Mathematical Assessment of PWM Control of A Induction Motor -

Outline Motivations Simple single coil AC motor model Three Phase Induction Motor Model and Simulation Sensorless control

Mathematical Assessment of PWM Control of A Induction 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

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1

Motivations

2

Simple single coil AC motor model

3

Three Phase Induction Motor Model and Simulation

4

Sensorless control


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Motivations

This project aims to develop PWM based schemes to control single phase AC induction to operate at different rotational speed with optimized motor performance in terms of: High torque output, particularly during start-up; High energy efficiency; Shortest spin-up time.


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


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Simple single coil AC motor model Voltage

Coil

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

(1)

-

(2)

J w' (t) = tm(t)

La

ia(t) Resistance

Ra

Back emf

eb(t)


tL,

eb(t) = Kb w(t),

(3)

tm(t) = Ki ia(t) .

(4)

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Governing equation and general solution From (1)-(4) and by eliminating 𝜔(𝑡), 𝑒𝑏 (𝑡) and 𝜏𝑚 (𝑡), we obtain 𝑖′′𝑎 (𝑡) +

𝐾𝑖 𝐾𝑏 𝑒′ (𝑡) 𝐾𝑏 𝜏𝐿 𝑅𝑎 ′ 𝑖𝑎 (𝑡) + 𝑖𝑎 (𝑡) = 𝑎 − = 𝑓 (𝑡), 𝐿𝑎 𝐿𝑎 𝐽 𝐿𝑎 𝐿𝑎 𝐽

(5)

The general solution of equation (5) is 𝑖𝑎 (𝑡) = 𝐶1 𝑒𝜆1 𝑡 + 𝐶2 𝑒𝜆2 𝑡 +

𝑒𝜆2 𝑡 𝜆2 − 𝜆1

∫

𝑡

0

𝑒−𝜆2 𝑠 𝑓 (𝑠)𝑑𝑠 −

𝑒𝜆1 𝑡 𝜆2 − 𝜆1

∫

𝑡

𝑒−𝜆1 𝑠 𝑓 (𝑠)𝑑𝑠.

0

(6)

At time 𝑡 = 0, we have 𝑖𝑎 (0) = 0,

𝑖′𝑎 (0) = 𝐶1 𝜆1 + 𝐶2 𝜆2 .


(7)

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

𝑇

(𝑖𝑜𝑎 (𝑡)

− ¯𝑖𝑎 )2 𝑑𝑡 = min

𝑒𝑎 (𝑡)

0

∫

𝑇

(𝑖𝑎 (𝑡) − ¯𝑖𝑎 )2 𝑑𝑡.

(8)

0

The following constraint condition is satisfied ∫

𝑇

𝑒𝑎 (𝑡)𝑖𝑎 (𝑡)𝑑𝑡 ≤ 𝛼0 .

(9)

0

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

𝑎0 ∑ 2𝜋𝑛 2𝜋𝑛 𝑒𝑎 (𝑡) = + [𝑎𝑛 cos( 𝑡) + 𝑏𝑛 sin( 𝑡)]. 2 𝑡 𝑡𝑝 𝑝 𝑛=1 Mathematical assessment of PWM control of a induction motor

(10)

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Simulation results (1) is a HtL t -2.5 -5 -7.5 -10 -12.5 -15

1

2

3

4

5

6

7

io a HtL t -2.5 -5 -7.5 -10 -12.5 -15

2

4

6

8

10

Eo a HtL 0.2 0.1 t -0.1 -0.2

2

4

6

8

10

Fig 2. Simulation results with 𝑅𝑎 = 0.1Ω, 𝐿𝑎 = 10−4 H, 𝐽 = 9 × 10−5 Kg ⋅m2 , 𝐾𝑖 = 𝐾𝑏 = 0.02, 𝜏𝐿 = 0.3 N⋅m. Mathematical assessment of PWM control of a induction motor

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Simulation results (2) s ia HtL

t -1 -2 -3 -4 -5

2

4

6

8

10

2

4

6

8

10

io a HtL t -1 -2 -3 -4

Eoa HtL 0.015 0.01 0.005 t -0.005

2

4

6

8

10

Fig 3. Simulation results with 𝑅𝑎 = 0.142Ω, 𝐿𝑎 = 1.1 × 10−3 H, 𝐽 = 0.088 Kg ⋅m2 , 𝐾𝑖 = 𝐾𝑏 = 0.628, 𝜏𝐿 = 3 N⋅m. Mathematical assessment of PWM control of a induction motor

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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: 𝜆𝑑𝑆 , 𝜆𝑞𝑆 , 𝜆𝑑𝑅 , 𝜆𝑞𝑅 ;

3

Resistance of stator and rotor: 𝑟𝑆 , 𝑟𝑅 ;

4

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

5

Magnetizing inductance: 𝑀 .


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Flux-current relationships are

The voltage equations are


The balance of the torques is

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𝑦1 = 𝜆𝑑𝑆 ,

𝑦2 = 𝜆𝑑𝑅 , 𝑦3 = 𝜆𝑞𝑆 , 𝑦4 = 𝜆𝑞𝑅 , 𝑦5 = 𝜔𝑚 . (11) ⎧ 𝑑𝑦1   = −𝑐1 𝑦1 − 𝑐2 𝑦2 + 𝜔𝑦3 + 𝑣𝑑𝑆 ,   𝑑𝑡     𝑑𝑦2   = −𝑐3 𝑦1 − 𝑐4 𝑦2 + 𝜔𝑦4 − 𝑝𝑦4 𝑦5 ,    ⎨ 𝑑𝑡 𝑑𝑦3 (12) = −𝜔𝑦1 − 𝑐1 𝑦3 − 𝑐2 𝑦4 + 𝑣𝑞𝑆 ,  𝑑𝑡    𝑑𝑦4   = −𝜔𝑦2 − 𝑐3 𝑦3 − 𝑐4 𝑦4 + 𝑝𝑦2 𝑦5 ,    𝑑𝑡     ⎩ 𝑑𝑦5 = 𝑐5 𝑦1 𝑦4 − 𝑐5 𝑦2 𝑦3 + 𝑐6 . 𝑑𝑡


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0.3

0.03

0.2

0.02 0.01

0

λdR

λdS

0.1

−0.1

−0.02

−0.3 −0.4

0 −0.01

−0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

−0.03

0.5

0

0.05

0.1

0.15

0.2

t 0.03

0.2

0.02

0.1

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

0.01

0

λqS

λqS

0.3

t

0.3

−0.1

0 −0.01

−0.2

−0.02

−0.3 −0.4

0.25

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

−0.03

0

0.05

0.1

0.15

0.2

0.25

t

t 50

ωm

0

−50

−100

−150

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t


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150

150

100

100

50

50

i dR

i dS


0 −50 −100 −150

0 −50 −100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

−150

0.5

0

0.05

0.1

0.15

0.2

t 150

100

100

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

50

0

i qR

iqS

50

−50

0 −50

−100

−100

−150 −200

0.25

t

150

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

−150

0

0.05

0.1

0.15

0.2

0.25

t

100

Te

50

0

−50

−100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t


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Sensorless control

Main ingredients: motion of the rotor induces a current back into the voltage of the 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 detector circuitry that industry would like to remove


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Position dependent mutual inductance Single applied stator voltage

𝑚𝑠1 (𝛿) =

∫ 𝛿+𝜋/2 𝛿−𝜋/2

Induced magnetic field

𝑙𝐵𝑎 (𝜃) 𝑑𝜃


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Induced voltage in the stator 𝑎 Induced voltage 𝑢𝑠𝑎 = 𝑙𝜎𝑎 𝑑𝑖 𝑑𝜏 One rotor bar

( 𝑙𝜎𝑎 = 𝑙𝑠 1 −

𝑚2𝑎1 𝑙𝑠 𝑙1

)

Two rotor bars

( 𝑙𝜎𝑎 = 𝑙𝑠 1 −

𝑚2𝑎1 +𝑚2𝑎2 𝑙𝑠 𝑙1

)

𝑙1 = 𝑙2 , 𝑚12 = 0 Mathematical assessment of PWM control of a induction motor

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Switching of the voltage to the stator The voltage differences are functions of these position dependent mutual inductances (4) 𝑢𝜎 = 𝑢𝑎 + 𝑢𝑏 + 𝑢𝑐 , 𝑢(1) 𝜎 − 𝑢𝜎 = 2𝑈𝑑


𝑙𝜎𝑎 𝑙𝜎𝑏 + 𝑙𝜎𝑎 𝑙𝜎𝑐 − 2𝑙𝜎𝑏 𝑙𝜎𝑐 𝑙𝜎𝑎 𝑙𝜎𝑏 + 𝑙𝜎𝑎 𝑙𝜎𝑐 + 𝑙𝜎𝑏 𝑙𝜎𝑐

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Construction of the position estimate

(4) 𝑝𝑎 = 𝑢(1) 𝜎 − 𝑢𝜎 ,

(6) 𝑝𝑏 = 𝑢(3) 𝜎 − 𝑢𝜎 ,

(2) 𝑝𝑐 = 𝑢(5) 𝜎 − 𝑢𝜎

𝑝𝛼 + 𝑗𝑝𝛽 = 𝑝𝑎 + 𝑒2𝜋𝑗/3 𝑝𝑏 + 𝑒−2𝜋𝑗/3 𝑝𝑐


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References

M. W. Degner and R. D. Lorenz, Using Multiple Saliencies for the Estimation of Flux, Position and Velocity in AC Machines, IEEE Transactions on Industry Applications, Vol, 34, No. 5, Sept/Oct. 1998, PP. 1097-1104. J.-I. Ha and S.-K. Sul, Sensorless Field-Oriented Control of an Induction Machine by High-Frequency Signal Injection, IEEE Transactions on Industry Applications, Vol, 35, No. 1, Jan/Feb. 1999, PP. 45-51. J. Holtz, Sensorless Control of Induction Motor Drives, Proceedings of the IEEE, Vol. 90, No. 8, Aug. 2002, pp. 1359-1394. M. Linke, R. Kennel, and J. Holtz, Sensorless Speed and Position Control of Permanent Magnet Synchronous Machines, IECON, 28th Annual Conf. of the IEEE Industrial Electronics Society, Sevilla/Spain, 2002, on CD ROM. M. Schroedl, Sensorless Control of AC Machines at Low Speed and Standstill based on the Inform Method, IEEE Industry Applications Society Annual Meeting, Pittsburgh, Sept. 30 - Oct. 4, 1996, pp. 270-277.


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Thank you!


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