Abstract—Field Weakening (FW) technique controls the

machine where it requires high torque at low speed and low torque at high speed. It is desirable to retain maximum torque capability in the FW region for many applications at high speed. In order to produce maximum torque that machine could possibly develop at FW region, the maximum inverter voltage and current must be appropriately utilized. This paper investigates the effect of leakage inductance in developing the torque of sensorless Field Oriented Controlled (FOC) induction machine in FW region. A drive system with Model Reference Adaptive System (MRAS) scheme based on Sliding Mode (SM) technique to estimate the rotor speed in the FW region using Space Vector Modulated (SVM) inverter is modeled. The drive system is simulated by MATLAB/Simulink and the speed-torque characteristics for various leakage inductances are compared with that of a machine with sensor.

. Keywords—Field weakening, Induction machine, Model

reference adaptive system, Sliding mode observer.

I. INTRODUCTION AXIMUM torque capability in the FW region is a desired property of FOC induction machine in applications such as washing machine, spindle drives

and gearless traction drives that require the achievement of wide speed range which extends up to several times of the rated speed. In sensorless control schemes, the motor is controlled without measuring its speed, making the drive system less expensive and unaffected by sensor failures. The estimation of the rotor flux and speed is an exigent task due to the high-order multiple variables, nonlinearity of induction motor dynamics and couplings of the system with the fact that some parameters can vary in a wide range [1]-[3]. Thus control and observation methodologies with strong robustness properties are required in sensorless control [4]. Model based methods, which seems to be a more practical approach, can be segregated into open loop estimators and closed loop

Nisha G. K. is with the Electrical Engineering Department, Government College of Engineering, Trivandrum, Kerala, India (email:nishacharu@gmail.com).

Lakaparampil Z. V. is with the Centre for Development of Advanced Computing (C-DAC), Trivandrum, Kerala, India. (e-mail: zvlakapara@cdac.in). Ushakumari S. is with the Electrical Engineering Department, Government College of Engineering, Trivandrum, Kerala, India.( email: ushalal2002@gmail.com).

observers, where closed loop observers are less susceptible to parameter variations than open loop estimators, but they are much more computationally rigorous. The two major classes of observers are deterministic observers such as MRAS, full or reduced-order adaptive observer, Luenberger observer, and stochastic observers such as extended Kalman filter theory [5]-[7]. MRAS based on rotor flux technique first proposed in [8] is the most popular scheme and researchers are focusing to improve the performance of this technique due to their relative simplicity and less computational effort [9], [10]. An attractive choice to purge on large computational time can be with the use of sliding-mode observers, which may ground fast response, insensitivity to parameter variations and robustness against external disturbance and simple implementation [11], [12]. During the past decade, several research papers were published to achieve the maximum torque production of the machine in the FW region [13]-[15]. In most of the approaches, maximum available inverter voltage is utilized to produce maximum torque in FW region when the excitation level is adjusted by closed loop control of the machine voltage. Field weakening has to be done without violating either current limit or voltage limit in such a manner that accelerating torque follows maximum torque trajectory. The variable level of the main flux saturation in the machine causes the variation of magnetic inductance [16]. Therefore, in model based approach, accurate speed estimation is possible only if the speed estimation algorithm is modified to account the variation of magnetic inductance in the FW region. In the literature, modeling of magnetizing inductance of induction machine has been discussed extensively in [17]. Majority of the study has been devoted to improve torque capability with the available inverter voltage and less attention has been given to alternative approaches. In this paper, a sensorless control scheme is developed based on FOC scheme and SM technique, to improve the torque capability in FW region by varying leakage inductance of the machine. The model is operated in FW region and the performance of the model is evaluated and compared with that of induction machine with sensor. In both, sensor and sensorless scheme, SVM fed inverter is used instead of conventional sin pulse width modulation inverter and the simulation results are generated by using MATLAB/Simulink.

Effect of Leakage Inductance on Torque Capability of Field Oriented Controlled

Induction Machine in Field Weakening Region Nisha G.K., Lakaparampil Z.V., and Ushakumari S.

M

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II. MATHEMATICAL MODEL OF SENSORLESS FOC SCHEME IN FW REGION

FOC offers a number of benefits including speed control over a wide range, precise speed regulation, fast dynamic response and operation above base speed [18]. In this control scheme, a complex current is synthesized from two quadrature components, one of which is responsible for the flux level in the motor and the other controls the torque production in the motor [19]. Fig. 1 shows the stator current space vector in rotating reference frame. The reference frame d-q is rotating with the angular speed equal to rotor flux vector angular speed, ωe. Using the space vector method the induction motor model is written as:

( )

( )

ss s s

rr r r

dV t R idt

dV t R idt

ψ

ψ

= +

= +

(1)

where, j

s s s m r

r r r m s

L i L i e

L i L i

εψ

ψ

= +

= +

(2) The complete set of motor equation is given by:

( )( )j

s rs s s s m

di d i eV t R i L Ldt dt

ε

= + +

(3) ( )( ) r s

r r r r m

di d iV t R i L Ldt dt

= + +

(4)

Fig. 1 stator current space vector in rotating reference frame

The electromagnetic torque is expressed by:

2( ) ( ) Im ( ) *3 2

jrd L m s r

d PT t J T t L i i edt

εω= + =

(5)

The machine dynamic voltages in matrix form is represented as,

( ) ( )( ) ( )

00

sds s e m m esd

sqe s s m e msq

rdm e r m r r e r r

rqe r m m e r r r r

iR pL Ls pL LViLs R pL L pLVipL L R pL LiL pL L R pL

ω ωω ω

ω ω ω ωω ω ω ω

+ − − + = − − + − − − − +

(6)

where,

Ls = Lm(1+σs) : Stator self Inductance Lm : Magnetizing Inductance σs : Stator leakage factor

Lr = Lm(1+σr) : Rotor self Inductance σr : Rotor leakage factor Rs : Stator resistance Rr : Rotor resistance Td : Electromagnetic torque ω : Angular speed

Although several schemes are available for sensorless FOC, the model reference adaptive system is the popular method because of its simplicity. The adaptive scheme for the MRAS estimator is designed based on Popov’s criteria for hyper stability concept [20]. MRAS estimators are usually observing mechanical speed with the two independent machine models, one of which is speed dependent. The difference between estimator outputs of magnetic flux is used for speed error reduction. Basic scheme of modified MRAS-SM speed estimator for FW is shown in Fig. 2.

Fig. 2 Bock diagram of modified MRAS-SM speed estimator for FW

The reference value and the adaptive values of rotor flux components are given as:

[ ]( )rrd sd s sd s sd

m

Lp v R i L piL

ψ σ= − −

(7)

1 ˆ 0ˆ ˆ

1 ˆˆ ˆ 0

ma arrd rd sdr r

aasqmrqrq

rr r

LiiL

ωψ ψτ τ

ψψ ωτ τ

− − = + −

(8)

The speed estimation is affected by noise and error, which causes a lagging in estimate of the speed and when this inaccurate estimate is fed back to the observer, the flux estimation accuracy deteriorates.

Sliding mode control is a control strategy in Variable Structure System (VSS) where the infnite feedback gain chosen generates the high frequency chattering and the discontinuous control input result in torque chattering and instigate mechanical resonance [21]. In SMC a control law is designed so as to bring the system trajectory on a predefined surface, where the switching surface satisfies the condition of having positive attraction, called the sliding surface.

The surface s(x) = 0 contains only endpoints and these points constitute a special trajectory representing motion called sliding mode. The resultant motion on the sliding surface shows that the point slides towards the equilibrium point [22]. The sliding mode control should be chosen such that the candidate Lyapunov function, V which is a scalar

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function of S and its derivative satisfies the Lyapunov stability criteria,

21( ) ( )2

V S S x= (9)

( ) ( ) ( )V S S x S x= (10)

The switching vector usw(t) for the stability condition should satisfy the equation given below and once the equivalent control input ueq(t) is known, sliding mode dynamics with the control vector u(t) can be written as:

( ) ( ( , ))swu t sign S x tη= (11) ( ) ( ) ( )eq swu t u t u t= +

(12)

The time varying sliding surface S(x) with reference to dynamic model of induction machine and the speed tuning signal is constructed. When the system reaches the sliding surface, the error dynamics at the sliding surface will be forced to exponentially decay to zero. Thus,

( ) 0S x e Ke dtω ω= + =∫ (13)

.( ) ( )V S S e Keω ω= + (14)

This can be attained when:

ˆr eq swu uω = + (15)

The drastic change of input is avoided by introducing a boundary layer with width, φ . By replacing sign(s) with sat(S/φ), then (11) becomes,

( / )swu sat Sη φ= (16)

A natural solution to reduce the chattering in the estimated speed is by means of a low-pass filter (LPF) as in (17).

1'1sw swu u

sµ=

+ (17)

Maximum mechanical speed adjusted by the induction machine is limited by DC link voltage and PWM inverter strategy and the maximum current to an induction machine is decided by the thermal limit of the inverter or the machine itself. Field weakening is used to allow operation of variable speed induction motor drives at high speeds where the rotor flux is getting reduced to below its rated value due to the increase of rotor speed than the base speed. In field-weakening control the d- and q-axis voltage references are used and the maximum phase voltage, Vsm, is decided by the DC link voltage, VDC, of the SVM inverter, in which Vsm is obtained in the linear control range as VDC/√3 for space vector modulation [23]. The reference current, Ism, should satisfy (18) regardless of the reference frame.

2 2 2ms sd sqI i i≥ + (18)

The induction machine can generate the maximum torque at the field weakening region under a given voltage limit, Vsm , should satisfy the equation as:

2 2 2ms sd sqV V V≥ +

(19)

In FW region the voltage limit ellipse can be obtained by neglecting current variations in the stator voltage equations and the voltage drop due to the stator resistance as:

( ) ( )22 2sd e s sq e s smi L i L Vω ω σ+ ≤

(20)

In constant torque region stator flux magnitude is constant and the rotor speed is less than the rated speed. The base speed, where the constant torque region ends is deduced as:

( ) ( )

2

2 2 2[ 1 ]

sm

sb

sm sd

VL

I iω

σ σ

=

− −

(21)

As the speed increases output power increases, when the applied voltage is at the maximum value, output power starts to decreases, the rate of decrease of developed power is related to drive parameters. Constant power region begins at base speed and ends at the speed where current limit is unattainable due to the voltage limit and the maximum torque is inverse proportion to mechanical speed. To achieve speed higher than base speed flux weakening algorithm has to be applied. The speed, where this region ends is deduced as:

2

1 2

(1 )2

sm

s sm

VL I

σωσ

+=

(22)

Constant slip frequency region extends beyond the first region with the voltage limit superseding the current limit and making a reduction in current and the maximum torque is equal to critical torque, so is inverse proportion to square of the mechanical speed. The intersection of the ellipse and the circle is shown in Fig. 3.

Fig. 3 Voltage constraint ellipse, current constraint circle and torque

locus

III. SVM INVERTER Space vector modulation is a sophisticated PWM method

that provides advantages such as higher DC bus voltage utilization and lower total harmonic distortion [24]-[26]. In SVM, three phase stationary reference frame voltages for each inverter switching state are mapped to the complex two phase orthogonal α-β plane. The mathematical transform for converting the stationary three phase parameters to the orthogonal plane is known as the Clarke’s transformation. The reference voltage is generated by two adjacent non-zero vectors and two zero vectors. The general idea of realizing the reference voltage vector, Vref is based on the sequential switching of active and zero vectors. An arbitrary target output

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voltage vector, Vref is formed by the summation of a number of these space vectors within one switching period.

A typical seven segment switching sequence for generating reference vector in sector three is shown in Fig.4.

Fig.4 Switching logic signals

IV. SIMULATION The Simulink/MATLAB simulation models for FOC

induction machine with and without sensors using SVM inverters are developed in [27]. Simulation model for sensorless induction machine using MRAS-Sliding mode observer to estimate the rotor speed is developed in [28]. The above model is extended from base speed region to FW region in [29]-[31]. The simulation models for FOC induction machine with and without sensors are developed based on the block diagrams as shown in Figs. 5 and 6. The induction machine used for the simulation is a 1.5 kW having the motor parameters as given in Table I. For comparing the performance of the developed drive systems, simulation is carried out for both models, sensorless and with sensor as Case-1 and Case-2 respectively.

TABLE I

SPECIFICATIONS OF INDUCTION MACHINE

Symbol Quantity Value

Vrated rated voltage 380 V Irated rated current 7.5 A

P output power 1500 W f frequency 50 Hz Rs stator resistance 4.850Ω Rr rotor resistance 3.805 Ω Ls stator inductance 264 mH Lr Rotor inductance 264 mH Lm mutual inductance 255mH

Fig. 5 Block diagram of FOC induction machine with sensor in

FW region

Fig. 6 Block diagram of sensorless FOC induction machine using

MRAS-SM speed estimator in FW region In the simulation, the motor starts from a standstill state to a

reference speed of 5500 rpm in no load condition. Figs. 7 (a), 7 (b) show the rotor speed response with time for leakage inductances ranging from 0.25 p.u to 1.5 p.u. for Case-1 and Case-2 respectively. The results show MRAS-SM observer estimates the rotor speed well and close to that of with sensors for low values of leakage inductance and speed performance deteriorates when leakage inductance increases in case-1. The maximum attainable rotor speed is increased from 3500 rpm to 4600 rpm by decreasing the leakage inductance from 1.0 p.u. to 0.5 p.u. in sensorless case. In Case-2, a slight improvement in rotor speed by decreasing the leakage inductance.

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(a) case – 1 (sensorless)

(b) case – 2 (with sensor) Fig. 7 Rotor speed vs. time

Figs. 8 (a), 8 (b) show the performance of electromagnetic torque with respect to time for both cases. Variation of torque with respect to rotor speed is presented in Figs. 9 (a) and 9 (b) for Cases 1 and 2 respectively for varying values of leakage inductance. Table II shows the speed and torque response for leakage inductances ranging from 0.5 p.u to 1.5 p.u. for Case-1 and Case-2 respectively. When leakage inductance decreases from rated value, torque capability is improved in the FW region as demonstrated in Table III.

(a) case – 1 (sensorless)

(b) case – 2 (with sensor)

Fig. 8 Torque vs. time

TABLE II

SPEED AND TORQUE RESPONSE (AT 4 SECS) Lls(p.u.) ωr (rpm) T (Nm)

Case-1 Case-2 Case-1 Case-2

0.5 4457 4573 2.60 2.79 1.0 3584 4400 1.72 2.66 1.5 2950 4230 1.30 2.35

TABLE III TORQUE CAPABILITY (P.U.)

Lls(p.u.) ωr = 2.0 p.u. ωr = 3.0 p.u.

Case-1 Case-2 Case-1 Case-2

0.5 0.496 0.485 0.300 0.315 1.0 0.400 0.441 0.198 0.305

1.5 0.190 0.410 0.089 0.270

(a) case – 1 (sensorless)

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(b) case – 2 (with sensor)

Fig. 9 Torque vs. Rotor speed

However, torque capability decreases sharply when leakage inductance increases from rated value for Case-1 while it decreases marginally for Case-2. Figs. 10 (a) and 10 (b) present the variation of motor power with respect to rotor speed in Case-1 and Case-2 respectively for various values of leakage inductance.

In FW region, power increases when leakage inductance decreases from rated value to 0.25 p.u. and decreases when leakage inductance increases from 1.0 p.u.to 1.5 p.u. for Case-1. It is observed that upper bound of constant power region is extended from ωr = 1.7 p.u. to 3.0 p.u. by decreasing leakage inductance from 1.0 p.u. to 0.5 p.u. in Case-1. In Case-2, dropping out of power from rated value is marginal while increasing leakage inductance.

(a) case – 1 (sensorless)

(b) case – 2 (with sensor)

Fig. 10 Power vs. Rotor speed

The variation of rotor magnetizing current (p.u.) with respect to rotor speed (p.u.). is shown in Figs. 11 (a) and 11 (b) for Case-1 and Case-2 respectively. The actual values of magnetizing current are in track with reference values in FW region for various values of leakage inductance for Case-1. In Case-2, slight decrease in actual values magnetizing current compared to reference values when leakage inductance increases from rated value to 1.5 p.u. in FW region.

(a) case – 1 (sensorless)

(b) case – 2 (with sensor)

Fig. 11 imr vs. Rotor speed

By observing the simulation results, it ensures that improvement of torque capability of sensorless induction machine in FW region is possible by decreasing the leakage inductance from 1.0 p.u. to 0.25 p.u. Maximum torque at FW region is obtained when the rated value of leakage inductance is reduced to 50%. A further reduction in leakage inductance does not improve the torque as well as the power. The actual values of magnetizing current is independent of variation in leakage inductance, which shows that torque capability is improved only because of the drop in leakage inductance.

V. CONCLUSION In this paper, MATLAB/Simulink simulation models for field oriented controlled induction machine with and without sensors are developed for FW region. SVM fed inverters are used to operate the models for comparing the performance of the FOC drives.

The maximum rotor speed and torque capability of sensorless FOC induction machine in FW region is improved to that of induction machine with sensor by decreasing the leakage inductance of the machine from rated value. Maximum rotor speed and torque at FW region is obtained

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when the rated value of leakage inductance is reduced to 50% of the rated value and a further reduction does not improve the torque as well as the rotor speed. The performance of the sensorless FOC induction machine in FW region is very much close to that of induction machine with sensor by dropping the leakage inductance to 50% of the rated value. The variation of leakage inductance has less effect on the magnetizing current of a sensorless induction machine than a sensor type machine. The leakage inductance is a sensitive parameter in the performance of a sensorless FOC induction machine in FW region in the sense that a slight increase in its value will sharply deteriorates the machine performance.

ACKNOWLEDGMENT The first author acknowledges support from SPEED-IT

Research Fellowship from IT Department of the Government of Kerala. The facilities extended by the Government College of Engineering Trivandrum, Kerala, India are gratefully acknowledged.

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[29] G. K. Nisha, Z. V. Lakaparampil and S. Ushakumari, “Sensorless field oriented control of SVM inverter fed induction machine in field weakening region using sliding mode observer”, Lecture notes in Engineering and Computer Science: in Proc. 7th World Congress on Engineering WCE, London, U. K., 2013WCE , pp. 1174-118.

[30] G. K. Nisha, Z. V. Lakaparampil and S. Ushakumari, “Performance Study of Field Oriented Controlled Induction Machine in Field Weakening using SPWM and SVM fed Inverters”, International

International Conference on Advances in Engineering and Technology (ICAET'2014) March 29-30, 2014 Singapore

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Review on Modelling and Simulation IREMOS, vol. 6, no. 3, pp. 741-752, June 2013.

[31] G. K. Nisha, Z. V. Lakaparampil and S. Ushakumari, “Four Quadrant Operation of Sensorless FOC induction machine in Field Weakening Region using MRAS–sliding mode Observer”, in Proc. IEEE International Conference on Control Communication and Computing, ICCC, Trivandrum, India, Dec. 2013, to be published.

Nisha G.K. was born in Trivandrum, India on 20th May 1976. She received the B.Tech. degree in Electrical and Electronics Engineering from TKM College of Engineering (University of Kerala) and M.Tech. degree in Electrical Machines from Government College of Engineering, Trivandrum (University of Kerala), Kerala, India, in 1997 and 2000 respectively. She is currently a Research Scholar in Government

College of Engineering (University of Kerala), Trivandrum, India. She has 3 publications in international journals and 7 international conferences at her credit. From 2001 onwards, she was with the University College of Engineering, Trivandrum as Lecturer and with the Mar Baselios College of Engineering and Technology, Trivandrum as Associate Professor in the Department of Electrical Engineering. Her research interest includes ac drives, pulse width modulation and field oriented control. Ms. Nisha received Certificate of Merit (student) Award for the 2012 IAENG International Conference on Electrical Engineering held at Hong Kong and Best Paper Award at the 2013 International Conference on Advance Engineering and Technology held at Mysore, India.

Dr. Z. V. Lakaparampil was born in Changanacherry, Kerala, India on 17th October 1956. He received BSc(Engg.) in Electrical Engineering from NIT Calicut, Kerala, India, in 1979, and DIISc in Electronics Design Technology and PhD in Electrical Engineering from Indian Institute of Science, Bangalore, India in 1980 and 1995 respectively. He is currently an Associate Director in Centre for

Development of Advanced Computing (C-DAC), Trivandrum, Kerala, India. From 1979 to 1988, he was with Keltron, Kerala, India. Since then he is with CDAC-T, His area of specialization are embedded controllers for power electronics, vector/ field oriented control, real-time simulator etc. He has 12 publications in international journals, 40 international/national papers, 2 patents and one copyright to his credit. PhD guide for 4 research scholars from different Universities. Dr. Lakaparampil is a Member of IEEE, life member of Institution of Engineers (India), Member of Engineering and Technology Programmes KSCSTE, Government of Kerala, India and Member for EV/HEV panel for Collaborative Automotive Research.

Dr. S. Ushakumari was born in Kollam, India on 15th May 1963. She received her B.Tech. Degree in Electrical Engineering from TKM college of Engineering (University of Kerala), Kollam, Kerala, India in 1985, M.Tech. in 1995 and Ph.D. in 2002 in the area of Control systems from Government College of Engineering (University of Kerala), Trivandrum, Kerala, India. She joined in the Department of Electrical

Engineering, Government College of Engineering, Trivandrum, Kerala, India in 1990, where she is currently an Associate Professor from 2002 onwards. She has 15 publications in international journals and 42 national and international conferences at her credit. Area of interest includes robust and adaptive control systems, drives, fuzzy logic, neural network etc. She is a reviewer of IEEE Transactions on Industrial Electronics, Elsevier International journal on Computers and Electrical Engineering and AMSE international journal on modeling and simulation. and editor of the international journal on Electrical Sciences being published. Dr. Ushakumari is a life member of ISTE and secretary of ISTE Trivandrum chapter.

International Conference on Advances in Engineering and Technology (ICAET'2014) March 29-30, 2014 Singapore

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