the induction machine handbook chuong (6)

Besides the magnetic energy related to the magnetization field investigated in Chapter 5, there are flux lines that encircle only the stator or only the rotor coils Figure 6.1.. end turn

Chapter 6 LEAKAGE INDUCTANCES AND RESISTANCES

6.1 LEAKAGE FIELDS

Any magnetic field (Hi, Bi) zone within the IM is characterized by its stored

magnetic energy (or coenergy) Wm

( )

2

ILdVHB2

1W

2 i i V

Equation (6.1) is valid when, in that region, the magnetic field is produced

by a single current source, so an inductance “translates” the field effects into

circuit elements

Besides the magnetic energy related to the magnetization field (investigated

in Chapter 5), there are flux lines that encircle only the stator or only the rotor

coils (Figure 6.1) They are characterized by some equivalent inductances called

leakage inductances Lsl, Lrl

end turn (connection)

leakage field lines

A

A

magnetisation flux lines zig - zag leakage flux lines

Figure 6.1 Leakage flux lines and components

There are leakage flux lines which cross the stator and, respectively, the

rotor slots, end-turn flux lines, zig-zag flux lines, and airgap flux lines (Figure

6.1) In many cases, the differential leakage is included in the zig-zag leakage

Finally, the airgap flux space harmonics produce a stator emf as shown in

Chapter 5, at power source frequency, so it should also be considered in the

leakage category Its torques occur at low speeds (high slips) and thus are not

there at no–load operation

6.2 DIFFERENTIAL LEAKAGE INDUCTANCES

As both the stator and rotor currents may produce space flux density

harmonics in the airgap (only step mmf harmonics are considered here), there

will be both a stator and a rotor differential inductance For the stator, it is

sufficient to add all L1mν harmonics, but the fundamental (5.122), to get Lds

1Km

;K

KgKpWL6

1 2 s

2 w c 1 2

2 1 e 0

νπ

τµ

1

ds 0 dS

K

KK

KL

As the pole pitch of the harmonics is τ/ν, their fields do not reach the back

cores and thus their saturation factor Ksν is smaller then Ks The higher ν, the

closer Ksν is to unity In a first approximation,

s st

That is, the harmonics field is retained within the slot zones so the teeth

saturation factor Kst may be used (Ks and Kst have been calculated in Chapter 5)

A similar formula for the differential leakage factor can be defined for the rotor

winding

µ

≠ µ

dr

K

KK

K

(6.5)

As for the stator, the order µ of rotor harmonics is

1m

K2 2±

=

m2–number of rotor phases; for a cage rotor m2 = Z2/p1, also Kwr1 = Kwrµ = 1

The infinite sums in (6.3) and (6.5) are not easy to handle To avoid this, the

airgap magnetic energy for these harmonics fields can be calculated Using (6.1)

ν

ν

ν= µ

s c m 0

g gK K

F

We consider the step-wise distribution of mmf for maximum phase A

current, (Figure 6.2), and thus

( ) ( )

2 m 1

N 1

2 j s 2

m 1

2 0 2 0 ds

F

FN

12F

d

=π

θθ

=σ

F( )θ

1

0.5 0.5

1

Figure 6.2 Step-wise mmf waveform (q = 2, y/τ = 5/6)

The final result for the case in Figure 6.2 is σdso = 0.0285

This method may be used for any kind of winding once we know the

number of turns per coil and its current in every slot

For full-pitch coil three-phase windings [1],

1Km

2q12

1q

2 1 w

2 1

2 2

12

1y1qy14

31qKm

2

q/y1y

2

2 2

2

2 1 w 2 1

−

−+

⋅π

=ε

(6.10)

In a similar way, for the cage rotor with skewed slots,

1'

K

12 1 r 2 skew 0

er r1

r er r er

skew

N

p2

;2/2/sin

;c

csin'

α

α

=ητ

⋅α







τ

⋅α

The above expressions are valid for three-phase windings For a phase winding, there are two distinct situations At standstill, the a.c field produced by one phase is decomposed into two equal traveling waves They both produce a differential inductance and, thus, the total differential leakage inductance (Lds1)s=1 = 2Lds

single-On the other hand, at S = 0 (synchronism) basically the inverse (backward) field wave is almost zero and thus (Lds1)S=0 ≈ Lds

The values of differential leakage factor σds (for three- and two-phase machines) and σdr, as calculated from (6.9) and (6.10) are shown on Figures 6.3 and 6.4 [1]

A few remarks are in order

• For q = 1, the differential leakage coefficient σds0 is about 10%, which means it is too large to be practical

• The minimum value of σds0 is obtained for chorded coils with y/τ ≈ 0.8 for all qs (slots/pole/phase)

• For same q, the differential leakage coefficient for two-phase windings is larger than for three-phase windings

• Increasing the number of rotor slots is beneficial as it reduces σdr0 (Figure 6.4)

Figures 6.3 do not contain the influence of magnetic saturation In heavily saturated teeth IMs as evident in (6.3), Ks/Kst > 1, the value of σds increases further

Figure 6.3 Stator differential leakage coefficient σ ds for three phases a.) and two-phases b.) for various q s The stator differential leakage flux (inductance) is attenuated by the reaction

of the rotor cage currents

Coefficient ∆d for the stator differential leakage is [1]

d 0 ds ds 1

mq 2

1

2

1 w

w r skew 0

ds d

1

;K

K'

K1

ν

ησ

−

≈

≠ ν

ν ν

Figure 6.4 Rotor cage differential leakage coefficient σ dr for various q s

and straight and single slot pitch skewing rotor slots (c/τ r = 0,1,…)

Figure 6.5 Differential leakage attenuation coefficient ∆ d for cage rotors

with straight (c/τ r = 0) and skewed slots (c/τ s = 1)

As the stator winding induced harmonic currents do not attenuate the rotor differential leakage: σdr = σdr0

A rather complete study of various factors influencing the differential leakage may be found in [Reference 2]

Example 6.1 For the IM in Example 5.1, with q = 3, Ns = 36, 2p1 = 4, y/τ = 8/9, Kw1 = 0.965, Ks = 2.6, Kst = 1.8, Nr = 30, stack length Le = 0.12m, L1m = 0.1711H, W1 = 300 turns/phase, let us calculate the stator differential leakage inductance Lds including the saturation and the attenuation coefficient ∆d of rotor cage currents

2 2

d st

s 0 ds

ds 0.92 1.541510

8.1

6.21016.1K

KK

Now the differential leakage inductance Lds is

H102637.01711.0105415.1LK

1 2 m

1 ts

s 0 dr

8.16.2108.2LK

K

σdr0 is taken from Figure 6.4 for Z2/p1 = 15, c/τr = 1, : σdr0 = 2.8⋅10-2

It is now evident that the rotor (reduced to stator) differential leakage inductance is, for this case, notable and greater than that of the stator

6.3 RECTANDULAR SLOT LEAKAGE INDUCTANCE/SINGLE

LAYER

The slot leakage flux distribution depends notably on slot geometry and less

on teeth and back core saturation It also depends on the current density distribution in the slot which may become nonuniform due to eddy currents (skin effect) induced in the conductors in slot by their a.c leakage flux

Let us consider the case of a rectangular stator slot where both saturation and skin effect are neglected (Figure 6.6)

bos

bs

Figure 6.6 Rectangular slot leakage

Ampere’s law on the contours in Figure 6.6 yields

( )

s s

s s

hhxh

; inbxH

hx0

; hxinbxH

The leakage inductance per slot, Lsls, is obtained from the magnetic energy

formula per slot volume

=

⋅µ

⋅

=

os os

s

s e 2 s 0 s e h

h

0

2 0 2

ms 2

sls

b

hb

hLnbLdxxH2

1i

2W

i

2

L

0s s

(6.18)

The term in square parenthesis is called the geometrical specific slot

permeance

( )1 310 mh

;5.25.0b

hb

os os

os s

s s

=

It depends solely on the aspect of the slot In general, the ratio hs/bs < (5−6)

to limit the slot leakage inductance to reasonable values

The machine has Ns stator slots and Ns/m1 of them belong to one phase So

the slot leakage inductance per phase Lsl is

qpLW2LmqmpLm

NL

1

s e 2 1 0 sls 1

1 1 sls 1

s sl

λµ

=

=

The wedge location has been replaced by a rectangular equivalent area on

Figure 6.6 A more exact approach is also possible

The ratio of slot leakage inductance Lsl to magnetizing inductance L1m is

(same number of turns/phase),

1 W

s c 2

m 1

sl

qK

KgK3L

Suppose we keep a constant stator bore diameter Di and increase two times

the number of poles

The pole pitch is thus reduced two times as τ = πD/2p1 If we keep the

number of slots constant q will be reduced twice and, if the airgap and the

winding factor are the same, the saturation stays low for the low number of

poles Consequently, Lsl/L1m increases two times (as λs is doubled for same slot

height)

Increasing q (and the number of slots/pole) is bound to reduce the slot

leakage inductance (6.20) to the extent that λs does not increase by the same

ratio Our case here refers to a single-layer winding and rectangular slot

Two-layer windings with chorded coils may be investigated the same way

6.4 RECTANGULAR SLOT LEAKAGE INDUCTANCE/TWO LAYERS

We consider the coils are chorded (Figure 6.7)

Let us consider that both layers contribute a field in the slot and add the

effects The total magnetic energy in the slot volume is used to calculate the

leakage inductance Lsls

( ) ( ) [H x H x] dx b( )xi

L2L

st h

0

2 2 1 0

1 H (x)1 2

x

mmfs

H (x)2fields (H(x))

x

bsb(x)bw

bosmutual field zone

Figure 6.7 Two-layer rectangular semiclosed slots: leakage field

os w i

su i sl i

k cu i cl

su i sl i

sl

su

i sl

s

k cu

s cl

i sl sl

s cl

sl sl

s cl

2 1

b

or bbwith

hhhfor x ;bcosInbIn

hhhxhhfor

;hhhxbcosInbIn

hhxhfor bIn

hx0for

;h

xbIn

xHx

>

γ+

++

(6.23)

The phase shift between currents in lower and upper layer coils of slot K is

γK and ncl, ncu are the number of turns of the two coils Adding up the effect of

all slots per phase (1/3 of total number of slots), the average slot leakage

inductance per phase Lsl is obtained

While (6.23) is valid for general windings with different number of turn/coil

and different phases in same slots, we may obtain simplified solutions for

identical coils in slots ncl = ncu = nc

+γ+

=µ

=

λ

os

os w

w s o 2 s

i

s k su s

su s

k 2 su sl e

2 c 0

sl sk

b

hb

hb

hcos1bh

bcoshb

hb

coshh4

1LnL

(6.24)

Although (6.24) is quite general–for two-layer windings with equal coils in

slots–the eventual different number of turns per coil can be lumped into cosγ as

Kcosγ with K = ncu/ncl In this latter case the factor 4 will be replaced by (1 +

K)2

In integer and fractionary slot windings with random coil throws, (6.24)

should prove expeditious All phase slots contributions are added up

Other realistic rectangular slot shapes for large power IMs (Figure 6.8) may

also be handled via (6.24) with minor adaptations

For full pitch coils (cosγK = 1.0) symmetric winding (hsu = hsl = hs′) (6.24)

becomes

( )

os

os w

w s

0 s

i s

s ' h h

h 0sk

b

hb

hb

hb

hb'hs sl su

Figure 6.8 Typical high power IM stator slots

6.5 ROUNDED SHAPE SLOT LEAKAGE INDUCTANCE/TWO

LAYERS

Although the integral in (6.2) does not have exact analytical solutions for

slots with rounded corners, or purely circular slots (Figure 6.9), so typical to

low-power IMs, some approximate solutions have become standard for design

o r , os

r , os 2 1

1 r , s r ,

b

bb

hb

hbb3

Kh

≈

2 1

y

y 2

y

y 2

y

y 2

K4

34

1K

21for

; 4

123K

3

23

1for

; 4

16K

1

y3

2for

; 4

31K

+

≈

≤β

≤+

β

−

≈

≤β

≤

−β

≈

≤τ

=β

≤β

w 1

o r , os

r , os 2 1

1 r , s r ,

bb

hb

hb

hbb3

Kh

or 1

or r

b

h66.0b

hb

b785

b

2 1 1

r r

b

hb

h66.0A8

b1b

h

+

−+

where Ab is the bar cross section

If the slots in Figure 6.9c, d are closed (ho = 0) (Figure 6.9e) the terms

hor/bor in Equations (6.29, 6.30) may be replaced by a term dependent on the bar

current which saturates the iron bridge

[m]

in b ;10bI

;I

10h12.13.0b

h

1 3 1 b 2 b

3 or or

h h b h

0.1b b

b 1

2 2

o

os s

os

b b

h h h

d.)

2

2 r

ho ho

e.)

b h h

b

r or or

2

f.)

Figure 6.9 Rounded slots: oval, trapezoidal, and round

This is only an empirical approximation for saturation effects in closed rotor slots, potentially useful for very preliminary design purposes

For the trapezoidal slot (Figure 6.9f), typical for deep rotor bars in high power IMs, by conformal transformations, the slot permeance is, approximately [3]

or or

or 2 or 2

or 2

2 or 2

or 2

2 or 2

h1bb

1b

blnb

b

1bb

b

b4

1b

bln

Finally for stator (and rotors) with radial ventilation ducts (channels)

additional slot leakage terms have to be added [8]

For more complicated rotor cage slots used in high skin effect (low starting

current, high starting torque) applications, where the skin effect is to be

considered, pure analytical solutions are hardly feasible, although many are still

in industrial use Realistic computer-aided methods are given in Chapter 8

6.6 ZIG-ZAG AIRGAP LEAKAGE INDUCTANCES

zig-zag stator leakage flux

zig-zag rotor leakage flux

b.)

Figure 6.10 Airgap a.) and zig-zag b.) leakage fields

The airgap flux does not reach the other slotted structure (Figure 6.10a)

while the zig-zag flux “snakes” out through the teeth around slot openings

In general, they may be treated together either by conformal transformation

or by FEM From conformal transformations, the following approximation is

given for the geometric permeance λzs,r [3]

rotorscagefor 1

;0.14

13b/gK45

b/gK5

y y

r , os c

r , os c r

e 1 0 zls

qpLW2

'K1a1a1N12

pL

s

2 1 2 1m

for the stator, and

=

'K2'K1a1aN

NN12

pL

r

2 s 2 s

2 1 2 1m

for the rotor with K′ = 1/Kc, a = bts,r/τs,r, τs,r = stator (rotor) slot pitch, bts,r–stator

(rotor) tooth-top width

It should be noticed that while expression (6.32) is dependent only on the

airgap/slot opening, in (6.34) and (6.35) the airgap enters directly the

denominator of L1m (magnetization inductance) and, in general, (6.34) and

(6.35) includes the number of slots of stator and rotor, Ns and Nr

As the term in parenthesis is a very small number an error here will notably

“contaminate” the results On the other hand, iron saturation will influence the

zig-zag flux path, but to a much lower extent than the magnetization flux as the

airgap is crossed many times (Figure 6.10b) Finally, the influence of chorded

coils is not included in (6.34) to (6.35) We suggest the use of an average of the

two expressions (6.33) and (6.34 or 6.35)

In Chapter 7 we revisit this subject for heavy currents (at standstill)

including the actual saturation in the tooth tops

Example 6.2 Zig-zag leakage inductance

For the machine in Example 6.1, with g = 0.5⋅10-3m, bos = 6g, bor = 3g, Kc =

1.32, L1m = 0.1711H, p1 = 2, Ns = 36 stator slots, Nr = 30 rotor slots, stator bore

Di = 0.102m, By = y/τ = 8/9 (chorded coils), and W1 = 300 turns/phase, let us

calculate the zig-zag leakage inductance both from (6.32 – 6.33) and (6.34 –

−

=

⋅π

−

=

=τ

−τ

=

−

−

858.0101.030105.031gDNb1

6628.0102.0

36105.061D

Nb1

ba

3

i r or

3

i s os

r , s

r , os r , s r , s

(6.37)

1.0632.11.045

101.06

32.1105.05

3 3

⋅

⋅

⋅+

101.03

32.1101.05

3 3

⋅

⋅

⋅+

The zig-zag inductances per phase Lzls,r are calculated from (6.33)

2

12.030010256.12

H10455.8187.03

2

12.030010256.12L

4 2

6

4 2

.02

7575.016628.016628.013612

21711

0

2 2

−+

7575.02

7575.01858.01858.030

363612

21711

6.7 END-CONNECTION LEAKAGE INDUCTANCE

As seen in Figure 6.11, the three-dimensional character of end connection field makes the computation of its magnetic energy and its leakage inductance per phase a formidable task

Analytical field solutions need bold simplifications [5] Biot-Savart inductance formula [6] and 3D FEM have all been also tried for particular cases

Y

Za.)

X

Zb.)

Figure 6.11 Three-dimensional end connection field

Some widely used expressions for the end connection geometrical permeances are as follows:

• Single-layer windings (with end turns in two “stores”)