The simulation of a multi-phase induction motor drive

Octavian Grigore-Müler, Mihai Barbelian

POLITEHNICA University of Bucharest/Faculty of Aerospace Engineering, Bucharest, Romania octavian_grigore@avionics.pub.ro, barbelian_m@yahoo.com

Abstract-Beginning with the development of the power electronics it was possible to generalize the induction motors, especially those with three phases, both for industry and for domestic appliance. The direct user of the multi-phase induction motor was the electric traction due to its special characteristics: simple and robust in structure, easy to maintain and very reliable. This paper presents the electric drive simulation of a commercial aircraft in the taxi phase using two 18-phase induction motors driven from single-controlled voltage source inverter.

I. INTRODUCTION

It is well known that the base of the electric drive was the DC motor. After the development of the variable speed electric drive systems the DC motors were replaced with three-phase induction motors [1]. These are a superior electric motor, simple and robust in structure, easy to maintain and very reliable; moreover, the supply device for this type of motors - the voltage source inverter - have been already generalized.

To increase the motor’s power per phase and to decrease its weight a multi-phase motors was used. Probably the first application of a multi-phase motor dates back to 1969 [2], a five-phase motor, followed by [3] a nine-phase motor (triple star), that was practically a cascade connection of some three-phase motors. All these papers show the advantages of the multi-phase motor over the correspondent three-phase one, much improved in reliability, and reduction of the power per inverter leg. Other advantages of the multi-phase motors are: the improvement of the noise [4], a possibility of reduction in the copper stator loss leading to an improvement in the efficiency [5] and, also, the improvement of the torque-speed characteristics by increasing the low speed torque more than 5 times than the three-phase induction motors.

Also, the last surveys of the state-of-the art in this area [6], [7], [8], [9] indicate an increasing interest of the scientific world-wide community within this domain.

II. MULTI-PHASE INDUCTION MOTOR MODEL

One of the most popular induction motor model - the dq model -, as shown in Fig. 1, was described in [10]. It has been introduced to facilitate the motor analysis and reduce the computational time. According to this model, a symmetrical induction machine is defined as a polyphase machine with:

i. uniform air gap; ii. linear magnetic circuit;

+ +− − sR rRmsls LLL −= mrlr LLL −=

sqi r

qi

sqv r

qv

e

sqs

q ω

ψ=φ

e

rqr

q ω

ψ=φ

sdωφ ( ) r

dr φω−ω

+ +− − sR rRmsls LLL −= mrlr LLL −= sdi r

di

sdv r

dv

e

sds

d ωψ

=φe

rdr

d ωψ

=φ

sqωφ ( ) r

qr φω−ω

Fig. 1. Dynamic or d-q equivalent circuit of an induction motor

iii. identical stator windings spatially distributed to produce a sinusoidal m.m.f. wave in space and arranged so that only one rotating m.m.f. wave is established by balanced stator currents.

iv. rotor coils or bars arranged so that, for any fixed time, the rotor m.m.f. wave can be considered to be a sinusoidal space with the number of poles equals to the stator m.m.f. wave.

In this case it’s assumed that the motor parameters are measured with respect to the stator windings. With the rotor variables referred to the stator windings and with self inductance separated into leakage inductance component and a magnetization inductance component, the voltage equations become: (1) s

qssd

sq

sq R isv +ωφ+φ=

(2) sds

sq

sd

sd R isv +ωφ+φ=

(3) ( ) rqrr

rd

rq

rq R isv +ω−ωφ+φ=

(4) ( ) rdrr

rq

rd

rd R isv +ω−ωφ−φ=

( mqsq

ls

sq X

1Ψ−Ψ=i ) (5)

( mdsd

ls

sd X

1Ψ−Ψ=i ) (6)

297978-1-4244-7020-4/10/$26.00 '2010 IEEE

2010, 12th International Conference on Optimization of Electrical and Electronic Equipment, OPTIM 2010

( mqr

qlr

rq X

1Ψ−Ψ=i ) (7)

( mdr

dlr

rd X

1Ψ−Ψ=i ) (8)

where: d: direct axis; q: quadrature axis; s: stator variable; r: rotor variable; is the flux linkage (i=q or d and j=s or r) and

;

jiΨ

ji

ji φω=Ψ e

: q and d - axis stator voltages; sd

sq , vv

: q and d - axis rotor voltages; rd

rq , vv

: q and d - axis magnetizing flux linkages; mdmq, ΨΨ

: stator resistance; sR : rotor resistance; rR : stator leakage reactance ; lsX ( )lsLω : rotor leakage reactance ; lrX ( )lrLω

: q and d - axis stator currents; sd

sq , ii

: q and d - axis rotor currents; rd

rq , ii

p: number of poles; n: number of phases; is the angular speed of arbitrary reference frame; ω is the stator angular electrical frequency; eω

is the rotor angular speed; rω

dtds ≡ : Laplace variable.

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ Ψ+

Ψ=+=Ψ

lr

rq

ls

sq

mqrq

sqmmq XX

XX ii (9)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ Ψ+

Ψ=+=Ψ

lr

rd

ls

sd

mdrd

sdmmd XX

XX ii (10)

where:

lrlsm

mdmq

X1

X1

X1

1XX++

==

Eliminating the currents (5) to (8) from (1) to (4) , as well as in (9) and (10), and if the resulting voltage equations are solved for and the following equations are obtained:

rq

sd

sq ,, ΨΨΨ r

dΨ

( ⎥⎦

⎤⎢⎣

⎡Ψ−Ψ+Ψ

ωω

−ω

=Ψ sqmq

ls

ssd

sq

sq X

R

e

e vs

) (11)

( ⎥⎦

⎤⎢⎣

⎡Ψ−Ψ+Ψ

ωω

−ω

=Ψ sdmd

ls

ssq

sd

sd X

R

e

e vs

) (12)

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ−Ψ+Ψ⎟⎟

⎠

⎞⎜⎜⎝

⎛ωω−ω

−ω

=Ψ rqmq

lr

rrd

rrq

rq X

R

e

e vs (13)

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡Ψ−Ψ+Ψ⎟⎟

⎠

⎞⎜⎜⎝

⎛ωω−ω

−ω

=Ψ rdmq

lr

rrq

rrd

rd X

R

e

e vs (14)

The equation of torque is:

( ,12p

2n r

qrd

rd

rq iiT

ee Ψ−Ψ⎟⎟

⎠

⎞⎜⎜⎝

⎛ω

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛= ) (15)

and the equation of the speed is:

( )

,J

p2

1r

e

le TTs

ω⎟⎟⎠

⎞⎜⎜⎝

⎛−

=ω (16)

where: J is the moment of inertia; is the load torque. lT

The equation of position is:

.1rr ω=θ

s (17)

III. REFERENCE - FRAME THEORY FOR A 18-PHASE INDUCTION MOTOR

R. H. Park was the first who introduced a revolutionary approach to electric machine analysis [11]. He defined a change of variables which replaced the variables (voltages, currents, and flux linkages) associated with the stator windings of a synchronous machine with variables associated with fictitious windings rotating with the rotor. So, he referred the stator variables to a frame of reference fixed in the rotor. Park's transformation has the unique property of eliminating all time-varying inductances from the voltage equations of the synchronous machine which occur due to electric circuits in relative motion and electric circuits with varying magnetic reluctance.

The following step was made by H. C. Stanley [12], who followed a change of variables in the analysis of induction machines. He demonstrated that the time-varying inductances in the voltage equations of an induction machine due to electric circuits in relative motion could be eliminated by

298

transforming the variables associated with the rotor windings (rotor variables) to variables associated with fictitious stationary windings. In this case the rotor variables are transformed to a frame reference fixed in the stator.

The next step was made by G. Kron [13], who introduced a change of variables that eliminated the position or time varying mutual inductances of a symmetrical induction machine by transforming both the stator variables and the rotor variables to a reference frame rotating in synchronism with the rotating magnetic field. This reference frame is commonly referred to as the synchronously rotating reference frame.

Finally, D. S. Brereton et al. defined Park's transformation applied to induction machines by employed a change of variables that also eliminated the time-varying inductances of a symmetrical induction machine by transforming the stator variables to a reference frame fixed in the rotor.

Park, Stanley, Kron, and Brereton et al. developed changes of variables, each of which appeared to be uniquely suited for a particular application. Consequently in induction machine analysis, each transformation was derived and treated separately in literature until it was unified in 1965 in one general transformation that eliminates all time-varying inductances by referring the stator and the rotor variables to a frame of reference that may rotate at any angular velocity or remain stationary. Thus all known real transformations may then be obtained by simply assigning the appropriate speed of rotation, which may in fact be zero, to this so-called arbitrary reference frame. It is interesting to note that this transformation is sometimes referred to as the “generalized rotating real transformation”, or ”generalized Park’s transformation” which may be somewhat misleading because the reference frame need not rotate. In any event, we will refer to it as the arbitrary reference frame [14].

The above transformations can be defined with the correlations:

rr

dq ii C= (21)

ss

dq Ψ=Ψ C (22)

rr

dq Ψ=Ψ C (23)

In equations (18) to (23) the matrix C is as follows:

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

α⎟⎠⎞

⎜⎝⎛α⎟

⎠⎞

⎜⎝⎛α⎟

⎠⎞

⎜⎝⎛α⎟

⎠⎞

⎜⎝⎛

α⎟⎠⎞

⎜⎝⎛α⎟

⎠⎞

⎜⎝⎛α⎟

⎠⎞

⎜⎝⎛α⎟

⎠⎞

⎜⎝⎛

αααα

αααα

αααα

αααα

αααα

αααα

=

21

21...

21

21

21

218sin

2182sin...

2182sin

218sin0

218cos

2182cos...

2182cos

218cos1

.........................................................................

3sin6sin...6sin3sin0

3cos6cos...6cos3cos1

2sin4sin...4sin2sin0

cos4cos...4cos2cos1

sin2sin...2sinsin0

cos2cos...2coscos1

182C

(24)

where 18

2π=α is the angle between two consecutive phases

as shown in Fig.2.

ss

dq vv C= (18)

rr

dq vv C= (19)

ss

dq ii C= (20)

IV. SIMULINK IMPLEMETATION OF INDUCTION MOTOR

As shown in Fig. 3 the inputs of a squirrel cage 18-phase induction motor are the 18-phase voltages, their fundamental frequency and the load torque, and the outputs are the 18-phase currents, the electrical torque and the rotor speed.

The d-q model requires that all the 18-phase variables have to be transformed to the two-phase synchronously rotating frame. Thus, the induction motor model will have blocks which transforming the 18-phase voltages to the d-q frame and the d-q currents back to 18-phase.

The 18-phase induction motor model implemented in this paper consists of five important blocks: the Volts per Hertz control block, the 18-phase PWM inverter block, the Park’s transformation block which converts the 18-phase in two-phase variables, the d-q induction model block and the Park’s inversion block.

Fig. 2. Phase space vector of 18-phase inverter in d-q space

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A. The Volts per Hertz control block It is well known that the easiest way to vary the speed of an

induction motor is to use an AC drive to vary the applied frequency. Operating a motor at other than the rated frequency and voltage affect both motor current and torque. As long as a constant volts per hertz ratio is maintained the motor will have constant torque characteristics.

B. The 18-phase PWM inverter block

This block is required to drive the 18-phase induction motor. In fact it is the 18-phase PWM inverter which aims to

supply alternating current of variable voltage and variable frequency to the induction motor, allowing to control of synchronous speed and thus of motor speed. The inverter may also be used with AC induction generators, and can cause an AC induction motor to act as a generator for braking applications.

Th e V o l t z / H er t z B lock Th e 1 8 -ph a se P WM Inv e r t e r B lock The Pa r k’s Tra nsf o r ma t io n Bl oc k

Th

e d

-q I

nd

uct

ion

Mod

el B

lock

T

he

Par

k’s

In

vers

ion

Blo

ck

1 8 P W M I G B T

I n v e r t e r

Fig. 3. The 18-phase induction motor Simulink model

C. The Park’s transformation block The block is used to perform equations (18-19) and (24). Thus, this block converts 18-phase voltages in two-phase

stationary frame.

300

Fig. 4. The electric drive model implementation in Simulink

D. The d-q induction model block It’s solved the equations (5) to (17).

E. The Park’s inversion block This block performs the inversion of transformation made

by the Park’s transformation block to restore the initial variables.

V. SIMULATION RESULTS

In this section are presented the electric drive simulation of a commercial aircraft in the taxi phase using two 18-phase induction motors driven from a single-controlled voltage source inverter [16]. The induction motor will be built into the hub of the main gear wheel and it will give aircraft of all sizes full ground mobility (forward and reverse with steering) without engines operating or tugs (like is shown in Fig. 5).

The data needed for a Boeing B737NG (B737-700) are [17]:

− The weight used: − maxim take-off, W1 = 70.080 kg − maxim landing, W2 = 58.604 kg;

− maximum speed in the taxi flight phase, vmax = 20 mph;

− radius of main gear wheel, r = 0.5525 m;

− rolling resistance coefficient, ===5525.0005.0

r0.005μ

=0.009 (knowing that the main wheel pressure is between 117 and 205 psi [15]);

− gravity acceleration, g = 9.807 2s

m ;

− the gear box ratio, gr = 1/10. With these specifications it can be estimated the following

values: mechanical power (P), mechanical torque (Tl) and the speed of rotations (sr) at the wheel pin and electrical power (Pelmotor), motor torque (Te) and the motor’ speed of rotations (sre) needed by the two electrical motors:

the 2 induction 18-phase motors

rear E&E bay the location of

inverter

cockpit location of the

joystick

APU that provide

electricity supply

Fig. 5. The location of the systems’ components of the commercial aircraft electric drive

301

( )( ) kW55.3094.8)807.970080(0.009vgμW

Nm47.34175525.0)807.970080(0.009rgμW

rpm;5.154π2

605525.094.8

π260

rv

max1

1

max

=×××=×==×××=×=

=×=×=

PT

sr

rpm.1545105.154gr

Nm74.3411013417.47gr

HP20.37kW27.15255.30

2

=×==

=×=×=

====

srsr

TT

PP

e

e

elmotor

Thus, it can be chosen for electrical drive simulation a 60 Hz 18-phase induction motor with the following characteristics:

P = 14920 W (20 HP), sre = 1760 rpm, Vn = 460 V

(nominal voltage), Rs =0.2761 Ω, Ls =0.002191 H, Rr =0.1645 Ω,

Lr=0.002191 H, Lm =0.07614 H (mutual inductance),

Inertia and number of pole pairs: J =0.1 kgm2, p=2. It’s important to note that this calculation (e.g. the load of

the two induction motors) was done on-line by the B 737 Aircraft Ground Dynamic block explained below.

The Simulink electric drive model for a B737 aircraft, as shown in Fig. 4, consist of three main systems:

A. B 737 Aircraft Ground Dynamic

This includes the block which maintain the balance of the aircraft with its total weight, and the block which simulate the dynamic of the aircraft on the ground like a dynamic of a vehicle. B. The Gear Box

Performs the change of rotation movements’ parameters.

C. The model of the induction motors It is described in previous section. To control the aircraft’s speed the Simulink model has a

loop feedback on the rotor angular speed (which in fact is proportional with the source frequency). This control mode is represented in Fig. 6.

The simulation scenario is shown in Fig.7. To move the aircraft forward the pilot must shift both joysticks (left and right) on the same position and to move forward with the steering one of the joysticks must be offset. Thus, in the first case both induction motors are going exactly the same, so it was represented the main characteristics only for one motor.

In the second case it was represented only the

characteristics of the motor which is offset, and is compared with the first case. Also it was changed the load of the motor represented by the weight of aircraft at take-off, and the weight of aircraft at landing.

Fig. 6. The control scheme of the Simulink model

Fig. 7. The scenario for moving the B737 aircraft

a)

b)

Fig. 8. Stator Phase Voltage for a) W1 and sre; b) W1 and sre/2

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Fig. 11. Zoom image A of Stator Phase Voltage for W1 and sre/2

Fig. 9. Zoom image A of Stator Phase Voltage for W1 and sre

Fig. 10. Simulation results for W1 and sre

Fig. 12. Simulation results for W1 and sre/2

303

Fig. 13. Simulation results for W2 and sre

Fig. 14. Simulation results for W2 and sre/2

304

Simulation Tip: Use discrete integral to solve the systems (11) to (14), (15 and (16) instead of the continuous one because sinusoidal device which drive the PWM inverter work in discrete. Also, applied as continuous integral the systems give errors and increase the resources.

Fig. 9 and 11 shows that the control V/Hz which drive the motor through the PWM inverter works well (reducing the motor’ speed of rotations at a half the stator per phase voltage is also reducing to half to maintain constant the ratio V/Hz).

Comparing the representations of the characteristics for the maximum weight, corresponding to the aircraft weight at the take-off W1 (Fig.10) with those at the maximum weight at landing W2 (Fig. 13) it can concluded that the two 18-phase induction motors having each an electrical power of 14.920 kW, will have enough power to drive the B737NG aircraft (note that currents, torques and aircraft speeds are of the same values).

Moreover, from Fig. 13 and Fig. 14 it can be seen that for the same load, W2, but with other commands, sre and sre/2, the aircraft speed decreases with a corresponding decrease of the currents at the same torque of the motor. This is normal when the command was decreased and for the torque, which is equal with the load and the friction, that are the same for the two cases.

Fig. 10 illustrates that the aircraft accelerates and comes to steady state at about 6.5 s with a small slip (the torque increase) because of the inertia load. Also, when the steady state is achieved it can be seen a low frequency oscillation with a period around 1.8 s on the torque that explains the aircraft speed shape due to the friction load torque. This oscillation could not have important effects for various reasons such are: estimating the values of horizontal acceleration, ax and the g factor, n, obtained as it follows:

From Fig. 15 it can be seen that at the beginning v= 33 kph

with a dump of 0.5 kph after 0.45 s (4

s8.1= ):

g.031.0807.9

3.0g

;sm3.0

45.0360010005.0

2 ====×

=ΔΔ

= xx

an

tva

When the steady phase is reached (after approx. 50 s) the aircraft’s speed is 30 kph with a dump of 0.2 kph after 0.45 s:

g.012.0807.9123.0

g;s

m123.045.0360010002.0

2 ====×

=ΔΔ

= xx

an

tva

But for an ordinary car that reaches 100 kph in 7 s:

g,404.0807.996.3

g;

sm96.3

736001000100

carcar2car ====

×=

ΔΔ

=an

tva

acceleration and g factor that are supported very well by human body.

These show that for the B737NG aircraft the horizontal g factor is almost with one order of magnitude smaller or even less than the one for a car. Moreover, these variations, within the limits ± 0.5, ± 0.2 kph (representing in fact only 3% to 1.3% respectively) are not sensitive for the human senses, so it can be concluded that these oscillations might in fact not make passengers air sick, also it is unlikely that they will wear prematurely the equipment.

From the economical point of view and also taking into consideration issues connected to the protection of the environment, a few statistical data extracted from [18], [19] and [20] for an average taxi-out time of approx. 9.03 s pointed these results:

1. if the two induction motors had been supplied with electrical energy from the APU (Auxiliary Power Unit) the fuel consumption of the aircraft and also the noxious fumes will decrease as shown in Fig. 16.

2. if battery supplies will be used instead, knowing that the B737NG has two with a capacity of 36 Ah each and another extra two (for APU and engines) with 7.2 Ah each, an estimation of the electrical power from these is ( ) Wh8.2332V27Ah2.72Ah362 =××+× , this gives the real batteries energy: 80%(2332.8 Wh) =1866.24 Wh

⎪⎩

⎪⎨⎧

W5000 :is systemngconditioniair electrical

W14920:ismotorinduction:fornedeedpower the

Fig. 15. The aircraft speed for W1 and sre

305

Fuel flow (Ff) and emission data taxi-out time (tot) [20]: Ff NOx CO HC (min)

9,03 Nr.crt. Type Manufacturer

(kg/h) (kg/h) (kg/h) (kg/h) 1. Engine TR CFM56-7B24 [18] 392,4 1,73 8,63 0,98 2. APU Allied Signal-131-9(B)[18] 90 0,35 0,25 0,07

Driven scenario Dwell time/part Total dwell time Total Ff Total NOx Total CO Total HCNr.

crt. With Number (min) (min) (kg/h) (kg/h) (kg/h) (kg/h)

1. 1 0 9,03 9,03 59,06 0,26 1,30 0,15 2. Engine TR 1 1 9,03 5 14,03 91,76 0,40 2,02 0,23 3. APU 1 0 9,03 9,03 13,55 0,05 0,04 0,01 4. Batteries and APU 1 1 5,62 3,41 3,41 5,12 0,020 0,01 0,00

Fig. 16. Driven scenario and charts of the fuel flow and noxious fumes

The dwell time is then:

min,62.519920

0624.1866=

×

while the corresponding displacement of the aircraft will be:

km.09.360

62.533=

×

This result means that the four B737NG aircraft batteries might supply only one of the motors in the taxi-out flight phase. So to run as well the second motor there will be needed four more batteries. The remaining 20% of the batteries energy is needed for the starting the APU; also it was included the minimum capacity for batteries discharge. This way of supplying the induction motors is better especially because of the reduced noxious fumes (as represented in Fig. 16) even for the case when the maximum time to dwell the motors is smaller than the average taxi-out time. For this last mentioned situation it is necessary to start APU to supply the motors.

Therefore considering that in the taxi-out flight phase stage for the B737NG it is necessary the APU's function at least for the passenger’s cabin air conditioning systems, for pressurizing the aircraft, and also for the electrical power backup, and to supply the two motors in the final stage of the phase in the second mode, it will be chosen to power the two induction motors from APU.

VI. CONCLUSION

This paper presents a novel concept to drive a commercial aircraft in the taxi phase using a two 18-phase induction motors supplied by the aircraft’s APU instead of his turbojet engines. The concept was verified using a simulation and the results proved very satisfactory. The implications are important due to the reducing of the noxious fumes and noises from airport and around it.

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