Int. J. Elect. Enging. Educ., Vol. 36, pp. 163–172. Manchester U.P., 1999. Printed in Great Britain
MODELLING AND SIMULATION OF THE THREE-PHASE INDUCTION MOTOR USING SIMULINK
K. L . SHI, T . F. CHAN CHAN,, Y. K. WONG and S. L . HO Department of Electrical Engineering, Hong Kong Polytechnic University, Hong Kong generalized model of the three-phase induction induction motor and its ABSTRACT This paper describes a generalized computer simulation using MATLAB MATLAB/SIMULINK. Constructional details of various sub-models for the induction motor are given and their implementation in SIMULINK is outlined. Direct-online starting of a 7.5-kW induction induction motor is studied using the simulation model developed. KEYWORDS
MATLAB; modelling; simulation; SIMULINK; three-phase induction motor
LIST OF SYMBOLS L stator inductance s L mutual inductance m L rotor inductance r R stator resistance s R rotor resistance r R cable resistance c rotor speed v 0 P pole number V , V d-axis d-axis and q-axis components of the stator voltage vector V ds qs s V , V d-axis and q-axis components of the rotor voltage vector V dr qr r i ,i d-axis d-axis and q-axis components of the stator current vectors i ds qs s i ,i d-axis d-axis and q-axis components of the rotor current vectors i dr qr r J moment of inertia of rotor J moment of inertia of load L
1 INTRO INTRODU DUCT CTIO ION N Simula Simulatio tion n of the threethree-pha phase se induct induction ion machin machinee is well well docume documente nted d in the literat literature ure and a digita digitall comput computer er solutio solution n can be perfor performed med using using variou variouss methods, such as numeric programming, symbolic programming and the electromagnetic tromagnetic transient program ( EMTP)1,2. With the rapid rapid develo developme pment nt in computer hardware and software, new simulation packages which are faster and more user friendly are now available. This paper discusses the use of one such product, the SIMULINK software of MATLAB, in the dynamic modelling of the induction motor. The main advantage of SIMULINK over other programming softwares is that, instead of compilation of program code, the simu163
164
lation lation model model is built built up system systemati atical cally ly by means means of basic basic functi function on blocks blocks.. Through a convenient graphical user interface (GUI), the function blocks can be created, linked and edited easily using menu commands, the keyboard and an appropriate pointing device (such as the mouse). A set of machine di ff erential erential equations equations can thus be modelled modelled by interconnec interconnection tion of appropriat appropriatee function function bloc blocks ks,, each each of whic which h perfo perform rmin ing g a spec specifi ificc ma math them emat atica icall oper operat atio ion. n. Programming Programming eff orts orts are drastic drastically ally reduced reduced and the debugg debugging ing of errors errors is easy. easy. Since Since SIMULI SIMULINK NK is a model model operat operation ion progra programme mmer, r, the simulat simulation ion model model can be easily easily develop developed ed by additio addition n of new sub-mode sub-models ls to cater cater for various control functions. As a sub-model the induction motor could be incorporated in a complete electric motor drive system 3–5. 2
INDUCTION INDUCTION MOTOR MODEL CONSTRUCTED CONSTRUCTED USING SIMULINK A generalized dynamic model of the induction motor consists of an electrical sub-model to implement the three-phase to two-axis (3 /2) transformation of stator stator voltag voltagee and curren currentt calcul calculatio ation, n, a torque torque sub-mo sub-model del to calcul calculate ate the developed developed electromagnetic electromagnetic torque, and a mechanical mechanical sub-model sub-model to yield the rotor rotor speed. speed. In additio addition, n, a stator stator curren currentt output output sub-mode sub-modell is needed needed for calculating the voltage drop across the supply cables. 2.1 Electrical sub-model of the induction motor The three-phase to two-axis voltage transformation is achieved using the following equation 6: 1
−1/2
0
3/2
C D C V
ds
V
qs
=
−1/2 −3/2
D
agggbgggc
CD V
as
V
bs
V
cs
[A]
FIG. FIG. 1
Electrical Electrical model of an induction induction motor in SIMUL SIMUL INK. INK.
( 1)
165
where V , V , and V are the three-phase three-phase stator voltages, while V and V are as bs cs ds qs the two-axis components of the stator voltage vector V . s In the two-axis stator reference frame, the current equation of an induction motor can be written as 5,6:
C D GC i i i i
ds qs
=
dr
P t
0
t=
qr
L
0
L
0
0
L
0
L
L
0
L
0
0
L
0
L
s
m
m
s
m
r
m
r
D
1
−
aggbggc
[B]
AC D C
R
0
0
0
0
R
0
0
s
V
ds
×
V
qs
V
−
0
dr
V
qr
P
−
P 2
2
v L 0
m
s
v L 0
0
P
R
m
r
−
P 2
2
v L 0
r
v L 0
R
r
D C DBH i
r
i i i
ds
qs
dt
dr qr
aggggggggbggggggggc
( 2)
[C] [C ] As show shown n in Fig Fig. 1, Matr Matrix ix [A] [A] in Equa Equati tio on (1) and and ma matr trix ix [B] [B] in Equation (2) can be implemented by the ‘Matrix Gain’ block of SIMULINK 7, while matrix [C] in Equation (2) can be implemented by four ‘Fcn’ blocks of SIMULINK whose detail is illustrated in Fig. 2. In the electrical model, the three-phase voltage [V , V , V ] is the input and as bs cs the current vector [i , i , i , i ] is the output vector. The rotor voltage vector ds
FIG. FIG. 2
qs
dr
qr
Matrix [C [C] implemented implemented using four Fcn blocks of SIMU SIMULL INK. INK.
166
is normally zero because of the short-circuited cage rotor winding, i.e. V = 0 dr and V = 0. qr
2.2 T orque sub-model of induction motor In the two-axis stator reference frame, the electromagnetic T is given by6: T =
PL 3
m
(i i
dr qs
− iqr ids )
( 3)
Fig. 3 shows how the torque sub-model is realized in SIMULINK. 2.3 Mechanical sub-model of induction motor From the torque balance equations and neglecting viscous friction, the rotor speed v may be obtained as follows 8: 0
= 0
v
P
T − T
t
0
t=
J
L
( 4)
dt
where J is the moment of inertia of the rotor and load and T is the load torque. L Fig. 4 shows the implementation of the mechanical sub-model.
FIG. FIG. 3
FIG. FIG. 4
T orq orque ue sub sub-mo -model del..
Mechanical echanical sub-mo sub-model. del.
167
2.4 Stator current output sub-model The stator current output sub-model is used to calculate the stator current amplitude according to the following equation 6: 2 e )2 + (i e )2 | is | = √(i ds qs
( 5)
3
A SIMULINK ‘Fcn’ block is used to implement the above equation. The The elec electri trica call subsub-mo mode dell in Fig. Fig. 1, the the torq torque ue subsub-mo mode dell in Fig Fig.. 3, the the mechanical sub-model in Fig. 4, and the stator current output sub-model are grouped together to form the induction motor model as shown in Fig. 5. 3 SIMULATION SIMULATION SYSTEM OF INDUCTIO INDUCTION N MOTOR The complete simulation system of the induction motor includes the induction induction motor model in Fig. 5 and a power supply sub-model. 3.1 Power supply sub-model The voltage supply block consists of a three-phase sinusoidal voltage generator and a terminal-voltage calculation block which accounts for the voltage drop in the supply cable. The three-phase sinusoidal voltage generator is based on Equation (6) and one of the three phase voltages is modelled as shown in Fig. 6.
q
= |V | cos (vt + h ) V = |V | cos (vt − 2p/3 + h ) bs V = |V | cos( cos (vt + 2p/3 + h ) cs where | V | is the amplitude of the terminal voltage, V
as
( 6) v is the supply frequency,
and h is the initial phase angle. Due to the voltage drop in the supply cable, the terminal voltage is given by Equation (7):
| V | = E − Rc | is |
FIG. FIG. 5
( 7)
Induction nduction motor model in SIMUL SIMUL INK. INK.
168
FIG. FIG. 6
Modelling odelling one supply supply phase in SIMUL SIMUL INK. INK.
where E is the supply voltage and R is the cable resistance. Fig. 7 shows how c the equation is modelled in SIMULINK. Grouping the voltage generator block of Fig. 6 and terminal-voltage calculation block of Fig. 7, the power supply block is formed as shown in Fig. 8. 3.2 Simulation model of the induction motor The induction motor model in Fig. 5 and the power supply sub-model in Fig. 8 are grouped together to form the complete induction motor simulation model as show shown n in Fig. Fig. 9. The The XY-g XY-gra raph ph bloc block k7 is used used to disp displa lay y the the dyna dynamic mic torque/speed speed charac character teristi isticc of the induct induction ion motor, motor, while while the scope scope block block enables the speed, stator current, and stator voltage of the motor to be observed.
FIG. FIG. 7
T erminal-vol erminal-voltage tage calculati calculation on block. block.
FIG. FIG. 8
Power Power suppl supply y block. block.
169
FIG. FIG. 9
Simulation Simulation system system of an induction induction motor in SIMU SIMULL INK. INK.
4 SIMULATION SIMULATION RESULTS RESULTS The induct induction ion motor motor chosen chosen for the simulat simulation ion studie studiess has the followi following ng parameters: Type: Type: three-p three-phas hase, e, 7.5 kW, 6-pole, 6-pole, wye-co wye-conne nnected cted,, squirre squirrel-c l-cage age induct induction ion motor
= 0.288 V/ph L = 0.0425 V/ph s L = 0.0418 V/ph r J = 0.4 kg m 2 L R
s
= 0.158 V/ph L = 0.0412 V/ph m J = 0.4 kg m2 R
r
To illustrate the transient operation of the induction motor, a simulation study of direct direct-on -on-lin -linee startin starting g is demons demonstra trated ted.. At t = 0, the motor, motor, previo previousl usly y de-ene de-energi rgized zed and at stands standstill till,, is connec connected ted to a 220 V, 60 Hz threethree-pha phase se supply through a cable. The load torque, T , is constant at 20 N.m. Figs. 10 L to 15 show the results of computer simulation using the SIMULINK model. The result resultss are sim similar ilar to those those obtain obtained ed using using the tradit tradition ional al simulat simulation ion method involving di ff erentia erentiall equati equations ons.. It is noticed noticed that when when the supply supply cable has a large resistance, the torque oscillations in the torque /speed charac-
FIG. FIG. 10 10
Simulation Simulation results results with with cable resistan resistance ce R
c
= 0.2 V.
170
FIG. FIG. 11
Simulatio Simulation n results results with cable cable resistan resistance ce R
c
= 0.05 V.
FIG. 12
T orqu orquee/ speed speed characteristic with cable resistance R
FIG. 13
T orqu orquee/ speed speed characteristic with cable resistance R
c
c
= 0.2 V.
= 0.05 V.
teristic are reduced and decay more rapidly, but the run up time of the motor is longer. 5 CONCL CONCLUS USIO ION N SIMULIN SIMULINK K is a powerfu powerfull softwa software re packag packagee for the study of dynami dynamicc and nonlinear systems. Using SIMULINK, the simulation model can be built up
171
FIG. FIG. 14
Stator Stator phase current current with with cable cable resistance resistance R
= 0.2 V.
FIG. FIG. 15
Stator Stator phase current current with with cable cable resistance resistance R
= 0.05 V.
c
c
systematicall systematically y starting starting from simple sub-models. sub-models. The induction induction motor model developed may be used alone, as in the direct-on-line starting example presented, or it can be incorporated in an advanced motor drive system, e.g. fieldoriented control. The authors believe that SIMULINK will soon become an indispensable tool for the teaching and research of electrical machine drives. ACKNOWLEDGEMENT The work reported in this paper was funded by the Hong Kong Polytechnic University University research research grant V157. REFERENCES [1] [2 ] [3 ]
Krau Krause, se, P. C., ‘Simu ‘Simula latio tion n of symm symmet etric rical al induct induction ion mach machine inery’ ry’,, IEEE T rans. rans. Power Power Apparatus Systems, Systems, Vol. PAS-84, No. 11, pp. 1038–1053 (1965) Ghani, S. N., N., ‘Digital computer computer simulation simulation of three-phase three-phase induction induction machine machine dynamics dynamics — a generalized generalized approach’, approach’, IEEE T rans Industry Industry Appl., Appl., Vol. 24, No. 1, pp. 106–114 (1988) Wade, S., Dunnigan, Dunnigan, M. W. and William Williams, s, B. W., ‘Modelin ‘Modeling g and simulat simulation ion of induction induction machine vector control and rotor resistance identification’, IEEE IEEE T rans. Power Electronic Electronics, s, Vol. 12, No. 3, pp. 495–505 (1997)
172 [4 ]
[5 ]
[6] [7 ] [8 ]
Shi, K. L., Chan, T. F. and Wong, Wong, Y. K., ‘Model ‘Modelling ling of the three-pha three-phase se induction induction motor motor using SIMULINK’, SIMULINK’, Record Record of the 1997 IEEE IEEE Internati International onal Electric Machines and Drives Conference, USA, pp. WB3-6 (1997) Shi, K. L., Chan, T. F. and Wong, ong, Y. K., ‘Modell ‘Modelling ing and simulatio simulation n of direct direct self control control system’, IAST Pittsburgh, gh, USA, Conference: Modelling and Simulation Simulation, Pittsbur IAST ED International Conference: pp. 231–235 231–235 (May 1998) Trzyn Trzynadl adlows owski, ki, A. M., T he Field Field Orient Orientati ation on Princi Principle ple in Control Control of Inducti nduction on Motors otors, Kluwer (1994) Mathworkss Inc. (1997) Using SIMUL INK, INK, Dynamic System Simulation for MA MAT T L AB, The Mathwork Krause, P. C., C., Wasynczuk Wasynczuk,, O. O. and and Sudho Sudhoff , S. D., Analysis of Electric Machinery, Machinery, IEEE (1995)
ABSTRACTS – FRENCH, GERMAN, SPANISH Mode´ lisation et simulation d’un moteur triphase´ a` induction induction utilisant utilisant SIMULINK SIMULINK Cet article de´crit ´crit un mode`le `le ge´ne ´ne´ralise ´ralise´ d’un moteur triphase triphase´ a` induction induction et sa simulation simulation informatique utilisant MATLAB/SIMULINK. Les de´ tails constructifs de diff e´ rents sous-mode sous-mode` les du moteur a` induction induction sont donne mentation par SIMULINK SIMULINK est esquisse ´ s et leur imple´ mentation ´ e. Le de´marrage ´ma rrage direct dire ct d’ un moteur mo teur de 7. 5 kW est es t e´tudie ´tu die´ en utilis u tilisant ant l e mode mod e`le `l e de simulation simul ation de´velopp ´v eloppee´. ´. Modellieren Modellieren und Simulieren Simulieren der Drehstromindu Drehstrominduktionsm ktionsmotor otor mit SIMULINK SIMULINK Dieser Beitrag beschreibt beschreibt ein verallgemeinerte verallgemeinertess Modell der Drehstromindukt Drehstrominduktionsmas ionsmaschine chine und ihrer Computersimulie Computersimulierung rung mit MATLAB MATLAB/SIMULINK. SIMULINK. Konstruktive Konstruktive Einzelheiten Einzelheiten verschiedener verschiedener Untermo Untermodell dellee fu Induktio tions nsma masch schine ine werden werden angeg angegebe eben n und ihre ihre Durc Durchfu hfu hrung g in ¨ r die Induk ¨ hrun SIMULINK wird umrissen. Mit dem entwickelten Simulationsmodell wird direktes on-line Starten einer Induktionsmaschine von 7,5 kW studiert. Modelacio´ n y simulacio´ n de un motor de induccio ´ n de tres fases empleando SIMULINK Este artı´culo ´ culo describe describe un modelo modelo generalizad generalizado o de un motor motor de induccio induccio ´n ´ n de tres tres fases fases y su simulacio´ n por computador empleando MATLAB/SIMULINK. Se muestran detalles constructivos de varios submodelos del motor de induccio ´ n y se destaca su implantacio ´ n en SIMULINK. Se estudia estudia la puesta puesta en marcha marcha de un motor motor de induccio induccio ´n ´ n de 7,5 kW emplean empleando do el modelo modelo de simulacio´ n desarrollado.
