Analysis and Prediction of Inverter Ewitching Frequency in Direct Torque Control of Induction...

8/8/2019 Analysis and Prediction of Inverter Ewitching Frequency in Direct Torque Control of Induction Machine Based Opn Hy

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 3, JUNE 2001 545

Analysis and Prediction of Inverter SwitchingFrequency in Direct Torque Control ofInduction Machine Based on Hysteresis

Bands and Machine ParametersJun-Koo Kang, Member, IEEE, and Seung-Ki Sul , Fellow, IEEE

AbstractIn this paper, the influences of the hysteresis bandson the direct torque control (DTC) of an induction motor areanalytically investigated, and the switching frequency of theinverter is predicted based on the analysis. The flux and torquehysteresis bands are the only gains to be adjusted in DTC, andthe inverter switching frequency and the current waveformare greatly influenced by them. Therefore, the magnitude ofthe hysteresis band should be determined based on reasonableguidelines which can avoid excessive inverter switching frequencyand current harmonics in the whole operating region. This paperpredicts the inverter switching frequency according to torque andflux hysteresis bands based on induction machine parameters andcontrol sampling period, and investigates the effect of hysteresisbands to line current harmonics. The simulated and experimentalresults provethe usefulness and feasibility of the proposed method.

Index TermsDirect torque control, distortion factor, hysteresisband, induction motor, total harmonic.

I. INTRODUCTION

S

INCE THE concept of direct torque control (DTC) was de-

veloped in the mid 1980s [1], [2], it has been used in manyac drives applications because it provides a fast torque response

and robustness against machine parameter variations without

speed sensor. DTC is also very simple in its implementation be-

cause it needs only two hysteresis comparators and switching

vector table for both flux and torque control. Therefore, the only

gains to be adjusted are the amplitudes of the hysteresis band.

The amplitude of the hysteresis band greatly influences the drive

performance such as flux and torque ripples, inverter switching

frequency, and current harmonics. Many papers have concerned

the improvement of the performance of DTC [3], [4], however,

there have been few papers about the influences of the hysteresis

band. The effects of the flux and torque hysteresis bands to the

system performance are shown in [5] and [6], but in these pa-pers, there is no theoretical analysis about the variation of in-

verter switching frequency according to the machine parameters

and the operating speed condition.

Manuscript received October 6, 1998; revised February 21, 2001. Abstractpublished on the Internet February 15, 2001.

J.-K.Kang is with Yaskawa Electric Corporation, Kitakyushu-City 803-8530,Japan.

S.-K. Sul is with the School of Electrical Engineering, Seoul National Uni-versity, Seoul 151-742, Korea.

Publisher Item Identifier S 0278-0046(01)03973-9.

In the DTC of induction motors, even though hysteresis bands

are set to be constant, the switching frequency varies according

to the operating conditions such as motor speed, flux level, and

output torque. This makes the hysteresis bands large enough not

to exceed the upper limit of inverter switching frequency that is

predetermined by hardware thermal restriction. Since the hys-

teresis bands are set to cope with the worst case, the systemperformance is degraded, especially in the low-speed region.

In this paper, design procedures of flux and torque hysteresis

bands are suggested for considering switching frequency and

harmonic distortion of currents. To predict the switching fre-

quency, theoretical and experimental investigations have been

performed on the variations of the flux and torque by the se-

lected voltage vector in DTC. In what follows, basic flux and

torque equations of induction motor are briefly discussed first.

Then, the switching frequency of the inverter is obtained with

the hysteresis bands and motor equations. Finally, experimental

results are presented and compared with simulation results.

II. FLUX AND TORQUE EQUATIONS FOR DTC ALGORITHM

An induction motor can be modeled with stator and rotor

fluxes as state variables by the following equation:

V

(1)

where and represent stator and rotor flux vectors, V is

the input voltage vector, and are stator and rotor resis-

tances, , , and are stator, mutual, and rotor inductances

respectively, and is a rotor angular speed expressed in elec-trical radians and leakage factor .

In the DTC algorithm, electromagnetic torque and stator flux

are used as control quantities. For a digital implementation, a

discrete form of torque can be expressed in terms of stator and

rotor flux vectors, or stator flux and current vectors as

Im

Im I (2)

02780046/01$10.00 2001 IEEE
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546 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 3, JUNE 2001

(a)

(b)

Fig. 1. Control sequence of DTC algorithm with switching-voltage space vectors. (a) Control sequence. (b) Switching-voltage space vector.

where is the number of poles, I is a stator current vector,

and denotes the complex conjugate.

A discrete form of stator flux at sampling instant

can be written in terms of applied voltage and current as

V I (3)

where is a sampling period. The realization of the DTC

scheme is outlined in Fig. 1 where and are flux and

torque commands, respectively. Procedures i)viii) are repeated

at every sampling period. At the beginning of each sampling

period, the controller reads in the stator voltage and current.

The flux and torque are then calculated using (2) and (3). Flux

magnitude error and torque error are then calculated,

which are used as inputs of hysteresis comparators. The output

voltage vector is selected based on Table I where sector is

TABLE IVOLTAGE-VECTOR TABLE FOR DTC ALGORITHM

determined according to the location of flux vector . The

switching signals of selected voltage vector are sent to the gate

drive circuit. The typical waveform of by the torque hys-

teresis controller of DTC is illustrated in Fig. 2 where torque


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KANG AND SUL: INVERTER SWITCHING FREQUENCY IN DIRECT TORQUE CONTROL OF INDUCTION MACHINE 547

Fig. 2. Typical waveform of the output torque in the DTC of induction machine.

rising time and falling time are integer multiples of ,

and is a torque hysteresis band. In the steady state, torqueincreases with a slope by nonzero voltage vectors and de-

creases with a slope by zero voltage vectors.

If is small enough to assume and are

constant during sampling time, stator and rotor fluxes at sam-

pling instant are given by following equations [10]:

V (4)

(5)

By substituting (4) and (5) into (2), and neglecting the square

of , the torque variation during by nonzero vector

V can be given by

Im V

(6)

Similarly, torque variation by zero vector can be written

as

Im (7)

From (3), the flux variation during at the th

sampling instant can be expressed as

V I (8)

III. TORQUE AND SWITCHING FREQUENCY

A. Instantaneous Torque Slope Equations

The switching frequency caused by the torque hysteresis con-

troller of DTC is closely related to the parameters and the op-

(a) (b)

(c)

Fig. 3. A relativemagnitude ofthreecomponents whichconsistof torqueslopeh . (a) h . (b) h . (c) h .

erating condition of an induction motor. Positive and negativeslopes of the torque by an applied voltage vector can be calcu-

lated from (6) and (7), respectively. In (6), ascending slope

can be divided into three parts as

(9)

where

Im V

Im
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(a)

(b)

Fig. 4. Variation of motor torque according to the selected voltage vector insector 1. (a) ~V is selected as an output voltage. (b) ~V is selected as an outputvoltage.

Fig. 5. Torque ripple around the torque command by hysteresis controller.

In Fig. 3, to observe the relative magnitude of , , and ,

150% step torque response of DTC is simulated with a 7.5-kW

induction motor until it reaches to 1200 r/min (simulation pa-

rameters are listed in Appendix). As can be seen from Fig. 3,

(a)

(b)

Fig. 6. Switching frequency of torque hysteresis controller with differenthysteresis band . (a) 7.5 kW. (b) 240 kW.

is negligibly small compared to and . Thus, the instanta-

neous torque slope equation can be approximated as

(10)

In the same manner, the descending slope in (7) is

(11)

B. Average Torque Slope Equations

In order to get an average switching frequency caused by the

torque controller, an average torque slope should be calculated.

To increase the torque in the sector m, voltage vector V or

V is selected based on the voltage vector table in Table I.

When the stator flux vector is in sector 1, either V or V is

selected regarding the sign of flux error. First, suppose that V


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TABLE IISPECIFICATIONS OF INDUCTION MACHINE AND INVERTER

is selected to increase stator flux as shown in Fig. 4(a). Then,

, the angle between voltage vector V and the stator flux ,

decreases from to as the stator flux rotates. There-fore, torque slope changes according to the decrease of .

The average torque slope in sector 1 can be obtained using (10)

as

VV V

(12)

where and is the angle be-

tween the stator and rotor flux vector. If V is selected to de-

crease stator flux as shown in Fig. 4(b), the average slope in

sector 1 is

VV V

V

(13)

Under the assumption that V and V are equally selected in

sector 1, the resulting average torque slope becomes

V V(14)

The average descending slope can be obtained as

(15)

Since can be approximated as at no-load con-

dition, (14) and (15) yield

V (16)

(a)

(b)

Fig. 7. Change of stator flux by the selected voltage vector in the space-vectorplane. (a) ~V is selected. (b) ~V is selected.

(17)

where and are torque slopes defined at no-load con-

dition and is the amplitude of stator flux. As can be seen


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Fig. 8. Simulated trajectory of stator flux according to various flux hysteresis bands.

from (16) and (17), the torque slopes are the functions of stator

flux, motor speed, and dc-link voltage. In particular, they are

strongly affected by motor speed.

C. Calculation of Switching Frequency Caused by Torque

Hysteresis Band

To calculate , a switching frequency due to the torque hys-

teresis controller, the torque rising time , and the falling timeare divided into several subregions as shown in Fig. 5. Using

no-load torque slopes and , and in Fig. 5 can

be expressed as

V

(18)

(19)

Even though and are slightly different from and

according to load conditions, it is observed that it does not

make a large difference for our application purpose. A control

delay time is an elapsed time from the point at which output

torque exceeds hysteresis band to thepointat which torque slope

is changed by the updated voltage vector. In digital implemen-

tation, a longer sampling period results in a longer delay time,

that means larger overshoot of torque and lower switching fre-

quency. Thus, the sampling period is an important factor deter-

Fig. 9. Number of voltage vector changes versus hysteresis band withdifferent sampling periods.

mining control performance and system switching frequency.

The overshoot interval and can be expressed as

(20)

In the same manner, and can be expressed as

(21)

The resulting switching frequency by the torque hysteresis

controller is given by

(22)


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period affect the switching frequency of the flux hysteresis con-

troller, thevariation of versushysteresis bands with different

sampling periods is shown in Fig. 9. It can be seen from Fig. 9

that, if is larger than 0.04, is not strongly affected by the

, and if is larger than 0.07, is constant as 3. Using

in Fig. 9, the resulting average switching frequency of the flux

hysteresis controller, can be obtained as

sectors arms Hz (25)

where is a frequency of stator flux.

C. Total Inverter Switching Frequency

The resulting inverter frequency can be obtained by com-

bining (22) and (25) as

(26)

Fig. 10 shows the configuration of the proposed block which

predicts inverter switching frequency. In Fig. 10, (26) is used to

predict maximum switching frequency of the inverter in DTC.

If exceeds the maximum switching frequency , the

hysteresis bands and should be redesigned. The design

of hysteresis bands will be discussed in the next section.

V. EXPERIMENTAL VERIFICATION

Experimental tests have been carried out to verify the anal-

ysis on the effect of the flux and torque hysteresis bands to

the switching frequency. The experimental setup consists of a

7.5-kW squirrel-cage induction machine, insulated gate bipolartransistor (IGBT) inverter, and digital-signal-processor (DSP)

(TMS320C40)-based controller. The specifications and param-

eters of the induction machine are listed in Table II.

Fig. 11(a) shows the switching frequency of the torque con-

troller versus motor speed under the no-load steady state

where solid lines indicate simulated results from (22). The mag-

nitude of the normalized torque hysteresis band is varied

from 0.005 to 0.1 with flux hysteresis band . In

Fig. 11(a), the torque hysteresis controller has itspeak switching

frequency at around 1100 r/min. It means that the maximum

switching frequency of the inverter should be designed consid-

ering that region. From Fig. 11(a), it can be seen that the exper-

imental result is consistent with the simulation result.Fig. 11(b) shows the switching frequency of the flux con-

troller versus motor speed under the steady state where solid

lines indicate simulated results from (26). The magnitude of the

flux hysteresis band is varied from 0.003 to 0.09 of the rated

flux for no-load operation with . It can be seen from

Fig. 11(b) that is proportional to motor speed because the

number of selected voltage vector is also increased as the stator

flux rotates. In the case of and , there

is no difference in switching frequency. This is well in accor-

dance with the simulation results in Fig. 8 where the hysteresis

band larger than about does not contribute to reduce

switching frequency of flux. Through the result, it is proved

(a)

(b)

Fig. 11. Switching frequency versus motor speed with different hysteresisbands. (a) Torque hysteresis controller. (b) Flux hysteresis controller.

that (22) can be used for predicting switching frequency ac-

cording to various s.

In determining the amplitude of hysteresis bands, both

switching frequency and total harmonic distortion factor

(THD) should be considered. Fig. 12(a) shows the effects

of flux and torque hysteresis bands on the THD of motor

current. It can be seen that THD is strongly dependent on

and moderately affected by . It is desirable to keep

small and adjust for switching frequency control because

an increase of causes rapid increase of THD without an

effective decrease of the switching frequency. From Fig. 11(b)

and Fig. 12(a), it can be observed that smaller than 0.01


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(a)

(b)

Fig. 12. (a) THD of motor current versus hysteresis bands using direct torquecontrol algorithm. (b) THD of motor current versus switching frequency usingcurrent-controlled space-vector PWM.

increases rapidly, but the THD of current is not lowered

proportionally. Therefore, smaller than 0.01 does not

seem to be effective in improving the performance. However,

it should be noted that a longer sampling period makes the

system sluggish, even with the same hysteresis band, as can

be seen in Fig. 9. Since Fig. 12(a) is obtained with a 50- s

sampling period, THD results can be adapted depending on

sampling period, based on the relationship in Fig. 9. Fig. 12(b)

shows the THD of motor current versus switching frequency

using current-controlled space-vector pulsewidth modulation

(PWM). By comparing Fig. 12(a) and (b), it can be seen that,

in DTC, since THD is strongly affected by , THD can belarge even if switching frequency increases with small ,

whereas in current-controlled space-vector PWM, THD is

simply proportional to switching frequency.

VI. CONCLUSIONS

This paper has presented the results of an investigation into

the relationship between the hysteresis bands of DTC and the

switching frequency of the inverter. The switching frequency

of the inverter is predicted using the proposed average torque

slope equations of induction machines and the predetermined

voltage-vector change versus the flux control hysteresis band.

The results show that the switching frequency of the torque con-

troller has a peak value at medium speed due to the effect of

back EMF, while that of the flux controller is proportional to

operating speed. Analysis results can be a predictive tool for de-

signing the hysteresis bands of the inverter which make it pos-

sible to limit maximum switching frequency of power devices.

The proposed approach is also expected to be applicable to other

ac motors such as permanent-magnet synchronous motors andsynchronous reluctance motors.

REFERENCES

[1] I.Takahashi and T. Noguchi, A newquick-response and high-efficiencycontrol strategy of an induction motor, IEEE Trans. Ind. Applicat., vol.22, pp. 820827, Sept./Oct. 1986.

[2] M. Depenbrock, Direct self-control (DSC) of inverter-fed inductionmachine, IEEE Trans. Power Electron., vol. 3, pp. 420429, July 1988.

[3] D. Casadei, G. Grandi, G. Serra, and A. Tani, Effects of flux andtorque hysteresis band amplitude in direct torque control of inductionmachines, in Proc. IEEE IECON94, 1994, pp. 299304.

[4] T. G. Habetler and F. Profumo, Direct torque control of induction ma-chines using space vector modulation, IEEE Trans. Ind. Applicat., vol.28, pp. 10451052, Sept./Oct. 1992.

[5] T. Noguchi and I. Takahashi, High frequency switching operation ofPWM inverter for direct torque control of induction motor, in Conf.Rec. IEEE-IAS Annu. Meeting, 1997, pp. 775780.

[6] M. P. Kazmierkowski and A. B. Kasprowicz, Improved direct torqueand flux vector control of PWM inverter-fed induction motor drives,

IEEE Trans. Ind. Electron., vol. 42, pp. 344350, Aug. 1995.[7] High frequency switching operation of PWM inverter for direct torque

control of induction motor, in Conf. Rec. IEEE-IAS Annu. Meeting,1997, pp. 775780.

[8] P. Tiitinen,The next motor control method, DTCdirect torque control,in Proc. Int. Conf. Power Electronics, Drives and Energy Systems for

Industrial Growth, 1996, pp. 3743.[9] G. Buja and D. Casadei, DTC-based strategies for induction motor

drives, in Proc. IEEE IECON97, 1997, pp. 15061516.[10] D. Casadei and G. Serra, Analytical investigation of torque and flux

ripplein DTCschemes forinduction motors, in Proc. IEEE IECON97,1997, pp. 552556.

Jun-Koo Kang (S96M01) received the B.S.,M.S., and Ph.D. degrees in electrical engineeringfrom Seoul National University, Seoul, Korea, in1986, 1988, and 1999, respectively.

He was with the Power Electronics Laboratory,R&D Center, LG Industrial Systems Company,as a Research Engineer from 1988 to 1993 and asa Senior Research Engineer from 1993 to 1997.Since 1999, he has been with Yaskawa ElectricCorporation, Kitakyushu-city, Japan, where he iscurrently a Senior Engineer. His research interests

include the analysis and control of power electronics, matrix converters, andhigh-performance ac drives.

Seung-Ki Sul (S78M80SM98F00) was bornin Korea in 1958. He received the B.S., M.S., andPh.D. degrees in electrical engineering from SeoulNational University, Seoul, Korea, in 1980, 1983, and1986, respectively.

He was with the Department of Electrical andComputer Engineering, University of Wisconsin,Madison, as an Associate Researcher from 1986 to1988. He then was with Gold-Star Industrial SystemsCompany as a Principle Research Engineer from1988 to 1991. Since 1991, he has been a member of

the faculty of the School of Electrical Engineering, Seoul National University,where he is a Full Professor. His current research interests are power electroniccontrol of electric machines, electric vehicle drives, and power convertercircuits.

Analysis and Prediction of Inverter Ewitching Frequency in Direct Torque Control of Induction...

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