2. Determination of main sizes and calculation of asynchronous machine

The principal sizes of asynchronous machine (diameter and length of armature) are determined by operation mode and value of external load. If an asynchronous machine is designed for operation in motor and generator modes with commensurable powers, then a calculation must be conducted for these two modes and the sizes of machine get out those which turned out most. If load in the motor mode there is the far less load of the generator mode, then its sizes are determined by the generator mode. Therefore at the choice of parameters all requirements are taken into account on providing of normal motor mode. If an asynchronous machine operates only in the generator mode, then its sizes are determined by the generator mode.

Fig. 4

The principle sizes of asynchronous machine can be defined by means of the known calculation equation:

D²l_i = (6,1∙10⁴P_el) / α_i∙k_sh∙k_w1∙A∙B_δ∙n (6)

Here Р_el − electromagnetic power, V∙A. Electromagnetic power is determined:

а) for the motor mode

Р_el = P₁E_ph1 / U_nom (7)

where Р₁ − power, consumed from a network; η – efficiency of motor; соs φ_M − power-factor of motor; k_E = Е_ph1/U_nom − coefficient, taking into account a voltage drop across active and inductive resistances of motor; k_E = 0,8÷0,9 − value, dependent on the number of pole pairs; large values behave to the two-poles motors; Р₂ − power at the output (on a shaft).

The rated value of primary current can be preliminary defined by a formula

I_1nom = P₁ / (m₁U_nom)

b) for the generator mode

Р_el = т₁Е_ph1 /I₁ = m₁ U_nom I_1nom k_Е; (8)

it is preliminary possible to accept k_E = 1.

Unlike synchronous machines an armature current value for the asynchronous generators is not known, because it depends on the values of currents: through an excitative capacitor and magnitizing current. The exact value of stator current can be defined from the plotting of vector diagram of voltage at a test calculation. Preliminary the value of stator current can be defined by means of expression (for nominal power)

I_1nom = I_nom cosφ / cos φ_G (9)

where I_nom − current of load (nominal); соs φ – power factor of load; φ_G − phase shift angle between voltage and armature current I₁.

For generators by power about 1 kV∙A

tg φ_G = 0,5÷ 0,7; cos φ_G = 0,86÷0,81.

The values of angles φ_G depend on generator load in a strong degree. The less load current, the anymore angle φ_G, and the armature current increases.

If in initial equation (6) to enter the relation λ_i = l_i/D, then

D = ³√(6,1∙10⁴P_el) / (α_i∙k_ph∙k_w1∙A∙B_δ∙n∙λ_i) (10)

At the choice of the electromagnetic load A and B_δ as it applies to asynchronous machines it is necessary to take into account the following.

1.Choice of large values B_δ results in the increase of magnetizing current

I_m = pF₂ / 0,9m₁k_w1w_ph, (11)

I_m ≡ B_δ² (12)

because F_∑ ≡ B_δ; w_ph1 ≡ 1/ B_δ; F_∑ − total MMF of magnetic circuit.

2. With the increase of B_δ the power-factor соs φ_M goes down, dispersion diminishes, the short circuit current increases.

3. The value of excitant capacity increases with the increase of B_δ, because the magnetizing current I_m increases. General mass of the system increases.

4. Not only the numerical value of product A∙B_δ, but also their correlation A / B_δ have influence on the value of inductive resistances of short circuit Х_sc and magnetizing Х_m, and also on the value of magnetizing current I_m.

Increase of ratio A / B_δ increases a value Х_sc, because

X_sc* = X_sc I_1nom / U_nom ≡ Λ А/В_δ, (13)

where X_sc = Х_s1+ Х’_s2 − total inductive resistance of dispersion; Λ − coefficient, characterizing dispersion.

The less value of X_sc, the more torque, the more overload of motor. Therefore, if it is need, for example, to improve М_max, it is necessary to increase B_δ and to decrease A.

If to accept F_∑ =1,6∙k_δ1∙ k_δ2∙В_δ∙δ∙k_μ and to take into account that (m₁∙w_ph1 / p) I_1nom =Аτ, that

I_m /I_1nom = 1,78∙k_μ(k_δ1∙ k_δ2) / (δ/τ)(В_δ/A)10⁴ (14)

X_m = 0,56(k_w1/k_μ)( δ/τ’)(A/В_δ)(U_nom/I_1nom)10^-4 (15)

i.е. at increase of B_δ and diminishing of A a relative value of I_m /I_1nom increases and, so, соsφ_M diminishes.

For asynchronous machines at the choice of values A and B_δ it is necessary to pay attention to the got values of magnetizing current I_m, and so, соsφ_M, and also on the coefficient of overload k_ovl, determined by Х_sc.

At A∙B_δ = const for the increase of power-factor соsφ_M it is necessary to diminish a value B_δ and to increase a value A, and for the increase of overload ability k_ovl it is necessary to increase a value B_δ and to diminish a value A.

This contradiction is solved by the choice of such ratio between A and B_δ, at which the most value of power-factor is provided at the given value of overload ability.

Recommended values of B_δ and A for asynchronous machines, operating in the continuous duty, driven accordingly in the table. 1 and 2 (f = 400 Hz, m₁= 3, 2p = 4).

Tabl. 1

P2, kW	0,1	0,2	0,5	0,75	1,0	2,0	5,0	10
Вδ, T	0,36	0,48	0,55	0,57	0,6	0,62	0,64	0,66

Tabl. 2

P2, kW	0,1	0,2	0,5	0,75	1,0	2,0	5,0	10
A, A/cm	160	180	220	235	250	270	290	310

For asynchronous machines, operating in the individual systems of electrical equipment, it is rationally to choose frequency of rotation most. With the increase of frequency of rotation mass of machine diminishes. So, for example, mass of machine by power of 1 kW at n = 6000 r/m is approximately in two times more as compared to the same machine at n = 12 000 r/m. For individual turbine-generator sets at f = 400 Hz usually frequency of rotation is got out equal to 24 000 r/m (2р = 2). At the choice of frequency of rotation it is necessary to check a machine for mechanical durability. For asynchronous machines a permissible linear speed equals to 80m/s.

Data of tables. 1 and 2 for a choice of optimal electromagnetic load A and B_δ are correct also for bipolar asynchronous machines with n = 24 000 r/m, what the curves of specific mass are evidence in a fig. 5.

I f the designed asynchronous machine in the traction mode operates on flyweight with the great moment of inertia, then in order to avoid its overheat during acceleration it is necessary considerably to reduce the linear load (up to 80÷100 A/cm at power about 1÷2 kW).

With the increase of rotation frequency and power of asynchronous machine an efficiency η and power factor Fig. 5 соsφ_M are increased. Thus the less a power of machine, the stronger influence of rotation frequency on values of η and соsφ_M, that is explained by the known correlations for specific losses and magnetizing current:

ΔP =P/P_el ≡ M_am/P_el ≡ (P_el/n)^3/4/P_el = 1/(P_el^1/4n^3/4), (16)

i_m = I_m/I_1nom = l/l² = ⁴√(1/P_el) = 1/P_el^1/4 (17)

where М_am − mass of active materials.

Specific losses on unit of power (at B_δ = const and j_а = const) are proportional to 4-th root of power and 3-d root of frequency of rotation, i.e. with the increase of rotation frequency the specific losses fall and efficiency increases.

A magnetizing current is directly proportional to linear dimensions, and the load current is directly proportional to the second degree of linear dimensions.

So, relative value of magnetizing current i_m = I_m/I_1nom is inversely proportional to linear dimensions, i.е. a relative magnetizing current goes down with increasing power, that corresponds to growth of соsφ_M for asynchronous machines. The increase of rotation frequency acts just the same, if to take into account that with the increase of n at f = const a ratio δ'/τ diminishes.

If operation of asynchronous machine is provided for traction mode, then it have to set oneself preliminary by values of efficiency and power-factor соsφ_M. Recommended values η and соsφ_M for three-phase asynchronous motors by frequency of f = 400 Hz and number of poles 2p = 4 in the function of power driven to the table. 3.

Tabl. 3

P2,W	25	50	100	200	500	750	1000
η	0,40	0,53	0,63	0,70	0,76	0,77	0,78
соsφM	0,53	0,62	0,70	0,76	0,77	0,77	0,78

At р = 1 a value of η and соsφ_M more than in a table. So, for example, at Р₂ = 1000 W: η = 0,79, соsφ_M = 0,86.

For asynchronous machines attitude of length of machine toward the diameter λ = l/D ≈ λ_i gets out greater, than for synchronous machines, i.е. asynchronous machines are executed relatively longer, than synchronous, at the same quantity of poles.

There is an area of optimal value of λ_i which gives the least value of mass and fabrication cost at given energy rates. On the basis of theoretical researches and their practical confirmation for asynchronous machines next optimal values are set for λ_i = f (р) at condition that surface speed and flywheel moment (GD²) does not limit the choice of diameter value:

λ_i = l_i / D = 1,6p^-2/3 (18)

λ’_i = l_i / τ = ³√p (19)

The optimal value of λ_i at given number of pole pairs depends also on the diameter (powers) of machine – is decreased at the increase of diameter. The optimum of ratio λ_i diminishes with increasing of power. The choice of ratio λ_i in defiance of geometric similarity is often determined by the terms of machine cooling.

As it applies to the asynchronous generators of small power dependence of specific mass of attitude of armature length toward its diameter illustrated by curves, brought around to a fig. 6.

A s researches show, asynchronous generators must to have less values λ_i, then asynchronous motors, for providing of the small nominal sliding.

By the chosen value λ_i a calculation armature length is determined: l_i = λ_i D (20)

Checking of this value is thus conducted under the required value of E.M.F. in generator mode: l_i = (E_ph1I_1nom ∙p∙m₁) / (k_sh∙k_w1∙f∙π²∙α_i∙A∙B_δ∙D²) (21)

Length of active part of armature in the first approaching it is possible to assume by an Fig. 6 equal to calculation.