THE DOUBLE CAGE BEHAVES LIKE A DEEP BAR CAGE

In some applications, very high starting torque–Tstart/Trated ≥ 2.0–is required. In such cases, a double cage is used. It has been proved that it behaves like a deep bar cage, but it produces even higher starting torque at lower starting current. For the case when skin effect can be neglected in both cages, let us consider a double cage as configured in Figure 9.18. [8]

image

 

image

 

Figure 9.18 Double cage rectangular-shape geometry a.) and equivalent circuit b.)

The equivalent single bar circuit is given in Figure 9.18b. For the common ring of the two cages

Rr = Rring, Rbs = Rbs upper bar, Rbw = Rbw lower bar (9.49) For separate rings

Rr = 0, Rbs = Rbs + Rerings, Rbw = Rbw + Reringw (9.50)

The ring segments are included into the bar resistance after approximate reduction as shown in Chapter 6. The value of Lring is the common ring inductance or is zero for separate rings. Also for both cases, Lee) refers to the slot neck flux.

Le (Φe )= µ0lstack ha44 (9.51)

We may add into Le the differential leakage inductance of the rotor. The start (upper) and work (lower) cage inductances Lbs and Lbw include the end ring inductances only for separate rings. Otherwise, the bar inductances are

Lbs = µ0lstack 3hass

Lbw = µ0lstack  3haww + ha + hass  (9.52) There is also a flux common to the two cages represented by the flux in the starting cage. [3]

Lml = µ0lstack 2hass (9.53)

In general, Lml is neglected though it is not a problem to consider in solving the equivalent circuit in Figure 9.18. It is evident (Figure 9.18a) that the starting (upper) cage has a large resistance (Rbs) and a small slot leakage inductance Lbs, while for the working cage the opposite is true.

Consequently, at high slip frequency, the rotor current resides mainly in the upper (starting) cage while, at low slip frequency, the current flows mainly into the working (lower) cage. Thus both Rbe and Xbe vary with slip frequency as they do in a deep bar single cage (Figure 9.19).

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Figure 9.19 Equivalent parameters of double cage versus slip frequency 

LEAKAGE FLUX PATH SATURATION–A SIMPLIFIED APPROACH

Leakage flux path saturation occurs mainly in the slot necks zone for semiclosed slots for currents above 2 to 3 times rated current or in the rotor slot iron bridges for closed slots even well below the rated current (Figure 9.20).

Consequently,

a ‘os = aos + (τss µsaos )µ0; τss = πNDsi

a ‘or aor + sr µraor )µ0; τsr = π(DNi s 2g) (9.54) (τ −

= a ‘or = bµorbrµb

The slot neck geometrical permeances will be changed to: a′os/hos, a′or/hor, or a′or/hor dependent on stator (rotor) current.

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Figure 9.20 Leakage flux path saturation conditions

With ns the number of turns (conductors) per slot, and Is and Ib the stator and rotor currents, the Ampere’s law on Γs, Γr trajectories in Figure 9.20 yields

Hts (τss − aos )+ H aos os ≈ n Is s 2; µsHts = Bts = µ0Hos

clip_image001 Htr (τsr − aor )+ H aor or ≈ Ib 2; µrHtr = Btr = µ0Hor (9.55)

H btr or = Ib 2; µbr = HBtbtb

The relationship between the equivalent rotor current Ir′ (reduced to the stator) and Ib is (Chapter 8)

Ib = I ‘r 6p1qnNsrKw1 = I ‘n Kr s w1 NNsr (9.56)

Ns–number of stator slots; Kw1–stator winding factor.

When the stator and rotor currents Is and Ir′ are assigned pertinent values, iteratively, using the lamination magnetization curves, Equations (9.55) may be solved (Figure 9.20) to find the iron permeabilities of teeth tops or of closed rotor slot bridges. Finally, from (9.54), the corrected slot openings are found. With these values, the stator and rotor parameters (resistances and leakage inductances) as influenced by the skin effect (in the slot body zone) and by the leakage saturation (in the slot neck permeance) are recalculated. Continuing with these values, from the equivalent circuit, new values of stator and rotor currents Is, Ir′ are calculated for given stator and voltage, frequency and slip. The iteration cycles continue until sufficient convergence is obtained for stator current.

Example 9.4. An induction motor has semiclosed rectangular slots whose geometry is shown in Figure 9.21.

The current density design for rated current is jCo = 6.5 A/mm2 and the slot fill factor Kfill = 0.45. The starting current is 6 times rated current. Let us calculate the slot leakage specific permeance at rated current and at start.

Solution

The slot mmf nsIs is obtained from

n Is s = (h a Ks s ) fill Coj =10 35⋅ ⋅0.45⋅6.5 =1023.75Ampereturns

Making use of (9.54),

clip_image001[14] Hts (18− 4.5)+ Bµts0 4.5⋅103 =1023.75 2

Bts clip_image003[4]ts = − H 3ts µ0

It is very clear that for rated current Bts < 0.4029, so the tooth heads are not saturated. In (9.54), aos′ = aos and the slot specific geometrical permeance λs is

clip_image005[4]

clip_image0063⋅10 4.5+10 4.5

For the starting conditions,

clip_image007 Htstart (18− 4.5)⋅10−3 + Bµ0ts 4.5 ⋅10−3 = 6142.5 2

Btstart = 2.41− Htstart 3µ0

Now we need the lamination magnetization curve. If we do so we get Btstart

≈ 2.16T, Htstart ≈ 70,000 A/m. The iron permeability µs is

clip_image008 µs = Btstart µ0−1 = 70000⋅2.1.16256 10⋅ −6 = 24.56

µ0 Htstart

Consequently, aos ‘= aos + (τ −s µsaos )µ =0 4.5+ (1824.564.5) = 5.05mm

The slot geometric specific permeance is, at start,

clip_image010

clip_image011

The leakage saturation of tooth heads at start is not very important in our case. In reality, it could be more important for semiclosed stator (rotor) slots.

Example 9.5. Let us consider a rotor bar with the geometry of Figure 9.21. The bar current at rated current density jAlr = 6.0 A/mm2 is

Ibr = jAlrhr clip_image013A

 

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Figure 9.22. Closed rotor slots

Calculate the rotor bridge flux density for 10%, 20%, 30%, 50%, 100% of rated current and the corresponding slot geometrical specific permeance.

Solution

Let us first introduce a typical magnetization curve.

B(T)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

H(A/m)

22.8

35

45

49

57

65

70

76

83

90

B(T)

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

H(A/m)

98

106

115

124

135

148

162

177

198

220

B(T)

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

H(A/m)

237

273

310

356

417

482

585

760

1050

1340

B(T)

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2.0

H(A/m)

1760

2460

3460

4800

6160

   

8270 11170 15220 22000 34000

   

Neglecting the tooth top saturation and including only the bridge zone, the Ampere’s law yields

clip_image001[16] = I ;b H br = aIbrb ; a br =11mm

H br a br

For various levels of bar current, Hbr and Bbr become

Ib (A)

120

240

360

600

1200

Hbr

10909

21818

32727

54545

109090

Bbr

1.84

1.95

1.98

2.1

2.2

µ

µrel = µbr0

134.28

71.16

48.165

30.653

16.056

aor ‘= µarelbr (mm)

0.0819

0.1546

0.2284

0.3588

0.6851

λslot = 3(a r2+hrabr ) + ahorbr

14.293

75.1

5.42

3.8287

2.5013

Note that at low current levels the slot geometrical specific permeance is unusually high. This is because the iron bridge is long (abr = 11mm); so Hbr is small at small currents.

Also, the iron bridge is 1mm thick in our case, while 0.5 to 0.6 mm would produce better results (lower λslot). The slot height is quite large (hr = 25 mm) so significant skin effect is expected at high slips. We investigated here only the currents below the rated one so Sf1 < 3 Hz in general. Should the skin effect occur, it would have reduced the first term in λslot (slot body specific geometrical permeance) by the factor Kx.

As a conclusion to this numerical example, we may infer that a thinner bridge with a smaller length would saturate sooner (at smaller currents below the rated one), producing finally a lower slot leakage permeance λslot. The advantages of closed slots are related to lower stray load losses and lower noise.

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