Reference No 8

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708 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 50, NO. 4, AUGUST 2003

Parameter Estimation Scheme for Low-SpeedLinear Induction Motors Having Different

Leakage InductancesGubae Kang, Student Member, IEEE, Junha Kim, Student Member, IEEE, and Kwanghee Nam, Member, IEEE

AbstractLinear induction motors (LIMs) are characterized bya large air gap and, as a result, large leakage inductances. More-over, due to its unslotted structure and the absence of end rings inthe secondary part, the primary leakage inductance is much largerthan the secondary leakage inductance. Such differences preventus from using parameter estimation methods developed for the ro-tary motors. We propose a parameter estimation scheme for a LIMthat utilizes a pulsewidth-modulation inverter. It yields mutual in-ductance by numerically solving a third-order polynomial. Directestimation of mutual inductance enables us to calculate the leakage

inductances separately. The proposed estimation scheme is testedwith various example models and with a real 20-kW single-sidedLIM.

Index TermsLinear induction motor (LIM), parameter esti-mation, pulsewidth-modulation (PWM) inverter, rotary inductionmotor (RIM).

I. INTRODUCTION

E XACT knowledge of parameters is essential for thefield-oriented control of induction motors, since theperformance of the controllers depends on the accuracy of the

motor parameters. Parameter estimation methods are classified

into two categories: online estimation [1][3] and offline

estimation [4][9]. Holtz and Thimm [1] proposed an online

parameter estimation method in which the steepest gradient

method was utilized to adjust motor parameters so that the

current trajectories of the model and real motor output match.

Zai et al. [2] and Rabelo and Silvino [3] utilized the extended

Kalman filter to identify the magnetizing inductance and the

rotor time constant. In these schemes parameter initialization is

very important, since some erroneous initial values may lead

to the divergence of parameter estimates. Therefore, a prior

commissioning procedure is required to use the online schemes.

Various offline estimation methods have been proposed

[4][6]. Willis et al. [4] derived an equivalent circuit model

from a second-order transfer function for induction motors byutilizing a standstill frequency response test. This requires a

frequency response analyzer and a power amplifier to obtain

Manuscript received December 6, 2001; revised October 8, 2002. Abstractpublished on the Internet May 26, 2003.

G. Kang and K. Nam are with the Department of Electrical Engineering,Pohang University of Science and Technology, Pohang 790-784, Korea(e-mail:[email protected]).

J. Kim was with the Department of Electrical Engineering, Pohang Univer-sity of Science and Technology, Pohang 790-784, Korea. He is now with SeohoElectric Company, Ltd., Anyang 430-817, Korea (e-mail: [email protected]).

Digital Object Identifier 10.1109/TIE.2003.814864

current and voltage spectra. Moon and Keyhani [5] also utilized

the same transfer function, but they applied the maximum-like-

lihood method to find the parameters of a second-order transfer

function model. Stankovic et al. [6] proposed a static dc

excitation method to estimate a magnetizing inductance. The

above three methods require one- or two-phase excitation and

the use of special equipment.

Other parameter estimation schemes were developed that

utilized only an inverter [8], [9]. Moveover, they assumed that

stator leakage inductance was equal to rotor leakage induc-

tance. That assumption may be true for rotary induction motors

(RIMs), but in the case of linear induction motors (LIMs), the

secondary leakage inductance is much smaller than the primary

leakage inductance, due to the unslotted structure and the

absence of end rings. This structural difference makes many

parameter estimation methods for RIM unsuitable for a LIM.

In this work, we are considering an offline LIM parameter

estimation method which requires only an inverter. The end ef-

fect of the LIM is not considered here, so that the LIM model

is the same as the RIM model and, as a consequence, the result

is valid for low-speed LIMs. Our method follows the classical

method in utilizing a dc current test and a no-load test to obtainthe primary parameters. However, in the secondary parameter

estimation, we adopt the method of injecting alternating -axis

current while letting the -axis current be equal to zero [8], [9],

instead of the locked mover test.

A distinct feature of this work lies in the fact that the mutual

inductance is calculated by solving a third-order polynomialwhich was derived from the total equivalent inductance. Such a

method of obtaining allows us to calculate the leakage in-

ductances of the primary and secondary windings separately,

along with the secondary resistance. Another technical point

in this approach is that the peak values of power, current, and

voltage are obtained through low-pass filtering their absolute

values. The proposed estimation method has been tested withvarious example models and a real 20-kW single-sided LIM.

II. PRIMARY AND SECONDARY LEAKAGE INDUCTANCES

OF A LIM

In this section, we use a simple finite-element method (FEM)

analysis to show that the leakage inductance of the primary

winding is much larger than the secondary leakage inductance.

We performed FEM analysis with the specification of the LIM

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Fig. 1. Flux pattern, total flux , , and flux through air gap , (a) when only the primary winding is conducting and no current flows inside thealuminum plate and (b) when a current sheet is distributed sinusoidally along aluminum plates and no primary winding current flows.

flows through the secondary circuit. Hence, it is necessary to

minimize the slip by sychronizing the speed. This means that

the primary inductance can be estimated only when the LIM is

moving. However, a problem with the LIM is that the stroke is

limited in achieving the steady state. Therefore, in estimating

with variable-voltage variable-frequency (VVVF) control

mode, it is better to keep the V/F ratio large so that the speed

is low.

Under a no-load or a low slip condition, the secondary circuit

is not seen from the source. Hence, an approximate equation

follows, such that

(6)

where is the exciting angular frequency. Since the reactive

power is given by

, it follows from (6) that

(7)

To reduce end effect, frequency needs to be se-

lected less than 18 Hz, since end effect is negligibly small for

Hz [12], [13]. In practical applications, it is better to

use the average of the estimated values obtained from various

sample points.

B. Estimation of and

For obtaining and , a large current must flow through

the secondary circuit, i.e., current path through the secondary

circuit must be dominant. To provide a large current flow

through the secondary circuit, the locked test is normally

used. However, with an inverter it is possible to flow a large

current through the secondary circuit without locking the

motor [8], [9]. It can be realized by letting the -axis current

be equal to zero while applying an alternating current to the

axis, i.e., , , where is a positive

constant. Such a current pair is obtained by letting the phase

currents be , , and

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KANG et al.: PARAMET ER ES TIMAT ION SCHEM E FOR LOW -S PE ED LIMs HAVING DIF FE RE NT LEAKAGE INDUCTANCES 71 1

Fig. 2. -axes equivalent model of a LIM with PWM inverter in the stationary frame.

Since at standstill, the LIM equivalent circuit

shown in Fig. 2 can be represented as a series circuit, such

that

(8)where

(9)

(10)

and need to be estimated to obtain the estimates of the

secondary parameters.

For , the -axis voltage is obtained in the

steady state, such that

(11)

where . Based on (11), numerous

methods can be used to estimate and . In the fol-

lowing, however, we illustrate estimation methods that utilize

low-pass-filtered values instead of instantaneous measure-

ments.

The average real power is given by

(12)

(13)

The peak current may be obtained by a peak detection

method. It requires a special circuit or an algorithm finding a

maximum value. However, such methods are sensitive to noise.

Hence, we are using a different filtering method. Utilizing the

Fourier series expansion, it follows that

(14)

We denote by the filteredoutput of by a low-pass

filter that has a unity dc gain, i.e.,

(15)

where is the differential operator and is a positive constant

satisfying . Hence, we obtain from (13) and (15) that

(16)

where is calculated by integrating numerically

according to (12).On the other hand, the -axis voltage can be written from (11)

such that

(17)

We take absolute values on both sides of (17) and filter them

with the low-pass filter having unity gain. Then, it follows that

(18)

Since , we


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Fig. 6. Estimation errorsof , , and fora group of imaginary LIMs inwhich the leakage ratios are different along with s, while other parametersremain the same.

Fig. 6 shows that the proposed method gives much more accu-

rate estimation results for various motors. Specifically, the pro-

posed method performs well even when the primary and sec-

ondary leakage inductances are different, and this is the major

important difference from the conventional estimation method.

C. Effects of and on the Accuracy of and

Estimating and is, of course, dependent on the accu-

racy of the pre-obtained and . To see the sensitivity of

the estimation methods on the errors of and , and

are calculated via two methods for variousvalues of and. Fig. 7 showsestimation errorsof and depending

on and . Notethat the proposedmethod isless sensitiveto

the errors of and , i.e., the proposed method yields more

accurate results than the conventional method.

D. Statistics

For the statistics of the estimation performance, we applied

the proposed and the conventional methods to 11 real motor

models (seven RIMs and four LIMs). In these simulation

studies, we let for RIMs, for LIMs,

rad/s, and assume that and have 5%

(a)

(b)

Fig. 7. Error dependency of and on and .

Fig. 8. Statistics of estimation errors derived from the test results in Table II( 5% errors are assumed in and ).

errors. Table II shows the real motor parameters and the

estimation results of the proposed method. Fig. 8 shows the

bar graph of estimation errors when both the proposed and

conventional methods are applied to the 11 motor models in

Table II. Note that the proposed method is much better in the

leakage inductance estimation of LIM than the conventional

method. However, the performance of each of the two methods

is about the same in the case of RIMs. Table III summarizes the

possible estimation error sources and methods to reduce errors.

To obtain more accurate estimation, it is better to use high

frequency, . However, if is very large, then eddy-current

loss may degrade the performance.


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KANG et al.: PARAMET ER ES TIMAT ION SCHEM E FOR LOW -S PE ED LIMs HAVING DIF FE RE NT LEAKAGE INDUCTANCES 71 5

TABLE IIIPOSSIBLE ESTIMATION ERROR SOURCES

Fig. 9. Experimentalresult: variation whenthe frequencyof -axiscurrentvaries while .

VI. EXPERIMENTAL RESULTS

The proposed estimation algorithm is implemented in an in-

verter that utilizes a 32-bit processor TMS320C32 and insulated

gate bipolar transistors (IGBTs) (100 A, 1200 V). The PWM

switching frequency was chosen as 3 kHz, and the dead time

was set to be 3 s. Dimensions of the single-sided LIM under

test are listed in Table I. The electrical ratings of the LIM are as

follows: rated power 20 kW, rated line voltage 440 V, rated

frequency 60 Hz, and number of poles 4.

is estimated as 0.88 by the dc current test. To

obtain , we applied the VVVF control at a low speed (0.864

m/sec) by making the V/F ratio large. With this method, is

estimated as 59 mH. , , and are estimated at stand-

still by supplying and . By solving

the third-order equation (29), is estimated as 48.3 mH. Cor-

respondingly, is calculated as 52.7 mH from (27). In solving

the third-order equation (29), we choose a step size, such as.

The secondary resistance is estimated by utilizing (30). We

supplied with different exciting frequency,

Hz while keeping . Since

no -axis current flows, no propagating magnetic wave is de-

veloped. Then, the mover does not move at all, while a large

alternating -axis current flows through the secondary circuit.

Fig. 9 shows the estimated values based on experimental re-

sults, in which increases as the frequency increases. This is

thought to be caused by the decrease in skin depth of the alu-

minum plate. Note that we are utilizing 3-mm-thick aluminum

plate and that its skin depth is about 16 mm at 30 Hz.

Fig. 10. Experimental result: plots of speed and -axis current in thesynchronous reference frame when we let the secondary time constant in thecurrent controller be (a) , (b) , and (c) .

Fig. 10 shows the plots of speed and the primary -axis cur-

rent in the synchronous reference frame for three different

secondary time constants , , and .

Fig. 10(a) shows the case when the estimated value is

utilized directly for the slip calculation. However, Fig. 10(b) and

(c) show the cases when and are utilized,

respectively. Note that is constant during the acceleration


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period for the case in Fig. 10(a), while it slopes for the cases in

Fig. 10(b) and (c). This fact states that the case in Fig. 10(a) is

tuned correctly, while the cases in Fig. 10(b) and (c) are detuned

cases [11]. In other words, these experimental results support the

conclusion that the proposed estimation scheme yields accurate

secondary parameters.

VII. CONCLUDING REMARKS

We developed a parameter estimation scheme for the LIM, in

which the primary and secondary leakage inductances are dif-

ferent. While estimating the parameters of the primary winding

is the same as the existing method, to obtain the mutual in-

ductance we solved a third-order polynomial which was de-

rived from the total equivalent inductance. Once the mutual in-

ductance is obtained, the remaining parameters follow directly.

Simulation study shows that the proposed method outperforms

the conventional method in estimating the parameters of LIMs,

and looks robust for possible measurement errors. Our scheme

does not require extra voltage sensors and specific test equip-

ment other than an inverter. Therefore, the proposed schemeis suitable for practical use in the parameter estimation of the

LIM, as well as the RIM. However, since the end effect is not

considered in this LIM modeling, the result will be limited to

low-speed LIMs.

REFERENCES

[1] J. Holtz and T. Thimm, Identification of the machine parameters in avector-controlled induction motor drive, IEEE. Trans. Ind. Applicat.,vol. 27, pp. 11111118, Nov./Dec. 1991.

[2] L. Zai, C. L. DeMacro, and T. A. Lipo, An extended Kalman filterapproach to rotor time constant measurement in PWM induction motordrives,IEEE. Trans. Ind. Applicat., vol. 28, pp.96104,Jan./Feb.1992.

[3] B. C. Rabelo and J. L. Silvino, Alternative methods of estimating themagnetising characteristics and the rotor time constant for the induction

machine vector control, in Proc. IEEE PESC98, 1998, pp. 973978.[4] J. R. Willis, G. J. Brock, and J. S. Edmonds, Derivation of induction

motor models from standstill frequency response tests, IEEE. Trans.Energy Conversion, vol. 4, pp. 608615, Dec. 1989.

[5] S. Moon and A. Keyhani, Estimation of induction machine parametersfrom standstill time-domain data, IEEE. Trans. Ind. Applicat., vol. 30,pp. 16091615, Nov./Dec. 1994.

[6] A. Stankovic, E. R. Benedict, V. John, and T. A. Lipo, A novel methodfor measuring induction machine magnetizing inductance, in Conf.

Rec. IEEE-IAS Annu. Meeting, 1997, pp. 234238.[7] R. J. Kerkman, J. D. Thunes, T. M. Rowan, and D. W. Schlegel, A fre-

quency-based determination of transient inductance and rotor resistancefor field commissioning purposes, IEEE. Trans. Ind. Applicat., vol. 32,pp. 577584, May/June 1996.

[8] M. Bertoluzzo, G. S. Buja, and R. Menis, Inverter voltage drop-freerecursive least-squares parameter identification of a PWM inverter-fedinduction motor at standstill, in Proc. IEEE ISIE97, vol. 2, 1997, pp.

649654.[9] A. Gastli, Identification of induction motor equivalent circuit parame-

ters using the single-phase test, IEEE. Trans. Energy Conversion, vol.14, pp. 5156, mar. 1999.

[10] E. Levi and S. B. Vukosavic, Identification of the magnetizing curveduring commisioning of a rotor flux oriented induction machine, Proc.

IEEElect. Power Applicat., vol. 146, no. 6, pp. 685693, 1999.[11] J. Seok and S. Sul, Induction motor parameter tuning for high-perfor-

mance drives, IEEE. Trans.Ind. Applicat., vol. 37, pp. 3541, Jan./Feb.2001.

[12] J. F. Gieras, G. E. Dawson, and A. R. Eastham, A new longitudinal endeffect factor for linear induction motors, IEEE. Trans. Energy Conver-sion, vol. 22, pp. 152159, Mar. 1987.

[13] J. F. Gieras, Linear Induction Drives. Oxford, U.K.: Clarendon, 1994.[14] D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of ACDrives. Oxford, U.K.: Clarendon, 1996.

[15] S. Nakamura,Applied Numerical Methods in C. Englewood Cliffs,NJ:Prentice-Hall, 1995.

Gubae Kang (S96) was born in Uiseong, Korea, in1973. He received the B.S. degree in electronic andelectrical engineering in 1996 from Kyungpook Na-tional University, Daegu, Korea, and the M.S. degreein electrical engineering in 1998 from Pohang Uni-versity of Science and Technology, Pohang, Korea,where he is currently working toward the Ph.D. de-gree.

His research interests are ac motor control, EVmotor drives, and power converter/inverter systems.

Junha Kim (S97)was born in Uljin,Korea, in 1970.Hereceivedthe B.S., M.S., andPh.D. degrees in elec-tronic and electrical engineering from Pohang Uni-versity of Science and Technology, Pohang, Korea,in 1997, 1999, and 2003, respectively.

He is currentlywith Seoho Electric Company, Ltd.,Anyang, Korea. His research interests are ac motorcontrol, electric vehicle, and PWM converters.

Kwanghee Nam (S83M86) received the B.S.and M.S. degrees in chemical technology andcontrol and instrumentation engineering from SeoulNational University, Seoul, Korea, in 1980 and1982, respectively, and the M.A. and Ph.D. degreesin mathematics and electrical engineering from theUniversity of Texas, Austin, in 1986.

He is currently a Professor in the Departmentof Electrical Engineering, Pohang University ofScience and Technology (POSTECH), Pohang,Korea. He served as Director of the POSTECH

Information Research Laboratories and as Dean of the Graduate School of

Information Technology from 1998 to 2000. His main interests are ac motorcontrol, power converters, computer networks, and nonlinear systems analysis.

Prof. Nam received an IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICSBest Paper Award in 2000.
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