For DC motor

(3.43)

where R1 - is a full resistance of armature circuit, Ohm;

(3.44)

where І_st – is a starting current:

(3.45)

where λ – is an overloaded ability by a current (moment),

λ = (2 ÷ 2.3);

І_r – is a rated current of the motor, A (appendix Б); R_a – is a resistance of armature circuit, Ohm (see the formula 3.25).

After determination of a factor J we can find the switching moment M2 for a motor

(3.46)

where М₁ = λ·М_r – is a starting moment, N m.

In asynchronous motors the developed moment is proportional to the square of the mains supply voltage, so even insignificant decrease of voltage at starting considerably decreases the starting moment. So, chosen moment at start should be in (25-30) % less than maximal moment, developed by the motor at rated voltage, i.e.

(3.47)

Value М₁for DC motor is determined by taken switching stage and overload capacity and shouldn`t exceed 2.7.

For reliable motor start at rated load it should to choose the moment that is created at switching of starting rheostat speed, by 10-20 % higher than total (rated) static resistance moment.

(3.48)

But some cases can take place in given course work, when accordingly to output data, chosen motor power will exceed static motor power, correspondently to set value of final load (from the overload capability testing). In this case to obtain set quantity of switching steps of starting rheostat it is necessary to define graphically the value of switching moment М₂.

Resistance of starting rheostat (see fig. 3.3) for DCM:

for asynchronous motor:

Resistance of starting rheostat sections:

Total resistance of starting rheostat for DCM:

for asynchronous motor:

(3.49)

(3.50)

(3.51)

(3.52)

(3.53)

In case of graphic calculation, R₁ is found previously (formulas 3.41 and 3.44); the scale of resistance is defined previously (see fig. 3.1.20).

[Ohm/cm] (3.54)

Then resistance of sections and the total resistance of starting rheostat are calculated:

(3.55)

3.1.21 Calculation and plotting of braking characteristics.

Plotting of regenerative braking characteristic is realized by two points (see fig. 3.3). (ω_s; М_т), (ω₀; М = 0). Let us consider that ratio of maximal moment at regenerative braking is equal to ratio of maximal moment at starting (see formulas 3.40; 3.43) i.e. М_Т = М_s3.

Step regenerative resistance is defined by graph on fig. 3.3.

(3.56)

(3.57)

Section regenerative resistance R_regs

Plotting of dynamic braking characteristic for DCM is realized by two points (ω = 0; М = 0), (ω_d = 0,3*ω_rat; М_d = 0,8*М_rat). Coordinates of a second point are set randomly.

Dynamic braking resistance R_d

_(3.58)

where gk – is found from fig. 3.3 previously, for this half line ОК is drawn in parallel to natural characteristic.

To describe processes acting at lowering of loaded (empty) wagon, i.e. shifting of point, that characterizes regime, by virtue of fig. 3.3.

3.1.22 Calculation of transient processes at motor starting.

Calculation includes the residence time defining on each step and view of current (moment) changing exponential curve at transition from step to step. To execute the calculation for loaded wagon. To combine current (moment) changing graphs at starting with speed diagram.

For DCM run time of motor t_х on considered step:

(3.59)

or

(3.60)

(3.61)

where Т_mx – el. mechanical time constant for the same step

Іs (М _s) – rated current (moment) of load, А

І_red – reduced inertia moment (kg/m²) (see formula 3.8)

М_shcх – short circuit moment (N·m) of considered step "х"

(3.62)

(3.63)

(3.64)

where R_ΣX – resulting resistance of armature circuit "х".

It can be seen from formulas (3.40 - 3.41) that with decreasing of armature circuit resistance, residence time of motor on considered step decreases. At transition of motor to natural characteristic time of achieving the constant speed value is accepted to be equal (3.4) to time constant value Т_m on considered step.

The character of exponential dependence of current (moment) changing is defined by

(3.65)

For AC motor time tх of motor run on step "х"

(3.66)

where Т_mх - el. mechanical time constant of the same step.

where І_red – reduced inertia moment (kg/m²), (see formula

3.1.7.1);

∆ω_х – speed increment on each step of lowering.

∆ω_х – is defined from fig. 3.3. So:

- for the first step of starting ∆ω_х = ∆ω₁ = ω₁;

- for second ∆ω_х = ∆ω₂ = ω₂ -ω₁;

- for third ∆ω_х = ω₃ –ω₂;

- for natural ∆ω_х = ω₄ –ω₃.

Run time from ω₄ to ω_s to accept equal (34)Tmx, defined for natural characteristic.

Exponential curve character is defined by value

(3.68)

3.1.23 Designing of principal scheme for starting control and for motor braking.

Principal scheme have to allow executing of gradual motor starting, reversing, braking regimes, residence on regulation characteristic; el. magnet brakes control must be involved in scheme.

Task details for principal scheme designing are considered by teacher with each student independently.

Explanatory note have to contain description of scheme operation.