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Computation of Air-Gap Flux Density of Three-Phase Line Start Permanent Magnet Synchronous Motors Nguyen Vu Thanh", Bui Dinh Tieu, Pham Hung Phi, Nguyen Thanh Son Hanoi University of S

Computation of Air-Gap Flux Density of Three-Phase Line Start

Permanent Magnet Synchronous Motors

Nguyen Vu Thanh", Bui Dinh Tieu, Pham Hung Phi, Nguyen Thanh Son

Hanoi University of Science and Technology

No 1 Dai Co Viet Sir Ha Noi Viet Nam

Received: January 10, 2014; accepted: April 22, 2014

Abstract

When converting the design of phase squirrel cage rotor induction motors into the design of three-changed In this case, the pieces of permanent magnet have been inserted in the squirrel cage rotor Thus, the gap flux is mainly produced by the magnet Hence, it is very important to determine exactly the air-gap flux density This paper presents a novel method for a detailed analysis of the air-air-gap flux density determination related to the steel lamination B-H characteristics Finally, the results are verified by the FEIvl tool m Ansoft Maxwell software

Keywords: LSPMSMs, air-gap dux density, magnetomotive force, size of permanent magnet

1 Introduction

According to [1-5], the design of the Ime start

permanent magnet synchronous motors (LSPMSMs)

IS based on the induction motors (IMs) design

procedures as the initial work In the design process of

IMs, the magnitude of the air-gap flux density

fundamental (Bg) is usually chosen in advance [15,16]

However, in the case of LSPMSMs, choosing the

magnitude is unreasonable It is caused by the

presence of permanent magnets buried in the rotor In

this case, the air-gap flux is mainly produced by the

operatmg point flux of permanent magnet (PM) Thus,

if the Bg is chosen in the LSPMSMs design process,

usmg the FEM method to verify the Bg causes a

significant enor and accuracy of design parameters in

the next steps In references [6-12], the determinahon

of the air-gap flux based on the analysis method often

charactenstics and only takes the PM and au^-gap into

account Thus, analysis results become linear and

inaccurate In this paper, the analysis method is

introduced to determine the magnitude of the

fundamental of the air-gap flux density with non-linear

effect of steel lamination B-H curves in the steady

FEM tool (Maxwell 2D) and CAD tool (RMxprt) of

Ansoft Maxwell software is used due to the advantages

of the software including the high accuracy and

processing speed

2.1 Analysis of permanent magnet fiux density (Bm)

at the operating point

The Bm plays a vital role in analyzing the Bg, The method of Bm determmation is shown in this section

The model of the LSPMSM in this research is

shown in Fig I

2 Computation of the peak value

fundamental of the air-gap flux density

'f the

Fig 1 Mam flux path in LSPMSM

ah IS the length of stator yoke (Isy), ah and be are the height of stator tooth (lu), hg and cd are the height of

is the length of rotor yoke (Iry), length of magnet (Lm), width of magnet (Wm), rip distance (Wf)

* Conesponding Author Tel (+84)3869,2511,

To determine the operating point of permanent

magnet conesponding to the demagnetization

characteristics, the mam flux path is considered in the

motor The flux goes from the north pole of the magnet

to teeth and yoke of stator through an air-gap and

through rotor teeth and yoke and back to the south

pole, via an air gap In this process, the flux crosses the

permanent magnet twice, the stator and rotor tooth

length twice, the air gap twice and the stator and rotor

yoke length once, as shown m Fig 1

In Fig 1, the magnetic circuit is considered

with the permanent magnet source Bg is determined

taking the nonlinear B-H curve of the steel into

account

^ Hdl = 2Hn,Ln, 2F,r 4- 2Fts 4- 2Fg

-I-F.v+F,3, = 0 (I)

After some manipulations, we obtain

„ _ l-pHnit-n, lip A

(2)

where

ge' Effective air gap [7,11]

E - K , g

Kc = 1 - ^ + ^ l n H

Ho' Permeability of air (po- 4Tt,10'' Tm/A)

Tj! Tooth pitch (m)

A = 2H„(B„)l,r + 2Hts(Bts)lts

-I-Hsy(Bsy}lsy + H r y ( B r y ) l r y

where

Hm Operatmg point permanent magnet field

intensity, (A/m)

Hn: Rotor tooth field intensity, (A/m)

His' Stator tooth field mtensity, (A/m)

Hiy: Rotor yoke field intensity, (A/m)

B^: Tooth flux density of stator, (T)

BIT: Tooth flux density of rotor, (T)

Bsy, Yoke flux density of stator, (T)

B,y: Yoke flux density of rotor, (T)

Fu' Mmf of rotor tooth, (A,T)

F^; Mmf of stator tooth, (A.T)

¥ry: Mmfof rotor yoke, (A.T)

Fg Mmf of air-gap (A.T)

On the other hand' (!>„, = Kim'Sg

where

Kin,: Leakage flux factor [17],

-^ Bn,Sn, = Kl^BgSg (•*)

where Sm: Area of permanent magnet Sg' Area of air-gap under pole pitch Substituting equation (2) into equation (4), the permanent magnet flux density of operating point is derived as

lol-mKlmSe , , lioKimSgA

2Sn,ge

In addition, according to [6-12], the demagnetization line charactenstics can be written as follows:

B = Br + poHmH where

fimi' Relative permeability of the magnet

As the operating point of permanent magnet is the intersection of the air-gap Ime with the second quadrant demagnetization curve (demagnetization line), so we can obtain

Substituting equation (6) into equation (5) with some manipulations, we have

airgap line

Operating point (Bm, HM

DemagnetizaJiertiT i ne

B,

B

Fig 2 The operating point of permanent magnet (Bri Remanent flux density, HQ: Intrinsic coercive force) where

(8)

where

T: pole pitch (m)

a,: The flux density shape factor

According to electnc machine design matenals

[12, 15, 16], the factor a, is usually determined by

this factor depends on the saturation level of the steel

lammation and the shape of distribution of air-gap flux

density

From expression (4),

transformation, Bg becomes

after

2.2 Determining the yoke and tooth flux density of

stator and rotor

•;• The loath flux density ofslalor and rotor

Assuming that the air-gap flux under the tooth

pitch of stator (d>g') and rotor (<Sg'") passes through the

stator tooth (fts) and rotor tooth ( * „ ) [12, 15, 16],

In the case of stator: <t>g' — <I>ts

where

Ls: Stack length (m)

kfc- Influence factor of steel lamination

thickness

K: Stator tooth width (m)

Substituting equation (II) into equation (12)

•Jts = ^"l'\ B„ -^ B,s= k„Bm (13)where •.'Kin, I

In the case of rotor '

HrkpeLs btrl<FB

(14)

(15)

Where

bn: Rotor tooth width (m)

Substituting expression (11) into expression (15)

^B,f = k™Bn, (I6)where

i,iK|n,b,

Assummg that the half of air-gap flux under the pole pitch (fl>g) passes through the stator yoke (*I*sy)

In the case of stator: "l^g = <tsy

i e i L s - SyhsyLskpe — - B g T L j

Where Siy: Area of stator yoke hsy: Height of stator yoke Substituting equation (11) into equarton (18)

•'•y',K,„Xi.,.°-^°'>''''^y-°- ('"

where

In the case of rotor: $g = i^ty

BrySry - - B g T L s - * BryhryLskpe = " B g l L s

- ^ B , y h , ^ k p e = i B g t (21)

Where S,j: Area of rotor yoke h,y: Height of rotor yoke Substituting equation (II) into equation (21)

^^y ^ 2K Th k B'"->Bry = k,y„B^ (22)

where

(23)

2.3 Determining the A parameter

In Section A, the parameter A is found, as follows:

A ^ 2HEr(Btr)ltr + 2 H t s ( B t s ) l i s +

H;y(Bsy)lsy-I-Hr.y(Bry)lryFrom Fig 1, It is noted that the relative permeability of permanent magnet is approximately equal to the relative permeahiHty of air-gap Thus, two lengths of permanent magnet are similar to two air-gaps So the length of rotor yoke is only lry-2U>

A -2H„(B„)ltr + 2Ht5(Bt5)i,5 -i- H,y{Bsy)l,y -1-Hry(Bry)0o'-2L^) (24) From Section B, it is easily found that the flux

density of tooth and yoke is a function of the

2.4 Calculating the operating point flux density of

permanent magnet ( B ^

From the Sections A, B and C, the algorithm of

determinmg Bj^ is shown in Fig 12 in which Bn, is the

chosen flux density of permanent magnet, B ^ 'S the

calculated flux density of permanent magnet If the

relative enor between Bn, and B^^ is withm the range

+ 1%, the B ^ value is accepted If the relative enor is

without the range ± I %, B™ value is

re-selected so that Bm value is near the previous value

ofBS

3 Results and conclusions

We investigate three cases for 2.2kW motor In

these cases, the width (Wm) and the length (Lm) of

magnet are varied

> Case 1: L„, - 2 mm , W„, = 54mm

Case 2: L„^3mm,W„ = 51mm

Case 3: L„ = 8 mm; W^ - 40mm

The analytical results for the three cases are

shown in table I

The results companng the analytical method

(AM) to the FEM method (FM) are shown in Table 2

FEM verification ofBm and B\ of2.2kW

Bg is the magnitude of the first fundamental of the

air-gap flux density Bg value is found by Fast Fourier

Transform (FFT) method

"r Case 1: L^^ 2 mm, Wm - 54mm,

shown w Fig 4, Fig 5 and Fig 6

>• Case 2 L„ = 3 mm; W„ = 51mm

shown in Fig 7 Fig S and Fig 9

Case 3: L„, = 8 mm fr„ - 40mm

shown in Fig 10, Fig 11 Fig 12

Table 1 The t 2.2kWmotor lalytical results for the three c

Parameters Kta

K,

gc k,n

km

k , k,,n,

a

b

c B.(T) B,.(T)

B , ( T ) B.,(T) H„ (A/m) H„ (A/m) H,(A/m) H„(A/m) cti B„(T) BS(T) B,(T)

C a s e l 1,529 1,347 0,0006 1,132 0,940 1,617 0,962 6,609 6,395 0.0018 0,836 1,008 0,856 1,439

450

660

490

2450 0,75 0,89 0,891 0,485

Case 2 1,389 1,347 0.0006 1,083 0,871 1,547 0,921 9,794 10,026 0,0019 0,827 1,029 0,875 1,470

550

840

590

4390 0,815 0,95 0,952 0,496

Cases 1,161 1,347 0,0006 0,875 0,726 1,250 0,74t 30,019 33,082 0,0024 0,785 0,945 0,804 1,350

460

700

500

2850 0,94 1,01 1,084 0,4SS

AM

FM

e(%

Table 2 Comparative results for 2 2kW

Bm(T) 0,89

0,92

2 3.47

9

0,95

2 0,98

5 3,46

6

1,08

4 1,06

8 1,47

6

Bs(T) 0,48

5 0,48

8 0,61

8

0,49

6 0,49

4 0,40

3

0,45

5 0,45

2 0,65

9

Fig 3 Flux lines and air-gap flux density for case 1

Fig 4 FFT for ak-gap flux density for case 1, with Bg = 0,488 (T)

Fig 10 FFT for air-gap flux density for case 3, with Bg - 0,452 (T)

-F

Fig, 11 Magnet flux density at the operating pomt for case 3, with Bn, = 1,068 (T)

Finding L™ and Wm of the

magnet, accordmg to a

certain Vm

Calculating the parameters b^, b,r, 1«, I„, g

Calculatmg the parameters

ksm, k™, kysn,, kyn., K t a

Calculatmg d,e and c ii expression (8)

Preset the magnet flux density, B™

i

Determine the flux density

B« B„, Bsy, B^

X

Pick up H from B-H curve

of steel, we get H^, Hlr, Hys, Hy,

Calculating A parameter from above information

X

Calculating B^ from expression (8)

Fig 12 Computmg algorithm ofBj

From the obtained results in the table 2, it is

easy to recognize that the analysis enor of air-gap flux

density between the AM and the FM is less than 1%

when taking steel lamination B-H curves into account

The obtamed results are quite accurate when applying

include'

1) Fmding the some factors (k™, k™, ksym,

kiym) which are used determine the relationship

between B^, BE, B.^, B^y and Bn,

2) Proposing a computing algorithm to

determine the operating pomt flux density of

permanent magnet

Fig 13 Rotor slot

(ho = 0.4 mm,

hi = 1.4 mm,

dl - 3 8 9 mm,

d 2 - 1 9 9 mm,

hT= 12 mm)

Fig 14 Stator slot

(hos = 1 mm,

h l 2 = l 2 4 m m ,

bos = 2.9 mm,

d l - 3 7 1 mm,

d2 - 5.3 mm,

hs= 13.6 mm)

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