9. Losses of energy and efficiency of asynchronous machine

In an asynchronous machine there are next losses of energy: in windings of stator and rotor Р_cp1, Р_cp2; in steel of stator and rotor Р_st1, and Р_st2; mechanical Р_mech; additional Р_ad; in the excitant and regulative devices Р_ex,c.

Total losses

∑P = P_m1 + P_M2 + P_st1 + P_st2 + P_mech + P_ad + Р_ex,c. (61)

There are losses Р_ex,c in the generator mode. Losses are in windings of stator and rotor, conditioned by its currents:

Р_{cp 1} = m₁∙I₁²∙r₁, Р_{cp 2} = m₂∙I₂²∙r₂ = m₁∙I '₂²∙r '₂ (62)

Losses in stator steel are determined as a sum of losses in toothing Р_z1, in the back of armature Р_j1 and pulsation losses Р_pl1 :

Р_st = P_z1 + P_j1 + P_pl,

P_z1 = k_T∙[k_h∙σ_h∙f₁ / 400 + k_ed∙σ_ed (f₁ / 400)²]∙B²_z1av∙M_z1, (63)

P_j1 = k_T∙[k_h∙σ_h∙f₁ / 400 + k_ed∙σ_ed (f₁ / 400)²]∙B²_j1∙M_j1, (64)

P_pl1 = k_T∙σ_ed (f_z1∙B_pl1 / 100)²∙M_z1, (65)

where k_T − technological coefficient, k_T ≈ 2 for the toothing of armature and k_T ≈ 1,4 for the yoke of armature; σ_h and σ_ed − coefficients, depending on the brand and the thicknesses of steel sheet; k_h − coefficient, taking into account the unevenness of distribution of magnetic induction on the thickness of sheet; at frequency 400 Hz magnetic induction practically does not change on the thickness of sheet and it is possible to accept k_h = 1; B_z1av and B_j1 − magnetic inductions accordingly in a tooth and in a back of stator armature; В_pl1 − amplitude of pulsations of magnetic induction in the stator toothing,

В_pl1 = γ_z2∙δ∙B_zav / (2t_z1min),

where γ_z2 = [(b_sl1/δ)²]/(5 + b_sl2/δ); M_z1 and М_j1 is a mass of toothing and back of stator armature.

The pulsation losses Р_pl arise up because of toothed structure of rotor.

For the count of losses in rotor steel it is possible to apply the same formulas, what for stator; in these formulas, however, instead of f₁ it is necessary to put frequency of rotor current f₂. Because frequency f₂ is small, then losses in a rotor at normal frequency of rotation are ussualy small, what these losses are usually ignored because of.

Losses of a friction in bearing and convection losses behave to mechanical. The relative value of mechanical losses depends on frequency of rotation and power of machine. At a speed-down and increasing power of asynchronous machine the relative value of its mechanical losses diminishes. In asynchronous motors at frequency 400 Hz and power from 0,5 to 5 kW relative values of mechanical losses

Р_mech/Р_nom = 0,1 ÷ 0,03.

Losses from a friction in rolling bearing of asynchronous machines, losses of air friction and losses on a ventilator it is possible to count up by formulas (2.303) ÷ (2.306), [1]. The value of mechanical losses for asynchronous generators it is possible to accept from 0,03÷0,02 up to 0,015÷ 0,01 of a nominal active-power. The first values of losses are typical for generators by power of units kVA, and second − for generators by power in ten kVA and f = 400 Hz.

Additional losses are conditioned by the ultraharmonics of MMF, pulsations of main magnetic flux, by the presence of massive details in the construction and other. Additional losses are usually accepted by equal 0,01 of the active-power of generator.

Losses Р_ex,c consist of losses in the capacitors P_ex and losses in the control circuit Р_c.

Losses in capacitors are determined by the angle of losses tgδ:

Р_c = Р_ex∙tgδ = 2π∙10^-6∙m₁∙f₁/С_c∙U²_c∙tgδ. (66)

If capacitors are connected in a star, then U_c = U_ph1, and if in a triangle, then U_c = √3∙U_ph1. But because at the set reactive-power of capacitors Р_ex their capacity diminishes in three times, then losses in capacitors turn out identical.

The value of losses angle tgδ increases with the increase of frequency. In the range of frequencies 400÷1000 Hz it is growth small. For the capacitors К75-10 a value of tgδ at a temperature 100°C not exceed 0,008; for the capacitors К71-4 at a temperature 85°C a tgδ ≤ 0,002; for the capacitors МБГ4 at a temperature 70°C a tgδ ≤ 0,015.

Losses in the control circuit depend on the system of adjusting. At the choice of the control system with bias of armature back

P_c = m₁∙U_ph1∙k_I∙I_b,max / η_rc,

where a k_I − coefficient of current convertion in the rectification circuit; I_{b, max} − maximal value of current in bias winding; η_rc – efficiency of rectifier, accepted η_rc ≈ 0,85.

Efficiency of asynchronous machine:

а) for the tractive mode

η_M = [(P₁ − ∑P)/ P₁]∙100; (68)

b) for the generator mode

η_G = 1 − ∑Р/(Р∙соs φ + ∑P), (69)

where Р − full output power of generator.