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
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
Sensorless control
1 Motivations
2 Simple single coil AC motor model
3 Three Phase Induction Motor Model and Simulation
4 Sensorless control
Mathematical assessment of PWM control of a induction motor 2 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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.
Mathematical assessment of PWM control of a induction motor 3 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
Sensorless control
Basic scheme of field orientated control for 3-phase AC-motors
Mathematical assessment of PWM control of a induction motor 4 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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)
Mathematical assessment of PWM control of a induction motor 5 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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)
Mathematical assessment of PWM control of a induction motor 6 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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)
Mathematical assessment of PWM control of a induction motor 7 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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.
Mathematical assessment of PWM control of a induction motor 8 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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.
Mathematical assessment of PWM control of a induction motor 9 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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 .
Mathematical assessment of PWM control of a induction motor 10 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
Sensorless control
Flux-current relationships are
The voltage equations are The balance of the torques is
Mathematical assessment of PWM control of a induction motor 11 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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)
Mathematical assessment of PWM control of a induction motor 12 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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
Mathematical assessment of PWM control of a induction motor 13 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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
Mathematical assessment of PWM control of a induction motor 14 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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
Mathematical assessment of PWM control of a induction motor 15 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
Sensorless control
Position dependent mutual inductance
Single applied stator voltage Induced magnetic field
ms1(�) =∫ �+�/2�−�/2 lBa(�) d�
Mathematical assessment of PWM control of a induction motor 16 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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
Mathematical assessment of PWM control of a induction motor 17 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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
Mathematical assessment of PWM control of a induction motor 18 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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
Mathematical assessment of PWM control of a induction motor 19 / 21
OutlineMotivations
Simple single coil AC motor modelThree Phase Induction Motor Model and Simulation
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.
Mathematical assessment of PWM control of a induction motor 20 / 21
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