the induction machine handbook chuong (15)

It has an aluminum cast cage rotor and, in general, random wound stator coils made of round magnetic wire with 1 to 6 elementary conductors diameter ≤ 2.5mm in parallel and 1 to 3 curren

15.1 INTRODUCTION

The power of 100 kW is traditionally considered the border between a small and medium power induction machine In general, sub 100 kW motors use a single stator and rotor stack (no radial cooling channels) and a finned frame washed by air from a ventilator externally mounted at the shaft end (Figure 15.1) It has an aluminum cast cage rotor and, in general, random wound stator coils made of round magnetic wire with 1 to 6 elementary conductors (diameter

≤ 2.5mm) in parallel and 1 to 3 current paths in parallel, depending on the number of pole pairs The number of pole pairs 2p1 = 1, 2, 3, … 6

Figure 15.1 Low power 3 phase IM with cage rotor

Induction motors with power below 100 kW constitute a sizable portion of the world electric motor markets Their design for standard or high efficiency is

a nature mixture of art and science, at least in the preoptimization stage Design optimization will be dealt with separately in a dedicated chapter

For the most part, IM design methodologies are proprietory

Here we present what may constitute a sample of such methodologies For further information, see also [1]

15.2 DESIGN SPECIFICATIONS BY EXAMPLE

Standard design specifications are

• Phase connections: star

• Targeted power factor: cosϕn = 0.83

• Targeted efficiency: ηn = 0.895 (high efficiency motor)

• p.u locked rotor torque: tLR = 1.75

• p.u locked rotor current: iLR = 6

• p.u breakdown torque: tbK = 2.5

• Insulation class: F; temperature rise: class B

• Protection degree: IP55 – IC411

• Service factor load: 1.0

• Environment conditions: standard (no derating)

• Configuration (vertical or horizontal shaft etc.): horizontal shaft

15.3 THE ALGORITHM

The main steps in IM design are shown in Figure 15.2 The design process may start with (1) design specs and assigned values of flux densities and current densities and (2) calculate in the stator bore diameter Dis, stack length, stator slots, and stator outer diameter Dout, after stator and rotor currents are found The rotor slots, back iron height, and cage sizing follows

All dimensions are adjusted in (3) to standardized values (stator outer diameter, stator winding wire gauge, etc.) Then in (4), the actual magnetic and electric loadings (current and flux densities) are verified

If the results on magnetic saturation coefficient (1 + Kst) of stator and rotor tooth are not equal to assigned values, the design restarts (1) with adjusted values of tooth flux densities until sufficient convergence is obtained in 1 + Kst Once this loop is surpassed, stages (5) to (8) are traveled by computing the magnetization current I0 (5); equivalent circuit parameters are calculated in (6), losses, rated slip Sn, and efficiency are determined in (7) and then power factor, locked rotor current and torque, breakdown torque, and temperature rise are assessed in (8)

In (9) all this performance is checked and if found unsatisfactory, the whole process is restarted in (1) with new values of flux densities and/or current densities and stack aspect ratio λ = L/τ (τ – pole pitch)

The decision in (9) may be made based on an optimization method which might result in going back to (1) or directly to (3) when the chosen construction and geometrical data are altered according to an optimization method (deterministic or evolutionary) as shown in Chapter 18

All construction and geometrical data are known and slightly adjusted here

Sizing the electrical &

magnetic circuits

A =I/J

A =toothCoΦ toothCo/Bt

Verification of electric and magnetic loadings:

J =I/A

B = /A t Φtooth toothfCof Cof

Design specs electric &

magnetic loadings:

J , J , B

B , B , λ Cos Cor g

t c start

seeking convergence

in teeth saturation coefficient

1 + Kst1

3

Computation of magnetisation current

Computation of equivalent circuit electric parameters

R , X , R' , X' ,X

6 Co

s sl r rl m

Computation of loss, S (slip), efficiencyn 7

Computation of power factor, starting current and torque, breakdown torque, temperature rise

8

is performance satisfactory? 9NO

YES END

Figure 15.2 The design algorithm

So, IM design is basically an iterative procedure whose output–the resultant machine to be built–depends on the objective function(s) to be minimized and

on the corroborating constraints related to temperature rise, starting current (torque), breakdown torque, etc

The objective function may be active materials or costs or (efficiency)-1 or global costs or a weighted combination of them

Before treating the optimization stage in Chapter 18, let us perform here a practical design

15.4 MAIN DIMENSIONS OF STATOR CORE

Here we are going to use the widely accepted Dis2L output constant concept

detailed in the previous chapter For completely new designs, the rotor

tangential stress concept may be used

Based on this, the stator bore diameter Dis (14.15) is

97.0p005.098.0K

;C

Sfpp

0 gap 1

=λϕη

D

2pL

;cosPKS

is 1 n

n n E

From past experience, λ is given in Table 15.1

Table 15.1 Stack aspect ratio λ

λ 0.6 – 1.0 1.2 – 1.8 1.6 – 2.2 2 -3

From (15.2), the apparent airgap power Sgap is

VA8.718183.0895.0105.597.0S

8.71816015

222

⋅

⋅

=The stack length L (from 15.2) is

m1315.0221116.05.1

⋅

⋅π

=

m0876.022

1116

⋅

⋅π

=τThe number of stator slots per pole 3q may be 3⋅2 = 6 or 3⋅3 = 9 For q = 3,

the slot pitch τs will be around

m10734.933

0876.0q

3 s

=

In general the larger q gives better performance (space field harmonics and

losses are smaller)

The slot width at airgap is to be around 5 to 5.3 mm with a tooth of 4.7 to

4.4 mm which is mechanically feasible

From past experience (or from optimal lamination concept, developed later

in this chapter), the ratio of the internal to external stator diameter Dis/Dout,

bellow 100 kW for standard motors is given in Table 15.2

Table 15.2 Inner/outer stator diameter ratio

m18.062.0

1116.0K

DD

22pfor m10P02.01.0g

1 3

3 n

1 3

3 n

≥

⋅

⋅+

=

=

⋅

⋅+

As known, too small airgap would produces large space airgap field

harmonics and additional losses while a too large one would reduce the power

factor and efficiency

15.5 THE STATOR WINDING

Induction motor windings have been presented in Chapter 4 Based on such

knowledge, we choose the number of stator slots Ns

363322qmp

A two layer winding with chorded coils: y/τ = 7/9 is chosen as 7/9 = 0.777

is close to 0.8, which would reduce the first (5th order) stator mmf space

harmonic

The electrical angle between emfs in neighboring slots αec is

936

22N

p2

s

1 ec

π

=π

=π

=

The largest common divisor of Ns and p1 (36, 2) is t = p1 = 2 and thus the

number of distinct stator slot emfs Ns/t = 36/2 = 18 The star of emf phasors has

18 arrows (Figure 15.3a) and the distribution of phases in slots of Figure 15.3b

18 ,3617 ,35 16,3415,33 14,32

13,31

12,30

11,29

9, 27

8,267,25 6,24 5,23 4,22

3,21

2,20

AB’

C’ C’ C’ B B B A’ A’ A’ C C C B’ B’ B’

A C’ C’ C’ B B B A’ A’ A’ C C C B’ B’ B’ A A A C’ C’ C’ B B B A’ A’ A’ C C C B’ B’ B’ A A

Figure 15.3 A 36 slots, 2p 1 = 4 poles, 2 layer, chorded coils (y/τ = 7/9) three phase winding

The zone factor Kq1 is

9598.018sin3

5.0q

sinq6

72sin

y2sin

The number of turns per phase is based on the pole flux φ,

g

iτLBα

=

The airgap flux density is recommended in the intervals

( )

(0.75 0.85)T for 2p 8B

62pfor T82.07.0B

42pfor T78.065.0B

22pfor T75.05.0B

1 g

1 g

1 g

1 g

The pole spanning coefficient αi (Chapter 14, Figure 14.3) depends on the

tooth saturation factor 1 + Kst

Let us consider 1 + Kst = 1.4 with αi = 0.729, Kf = 1.085 Now from (15.10)

with Bg = 0.7T:

Wb10878.57.01315.00876.0729

=φ

The number of turns per phase W1 (from Chapter 14, (14.9)) is:

10878.560902.0085.14

3

46097.0f

The number of conductors per slot ns is

qpWan

1

1 1

where a1 is the number of current paths in parallel

In our case, a1 = 1 and

33.31328.1861

⋅

⋅

It should be an even number as there are two distinct coils per slot in a

double layer winding, ns = 30 Consequently, W1 = p1qns = 2⋅3⋅30 = 180

Going back to (15.12), we have to recalculate the actual airgap flux density

Bg

T726.01808.1867.0

The rated current I1n is

A303.946073.183.0895.0

5500V

3cos

PI

1 n n

As high efficiency is required and, in general, at this power level and speed,

winding losses are predominant from the recommended current densities

J

2,42pfor mm/A74J

1 2 cos

1 2 cos

=

=

=

=K

we choose Jcos = 4.5A/mm2

The magnetic wire cross section ACo is

2 1

cos

n

15.4303.9aJ

Table 15.3 Standardized magnetic wire diameter

Rated diameter [mm] Insulated diameter [mm]

0.3 0.327 0.32 0.348 0.33 0.359 0.35 0.3795 0.38 0.4105 0.40 0.4315 0.42 0.4625 0.45 0.4835 0.48 0.515 0.50 0.536 0.53 0.567 0.55 0.5875 0.58 0.6185 0.60 0.639 0.63 0.6705 0.65 0.691 0.67 0.7145 0.70 0.742 0.71 0.7525 0.75 0.749 0.80 0.8455 0.85 0.897 0.90 0.948 0.95 1.0 1.0 1.051 1.05 1.102 1.10 1.153 1.12 1.173 1.15 1.2035 1.18 1.2345 1.20 1.305 1.25 1.305 1.30 1.356 1.32 1.3765 1.35 1.407 1.40 1.4575 1.45 1.508 1.5 1.559

With the wire gauge diameter dCo

mm622.106733.24A4

In general, if dCo > 1.3 mm in low power IMs, we may use a few conductors

in parallel ap

mm15.1206733.24aA4'd

p

Co

⋅π

⋅

=π

Now we have to choose a standardized bare wire diameter from Table 15.3

The value of 1.15 mm is standardized, so each coil is made of 15 turns and

each turn contains 2 elementary conductors in parallel (diameter dCo’ = 1.15

mm)

If the number of conductors in parallel ap > 4, the number of current paths

in parallel has to be increased If, even in this case, a solution is not found, use

is made of rectangular cross section magnetic wire

15.6 STATOR SLOT SIZING

conductors in parallel ap with the wire diameter dCo’, we may calculate the

useful slot area Asu provided we adopt a slot fill factor Kfill For round wire, Kfill

≈ 0.35 to 0.4 below 10 kW and 0.4 to 0.44 above 10 kW

2 2

fill

s p 2 Co

40.04

30215.1K

4

na'd

Figure 15.4 Recommended stator slot shapes

For such slot shapes, the stator tooth is rectangular (Figure 15.5) The

variables bos, hos, hw are assigned values from past experience: bos = 2 to 3 mm ≤

8g, hos = (0.5 to 1.0) mm, wedge height hw = 1 to 4 mm

The stator slot pitch τs (from 15.3) is τs = 9.734 mm

Assuming that all the airgap flux passes through the stator teeth:

Fe ts ts s

g L B b LK

KFe ≈ 0.96 for 0.5 mm thick lamination constitutes the influence of lamination

insulation thickness

hb

bs2

os s

Figure 15.5 Stator slot geometry

With Bts = 1.5 – 1.65 T, (Bts = 1.55 T), from (15.22) the tooth width bts may

be determined

m1075.496.055.1

10734.9726.0

bN

hhDb

3 3

3

ts s

w os is 1 s

⋅+π

=

=

−++π

s su

+

Also,

s s 1 s 2 s

Ntanhb

(15.25) From these two equations, the unknowns bs2 and hs may be found

s su 2 1 s 2 2 s

NtanA4b

(15.26)

A

4

1 s s su

The slot useful height hs (15.24) writes

m1036.211016.942.572.1552bbA2

2 s 1 s

su s

−

⋅+

⋅

=+

assuming that stator and rotor tooth produce same effects in this respect

mg

mtr mts st

FFF1K

(15.29) The airgap mmf Fmg is

Aturns77.24210256.1726.01035.02.1

Bg2

⋅

⋅

with Bts = 1.55T, from the magnetization curve table (Table 15.4), Hts = 1760

A/m Consequently, the stator tooth mmf Fmts is

w os s

99.414.0FFK

As this value is only slightly larger than that of stator tooth, we may go on

with the design process

However, if Fmtr << Fmts (or negative) in (15.31) it would mean that for given 1 + Kst, a smaller value of flux density Bg is required

Consequently, the whole design procedure has to be brought back to Equation (15.10) The iterative procedure is closed for now when Fmtr ≈ Fmts

As the outer diameter of stator has been calculated in (15.4) at Dout = 0.18m,

the stator back iron height hcs becomes

2

hhh2DD

cs

=+++

−

=

=+++

10878.5Lh

2

3 cs

Evidently Bcs is too large There are three main ways to solve this problem

One is to simply increase the stator outer diameter until Bcs ≈ 1.4 to 1.7 T The

second solution consists in going back to the design start (Equation 15.1) and

introducing a larger stack aspect ratio λ which eventually would result in a smaller Dis, and, finally, a larger back iron height bcs and thus a lower Bcs The

third solution is to increase current density and thus reduce slot height hs

However, if high efficiency is the target, such a solution is to be used cautiously

Here we decide to modify the stator outer diameter to Dout’ = 0.190 m and

thus obtain

1034.1016.22

180.0190.0b

b16

.2

⋅

⋅

=

−+

This is considered a reasonable value

From now on, the outer stator diameter will be Dout’ = 0.190 m

15.7 ROTOR SLOTS

For cage rotors, as shown in Chapters 10 and 11, care must be exercised in

choosing the correspondence between the stator and rotor numbers of slots to

reduce parasitic torque, additional losses, radial forces, noise, and vibration Based on past experience (Chapters 10 and 11 backs this up with pertinent

explanations), the most adequate number of stator and rotor slot combinations

are given in Table 15.5

Table 15.5 Stator / rotor slot numbers

2p1 Ns Nr – skewed rotor slots

2 24

36

48

18, 20, 22, 28, 30, ,33,34 25,27,28,29,30,43 30,37,39,40,41

6 36

54

72

20,22,28,44,47,49 34,36,38,40,44,46 44,46,50,60,61,62,82,83

8 48

72

26,30,34,35,36,38,58 42,46,48,50,52,56,60

12 72

90

69,75,80 86,87,93,94

For our case, let us choose Ns ≠ Nr = 36/28

As the starting current is rather large–high efficiency is targeted–the skin

effect is not very pronounced Also, as the locked rotor torque is large, the

leakage inductance will not be large Consequently, from the four typical slot

shapes of Figure 15.6, that of Figure 15.6c is adopted

Figure 15.6 Typical rotor cage slots

First, we need the value of rated rotor bar current Ib,

n r

1 w 1 I

NKmW2K

with KI = 1, the rotor and stator mmf would have equal magnitudes In reality,

the stator mmf is slightly larger

864.02.083.08.02.0cos8.0

From (15.34), the bar current Ib is

303.99019.018032864.0

For high efficiency, the current density in the rotor bar jb = 3.42 A/mm2

The rotor slot area Ab is

2 6 6

b

b

1042.36.279J

2sin2

6.279N

psin2

II

r 1

The current density in the end ring Jer = (0.75 – 0.8)Jb The higher values

correspond to end rings attached to the rotor stack as part of the heat is

transferred directly to rotor core

With Jer = 0.75⋅Jb = 0.75⋅342⋅106 = 2.55⋅106A/m2, the end ring cross section,

Aer, is

2 6 6

er

er

10565.2255.628J

Figure 15.7 Rotor slot geometry

The rotor slot pitch τr is

28

107.06.111N

g

r

is r

=

−π

=

With the rotor tooth flux density Btr = 1.60T, the tooth width btr is

m1088.510436.126.196.0726.0B

K

B

r tr Fe

g tr

1 or

Ndh

;m1070.528

1088.52817.06

111

NbNhDd

3 or

3 3

r tr r or re 1

π

−

−π

2 2 1 b

+++π

r r 2 1

Ntanhd

102

70.52.15.02024602

ddhh

H

or r

This is rather close to the value of Vmtr = 55.11 Aturns of (15.39) The

design is acceptable so far

If Vmtr had been too large, we might have reduced the flux density, thus

increasing tooth width btr and the bar current density Increasing the slot height

is not practical as already d2 = 1.2⋅10-3m This bar current density increase could

reduce the efficiency below the target value We may alternatively increase 1 +

Kst, and redo the design from (15.10)

When the power factor constraint is not too tight, this is a good solution To

maintain same efficiency, the stator bore diameter has to be increased So the

within bounds

When Vmtr is too small, we may increase Btr and return to (15.40) until

sufficient convergence is obtained The required rotor back core may be