ACTIVE AND INDUCTIVE RESISTANCES OF STATOR AND ROTOR WINDING

 

Calculation of active resistances.Activee resistance of phase of stator winding is calculated, according to a formula (2.3). Resistance r₁ is in the heated state

r₁ = r_1ϑ = r_{1 20}⁰ [1 + 0,004(ϑ + ϑ_env − 20°)], (25)

where r_{1 20}⁰ = (w_ph1∙l_{ar av}) / (57S_ar∙a₁∙a₂) − resistance of stator phase at 20°C, Ω; ϑ − the expected overheat of winding, °C; ϑ_env − temperature of environment; l_{ar av} − average length of loop of armature winding, m.

Active resistance of phase of squirrel-cage rotor in the heated state is (fig. 4.9)

r₂ = r_2ϑ = (1/p)(r_b + r_r)[1 + α∙(ϑ + ϑ_env − 20°)], (26)

where р − number of poles pair; r_b − active resistance of rotor bar (at 20° С); r_r − active resistance of s.c. rings (at 20° С); α − temperature coefficient of material resistance of rotor winding.

Active resistance of rotor bar

r_b20⁰ = (l_b / S_b)∙ρ, (27)

where l_b − length of rotor bar, m; S_b – cross-section area of bar, mm²; ρ − specific electric resistance of bar material at 20° С.

For rotors with an aluminium pouring for a slot, represented on a fig. 4.9, c, the area of cross-section of bar is determined by a formula (mm²)

S_b = π(b₁ − 0,2)² / 8 + π(b₂ − 0,2)² / 8 + (b₁ + b₂ − 0,4)∙h₁ / 2 (28) Active resistance of s.c. rings, reduced to the bar (at 20° С),

r_r20о = 2π∙D_r / [z₂∙S_r∙(2sin π∙p/z₂)²]∙ρ, (29)

where D_r − diameter of s.c. ring, m; S_r − area of cross-section of s.c. ring, mm².

Active resistance of phase of rotor winding, resulted to the stator winding,

r'₂ = r₂∙k_rd, (30)

where k_rd − coefficient of reduction;

k_rd = (m₁/m₂) [w_ph1∙k_w1 / (w_ph2∙ k_w2)]² = 4p∙m₁∙w_ph1²k_w1² /z₂ (31)

At m₁ = 3

k_rd = w²_ph1∙k²_w1 / (w²_ph2∙ k²_w2) = 12∙p∙ w²_ph1∙k²_w1 / z₂. (32)

Values of specific resistances ρ and temperature coefficients α for materials, applied for shortcircuited windings, driven to the table.7.

 

Tabl. 7

Material	ρ, Ω∙ mm2/m	α	γ
Copper M-1	0,0175=1/57	0,0040	8,9
Brass ЛС-59-1	0,065	0,0026	8,5
Brass Л-62	0,071	0,0017	8,5
Aluminium	0,035=2/57*	−	2,6

* For an aluminium a value ρ takes into account the presence of emptinesses at pouring.

 

Calculation of inductive resistances.Inductive resistance of stator winding phase (Ω)

X_s1 = [4π∙f₁∙w²_ph1∙l₁ / (p∙q₁)] ∑λ₁∙10^-8 (33)

where q₁ = z₁/ (2p∙m₁)– a number of slots on a pole and phase of stator winding; l₁ − active length of stator; f − frequency of network; λ₁ − total specific conductivity for the leakage fluxes of stator winding.

Inductive resistance of phase of shortcircuited rotor

X_s2 = (2π∙f₁∙l₂ /p)∙∑λ₂∙10^-8(34)

where l₂ − active length of rotor; ∑λ₂ − total specific conductivity for the leakage fluxes of rotor

∑λ₂ = λ_sl2 + λ_δ2 + λ_ec2 (35)

where λ_sl2 − specific conductivity of slot dispersion of rotor; λ_δ2 − specific conductivity of dissipation in an air-gap; λ_ec2 − specific conductivity of dissipation of end coils.

Specific conductivity of rotor slot dissipation λ_sl2 equals to:

а) for a round slot (fig. 9, а)

λ_sl2 = 1,25(0,66 + h_lm2 / b_g2); (36)

b) for a pyriform slot (fig. 9, b)

λ_sl2 = 1,25 [(2/3)∙h₁/ (b₁ + b₂) + 0,66 + h_lm2 / b_g2]; (37)

c) for pyriform closed slot (fig. 9, c)

λ_sl2 = 1,25 [(2/3)∙h₁/ (b₁ + b₂) + λ_sb],

where λ_sb − specific conductivity of leakage fluxes through the bridge of slot, determined graphicaly.

Specific conductivity of dissipation in an air-gap λ_δ2 for motors with a shortcircuited rotor

λ_δ2 = t_z2/(9,5δ∙k_δ) (38)

 

Fig. 9

For motors of very small power with a shortcircuited rotor a specific conductivity A,_к2 can be count up by a formula

λ_δ2 = (t_z1 – b_g1 − b_g2) / (12,8∙δ), (39)

where b_g1 and b_g2 − width of stator and rotor slot slit respectively.

Specific conductivity of dissipation of end coils of s.c. rotor

λ_ec2 = 2,89D_r / [z₂∙l₂∙(2sin πρ / z₂)²]∙lg2,36 D_{r av} / (a + b),(40)

where D_{r av} − average diameter of s.c. ring; (a + b) − half of perimeter of section of s.c. ring.

Inductive resistance of winding phase of s.c. rotor, reduced to the stator winding,

X ^‘_s2 = X_s2∙k_rd = X_s2(4p∙m₁∙w_ph1²∙k_w1²) / z₂ (41)

5. CHOICE OF EXCITATION CAPACITOR

Total reactive-power of capacitors Р_ex, required for excitation of three-phase asynchronous generator and compensation of phase-shift of loading,

Р_ex=Р_al∙(tg φ_G + tg φ), (42)

where Р_al − active-power, consumed by load (on three phases); φ_G, φ − angles of phase-shift between voltage and current of generator and load respectively.

As stated above, for generators of small power (up to 1 kW) a tgφ_G = 0,5÷0,7, and a power-factor of load usually is within the limits of 0,85÷0,95, i.е. tgφ = 0,3÷0,6.

Thus, necessary power of excitation capacitors is approximately equal to the consumable active-power, i.е.

Р_ex ≈ Р_al (43)

Because power of excitation

Р_ex = 2π∙f∙С_cb∙U_c², (44) I" (4.44)

then the capacity of capacitors block diminishes with the increase of frequency and voltage at a capacitor:

С_cb=Р_ex / (2πfU_c²) (45) : (4.45)

However with the increase of operating voltage a mass of capacitors increases, and with the increase of frequency it is necessary to reduce operating voltage because of internal losses growth in capacitor.

For the capacitors of type К75-10, being the best for operating in the alternating current circuits, dependence of mass М_C of operating voltage

М_C≡k_cU_op²С_cb (46)

where k_c − coefficient of proportionality, g/(V²F).

The value of k_c depends on mass of capacitor and is span:

k_c =(0,8÷1,2)∙10³.

Dependence of permissible voltage U_per at the capacitor is determined by technical requirements. For capacitors К75-10 this dependence can be presented by expression:

U_per / U_op = 1,9÷0,45lg f (47)

The graphic image of this dependence is presented on a fig. 10.

Fig. 10 From(47) it is possible to find expressions for specific mass of capacitors

М_с/Р_ex = (k_с/2π)1/[f (1,9÷0,45lg f)]. (48)

The graphic image of dependence of capacitor specific mass on frequency is presented on a fig. 11. The got result (48) shows that specific mass of capacitors does not depend on voltage and is minimum on frequency 2400 Hz. A circuit of capacitors connection does not import also: a star or triangle − mass of capacitors is identical. It is important only, that capacitors were used at their voltage rating.

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If a generator operates at environment temperature below 70°C, a voltage at capacitor can be increased on 20%, here specific mass of capacitors diminishes yet on 40%.

At star connection of capacitors, when voltage at capacitor equals to phase one U_с Fig. 11 = U_ph1 a capacity of capacitors on a phase is

С_λ = [1/(2π∙f∙U_ph1²)]∙Р_ex/3. (49)

If capacitors are connected in a triangle, then voltage at them increases in √3 times as compared to voltage at connection of capacitors in a star, and a capacity diminishes in three times

C_Δ ={1/[2π∙f∙ (√3∙U_ph1)²]}∙P_ex/3. (50)

For the optimal use of capasitors at their triangle connection it is necessary to choose them on heightened operating voltage, not to exceed the permissible specific losses and field intensity in the dielectric of capacitor.

At the known design parameters of generator and given voltage and frequency a capacity of capacitors at their star connection equals to:

С_Y = 1/(ω₁²∙L₁) = 1/ [ω₁²(X₁ + X_m0) / ω₁] = 1/ [ω₁(X₁ + X_m0)], (51) where X_m0 = Е_phо/I_mо − inductive resistance of magnetizing contour at o.c. of generator (point A on a fig. 1).