A synchronized sinusoidal PWM based Rotor Flux Oriented Controlled Induction Motor Drive for Traction Application Saroj Kumar Sahoo1 and Tanmoy Bhattacharya2
Aravind M.
Department of Electrical Engineering Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India, Pin - 721302 1
[email protected],
[email protected]
ABB-GISL Grid Systems R&D ABB Global Industries and Services Limited Chennai, INDIA
[email protected]
Abstract— This paper proposes a vector controlled induction machine drive with a wide field weakening zone. The proposed scheme integrates methods of field weakening, synchronized pulse width modulation, overmodulation and harmonic current estimation and proposes a consolidated scheme which ensures good dynamic response with wide speed variation. A triangular carrier comparison based Synchronized Sinusoidal Pulse Width Modulation (SPWM) for medium voltage inverters with low switching frequency (less than 500Hz) is proposed in this paper. To have very good dynamic behavior of the synchronization scheme, the triangular carrier is generated from the instantaneous voltage references in a phase locked manner. An overmodulation scheme which ensures linearity between the reference voltage and the fundamental motor terminal voltage is proposed. An inverse gain based linearization method is used to match the reference and inverter output voltage fundamental. This overmodulation strategy fails at the zone of high values of modulation index and a reference modification approach is used in that zone. The proposed PWM and overmodulation schemes integrate methods of field weakening and harmonic current estimation available in the literature and propose a consolidated scheme which ensures good dynamic response with wide speed variation for rotor flux oriented induction motor drive. The scheme is experimentally verified and the results are presented.
I.
INTRODUCTION
Vector controlled induction motor drives are widely used in many application areas. With the switching capabilities of present day power semiconductors such as MOSFETs and IGBTs and the processing speed of microprocessors and digital signal processors, switching frequencies of the order of several kHz are generally employed in these drives, especially in lower power applications with ratings up to tens of kilowatts. Good bandwidth for torque current control can easily be achieved in these drives. In addition, even if the speed range of the drive extends into the voltage limited field weakening region, the inverter is still maintained in the PWM mode, especially if the dc bus voltage is derived through a UPF boost front end converter.
There are, however, applications such as traction, electric vehicles etc., where the requirements are somewhat different and more demanding. For example, in traction drives, the inverter ratings are of the order of 200-500 kW, depending on whether the inverter is driving a single traction motor or two in parallel, which is a common arrangement. Because of the limitation in switching speed of the feasible power semiconductor devices at these inverter ratings, the switching frequency is limited to a few hundred hertz only. For lower switching frequency of operation, the ratio between the switching frequency to the fundamental frequency is very low especially in the higher fundamental frequency of operation. Hence, synchronization between the fundamental and the carrier has to be maintained to avoid subharmonics [1]. Moreover, the impedance offered by the induction motor to the subharmonic frequency components is very less and its presence can induce a very large subharmonic current in the machine windings. The PWM waveforms must possess 3-phase symmetry, quarter wave symmetry and half wave symmetry to minimize the harmonics [1]. Many synchronized PWM strategies are available in the literature addressing the synchronization in a steady state situation. Reference [2] proposes an optimal PWM scheme to reduce current harmonics and the analysis is given for a synchronous case. Reference [3] proposes a bus clamping synchronized PWM strategy based on space vector approach. The previous two schemes describe only steady state pulse generation techniques. But, the strategies to handle dynamic frequency variations and pulse ratio changing are not addressed. Smooth pulse ratio changing for space vector modulation is addressed in [4]. But, this PWM scheme also does not address the issue of fast changes in the voltage references when used in a vector controlled system. The pulse width modulation strategy should be such that it smoothly transits into the overmodulation zone upto the six step mode of operation while maintaining synchronization throughout. Moreover, the overmodulation algorithm has to be
This research is funded in part by ISIRD, SRIC,IIT Kharagpur and EE Department, IIT Kharagpur.
978-1-4673-4355-8/13/$31.00 ©2013 IEEE
797
carrier based for smooth transition from linear modulation zone. Linear relation between fundamental reference and the fundamental content of the inverter output is also expected. A carrier based overmodulation scheme with modified reference based linearization approach is presented in [5]. But, the scheme is developed for high frequency carrier and cannot be used with low frequency carrier based PWM strategies in its present form. Reference [6] discusses different carrier based overmodulation strategies originally developed for high frequency carrier based application. Hence, in their present form, these strategies are not suitable to be used in a low frequency carrier based application.
II.
With low switching frequencies and the requirement for overmodulation, the ripple in the motor current can be appreciable. The sampling of the current for feeding back into the vector control system has to be done carefully. Filtering may be required in the d-q co-ordinates, thereby limiting the achievable bandwidth for current control. But use of filter will reduce the achievable bandwidth of the current control loop. A fundamental current extraction algorithm is proposed in [7] for vector control in the overmodulation zone. But this algorithm is in synchronous reference frame and cross coupling terms are required for accurate estimation. Moreover, this scheme is active only in the overmodulation zone, whereas, current ripple is dominant even in the linear modulation zone with low switching frequency.
SYNCHRONIZATION STRATEGY
Figure 1. Schematic block diagram of induction motor vector control
It is mandatory in traction application to have a wide field weakening range of operation. The field weakening algorithm of [8] ensures maximum possible torque with fixed DC link voltage with the assumption that the current controllers are not saturated. This assumption does not allow the inverter to go into the six step mode of operation. Hence, in the field weakening zone, the inverter should be operated deep into the overmodulation region where the inverter switching frequency is limited to three times the fundamental frequency. But a safe margin from the six step mode of operation should be kept to avoid current controller saturation. In this paper, the authors propose a carrier based synchronous PWM scheme where the carrier is generated from the instantaneous voltage references in a phase locked manner and attains synchronization at all operating conditions without being parameter dependent. An inverse gain based linearization method in the overmodulation zone similar to [6] is used in this paper. But, because of low switching frequency, this linearization scheme cannot be extended till the six-step mode of operation. Therefore, a hybrid overmodulation scheme with two zones of operation is developed. The fundamental current extraction algorithm similar to [7] combined with the field weakening algorithm of [8] complete the overall scheme. This paper is hence arranged as follows. Section II describes the strategy developed for generating synchronous PWM. Section III explains the overmodulation strategy. Section IV explains the fundamental current extraction algorithm. Section V presents the field weakening algorithm along with the overall block diagram. Section VI presents some simulation and experimental waveforms. Finally the paper is concluded in section VII.
Figure 2. Variation of switching frequency vs reference frequency
Figure 3. Carrier generation in a phase locked manner from fundamental voltage references
In high power traction drives, because of the limitation to the maximum switching frequency, the frequency ratio i.e. the ratio of carrier frequency to fundamental frequency decreases. Thus, sub-harmonics get introduced in the PWM pulses and
798
As the reference frequency increases from 0 to 50Hz, the carrier changes as constant 500Hz to 21 times the fundamental frequency to 15 times the fundamental frequency and finally to 9 times the fundamental frequency. Also, a hysteresis band is introduced at each of the carrier changeover frequencies to avoid multiple changeovers at the boundaries. Figure 2 gives the information about the carrier changeover frequencies and the hysteresis bands. Figure 3 shows the block diagram of the proposed carrier generation method. The vector controller generates α and β axis voltage references. The phase angle information is extracted from the reference, which is a ramp at steady state. This phase angle is now multiplied with 9, 15 and 21 in order to generate carriers having 9, 15 and 21 times the fundamental reference frequencies respectively. These ramps are then converted to triangular carriers of unity peak. In order to ensure that the carrier changeover happens at their zero crossings, the changeover is allowed to happen at the zero crossing of the third harmonic component. It is to be noted that the carrier of three times the fundamental frequency is generated only to facilitate smooth carrier changeovers and it is never used to generate PWM by comparing it with the fundamental voltage references.
III.
OVERMODULATION SCHEME
modified reference y1 y2
1
voltage reference volt-sec lost volt-sec compensated
0
-1
phase angle
high low Figure 4. Loss of volt-sec and compensation technique for modulation index 1.112 in the overmodulation zone 1
For an overmodulation algorithm to be useful in a vector control scenario, the main requirement is to retain the synchronization and linearity between the input reference and output fundamental. The overmodulation scheme can be implemented either by reducing the height of the carrier or by increasing the height of the voltage reference. Keeping the carrier height constant makes it easier to generate the synchronous carrier from the reference voltage. To maintain linearity, the fundamental component of input voltage reference must be conserved in the output PWM voltage. There is a loss of volt-second in the portion of reference that lies above carrier as shown in Figure 4. This causes non linearity between the input and output. If this volt-second is compensated for the fundamental component of the reference, the linearity can be maintained. Hence, the reference voltage is multiplied by a suitable factor to compensate for the voltsecond lost and thus maintaining the linearity in the overmodulation zone. But this strategy is not applicable over the entire region of overmodulation. The overmodulation scheme is divided into two zones as explained below. During the entire overmodulation region, the carrier frequency is kept 9 times the fundamental reference and in synchronism with it.
799
3
Compensation Gain
finally in the inverter output voltage. These sub-harmonics become dominant when the frequency ratio reduces. The impedance offered by the induction machine windings to these sub-harmonics is very less and hence the sub-harmonic currents can become huge even at the presence of very small sub-harmonic voltage. These currents generate huge loss and very low frequency torque pulsations. When the voltage reference and the carrier are synchronized, the sub-harmonics are eliminated even at very low switching frequency. Hence, synchronous PWM becomes necessary for high power traction drives. Figure 1 shows the position of the PWM block in a vector controlled induction motor drive system. The magnitude, frequency and phase of these voltage references can vary instantaneously. To maintain synchronization, the frequency and phase of the carrier should also vary instantaneously. In this paper phase angle information of the fundamental voltage reference is used to generate a carrier in a phase locked manner with the instantaneous voltage references. Further, to maintain three phase symmetry, half wave symmetry and quarter wave symmetry, the carrier frequency is maintained at 9, 15 or 21 [3(2n±1), n=1,2,3,….] times the fundamental frequency. The ratio of triangular carrier frequency to the fundamental frequency is termed here as the frequency ratio. Maximum carrier frequency should be lesser than or equal to the maximum allowable switching frequency of the inverter. As a specific case, maximum allowable switching frequency is taken to be 500Hz in this paper. When the fundamental frequency increases, the carrier frequency also increases proportionally for a fixed frequency ratio. If the carrier frequency exceeds 500Hz then the frequency ratio is reduced to the next lower permissible value, i.e. from 21 to 15 or from 15 to 9. For lesser fundamental frequency (less than 15Hz) a constant 500Hz carrier is used. Since the frequency ratio is large, the problem of subharmonics will not be predominant below 15Hz range. Hence, a constant 500Hz carrier and synchronous carriers with frequency ratio of 21, 15 and 9 are generated to accommodate a fundamental frequency range of 0 to 50Hz.
2.5 2
1.5
1 1
1.05
1.1 1.15 Modulation Index
1.2
Figure 5. Modulation Index vs Compensation Gain variation
A. Overmodulation Zone 1 In this zone, the linearity is maintained between the input reference and the output PWM by multiplying the voltage reference by a gain varying in a non-linear fashion. Figure 4 shows the loss of volt-second and the compensation technique for modulation index, mi =1.112. To compensate for the lost volt-second in voltage reference, a standard approach is to multiply it by a compensation gain (1.2 for mi=1.112) to get modified reference. This modified reference is used for generating the PWM pulses whose fundamental is equal to the reference modulation index (=1.112). It is to be noted that for a particular modulation index, the value of compensation gain varies for different frequency ratio. For frequency ratio = 9 and in the overmodulation zone, there occurs a maximum of three intersection points between the reference and the carrier in the first quarter cycle. These intersection points can be found out by solving the equation of carrier and modified reference. From Figure 4,
1.2732. As the frequency ratio increases, one can go closer to 1.2732 but still cannot trace the entire overmodulation zone using this technique. So, a different technique, as explained in the following section, is implemented to obtain any modulation index beyond 1.234. In order to keep a safe margin, the overmodulation zone 1 extends from modulation index 1 to 1.22. B. Overmodulation Zone 2
1.5 1 0.5
y1 = m × θ + c
(1)
-0.5
y 2 = b × sin θ
(2)
-1
b = a × mi
(3)
4
⎡ 2 ( cos θ 1 − cos θ 2 + cos θ 3) − 1⎤⎦ π⎣
6
low
θ
0
π/5
2π/5
3π/5
π−θ π 4π/5
Figure 7. Generation of pulse width modulated wave in the overmodulation zone 2
(4)
5.7
0.3
Voltage Reference
Clamp Height
Modified Reference
4
2
Phase angle
high
5
3
clamp height
0
Where ‘m’ is the slope, ‘c’ is the y-axis intercept of the straight line, ‘mi’ is the reference modulation index and ‘a’ is the compensation gain. Solving for y1= y2, θ1, θ2 and θ3 are found and used in (4) to find out the corresponding modulation index. mi =
voltage reference
carrier
Carrier 1.234
1
0.2
0.1
0 -1 0
π/5
2π/5
3π/5 Phase Angle
4π/5
π
0 1.18
Figure 6. Voltage reference and modified reference after multiplication with compensation gain for modulation index 1.234
Figure 8.
Using (3) the corresponding gain is found out. Thus the compensation gain for modulation index from 1 to 1.22 is found out and a look up table is formed as shown in Figure 5. This technique can be applied till a point where the slope of the carrier and the modified reference is same around zero crossover points of the modified reference. For frequency ratio 9, this happens at modulation index 1.234 (and compensation gain being 4.619) as shown in Figure 6. Any increase in reference modulation index would saturate the output to a square wave and the modulation index to 1.2732. Hence, it is not possible to trace any modulation index between 1.234 and
1.2
1.22 1.24 Modulation Index
1.26
Modulation Index vs Clamp Height variation
In the overmodulation zone 2, when the input voltage reference magnitude is greater than or less than the clamp height, it is clamped to +1 or -1 respectively while generating the PWM pulses directly as shown in Figure 7. The clamp height is calculated such that, the fundamental of the output PWM is equal to the fundamental of the input voltage reference. This zone extends from modulation index 1.18 to 1.2732. A hysteresis band (mi=1.18-1.22) is introduced between the two zones to avoid multiple changeovers between them. From the Fourier expression for the fundamental component of the PWM output, the clamp height ‘h’ for any
800
modulation index in the range of 1.18 to 1.2732 is calculated as follows,
⎡⎛ mi × π ⎞ 1⎤ + 1⎟ ⎥ 4 ⎝ ⎠ 2⎦ ⎣
θ = cos −1 ⎢⎜
h = mi × sin θ
L ir = − m is Lr Using (12) and (13), (11) can be written as
(5)
2 ⎛ Lm ⎞ d is = σ Ls + Rr ⎜ ⎟ is dt dt ⎝ Lr ⎠
dψ s
(6)
Where θ is the angle from which the reference is clamped to +1in the positive half cycle as shown in Figure 7. Thus, for any modulation index, the respective clamp height is calculated and the result is stored in a look-up table as shown in Figure 8. IV.
While implementing vector control, the stator currents of the motor have to be transformed to the rotor flux co-ordinate system. Since the switching frequency of the inverter is low, the stator currents contain considerable ripple and the ripple components appear on the d-q co-ordinates also. Since the vector control depends only on the dc value of the transformed signals – representing the fundamental component of the stator current – filters are required to eliminate the ripple. These filters can limit the bandwidth of the current loop. The problem becomes especially acute once the inverter enters deep into the overmodulation zone. An alternative way of eliminating the ripple in the motor currents from entering the vector control system is possible. This involves estimating the ripple currents and subtracting them from the motor currents, thereby extracting the fundamental component. The fundamental component is then transformed and used in the vector control system. The method is similar to that explained in [7]. In this method, the leakage model of the induction machine is used for harmonic current estimation. But, unlike the estimation in synchronous reference frame as in [7], this paper uses leakage model of the induction machine in stationary reference frame. The induction machine equations in the stationary reference frame can be expressed as
0 = Rr ir +
dψ r dt
dψ s dt − jωmψ r
Vs = Lh
(10)
V.
It can be assumed that at harmonic frequencies, rotor flux has negligible amplitude. Hence, (8) can be written as (11)
m
FIELD WEAKENING SCHEME
Above the rated frequency of the motor, the drive enters the field weakening mode, where the voltage remains constant at the rated motor voltage. The control of the drive has to take into account this limited voltage, as well as the limit on the output current of the inverter, while at the same time producing the highest possible torque at any speed. The field weakening scheme adopted in this work is similar to the scheme proposed in [8]. The stator voltage equations of the induction machine can be expressed using (16) and (17).
From (9) and (10) rotor flux can be expressed as
L ψ r = r [ψ s − σ Ls is ] L
2
Equation (15) implies that as far as the non-fundamental components of the applied voltage are concerned, the motor behaves like a simple R-L load. Equation (15) is sometimes referred to as the leakage model of the induction motor. Therefore, if the components of the motor voltage other than the fundamental can be extracted, then the non-fundamental components of the motor currents can be simply calculated as the response of a first order circuit according to (15). If the resulting current is then subtracted from the sensed motor current, the balance will be the fundamental current alone. The block diagram of this method of extraction of the fundamental current is shown in Figure 9. Note that, for this method to work properly, the leakage model has to be solved several times within a PWM switching interval. Otherwise the extraction of the ripple current will not be accurate.
(8)
ψ r = Lr ir + Lm is
dt
⎛L ⎞ Where Lh = σ Ls and Rh = Rs + Rr ⎜ m ⎟ ⎝ Lr ⎠
(15)
Figure 9. Fundamental current extraction method
(9)
dψ r
d is + Rh is dt
(7)
ψ s = Ls is + Lm ir
0 = Rr ir +
(14)
Using (14), (7) can be expressed as
FUNDAMENTAL CURRENT EXTRACTION
Vs = Rs is +
(13)
(12)
Again, by neglecting rotor flux, we can write from (10)
801
Vsd = Rs isd + σ Ls
disd di − σ Lsωisq + (1 − σ ) Ls mr dt dt
(16)
Vsq = Rs isq + σ Ls
disq + σ Lsωisd + (1 − σ ) Ls ωimr dt
(17)
From (20) and (24) it can be shown that for maximum applied voltage Vmax , Td will be maximum when i isq = sd . Hence, in the field weakening region, the
σ
minimum of
isd
σ
and
( imax )2 − ( isd )2
is used as the
limiting value of isq to satisfy both the current limit condition and maximum torque condition. The field weakening region 2 2 where ( imax ) − ( isd ) is minimum is called the field i weakening region I. The field weakening region where sd is
Figure 10. Control for field weakening
σ
Considering (16) and (17) in the steady state and neglecting resistance drop terms - which are small compared to stator voltage, especially for the motors rated in the hundreds of kilowatts range – the direct and quadrature components of the stator voltage can be obtained as: Vsd = −σ Lsω isq
(18)
Vsq = Lsωisd
(19)
The voltage limit condition is then given as
( Lsωisd )2 + (σ Lsωisq )
2
≤ (Vmax )
2
(20)
minimum is called field weakening region II or constant slip region. The block diagram for field weakening control is shown in Figure 10. To ensure that the current controllers are not saturated, Vmax is calculated such that the modulation index in the field weakening zone (voltage limiting zone) is at the midpoint of the overmodulation zone 2 (1.2266 approx). This value of Vmax also ensures that the switching frequency of the inverter is three times the fundamental frequency during the field weakening zone. The block diagram of the overall vector controlled system is shown in Figure 11. This overall system satisfies the following requirements of an induction motor drive for traction application
Similarly, the current limit condition is given as
( isd )2 + ( isq )
2
≤ ( imax )
2
(i) Limited switching frequency of the order of 500 Hz. (ii) Smooth transition from PWM to overmodulation mode.
(21)
Assuming imr = isd at steady state. From (20), the flux current reference isd ref can be calculated as
2 ⎛V ⎞ 2 isd ref = ⎜ max ⎟ − σ isq ⎝ ω Ls ⎠
(
(
)
(iii) Rotor flux oriented control of the induction motor based on the fundamental components of motor current. (iv) Control in the field weakening range.
(22)
isd ref calculated using (22) is clamped at the base value isd base below base speed. Above base speed, isd ref
)
becomes less than isd base and hence (22) gives the flux current reference. A compensation for the effect of stator resistance can also be provided as proposed in [8] to reduce the variation of modulation index with the variation of load. The maximum value of isq can be calculated by satisfying the condition from (21) as isq max =
( imax )2 − ( isd )2
(23)
The machine torque can be expressed as Td =
2 P Lm 2 isd isq 3 2 Lr
Figure 11. Block diagram of the overall vector control system.
(24)
802
VI.
SIMULATION AND EXPERIMENTAL RESULTS
The synchronous PWM, overmodulation scheme and the fundamental component extraction scheme are implemented in a Digilent make Spartan 3E 1600 FPGA platform. The digital realization of the PWM scheme is done using Xilinx System Generator. Figure 12(a) shows that the carrier changeover happens at the zero crossing of the third harmonic voltage. Figure 12(b) shows that the phase locking of reference and carrier is not lost even at the event of carrier changeover. Figure 13(a) and (b) show the steady state gate pulse, motor current and synchronous carrier for frequency ratio 15 and 9 respectively. Figure 13(c) and (d) show the steady state gate pulse, motor current and synchronous carrier for overmodulation zone 1 and 2 respectively.
Figure 13(e) and (f) show the gate pulse, motor current and synchronous carrier during carrier changeover from 15-21 and 15-9 respectively. Figure 14(a) and (b) show the α-axis actual and extracted fundamental current for modulation index 0.8 and 1.2732(six step) respectively. The complete rotor flux oriented control scheme including synchronous PWM, fundamental current extraction and field weakening is simulated in MATLAB/SIMULINK platform.
Figure 14. α-axis fundamental current extraction using leakage model of the induction machine at (a) 31.4Hz fundamental frequency and 0.8 modulation index (b) 50Hz fundamental frequency and 1.2732 modulation index 1.5
Figure 12. (a) Carrier changeover at the zero crossing of the third harmonic (b) Three phase voltage references and the synchronous carrier during carrier changeover.
Torque in pu
1
0.5 1
1
T
T 0
0
fc=15fr, fr=29Hz, mi=0.74
fc=9fr, fr=35.34Hz, mi=0.9 M
M +1
-0.5 2
+1 B
B -1
-1 (b)
(a) 1
1.5 T
0
fc=9fr, fr=43.6Hz, mi=1.11
overmodulation zone 1
B -1 1
T fc=15fr
fc=21fr
6
T 0
fc=15fr
Field Weakening Region II
0 8
(d)
(c)
0
Field Weakening Region I
0.5
overmodulation zone 2
B
1
ωm in pu
M +1
-1
1
fc=9fr, fr=49.1Hz, mi=1.25
M +1
ref
ωm in pu
1 T
0
0
Isq in A
fc=9fr
4 M
M
2
+1
+1 B -1
B -1
(e)
0
Isd in A
(f)
-2
Figure 13. (a) Experimental results showing gate pulse, motor current and synchronous tcarrier in the (a), (b)linear modulation zone (c), (d) overmodulation zone (e), (f) during carrier transition. Top waveform(T) is the gate pulse of the a-phase inverter leg top switch, Middle waveform(M) is the a-phase current and the Bottom waveform(B) is the synchronous triangular carrier.
2
2.5
3
3.5
4
Time in sec
4.5
5
5.5
6
Figure 15. Simulation result showing the torque, speed and d-q axis current response of the complete scheme for a step speed reference of 1.91pu.
803
Figure 15 shows the speed, torque and d-q axis current response for a speed reference of 1.91p.u. It can be seen that the d-q axis currents do not contain any high frequency components. This is because of the fundamental current extraction algorithm. Figure 16 shows the variation of modulation index and effective inverter frequency during this transient. 1.4
The contributions of the paper are the synchronization strategy and a compatible overmodulation algorithm. Fundamental current extraction algorithm and field weakening strategies, available in the literature, are suitably modified and incorporated to complete the scheme. The synchronization strategy, overmodulation algorithm and fundamental current extraction scheme are experimentally verified. The overall scheme is verified using simulation. The simulation and experimental results are presented.
1.2
REFERENCES
1
Modulation Index Mi
0.8
[1]
0.6
[2]
0.4 0.2
[3]
0 600
Inverter Switching frequency in Hz
500 400
[4]
300
[5]
200
[6] 100
2
2.5
3
3.5
4
4.5
5
5.5
6
Time in sec
Figure 16. Simulation result showing the variation of modulation index and inverter switching frequency for a step speed reference of 1.91pu.
[7]
VII. CONCLUSION This paper presents a rotor flux oriented control scheme for induction motor drive with synchronized sinusoidal PWM.
[8]
804
N. Oikonomou, and J. Holtz, “Closed-Loop Control of MediumVoltage Drives Operated With Synchronous Optimal Pulsewidth Modulation”, IEEE Trans. Ind. Appl., vol. 44, no. 1, pp. 115–123, Jan./Feb. 2008. Giuseppe S. Buja, and Giovanni B. Indri, “Optimal Pulsewidth Modulation for Feeding AC Motors”, IEEE Trans. Ind. Appl., vol. IA13, no. 1, pp. 38–44, Jan./Feb. 1977. G. Narayanan, and V. T. Ranganathan, "Two Novel Synchronized BusClamping PWM Strategies Based on Space Vector Approach for High Power Drives", IEEE Trans. Power. Electron., vol. 17, no. 1, pp. 84–93, Jan. 2002. V. Oleschuk, F. Blaabjerg, and B. K. Bose, "One-Stage and Two-Stage Schemes of High Performance Synchronous PWM with Smooth PulseRatio Changing", IAS Annual Meeting, 2002, vol 3, pp. 1974 - 1981, Oct. 2002. Vikram Kaura, "A New Method to Linearize Any TriangleComparison-Based PWM by Reshaping the Modulation Command", IEEE Trans. Ind. Appl., vol. 33, no. 5, pp. 1254–1259, Sep./Oct. 1997. Ahmet M. Hava, Russel J. Kerkman, and Thomas A. Lipo, "CarrierBased PWM-VSI Overmodulation Strategies: Analysis, Comparison, and Design", IEEE Trans. Power. Electron., vol. 13, no. 4, pp. 674– 689, July 1998. M. Khambadkone, and J. Holtz, “Compensated synchronous PI current controller in overmodulation range and six-step operation of spacevector-modulation-based vector-controlled drives”, IEEE Trans. Indus. Electron., vol. 49, no. 3, 2002 pp. 574-580. S. H. Kim, and S. K. Sul, “Maximum torque control of an induction machine in the field weakening region”, IEEE Trans. Indus. Appli., vol. 31, no. 4, 1995 pp. 787-794.
