CHAPTER 3 Classical Direct Torque Control of an Induction Motor

As stated in Chapter 1, classical direct torque control (DTC) for induction motors was first introduced in 1980’s. Classical direct torque control is a very simple control scheme with low computational requirement. A switching table is adopted to select one of eight basic voltage space vectors determined by the torque and stator flux errors and position of the stator flux vector. This classical direct torque control is a DTC with a Switching-Table (DTC-ST). The torque and stator flux are estimated by a voltage mode estimator in the stationary reference frame. Only stator resistance is involved in the calculation and no axis transformation is required for DTC-ST. In addition, there is no rotor velocity or position sensor required for the torque and stator flux control. This chapter describes the principle of the classical direct torque control scheme for induction motor. This chapter is organized as follows: section 3.1 presents the principle of classical direct torque control. Simulation results of the DTC-ST IM drive are given in section 3.2. Section 3.3 provides the experimental results. The drawbacks of the classical direct torque control have been discussed in section 3.4. 3.1 Classical direct torque control principle

3.1.1. Concept of DTC of an induction motor drive

In a DTC drive, current control schemes are not used, instead the stator flux linkage and electromagnetic torque are controlled directly and independently by the selection of the optimum voltage vectors. The selection rule is made to reduce the stator flux linkage and torque errors within specified hysteresis bands, to achieve the desired torque response under DTC which is faster than that of the FOC. The selection of the voltage space vector can be determined by the position of the stator flux linkage vector and the outputs of the two hysteresis comparators. Furthermore, the DTC algorithm is conducted in the stationary frame. Hence, the need for continuous position information which is necessary for rotor flux oriented control (RFOC) is eliminated. In stationary reference frame, the dynamic behavior of induction motor can be described as following equations:

ss s s

dv R idty

= + (3.1)

0 rs r r r

dR i jdty

w y= + - (3.2)

s s s m r

r m s r

L i L iL i L i

y

y

�๏= + ๏�๏= + ๏� (3.3)

( )3 3 sin ,2 23 sin ,2

m me s r s r s r

s r s r

ms r

s r

L LT P PL L L LLPL L

y y y y r rs s

y y gs

= ด = -

= (3.4)

21

where

2

1 ,m

s r

LL L

s = - (3.5)

where sR and rR are the stator and rotor resistance, ,s rL L and mL are the stator, rotor and mutual inductance, respectively. And rw is the rotor speed, P is the number of pole pairs, sr and rr are the angles of stator and rotor flux vectors, respectively, and g (equal to s rr r- ) is the angle between the stator and rotor flux vectors. The rotor flux vector changes slowly compared to the stator flux vector with a large time constant. So it can be assumed to be constant. The stator flux vector can be changed by applying proper stator voltage. Therefore, the torque can be rapidly changed by varying g in the required direction which is determined by the required torque reference. This is the basic idea of the classical DTC scheme. With voltage source inverter (VSI), the angle g can be easily changed by producing appropriate stator voltage vectors according to (3.1).

3.1.2. Control of the stator flux and torque with a two-level voltage source inverter

For a three-phase motor with balanced sinusoidally distributed stator windings, the stator voltage space vector in a b stationary reference frame can be represented as following

( )22 ,3s a b cv v av a v= + + (3.6)

where 2 /3ja e p= , ,a bv v and cv are the instantaneous values of the phase voltage referred to the ground, as shown in Figure 3.1.

�

�

A B C

aS bS cSdcV

��

Figure 3.1 An idealized 2 level voltage source inverter and three phase induction motor.

Figure 3.1 is an idealized 3 phase voltage source inverter, in which the non-linear effect of the inverter such as the dead-time and the forward voltage drop of the power

22

switch are all neglected. The terminal voltage of A, B and C are controlled by three ideal switches. The turn-on and turn-off times of the switches are neglected for now. If we define the switching function of the three phase terminals as ,a bS S and cS , which can only be either 0 or 1, the instantaneous output voltage vector generated by the ideal inverter can be written in (3.7).

( )223s dc a b cv V S aS a S= + + (3.7)

Considering the different combination of the ,a bS S and cS , there are eight voltage vector, as shown in Figure 3.2. There are six non-zero active voltage vector( )1 6V Vฎ and two zero voltage vectors( )0 7,V V . From equation (3.1), when a voltage space vector is applied to the induction motor. The stator flux linkage sy of an induction motor is related to the stator voltage vector sv by

.ss s s

d v R idty

= - (3.8)

Maintaining sv constant over a sample time interval sTD and neglecting the stator resistance sR , the integration of (3.8) yields

( ) ( ) ( ) ( ) ( ) .s

t

s s s s s s st T

t t t T v t d v t Ty y y t- D

D = - - D = = D๒ (3.9)

Equation (3.9) reveals that the stator flux vector is directly affected by variations on the stator voltage vector. On the contrary, the influence of sv over the rotor flux is filtered by the rotor and stator leakage inductances, and is therefore not relevant over a short time horizon. Since the stator flux can be changed quickly while the rotor flux reacts slower, the angle between both vectors g can be controlled directly by sv . A graphical representation of the stator and rotor flux behavior is illustrated in Figure 3.4. The tip of stator flux linkage vector sy moves at the same direction as the applied voltage vector ( )3V as shown in Figure 3.2. By how far the stator flux linkage vector moves is determined by the duration of this vector and amplitude of the voltage space vector which is proportional to the DC bus voltage of the inverter. The amplitude of the stator flux linkage vector can be regulated by applying a set of voltage space vectors, appropriately modulated. How fast the stator flux linkage rotates is determined by the sequence of the voltage space vectors. Hence, the stator flux linkage vector can be regulated by applying appropriate vectors.

23

�

�

��6 101V��5 001V

��4 011V ��1 100V

��2 110V��3 010V

��0 0007(111)

VV

3Vis applied

��s k�� �1s k� �

Figure 3.2 Eight voltage space vectors generated by a 2-level voltage source inverter.

��

��1�

2�3�

4�

5� 6�

s��

s�

Figure 3.3 Control of the stator flux linkage by applying voltage vectors.

To select the voltage vectors for controlling the amplitude of the stator flux linkage, the voltage vector plane is divided into six sector of 60o each, 1 6q q- , as shown in Figure 3.3. In each sector, only four adjacent voltage vectors are selected to keep the switching frequency to a minimum. Using (3.4), analyzing sector 1, for example, according to Figure 3.4, the voltage vectors may be selected to increase or decrease the amplitude of sy . For instance, voltage vector 2V and 3V are chosen to increase

24

�

�1�

2�

s�

r�

�s�

1V

2V3V

4V

5V 6V

or decrease the amplitude of sy when it is in region 1q and rotates in the clockwise direction. In this way, the amplitude of sy can be regulated in amplitude and direction as required by choosing the proper voltage vectors. Figure 3.3 shows how the voltage vectors are selected for keeping sy within a hysteresis band when sy rotation in the counter clockwise direction is rotating in the counter clockwise direction. The hysteresis band in sy is the difference in radii syD of two circles in Figure 3.3. The effects of the two zero voltage vector, 0V and 7V , are somewhat more complicated. It is seen from (3.9) that sy will stay at its previous position when zero voltage vector are applied. This is true for induction motors since the stator flux linkage is uniquely determined by the stator voltage, where the rotor voltages are always zero.

Figure 3.4 Illustration of voltage selection in sector 1.

3.1.3 Look-up table for DTC induction motor drive

The six voltage space vectors for controlling the toque and stator flux linkage are shown in Table 3.1. It is used for direction of operation of the drive. dy and edT are the outputs of the two hysteresis controllers for stator flux linkage and torque, respectively. 1dy = means the estimated flux linkage is smaller than the reference value, and a flux increasing vector should be chosen and vice versa. It is similar to

edT for torque control. 1 6q q- represent the region number defined in Figure 3.3, which implies the rough position information of stator flux linkage in a resolution of 60o electrical in the a b reference frame.

25

Table 3.1 Look-up table for stator flux linkage and torque control under DTC.

dy edT Sector 1

( )1q Sector 2

( )2q Sector 3

( )3q Sector 4

( )4q Sector 5

( )5q Sector 6

( )6q 1 ( )2 110V ( )3 010V ( )4 011V ( )5 001V ( )6 101V ( )1 100V 0 ( )7 111V ( )0 000V ( )7 111V ( )0 000V ( )7 111V ( )0 000V

1

-1 ( )6 101V ( )1 100V ( )2 110V ( )3 010V ( )4 011V ( )5 001V1 ( )3 010V ( )4 011V ( )5 001V ( )6 101V ( )1 100V ( )2 110V0 ( )0 000V ( )7 111V ( )0 000V ( )7 111V ( )0 000V ( )7 111V

0

-1 ( )5 001V ( )6 101V ( )1 100V ( )2 110V ( )3 010V ( )4 011V The digitized output signals of stator flux hysteresis comparator with two-level are defined as:

,1 s s ref sd ify y y y= ฃ - D (3.10)

,0 s s ref sd ify y y y= ณ + D (3.11) where syD is the error band of the flux comparator. The output of torque hysteresis comparator with three-level as:

,

,

,

1

: 0

1

:

e e e ref e

e e e ref

e e e ref e

dT if T T T

counterclockwise rotation dT if T T

dT if T T T

dT

clockwise rotation

๏ = < - D๏๏๏๏ = = ๏๏๏ = - > + D๏๏

,

,

,

1

0

1

e e e ref e

e e e ref

e e e ref e

if T T T

dT if T T

dT if T T T

๏๏๏๏๏๏๏๏๏๏๏๏ ๏ ๏ = > - D๏ ๏๏ ๏๏ ๏๏ ๏๏ = = ๏ ๏๏ ๏๏ ๏๏ = - < + D๏๏ ๏๏

(3.12)

where eTD is the error band of the torque comparator. 3.2 Modeling studies of the classical DTC induction motor drives

The diagram of the classical DTC drive is shown in Figure 3.5. The hysteresis controllers for torque and stator flux compare the torque and stator flux references with the estimated torque and stator flux, respectively. The comparator output is high (1) or low (0) according to whether the estimated values are lower or higher than their reference values. It may be noted that the switching points are arbitrary. Therefore, the switching frequency of the classical DTC is not fixed. It varies with different

26

bandwidths of the hysteresis controller, load toque and even with rotor speed. A detailed analysis of this is given in later section. The function of the switching logic is to select the appropriate stator voltage vector that satisfies both the torque and flux comparator outputs. The stator flux and torque estimators are very important in the classical DTC drive. The stator flux can be estimated by using an integrator, as shown in Figure 3.5. For a digitized controller, the stator flux estimator can be written as:

( ) ( ) ( ) ( )( )1s s s s s sk k v k R i k Ta a a ay y= - + - D (3.13)

( ) ( ) ( ) ( )( )1s s s s s sk k v k R i k Tb b b by y= - + - D (3.14)

( ) ( ) ( )2 2s s sk k ka by y y= + (3.15)

( )( )( )

1tan ss

s

kk

kb

a

yr

y-

ๆ ๖๗ ๗= ๗ ๗๗ ๘ (3.16)

Figure 3.5 System diagram of the look-up table classical DTC IM drive. The voltage vectors used in (3.13) and (3.14) can be calculated from (3.7). The a b components of the six available non-zero voltage vectors are listed in Table 3.2.

Table 3-2 The a b components of the non-zero voltage vectors.

1V 2V 3V 4V 5V 6V va 2

3 dcV 13 dcV 1

3 dcV- 23 dcV- 1

3 dcV- 13 dcV

vb 0 33 dcV 3

3 dcV 0 3

3 dcV- 33 dcV

s�

*eT

*s�

eT

ˆs�

aS

bS

cS

ai

ci

dcV

dcV

27

The machine parameters used in this thesis is shown in Table 3.3.

Table 3.3 Induction motor parameters.

3-phase 0.37 kW 30sR = W 50Hz 4-poles 31.49rR = W 230/400V 1.8/1.05A 1.0942sL H=

rN = 1360 rpm 1mL H= 1.0942rL H= The simulation results are shown in Figure 3.6 to Figure 3.9. As shown in Figure 3.6, shows the estimated torque, the actual rotor speed and the stator flux linkage for on-load from standstill to 900 rpm. Figure 3.7 shows the torque, stator flux linkage and the stator current waveform. The ripples in the torque and stator flux are found to be 0.5 Nm and 0.02 Wb respectively. The torque ripple is almost 20% of the rated torque, which is quite high. The stator flux ripple is also rather high and may cause extra iron loss in the stator. The stator flux waveform and stator flux trajectory are shown in Figure 3.8. In Figure 3.9 shows the torque transients step load change from no-load to full load, it is seen that the large ripples occur with the rated stator flux. Such high distortion in current is unacceptable for an industrial drive. In order to reduce the torque and stator flux ripple and the current distortion, high sampling frequency of 40 kHz has been used in an industrial drive product based on DTC for induction motor. However, it may be noted that such high sampling frequency was achieved with high speed specially designed IC. A DTC with advantages of low sampling frequency and low ripple violating is preferred. Figure 3.6 Simulation results: Estimated torque, stator flux amplitude and a b axis

stator current waveform at 900 rpm under full load.

eT��Nm

,s si ia b

time [s]

[ ]

ˆ

W b

sy

28

Figure 3.7 Simulation results: a b axis stator current trajectory at 900 rpm under full

load.

Figure 3.8 Simulation results: a b axis and stator flux trajectory at 900 rpm under

full load.

sia

sib

,s sa by y

sby

say

time [s]

29

Figure 3.9 Simulation results: Estimated torque response and stator flux amplitude when step load torque change from no-load to full load (0 to 2.6 Nm).

3.3 Experimental results

In order to verify the modeling results, the experimental work was carried out. The same IM for modeling was tested. The controller and inverter also have the same parameters, such as the DC bus voltage, sampling interval, bandwidth of the torque and stator flux linkage comparators and so on. The experimental setup mainly consists of a dSPACE DS1104 DSP controller board, a Pentium IV 1.5 GHz PC with Windows XP, a four-pole induction motor with parameters are given in Table 3.3 and the three-phase VSI inverter. The dc-link voltage and two line currents of the motor were sensed by isolated devices and fed back to the DSP for the computation of the stator flux and torque. A permanent DC machine was used as the dynamometer to load the IM. The DS1104 board is installed in Pentium IV PC. The control program is written in SIMULINK environment combined with the real-time interface of the DS1104 board. The main ingredient of the software used in the laboratory experiment is based on MATLAB/Simulink programs. The control law is designed in simulink and executed in real time using the dSPACE DS1104 DSP board. Once the controller has been built in Simulink block-set, machine codes are achieved that runs on the DS1104 TMS320F240 DSP processor. While the experimental is running, the dSPACE DS1104 provides a mechanism that allows the user to change controller parameter online. Thus, it is possible for the user to view the real process while the experiment is in progress. A dSPACE connector panel (CP1104) provides easy access to all input and output signals of the DS1104 board. The experimental set up is shown in Figure 3.10.

time [s]

eT��Nm

time [s]

[ ]

ˆ

W b

sy

30

eT��Nm

r�� �rpm

1000+ ฎ

0 ฎ

0 ฎ

3+ ฎ

5+ ฎ

ai cidcV r�

Figure 3.10 Experimental setup for the classical-DTC IM drive.

Experimental results are depicted in Figure 3.11 to Figure 3.17. The no-load starting transient performance is presented in Figure 3.11, the estimated torque and actual rotor speed are shown together when the motor runs at 900 rev/min. The top subplot is the estimated torque and the bottom subplot is the actual rotor speed.

Figure 3.11 Experimental results: Estimated torque and speed when the motor start-up to 900 rpm.

The steady-state operation of the DTC system is presented in Figures 3.12 and 3.13. The ripples of torque and stator flux are 0.7 Nm and 0.03 Wb, respectively, which are quite close to the values predicted in the modeling results in the previous part of this section as shown in Figure 3.7. In Figure 3.13, the estimated torque, stator flux and the stator current trajectory under full load are displayed. There are high ripples in the current waveforms as well, as shown in Figure 3.14. The a axis stator

31

ˆs� ��Wb

eT��Nm

3+ ฎ

0 ฎ

0 ฎ

1+ ฎ

0 ฎ

1+ ฎ

ˆs� ��Wb

eT��Nm

0 ฎ

3+ ฎ

phase current and line-to-line motor voltage waveforms with the inverter switching frequency is less than 5 kHz are shown in Figure 3.15 and the voltage spectrum is shown in Figure 3.16. The stator flux waveform and its circular trajectory, shows in Figure 3.17. In Figure 3.18, the dynamic response of torque is recorded. The regulation time of torque from 0 Nm to 2.6 Nm is very fast. Figure 3.12 Experimental results: Estimated torque and stator flux amplitude in the

no-load steady-state with the speed of 900 rpm. Figure 3.13 Experimental results: Estimated torque, stator flux amplitude at 900 rpm

under full load.

32

Figure 3.14 Experimental results: a b axis and stator current trajectory at 900 rpm under full load.

Figure 3.15 Experimental results: Stator current ( a - axis), line to line stator voltage,

at 900 rpm under full load.

[ ]VL Lv

6 0 0+ ฎ

[ ]Asi a

1 .5+ ฎ

1 .5- ฎ

6 0 0- ฎ

si�

si� 0.5 A/div

,s si ia b

1 .5+ ฎ

1 .5- ฎ

33

Figure 3.16 Experimental results: Spectrum of line to line voltage at 900 rpm under

full load.

Figure 3.17 Experimental results: a b axis and stator flux trajectory at 900 rpm under

full load.

s��

s�� 0.4 Wb/div

,s sa by y

0 .8+ ฎ

0 .8- ฎ

0

50

100

150

200

250

300

350

400

Frequency(Hz)0 1000 2000 3000 4000 5000 6000 7000 8000

34

Figure 3.18 Experimental results: Estimated torque response and stator flux amplitude when step load torque changes from no-load to full load (0 to 2.6 Nm).

3.4 Problems associated with DTC

The main problems with the DTC can be listed as the followings. 3.4.1 Stator resistance variation

In a DTC drive, estimation of torque and stator flux linkage is necessary. Therefore, according to (3.11), any difference between the actual stator resistance( )sR

and the resistance value used in the controller( )cR will give rise to errors in estimated stator flux linkage and torque. The system performance is degraded due to this estimation error. An estimator/observer for the sR is preferred for the DTC drive.

3.4.2 Non-linear effects of the inverter

So far, the non-linear of the inverter such as forward voltage drop, dead-time are neglected. At high speed, the voltage applied to the machine terminal is high. Hence, the voltage distortion caused by these effects is negligible. However, at low speed, the error caused by these effects can be accumulated to a large value and the effects cannot be negligible. They can cause significant errors in the estimated stator flux and torque and this is not desirable.

3.4.3 High ripples in torque and stator flux linkage, variation of switching frequency

Another drawback of the DTC is its high torque and stator flux ripples, even the controller works at very high sampling frequency. The ripple in torque gives rise to fatigue in the shaft, and also causes unsmooth operation. In order to reduce the torque and stator flux ripple, a higher sampling frequency with pure software should be used.

ˆs� ��Wb

0 ฎ

1+ ฎ

0 ฎ

3+ ฎeT

��Nm

35

The distortion in steady-state current of a DTC drive is also high. Variation of the switching frequency makes the output filter design difficult. In order to solve the problem of high torque ripple, Casadi, et al. have contributed works on torque ripple analysis in [24], torque ripple reduction with space vector modulation and discrete space vector modulation in [12] and speed sensorless drive in [25]. Martins, et al. contributed on torque and stator flux ripple analysis and reduction with multi-level inverter; a DTC switching table suitable for multi-level inverter was discussed in [26], [27]. Lascu, et al. in [21], [28] and Lai, et al. in [29] proposed an improved DTC to reduce torque and stator flux ripples by using space vector modulation, respectively. Noguchi, et al. proposed a dithering signal injection scheme to reduce the torque and stator flux ripple in [30], the switching frequency was increased with the proposed scheme. For zero speed operation has been achieved under a modified DTC scheme [21], [29]. In chapter 4, a modified DTC scheme is proposed and tested with a view to reduce the torque and stator flux ripples and flux ripples, and to make the switching frequency constant in a DTC drive.

3.4.4 Offset error in the current and voltage sensor

Both voltage and current sensors may suffer from offset errors. A very small offset in these signals makes the output of an integrator go to infinity in the steady-state, theoretically. However, if the integrator can be replaced by a low pass filter or filters which have similar property of a low pass filter, the DC component can be removed as reported in [31] by Hu, et al. 3.5 Conclusion

In this chapter, the application of the direct torque control has been described. The DTC is based on the fact that the increase of electromagnetic torque in an induction motor is proportional to the increase of the angle between the stator and rotor flux linkages and therefore fast torque response can be obtained by increasing the rotating speed of the stator flux linkage as fast as possible. The necessary conditions for successful application of the DTC to the IM have been discussed. The simulation and experimental results verify the DTC scheme. Problems associated with DTC scheme and their compensation techniques are discussed. One of improved DTC scheme with direct stator flux vector control is presented in the next chapter.