: )(tM output is the output function; )(tM input - is the

input function; )(zH - estimated transfer function; )(te - the estimation error.

Fig. 9. The Matlab/Simulink scheme used to estimate the inductance

)( ωjLq .

From relation (17) written in the amplitude domain, it is possible to determine the input and the output functions corresponding to the estimation block.:

[ ][ ]

.)2/3(

,)()2/3()(

)()2/3()(2

2

ω⋅⋅=

⋅ℑ⋅−ℑ+

+⋅ℜ⋅−ℜ=

uinput

suuvw

suuvwoutput

IM

RImUm

RIeUeM

(24)

The parameters of the estimation block are set to respect the transfer function for one rotor circuit and respectively for two rotor circuits.

Then, the relation for the inductance )( ωjLq , at given estimation error shown in Fig. 10, is determined : - For one rotor circuit:

.2512.010957.01

045.0)(⋅+⋅+

⋅=ωωω

jj

jLq (25)

- For tow rotor circuits:

.)000050008.01()24987.01()000048893.01()093591.01(045.0)(

⋅+⋅⋅+⋅+⋅⋅+⋅=

ωωωωω

jjjjjLq

(26)

Fig. 10. Estimation error of the transfer function )( ωjLq with two time

constants. The following values of the equivalent circuit parameters

result from equations (21) and (22): - For one rotor circuit:

.][01294.0,][2007.0

,][03748.0,][00752.0,][3.1

11 HLR

HLHLR

rr

qmt

ss

=Ω=

==Ω=

σ

σ

(27)

- For two rotor circuits:

.][2181.0,][45.4545,][01242.0,][1997.0

,][03748.0,][00752.0,][3.1

22

11

HLRHLR

HLHLR

rr

rr

qmt

ss

=Ω==Ω=

==Ω=

σ

σ

σ

(28)

Taking into account the estimated transfer functions, it is possible to determine with the help of Matlab/Simulink the variation of the amplitude of these transfer functions against frequency (Fig. 11).

In figure the graphs marked with 1 represent the first order transfer functions (25) and the graphs marked with 2 represent the second order transfer functions (26).

Fig. 11. Amplitude transfer function used to estimate the inductance )( ωjLq , versus frequency.

VI. CONCLUSION

There are several conclusions that can be drawn from the presented experimental method used to determine the parameters of the induction machine mathematical model.

The experimental method used to determine parameters of the mathematical model of the induction machine is a static method which can be easily achieved and with reduced costs. The method is useful for large and very large machines and can be used without affecting the driving system in which machine is embedded.

The paper presents a clear algorithm for determining and estimating the parameters of the induction machine mathematical model. The usage of Matlab/Simulink software produces low estimation errors and implies a low computation effort.

By analyzing the characteristics in Fig. 6 it can be observed that main inductivities have significantly different values than the differential magnetization inductivities, equally for low currents, as well as for high value currents. Therefore, in this case, to be able to consider magnetic saturation, both inductivities must be taking into account, both at low and high value currents.

From relations (27) and (28), as well as from graphic in Fig. 11 it can be seen that the influence of the second simulation circuit is small for the range of frequencies the experiments have been conducted and for the tested machine. Therefore, for the frequency interval 10-60 [Hz], the simulation of the skin effect can be achieved by using one single circuit, simplifying the equivalent circuit.

On the other hand some errors can be observed between measured and estimated characteristics (Fig. 8, Fig 10 and Fig. 11). These errors are mainly due to the low frequency used in testing, because of neglecting the high order harmonics in the voltage and current waveforms, and also because of the error produced by Matlab/Simulink estimation recursive block (OE).

REFERENCES [1] Bâlă, C., „Maşini electrice (Electrical Machines)”, E.D.P., Bucureşti,

1982; [2] Boldea, I., Nasar, S. A.,”Electric machine dynamics”, Macmillan,

NewYork, 1986; [3] Boldea, I., “Parametrii maşinilor electrice (Parameters of electrical

machines)”, Editura Academiei, Bucureşti, 1991; [4] Boldea, I., “Transformatoare şi maşini electrice (Transformers and

Electrical Machines)”, E.D.P., Bucureşti, 1994; [5] Cojan, M., Livadaru, L., Simion, AL.,”Încercările maşinilor electrice

(Electrical machines tests)”, Editura Shakti, 1997; [6] Dandeno, P.L, Poray, A.T., „Development of detailed turbogenerator

equivalent circuits from standstill frequency response measurement”, Toronto, Ontario Canada, 1998;

[7] Danilevici, IA. B., “Parametrii maşinilor de curent alternativ (AC electrical machines parameters)”, Editura tehnică, Bucureşti, 1968;

[8] Ghiţă, C., “Modelarea şi parametrii convertoarelor electromecanice (Modeling and parameters electromechanical converters)”, Editura Printech, Bucureşti, 2003;

[9] Ghiţă, C., Ilina, D.I., “Maşini electrice – Îndrumar de laborator (Electric Machines - Laboratory Handbook)”, Editura Printech, Bucureşti, 2003;

[10] Ilina I.D., „Contribiţii teoretice şi experimentale privind determinarea parametrilor masinilor electrice de curent alternativ (Theoretical and experimental contributions concerning the determination of the AC electrical machines parameters), Doctoral Thesis, UPB 2010;

[11] Ilina I.D., Ghiţă C., „Experimental method for estimating the parameters of a synchronous machine equivalent scheme, with consideration of skin effect influence”, Scientific Bulletin UPB, Series C, volume 73, pg. 165, Editura Politehnica Press, Bucureşti, 2011, ISSN: 1454-234x;

[12] Jerve G.K., “ Încercările maşinilor electrice rotative (Rotating electrical machines tests)”, Editura Tehnică, 1972;

[13] Koubaa Yassine, „Asynchronous machine parameters estimation using recursive method”, National Engineering School (ENIS), 2006;

[14] Mihalache, M., “Contribuţii asupra parametrilor maşinilor electrice de curent alternativ (Contributions to the AC electrical machines parameters)”, Doctoral Thesis, IPB, 1980;

[15] Park Gerald, „Parameter Identification of Salient-Pole Synchronous Machines Using the SSFR (Standstill Frequency Response) Test”, Queen’s University Kingston, Ontario, 1997;

[16] Repo Anna-Kaisa, „Numerical impulse response tests to identify dymamic inductioon-machine models”, Doctoral Dissertation, Helsinki University of Technology, Faculty of Electronics, Communications and Automation, Department of Electrical Engineering, 2002;

[17] Vas P., “Parameter estimation, condition monitoring, and diagnosis of electrical machines”, Oxford University Press, 2001;

[18] *** MATLAB and Simulink Guide, http://www.mathworks.com/.