Mathematical Assessment of PWM Control of A Induction Motor

OutlineMotivations

Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation

Sensorless control

Mathematical Assessment of PWM Control of AInduction Motor

Z. A. Aziz, C. S. Bohun, K. W. Chung,H. H. Dai, Y. H. Hao, D. Ho, H. X. Huang,

F. F. Wang, J. Wang, Y. B. Wang

December 11, 2009

Mathematical assessment of PWM control of a induction motor 1 / 21

OutlineMotivations


Sensorless control

1 Motivations

2 Simple single coil AC motor model

3 Three Phase Induction Motor Model and Simulation

4 Sensorless control


OutlineMotivations


Sensorless control

Motivations

This project aims to develop PWM based schemes to control singlephase AC induction to operate at different rotational speed withoptimized motor performance in terms of:

High torque output, particularly during start-up;

High energy efficiency;

Shortest spin-up time.


OutlineMotivations


Sensorless control

Basic scheme of field orientated control for 3-phase AC-motors


OutlineMotivations


Sensorless control

Simple single coil AC motor model

Voltage ea(t)

LaCoil

Resistance Ra

Back emf

ia(t)

ea(t) = La ia(t) + Ra ia(t) + eb(t),'

eb(t)

(1)

J ω(t)' = τm(t) - τL, (2)

eb(t) = Kb ω(t), (3)

τm(t) = Ki ia(t). (4)


OutlineMotivations


Sensorless control

Governing equation and general solution

From (1)-(4) and by eliminating !(t), eb(t) and �m(t), we obtain

i′′a(t) +Ra

Lai′a(t) +

KiKb

LaJia(t) =

e′a(t)

La− Kb�L

LaJ= f(t), (5)

The general solution of equation (5) is

ia(t) = C1e�1t + C2e

�2t +e�2t

�2 − �1

∫ t

0

e−�2sf(s)ds− e�1t

�2 − �1

∫ t

0

e−�1sf(s)ds.

(6)

At time t = 0, we have

ia(0) = 0, i′a(0) = C1�1 + C2�2. (7)


OutlineMotivations


Sensorless control

Control objectives

Find the optimized voltage input eoa(t), s.t.,∫ T

0

(ioa(t) − ia)2dt = minea(t)

∫ T

0

(ia(t) − ia)2dt. (8)

The following constraint condition is satisfied∫ T

0

ea(t)ia(t)dt ≤ �0. (9)

We consider the Fourier series expansion form of ea(t):

ea(t) =a02

+

∞∑n=1

[an cos(2�n

tpt) + bn sin(

2�n

tpt)]. (10)


OutlineMotivations


Sensorless control

Simulation results (1)

1 2 3 4 5 6 7t

-15-12.5-10-7.5-5

-2.5

iasHtL

2 4 6 8 10t

-15-12.5-10-7.5-5

-2.5

iaoHtL

2 4 6 8 10t

-0.2-0.1

0.10.2EaoHtL

Fig 2. Simulation results with Ra = 0.1Ω, La = 10−4 H, J = 9 × 10−5

Kg ⋅m2, Ki = Kb = 0.02, �L = 0.3 N⋅m.


OutlineMotivations


Sensorless control

Simulation results (2)

2 4 6 8 10t

-5-4-3-2-1

iasHtL

2 4 6 8 10t

-4-3-2-1

iaoHtL

2 4 6 8 10t

-0.005

0.0050.010.015

EaoHtL

Fig 3. Simulation results with Ra = 0.142Ω, La = 1.1 × 10−3 H,

J = 0.088 Kg ⋅m2, Ki = Kb = 0.628, �L = 3 N⋅m.


OutlineMotivations


Sensorless control

1 Current of stator and rotor in d- and q-coordinate: idS , iqS , idR, iqR;

2 Flux-linkage of stator and rotor in d- and q-coordinate:�dS , �qS , �dR, �qR;

3 Resistance of stator and rotor: rS , rR;

4 Self-inductance of stator and rotor: LS , LR;

5 Magnetizing inductance: M .


OutlineMotivations


Sensorless control

Flux-current relationships are

The voltage equations are The balance of the torques is


OutlineMotivations


Sensorless control

y1 = �dS , y2 = �dR, y3 = �qS , y4 = �qR, y5 = !m. (11)⎧⎨⎩

dy1dt

= −c1y1 − c2y2 + !y3 + vdS ,

dy2dt

= −c3y1 − c4y2 + !y4 − py4y5,

dy3dt

= −!y1 − c1y3 − c2y4 + vqS ,

dy4dt

= −!y2 − c3y3 − c4y4 + py2y5,

dy5dt

= c5y1y4 − c5y2y3 + c6.

(12)


OutlineMotivations


Sensorless control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t

λdS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.03

−0.02

−0.01

0

0.01

0.02

0.03

t

λdR

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t

λqS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.03

−0.02

−0.01

0

0.01

0.02

0.03

t

λqS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−150

−100

−50

0

50

t

ωm


OutlineMotivations


Sensorless control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−150

−100

−50

0

50

100

150

t

i dS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−150

−100

−50

0

50

100

150

t

i dR

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−200

−150

−100

−50

0

50

100

150

t

i qS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−150

−100

−50

0

50

100

150

t

i qR

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

−50

0

50

100

t

Te


OutlineMotivations


Sensorless control

Sensorless control

Main ingredients:

motion of the rotor induces a current back into the voltage ofthe stator

this signal can be recovered by sampling the stator voltage

the phase of this signal depends on the position of the rotor

this signal can be used in place of the position detectorcircuitry that industry would like to remove


OutlineMotivations


Sensorless control

Position dependent mutual inductance

Single applied stator voltage Induced magnetic field

ms1(�) =∫ �+�/2�−�/2 lBa(�) d�


OutlineMotivations


Sensorless control

Induced voltage in the stator

Induced voltage usa = l�adiad�

One rotor bar Two rotor bars

l�a = ls

(1 − m2

a1lsl1

)l�a = ls

(1 − m2

a1+m2a2

lsl1

)l1 = l2, m12 = 0


OutlineMotivations


Sensorless control

Switching of the voltage to the stator

The voltage differences are functions of these position dependentmutual inductances

u� = ua + ub + uc, u(1)� − u(4)� = 2Udl�al�b + l�al�c − 2l�bl�cl�al�b + l�al�c + l�bl�c


OutlineMotivations


Sensorless control

Construction of the position estimate

pa = u(1)� − u(4)� , pb = u(3)� − u(6)� , pc = u(5)� − u(2)�

p� + jp� = pa + e2�j/3pb + e−2�j/3pc


OutlineMotivations


Sensorless control

References

M. W. Degner and R. D. Lorenz, Using Multiple Saliencies for the Estimation ofFlux, Position and Velocity in AC Machines, IEEE Transactions on IndustryApplications, Vol, 34, No. 5, Sept/Oct. 1998, PP. 1097-1104.

J.-I. Ha and S.-K. Sul, Sensorless Field-Oriented Control of an InductionMachine by High-Frequency Signal Injection, IEEE Transactions on IndustryApplications, Vol, 35, No. 1, Jan/Feb. 1999, PP. 45-51.

J. Holtz, Sensorless Control of Induction Motor Drives, Proceedings of theIEEE, Vol. 90, No. 8, Aug. 2002, pp. 1359-1394.

M. Linke, R. Kennel, and J. Holtz, Sensorless Speed and Position Control ofPermanent Magnet Synchronous Machines, IECON, 28th Annual Conf. of theIEEE Industrial Electronics Society, Sevilla/Spain, 2002, on CD ROM.

M. Schroedl, Sensorless Control of AC Machines at Low Speed and Standstillbased on the Inform Method, IEEE Industry Applications Society AnnualMeeting, Pittsburgh, Sept. 30 - Oct. 4, 1996, pp. 270-277.


OutlineMotivations


Sensorless control

Thank you!


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Mathematical Assessment of PWM Control of A Induction Motor

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