EE595S: Class Lecture NotesChapter 2* / QD Models for Permanent

Magnet Synchronous Machines

S.D. Sudhoff

Fall 2005

*Analysis and Design of Permanent Magnet Synchronous Machines

S.D. Sudhoff, S.P. Pekarek B. Fahimi

Fall 2005 EE595S Electric Drive Systems 2

2.7 Extended Bandwidth QD Model

• Objectives

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2.7 Extended Bandwidth QD Model

• ABC Variable Model

pmabczabcsabc ,,, vvv += (2.7-1)

sabcsabczabc p ,,, )( iZv = (2.7-2)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

)()()()()()()()()(

)(,pZpZpZpZpZpZpZpZpZ

p

ssmm

mssm

mmss

sabcZ (2.7-3)

Fall 2005 EE595S Electric Drive Systems 4

2.7 Extended Bandwidth QD Model

• ABC Variable Model (Cont)

pmabcpmabc p ,, λv = (2.7-4)

(2.7-6)

(2.7-5)[ ]Trrrmpmabc )3/2sin()3/2sin()sin(, πθπθθλλ +−=

[ ] abcsrrrmePT i)3/2cos()3/2cos()cos(2

πθπθθλ +−=

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2.7 Extended Bandwidth QD Model

• Plan of Attack

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2.7 Extended Bandwidth QD Model

• Transformation of Voltage Equations

pmqds

zqds

sqd ,0,00 vvv +=

ssqdsqd

szqd ,0,0,0 iΖv =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

)(000)(000)(

)(

0

,0pZ

pZpZ

p s

s

sqdZ

)()()( pZpZpZ msss −=

)(2)()(0 pZpZpZ mss +=

(2.7-8)(2.7-7)

(2.7-9)

(2.7-10)

(2.7-11)

Fall 2005 EE595S Electric Drive Systems 7

Extended Bandwidth QD Model

• Transformation of PM Induced Voltages

pmqds

pmqd p ,0,0 λv =

[ ]Trrms

pmqd 0)cos()sin(,0 θθλ=λ

[ ]Trrmrs

pmqd 0)sin()cos(,0 θθλω −=v

spmqd

ssqds

ssqd pZ ,,, )( viv +=

(2.7-12)

(2.7-15)

(2.7-13)

(2.7-14)

Fall 2005 EE595S Electric Drive Systems 8

2.7 Extended Bandwidth QD Model

• Transformation of Torque Equation

( )rsdsr

sqsme iiPT θθλ sincos

223

−= (2.7-16)

Fall 2005 EE595S Electric Drive Systems 9

2.7 Extended Bandwidth QD Model

• Simulation Using Extended Bandwidth QD Model

Fall 2005 EE595S Electric Drive Systems 10

2.7 Extended Bandwidth QD Model

• Simulation Using Extended Bandwidth QD Model

Fall 2005 EE595S Electric Drive Systems 11

2.7 Extended Bandwidth QD Model

• Simulation Summary

rabcutrss

sqd ,vKv =

)cos( rmrsqs

sqs vu θλω−=

)sin( rmrsds

sds vu θλω+=

sqss

sqs upYi )(=

sqss

sqs upYi )(=

(2.7-17)

(2.7-20)

(2.7-21)

(2.7-18)

(2.7-19)

Fall 2005 EE595S Electric Drive Systems 12

2.7 Extended Bandwidth QD Model

• Final Note: Representing Transfer Function (Special Case – All Real Poles)

∑= +

=J

j j

js p

apY

1 1)(

τ (2.7-23)

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2.7 Extended Bandwidth QD Model

• Continuing…

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2.7 Extended Bandwidth QD Model

• Thus, for the all real pole case

jjwwjw xupx τ/)( ,, −=

∑=

=J

jjwj

sws xai

1,

(2.7-24)

(2.7-25)

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2.8 Parameter Identification of Extended Bandwidth QD Model

• Procedure for find P and λm

• Procedure for finding aj and τj

TJJaaa ][ 2121 τττθ =

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2.8 Parameter Identification of Extended Bandwidth QD Model

• Denote i’th frequency fi (radian frequency ωi)

• Measure impedance at i’th frequency Zm,i

• The admittance at the i’th frequency is Ys,i

imis Z

Y,

,1

23

=

Fall 2005 EE595S Electric Drive Systems 17

2.8 Parameter Identification of Extended Bandwidth QD Model

• Obtaining the parameters. Define the error at a given frequency error as

• Define the aggregate erroris

iscsic Y

YjYe

,

,,

),( −=

ωθ

( ) ∑=

=I

iicIc eE

1,

1θ

Fall 2005 EE595S Electric Drive Systems 18

2.8 Parameter Identification of Extended Bandwidth QD Model

• To solve for the parameters, let us maximize

)(1)(

cc E

Fθε

θ+

=

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2.8 Parameter Identification of Extended Bandwidth QD Model

• 3rd Order Fit

101

102

103

104

105

-80

-60

-40

-20

0m

agni

tude

, dB

101

102

103

104

105

-80

-70

-60

-50

-40

-30

phas

e, d

egre

es

frequency, Hz

Fall 2005 EE595S Electric Drive Systems 20

2.8 Parameter Identification of Extended Bandwidth QD Model

Table 2.8-1. Admittance Parameters with 3rd Order Transfer Function

1−Ωm ms

1a 4.14 1τ 0.134

2a 411 2τ 5.54

3a 0.394 3τ 0.00508

Fall 2005 EE595S Electric Drive Systems 21

2.8 Parameter Identification of Extended Bandwidth QD Model

• 6th Order Fit

101 102 103 104 105-80

-60

-40

-20

0m

agni

tude

, dB

101 102 103 104 105-80

-70

-60

-50

-40

-30

phas

e, d

egre

es

frequency, Hz

Fall 2005 EE595S Electric Drive Systems 22

2.8 Parameter Identification of Extended Bandwidth QD Model

Table 2.8-2. Admittance Parameters with 6th Order Transfer Function 1−Ωm ms

1a 1.10 1τ 0.0720

2a 0.203 2τ 0.0000198

3a 75.9 3τ 23.5

4a 383 4τ 5.58

5a 6.11 5τ 0.410

6a 0.476 6τ 0.0202