Examples of calculations for Section 5
Example 19
For an asynchronous motor with a phase rotor, calculate and construct natural and artificial mechanical characteristics: a) when a rotor is inserted into the chain ; b) with a reduced value of the supply voltage Engine data:
Solution
Calculation of the mechanical characteristics of the induction motor will be carried out according to Closs's revised formula:
where - the critical moment; - critical slip;
Coefficient of reduction of resistances:
The given values of rotor resistance:
Inductive short-circuit resistance:
The critical moment of the motor at the nominal value of the supply voltage:
where - with the ideal idle speed.
Critical slip on the natural characteristic:
Equation of natural mechanical characteristics:
Using this equation, the natural mechanical characteristic of an induction motor is calculated (Figure 10.27, curve 1). To construct the rheostatic mechanical characteristic at we define the total reduced resistance of the rotor chain:
where - the amount of additional resistance introduced into the rotor circuit.
Fig. 10.27. Mechanical characteristics of the induction motor:
1 - natural: 2, 3 - artificial
Critical slip on rheostat characteristic: Equation of rheostatic characteristic:
The rheostatic mechanical characteristic corresponding to this equation has the form (Figure 10.27, curve 2).
To construct a mechanical characteristic with Let's define the critical moment:
The equation of the artificial mechanical characteristic at В looks like:
The corresponding mechanical characteristic has the form (Figure 10.27, curve 3).
Example 20
For the engine considered in Example 19, calculate the resistance of the starting rheostat under the normal start-up mode in three steps and at , using the analytical method.
Solution
Since the starting mode is normal, the switching moment () must exceed the static by 10 ... 20%.
We accept . The ratio of the maximum moment at start-up () by the moment of switching is found by the formula (2):
We perform a check for the maximum (peak) moment, which should be less than the critical one:
Resistance of the stages of the starting rheostat when included in a single star:
Example 21
For the engine considered in example 19, calculate the resistance of the starting rheostat under the normal starting mode in three steps and using the approximate facsimile method.
Solution
When calculating the starting resistances, the approximate graphic method is based on the straightness of the mechanical characteristic, and the calculation is carried out as for a direct current motor with parallel excitation. In Fig. 10.28. The starting characteristics of the engine for the specified starting conditions are constructed.
Maximum starting torque:
Switching torque:
According to the graph in Fig. 9.28 determine the resistance of the stages of the starting rheostat:
Fig. 10.28. Starting characteristics of an asynchronous motor built in an approximate way
Comparing the results of the resistance calculations in examples 20 and 21, we see that the discrepancy is not more than 5%. Therefore, in practical calculations, you can use any of the methods considered.
Example 22
For an induction motor with a squirrel cage rotor, calculate and construct the natural and artificial mechanical characteristics for the two values of the stator current frequencies/= 35 Hz and/= 20 Hz, with the proportional law of the stator voltage:
Engine data:
Solution
We will calculate the mechanical characteristics according to the procedure given in [8, 16]. With the proportional law of frequency regulation the electromagnetic moment of the engine is determined by the formula:
where t is the number of stator phases; (/, "- phase nominal stator voltage at a frequency of 50 Hz; - relative frequency; is a coefficient dependent on the frequency .
The coefficient at a frequency of 50 Hz is equal to 1.03 and increases with decreasing frequency. At a frequency of 20 Hz . Without a large error, one can neglect the change in this coefficient.
Table 10.8
5 |
M, Nm |
(O, C-1 |
M, Hm |
CO, with_ | |
M, Hm |
(About, With "1"/ |
0.000 |
0 |
78.5 |
0 |
54.9 |
0 |
31.4 |
0.010 |
121 |
75.3 |
85 |
52.7 |
49 |
30.1 |
0.030 |
334 |
73.8 |
239 |
51.7 |
139 |
29.5 |
0.100 |
703 |
68.5 |
580 |
48.0 |
374 |
27.4 |
0.136 |
732 |
65.7 |
626 |
47.0 |
418 |
27.0 |
0.150 |
729 |
64.7 |
668 |
45.3 |
475 |
26.7 |
0.170 |
717 |
63.2 |
681 |
44.2 |
508 |
26 |
0.200 |
688 |
61.0 |
685 |
42.6 |
534 |
24.3 |
0.300 |
570 |
53.0 |
633 |
37.3 |
576 |
21.3 |
0.500 |
398 |
38.0 |
489 |
26.6 |
535 |
15.2 |
1,000 |
218 |
0 |
287 |
0 |
375 |
0 |
We calculate the motor speed at various frequencies using the formula:
In Table. 10.8 shows the results of calculations of the natural mechanical characteristic at Hz.
In Fig. 10.29 built the natural (curve 1) and artificial (curves 2 for Hz and 3 for Hz) the mechanical characteristics of the induction motor
Example 23
Calculate and build the angular characteristic of a synchronous, explicitly polar motor
Engine data:
Solution
The equation for the angular characteristic of a synchronous, pole-pole motor is:
Rated angular speed:
Rated motor EMF:
In Table. 10.9 shows the results of calculations of the angular characteristic of a synchronous motor, where the following notation is used:
Fig. 10.29. Mechanical characteristics of the induction motor:
1 - natural: 2 - artificial at /= 35 Hz; 3 - artificial at /= 20 Hz
• Synchronous torque component
• reactive torque component
The results of calculating the angular characteristics of a synchronous motor are summarized in Table. 10.9.
Table 10.9
0 |
0 |
0 |
0 |
15 |
2.88 |
2.03 |
4.92 |
30 |
5.57 |
3.52 |
9.1 |
45 |
7.88 |
4.07 |
11.95 |
60 |
9.65 |
3.52 |
13.2 |
75 |
10.7 |
2.04 |
12.74 |
90 |
11.1 |
0 |
11.1 |
105 |
10.7 |
-2.03 |
8.7 |
120 |
9.65 |
-3.52 |
6.13 |
135 |
7.88 |
-4.07 |
3.8 |
150 |
5.57 |
-3.52 |
2 |
165 |
2.89 |
-2.04 |
0.85 |
180 |
0 |
0 |
0 |
Fig. 10.30. Angular characteristic of a synchronous motor:
1 - explicit pole, 2 - non-pole pole, 3 - reactive torque component
In Fig. 10.30 shows the angular characteristic of a pole-pole synchronous motor , constructed in accordance with the data in Table. 10.9.
thematic pictures
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