Control of induction machine drives in overmodulation range

systems. Thus, the presence of the 5th, 7th, 11th, 13th order harmonics and so on should be 
considered in the output voltage of PWM inverter. The output voltage space vector (us) in the 
stationary reference frame is expressed as [1, 4] 
5 7 11 13
1 5 7 11 13( )
e e e e ej t j t j t j t j tsu u e K e K e K e K e
    − −= + + + + +
 (2) 
where u1 is the fundamental output voltage component and Kj (j = 5, 7, 11, 13, ) is the 
ratio of nth harmonic component and its fundamental. 
When the inverter output voltage is fed to the machine, then the harmonic currents in the 
machine caused by the harmonic voltage components are expressed as 
5 7 11 13
1 5 7 11 13( )
e e e e ej t j t j t j t j ts
i i i ii i e K e K e K e K e
    − −= + + + + +
 (3) 
where is is the machine current vector in the stationary reference frame, i1 is the 
fundamental current component, and Kij (j = 5, 7, 11, 13, ) is the ratio of n
th harmonic current 
components to its fundamental value. 
For vector-controlled machine drives, the machine current controller implemented in the 
rotating reference frame, in which the machine current vector (ie) is expressed as 
6 6 12 12
1 5 7 11 13(1 )
e
e e e e
j te s
j t j t j t j t
i i i i
i i e
i K e K e K e K e

   
−
− −
= 
= + + + + +
 (4) 
Equation (4) presents the DC quantity and multi-sixth order harmonic current 
components of the machine. Figure 5 shows the fast Fourier transform (FFT) analysis of the 
machine current and the inverter output voltage. As can be seen from the figure, the 
fundamental and fifth- and seventh-order components of stator and currents exist in the 
stationary reference frame and the DC component and the multi-sixth order harmonics of the 
stator current appear in the synchronous reference frame. 
Van Tan Luong, Doan Xuan Nam, Le Thanh Toi 
46 
(a) Stator voltage (V)
Fundamental
th
5 th7
(b) Stator current in stationary reference frame (A)
Fundamental
th
5 th7
(c) Stator current in synchronous reference frame (A)
DC
th
6
90
60
30
0
15
12
3
0
9
6
6
5
2
0
4
3
1
Figure 5. FFT analysis of stator voltage and current waveforms. 
4. MODELING AND CONTROL OF INDUCTION MOTOR 
4.1. Modeling of induction motor 
The dq-axis voltage equations in stationary reference frame are written as [10, 11] 
0
0
s s s
ds s ds ds
s s s
qs s qs qs
s s s
r dr dr r qr
s s s
r qr qr r dr
d
V R i
dt
d
V R i
dt
d
R i
dt
d
R i
dt


  
  
= +
= +
= + +
= + +
 (5) 
where , , ,
s s s
ds qs dsV V i and 
s
qsi are the dq-axis stator voltage and current, respectively, Rs, Rr 
are the stator and rotor resistances, respectively, ωr is the rotor speed, , , ,
s s s
ds qs dr   and 
s
qr are 
the dq-axis stator and rotor fluxes, respectively. The stator and rotor fluxes are expressed as 
s s s
ds s ds m dr
s s s
qs s qs m qr
s s s
dr r dr m ds
s s s
qr r qr m qs
L i L i
L i L i
L i L i
L i L i




= +
= +
= +
= +
 (6) 
where Ls=Lls+Lm, Lr=Llr+Lm in which Lls and Llr are the stator and rotor leakage 
inductance, Lm is the magnetizing inductance, 
s
dri and 
s
qri are the dq-axis rotor current 
components, respectively. 
Control of induction machine drives in overmodulation range 
47 
From the rotor flux linkage and the stator current, the machine torque (Te) is expressed as 
( )
3
4
s s s sm
e dr qs qr ds
r
LP
T i i
L
 = − (7) 
where P is the number of poles. 
4.2. Motor control 
By applying the rotor flux-oriented control ( 0qr = ), the rotor flux linkage exists on the 
d-axis only. Hence, the magnitude of the rotor flux ( ,r mag ) is expressed as 
,r mag dr m dsL I = = (8) 
where ,dr qr  and Ids are the dq-axis components of the rotor flux and stator current 
expressed in synchronous rotating reference frame. 
The machine torque is rewritten as 
3
4
m
e dr qs
r
LP
T I
L
= (9) 
where Iqs is the q-axis stator current in synchronous dq reference frame. 
From (8) and (9), the rotor flux linkage and the machine torque can be adjusted by 
controlling the d-axis stator current and the q-axis stator current, respectively. 
As shown Figure 5, the feed-forward terms ( ffdsV ,
ff
qsV ) in synchronous reference frame 
for the decoupling control can be expressed as 
*
*
ff
e s qsds
ff
qs e s ds
V L I
V L I
 

= −
=
 (10) 
where ωe is the synchronous angular frequency and 
2
1
L
m
L L
s r
 = − is the leakage factor. 
In the OVM range, the PI controller can not work well since the multi-sixth order 
harmonics are included in the machine currents which affect the performance of the control 
system considerably. To achieve the fast torque control of the vector-controlled drives, a large 
bandwidth of the current controller is required. In this case, the harmonic components in the 
feedback current will damage the performance of the PI current controller. 
To improve the performance of the current controller in OVM ranges, the multi-sixth 
order harmonics of the current in the synchronous reference frame needs to be removed. To 
do this, a band-pass filter is applied to extract only DC component in the measured current 
before it is fed to the current controller. The reference voltages obtained from the output of the 
current controllers are pre-processed by the static OVM strategy. Thus, the modified reference 
voltages are produced by the PWM inverter. As for the current controller, the controller gains 
(kp, ki) must be properly designed in OVM range to reduce the overshoot of the current 
controller response at the corner of the hexagon. Hence, the kp and ki are selected as 
2
2
cc
p tr
cc
i tr
k L
k R


=
=
 (11) 
Van Tan Luong, Doan Xuan Nam, Le Thanh Toi 
48 
diode rectifier PWM Inverter
IM
ias ibs
+
-
Source
~
+ -
+
+
SVPWM
Encoder
PI
+
- +
+
PI
T-1
+
+
BSF
Pre-
processor
 PWM in OVM+
-
PI
Figure 6. Block diagram of vector-controlled IM drives. 
where trR and trL are the equivalent stator resistance and transient stator inductance, 
respectively. cc is the cut-off frequency. Figure 6 shows the block diagram of the vector-
controlled induction motor drives. 
 5. SIMULATION RESULTS 
PSIM simulation has been carried out for a 3 kW-induction machine drive model under 
full-load condition to verify the effectiveness of the proposed method. The parameters of the 
induction motor are listed in the Table 1. The inverter input voltage is 331V, provided by a 
front-end diode rectifier to produce OVM operation modes. The switching frequency is 5 kHz. 
Table 1. Parameters of induction motor 
Parameter Value 
Rated power 3 kW 
Stator voltage/frequency 220 Vrms/50 Hz 
Stator resistance 0.533 Ω 
Rotor resistance 0.93 Ω 
Stator/rotor inductance 3 mH 
Magnetizing inductance 76 mH 
Number of poles 4 
Generator inertia 0.0033 kg.m2 
Control of induction machine drives in overmodulation range 
49 
(b) d-axis current (A)
(c) q-axis current (A)
Ids* Ids
Iqs*
Iqs
(d) Motor torque (Nm)
Te
vas vbs vcs
(a) Stator voltage (V)
Figure 7. Performance of current controllers without harmonic current elimination (m = 0.95). 
(b) d-axis current (A)
Ids* Ids
(c) q-axis current (A)
Iqs*
Iqs
(d) Motor torque (Nm)
Te
vas vbs vcs
(a) Stator voltage (V)
 Figure 8. Performance of current controllers with harmonic current elimination (m = 0.95). 
Figure 7 and 8 show the comparison of the current control performances between the two 
methods: the conventional method (without harmonic current elimination) and the proposed 
one (with harmonic current elimination) under the full load condition, in which the modulation 
index of the inverter is 0.95 at the operating speed of 2810 rpm. The current controllers with 
harmonic compensation give better performance than those without harmonic compensation. 
As can be clearly seen in Figure 8(b) and (c), the dq-axis currents in the proposed method have 
lower ripples than those in the conventional one (see Figure 7(b) and (c)). Also, the motor 
torque ripple (see Figure 8(d)) in the proposed method is significantly reduced, compared with 
torque in the conventional one, as shown in Figure 7(d). With the conventional method, the 
Van Tan Luong, Doan Xuan Nam, Le Thanh Toi 
50 
total harmonic distortion (THD) factors of the stator voltage in phase A, phase B and phase C 
in Figure 7(a), which are calculated thanks to the available THD function on Simview of PSIM 
software, are 23.905%, 27.502% and 30.833%, respectively. Meanwhile, with the proposed 
method, the THD factors of the stator voltage in phase A, phase B and phase C are 8.125%, 
9.877% and 10.205%, respectively. By comparison, the stator voltage in the proposed method 
is less distorted than that in the conventional one since the THD factors in the proposed method 
are about three times smaller than those in comparison with the conventional one. 
(a) Stator voltage (V) vas vbs vcs
(b) Stator current (A)
ias ibs ics
(c) d-axis stator current (A)
Ids* Ids
(d) q-axis stator current (A)
Iqs*
Iqs
(e) Motor speed (rpm)
r*
r
(f) Motor torque (Nm)
Te
Figure 9. Performance of the motor under overmodulation range (m = 0.986). 
By applying the proposed method, the performance of the machine under OVM range 
(m = 0.986) of the PWM inverter is shown in Figure 9. Due to the presence of harmonic 
components in the stator voltage, as shown in Figure 9(a), the dq-axis stator currents in Figure 
9(b) and (c) are much distorted. Despite this, the operation of the machine can be kept stable. 
During this operation mode, the measured machine speed follows its reference well, in which 
the percentage of the speed error is less than 1%, as shown in Figure 9(d). Figure 9(e) shows 
the machine torque, of which distortion is around 25%. 
Figure 10 shows the performance of the machine in transient state, in which the machine 
speed reference is changed from 2810 rpm at 2 s to 2940 rpm at 2.05 s. The machine still 
operates stably. In this condition, the operation mode of the PWM inverter is changed from 
Control of induction machine drives in overmodulation range 
51 
the OVM mode I (m = 0.95) to mode II (m = 0.989). It is observed in Figure 10(b) that the 
machine currents in OVM mode I are less distorted than those in OVM mode II. The q-axis 
machine current in Figure 10(d) is much increased at the moment of speed reference change. 
As shown in Figure 10(e), the machine speed is controlled to follow its reference well and the 
machine speed in OVM mode II are much more oscillated than that in OVM mode II. The 
machine torque is increased fast to accelerate the machine speed to its command, as shown in 
Figure 10(f). 
(a) Stator voltage (V)
(b) Stator current (A)
(c) d-axis stator current (A)
(d) q-axis stator current (A)
(e) Motor speed (rpm)
(f) Motor torque (Nm)
r* r
Te
Ids*
Ids
Iqs*
Iqs
vas vbs vcs
ias ibs ics
Figure 10. Performance of the motor in transient state. 
6. CONCLUSION 
In this research, a control performance of vector-controlled induction motor drives has 
been investigated in overmodulation range, in which the multi-sixth order harmonic 
components can be removed from the feedback current. For this, the PI current controller gives 
good performance regardless of the existence of the harmonic currents. Also, the speed and 
torque ripples of the machine are much mitigated. The proposed method is more effective for 
the maximum utilization of the machine torque. The validity of the proposed method is verified 
by the PSIM simulation results. 
Acknowledgements: This work was funded by Ho Chi Minh City University of Food Industry 
(Contract number 52 HD/DCT dated September 3, 2019). 
Van Tan Luong, Doan Xuan Nam, Le Thanh Toi 
52 
REFERENCES 
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inverters, IEEE Transactions Power Electronics 13 (6) (1998) 1144-1151. 
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in the overmodulation range including the six-step mode, IEEE Transactions on Power 
Electronics 8 (4) (1993) 546-553. 
3. Bolognani S. and Zigliotto M. - Novel digital continuous control of SVM inverters in 
the ovemodulation range, IEEE Transactions on Industry Applications 33 (2) (1997) 
525-530. 
4. Khambadkone A. M., Holtz J. -Compensated synchronous PI current controller in 
overmodulation range and six-step operation of space-vector modulation-based vector 
controlled drives, IEEE Transactions on Industrial Electronics 49 (3) (2002) 574-579. 
5. Venugopal S. and Narayanan G. - An overmodulation scheme for vector controlled 
induction motor drives, 2006 International Conference on Power Electronic, Drives 
and Energy System (2006) 1-6. 
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modulation and closed loop control of the stator flux vector in overmodulation into 
six-step mode, IEEE Transactions on Power Electronics 19 (3) (2004) 775-782. 
7. Tripathi A., Khambadkone A. M., Sanjib K. Panda - Direct method of overmodulation 
with integrated closed loop stator flux vector control, IEEE Transactions on Power 
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TÓM TẮT 
ĐIỀU KHIỂN ĐỘNG CƠ KHÔNG ĐỒNG BỘ TRONG VÙNG QUÁ ĐIỀU CHẾ 
Văn Tấn Lượng*, Đoàn Xuân Nam, Lê Thành Tới 
Trường Đại học Công nghiệp Thực phẩm TP.HCM 
*Email: luongvt@hufi.edu.vn 
Bài báo đề xuất một chiến lược điều khiển của động cơ không đồng bộ (IM) trong phạm 
vi quá điều chế (OVM) để cực đại việc sử dụng điện áp của bộ nghịch lưu nguồn áp (VSI). 
Trong vùng quá điều chế, việc điều chế độ rộng xung vector không gian (SVPWM) được tạo 
ra bởi điện áp tham chiếu bổ sung, từ đó có thể làm cho dòng điện động cơ bị méo dạng, dẫn 
đến phá hủy vận hành điều khiển bộ VSI. Để cải thiện vận hành điều khiển hệ thống, các thành 
phần họa tần dòng điện cần được loại bỏ trước khi đưa vào bộ điều khiển dòng điện dùng bộ 
tích phân-tỷ lệ (PI). Sự đáp ứng của bộ điều khiển dòng điện trong phạm vi quá điều chế được 
phân tích và cho vận hành tốt. Phương pháp đề xuất được xác minh bằng kết quả mô phỏng 
động cơ không đồng bộ công suất 3kW. 
Từ khóa: Máy điện không đồng bộ, quá điều chế, điều khiển vector.