Graphically expressed dependence of the electromagnetic moment on the slip is called mechanical characteristicasynchronous motor (Fig. 3.3).
Fig. 3.3. Mechanical characteristic of asynchronous motor
A simplified formula for calculating the electromagnetic moment of an induction motor (Kloss formula) can be used to construct the mechanical characteristic
In this case, the critical slip is determined by the formula
where λ m = M max / M nom is the overload capacity of the engine.
When calculating the mechanical characteristics, it should be borne in mind that when the slip values exceed the critical, the accuracy of the calculations is sharply reduced. This is due to the change in the parameters of the replacement circuit of an induction motor caused by magnetic saturation of the stator and rotor teeth, and an increase in the frequency of the current in the rotor winding.
The form of the mechanical characteristics of an induction motor largely depends on the values of the voltage applied to the stator winding U 1 (fig. 3.4) and active resistance of the rotor winding r"2 (Fig. 3.5).
Fig. 3.4. Stress effect U 1 on the mechanical characteristics of the asynchronous motor
The data provided in the catalogs for asynchronous motors usually do not contain information about the parameters of the equivalent circuit, which makes it difficult to use formulas for calculating the electromagnetic moment. Therefore, to calculate the electromagnetic moment, the formula
Fig. 3.5. Impact resistance r " 2 on the mechanical characteristics of the asynchronous motor
The performance properties of an induction motor are determined by its performance: the dependence of the speed n 2, shaft torque M 2, efficiency and power factor cosφ 1 from the engine payload R 2 .
When calculating the parameters for determining the performance of asynchronous motors, they use either a graphical method, which is based on the construction of a pie chart, or an analytical method.
The basis for performing any of the methods of calculating the performance characteristics are the results of the experiments of idling and short circuit. If the engine is designed, then this data is obtained in the process of its calculation.
When calculating the resistance of resistors r EBT used in stator or phase rotor circuits to limit the starting current or control the rotational speed, use the principle: for this particular asynchronous motor, slip sproportional to the active resistance of the rotor circuit of this engine. In accordance with this equality
(r 2 + r add) / s= r 2 / s Mr.,
where r 2 - active resistance of the rotor winding itself at the operating temperature; s -slip when a resistor is inserted into the rotor circuit r ext.
From this expression we obtain the formula for calculating the active resistance of the additional resistor g to 6, necessary to obtain the specified increased slip swith a given (nominal) load:
r ext = r 2 (s / s number - 1).
There are two methods for calculating starting resistors: graphical and analytical.
Graphic methodmore accurate, but requires the construction of a natural mechanical characteristics and the starting diagram of the engine, which is associated with the implementation of a large amount of graphic work.
Analytical methodthe calculation of starting rheostats is simpler, but less accurate. This is due to the fact that the method is based on the assumption of the straightness of the working section of the natural mechanical characteristics of an asynchronous motor. But when sliding close to critical, this assumption causes a noticeable error, which is even more significant, the closer the initial starting point M 1 to maximum moment M m ah. Therefore, the analytical method of calculation is applicable only for the values of the initial starting moment M 1 < 0,7 · M m ah .
Resistors on the starting rheostat steps:
the third r add3 = r 2 (λ m - 1);
second r ext2 = r add3 λ m;
the first r add1 = r Add2 λ m,
where r 2 - active resistance of the phase winding of the rotor of an induction motor,
where E 2 and I 2nom - catalog data on the selected engine size.
Starting resistors resistance on its steps:
first R PR1 = r add1 + r ext2 + r add3;
second R PR2 = r ext2 + r add3
third R PR2 = r add3
To limit the starting current of asynchronous motors with squirrel cageapply special schemes for their inclusion with elements limiting the starting current. All these methods are based on reducing the voltage applied to the stator winding. The most widely used schemes with the inclusion in the linear wires of the stator resistors or chokes (see. Fig. 3.14, b). The calculation of the required resistance of these elements for a given decrease in the starting current a, relative to its natural value, is carried out according to the formulas:
for resistors with resistance
R n = 
for chokes
x L = 
Motor impedance in short circuit mode Z ohm
Z k = U 1 /I P
Here x to and r k - inductive and active components of this resistance
R k = Z k cosφ k; x k =
Decrease in the artificial starting moment at inclusion R or L will make
α m = α 2 i
Table 3.1
Thus, if a value is set to α m, which determines the value of the artificial starting torque M ″ n, then to calculate the corresponding values R n or x Lyou can use the above formulas, substituting α 2 in them i, the value of α m
The electrical resistance of the motor windings listed in the catalogs usually correspond to a temperature of +20 ° C. But when calculating the characteristics and parameters of the motors, the resistance of their windings must be brought to the working temperature. In accordance with the current standard, the value of the operating temperature is taken depending on the heat resistance class of electrical insulation used in the engine: at the heat resistance class B, the working temperature is 75 ° C, and at the heat resistance classes F and H - 115 ° C. Recalculation of the resistances of the windings on the working temperature is performed by multiplying the resistance of the winding at a temperature of 20 ° C, the heating coefficient k t:
r= r 20 k t.
The values of this coefficient are taken depending on the purpose of the engines and their dimensions (height of the axis of rotation) (Table 3.1).
SYNCHRONOUS MACHINES
BASIC TERMS
A characteristic feature of synchronous machines is a hard link between the rotor speed n 1 and the frequency of the alternating current in the stator winding f 1:
n 1 = f 1 · 60 / r.
In other words, the rotating magnetic field of the stator and the rotor of the synchronous machine rotate synchronously,i.e. with the same frequency.
According to their design, synchronous machines are divided into over-polar and non-polar. In a clear-field synchronous machines, the rotor has pronounced poles, which are located on the field winding coils, fed by direct current. A characteristic feature of such machines is the difference in magnetic resistance along the longitudinal axis (along the axis of the poles) and along the transverse axis (along the axis passing in the interpolar space). Magnetic resistance to the flow of the stator along the longitudinal axis ddmuch less magnetic resistance to the flow of the stator along the transverse axis qq.In implicit polarized synchronous machines, the magnetic resistances along the longitudinal and transverse axes are the same, since the air gap of these machines along the stator perimeter is the same.
The design of the stator of a synchronous machine in principle does not differ from the stator of an asynchronous machine. In the stator winding during operation of the machine, an emf is induced and currents flow, which create a magnetomotive force (MDS), the maximum value of which
F 1 =0,45m 1 I 1 w 1 k ob1 / r
This MDS creates a rotating magnetic field, and in the air gap δ The machine creates a magnetic induction, the graph of which distribution within each pole division t depends on the rotor design (Fig. 4.1).
For a pole-synchronous machine, the stress equation is true:
Ú 1 =Ė 0 + Ė 1 d + Ė 1 q + Ė σ1 - İ 1 r 1
where Ė 0 - the main EMF of the synchronous machine, proportional to the main magnetic flux of the synchronous machine F 0 ; Ė 1 d - EMF response of the armature of the synchronous machine along the longitudinal axis, proportional to the MDS of the armature response along the longitudinal axis F 1 d; Ė σ1 - EMF reaction of the armature along the transverse axis, proportional to the MDS reaction of the armature along the transverse axis F 1 q; Ė σ1 - emf of scattering due to the presence of a magnetic flux of scattering F 0, the value of this EMF is proportional to the inductive dissipation resistance of the stator winding x 1
Ė σ1 = jİ 1 r
İ 1 r 1 - active voltage drop in the stator phase winding, usually this value is neglected when solving problems due to its small value.
Fig. 4.1. Distribution of magnetic induction along the transverse axis
implicit polar ( but) and the polar pole ( b) synchronous machines:
1 - mDS schedule; 2 - magnetic induction graph
For an implicit polar synchronous machine, the stress equation has the form
Ú 1 =Ė 0 + Ė c - İ 1 r 1
Here
Ė c = Ė 1 + Ė σ1
where Ė 1 - EMF of the armature response of an implicit polar synchronous machine. The stress equations considered correspond to stress vector diagrams. These diagrams have to be built to determine either the main EMF of the machine. E 0 or stator winding voltage U 1. It should be borne in mind that the voltage equations and the corresponding vector diagrams do not take into account the magnetic saturation of the magnetic circuit of a synchronous machine, which, as is well known, affects the value of inductive resistance, causing their decrease. Accounting for this saturation is a difficult task, so when calculating the EMF and voltages of synchronous machines are usually used practical chartEMF, which takes into account the state of saturation of the magnetic system, caused by the action of the armature reaction under the load of a synchronous machine. When constructing a practical EMF diagram, the magnetizing reaction force of the armature is not decomposed into longitudinal and transverse components; therefore, this diagram can be used both for calculating over-polar and implicit-polar machines.
When solving problems associated with either synchronous generators connected in parallel with the network, or with synchronous motors, use angular characteristicssynchronous machines, representing the dependence of the electromagnetic moment M from the load angle θ. It should be remembered that in pole-wise synchronous machines there are two points: the main M primary and reactive M p, and in non-polar machines - only the main point:
The load angle θ nom corresponds to the nominal moment M no. The maximum moment of a synchronous machine determines the overload capacity of a synchronous machine, which is important for both synchronous generators operating in parallel with the network and for synchronous motors. In implicit-pole synchronous machines, the maximum moment corresponds to a load angle θ = 90 °, in clear-pole machines θ kr< 90° и обычно составляет 60 - 80° в зависимости от соотношения основного и реактивного электромагнитных моментов этой машины.
To calculate the critical angle of load, which determines the overload capacity of pole-synchronous machines, you can use the expression.
The magnitude of the torque of an induction motor is greatly influenced by the phase shift between the current I 2 and e. d. E 2S rotor.
Consider the case when the inductance of the rotor winding is small and therefore the phase shift can be neglected (Fig. 223, a).
The rotating magnetic field of the stator is replaced here by the field of the N and S poles rotating, suppose clockwise. Using the rule of the right hand, we determine the direction of e. d. and currents in the rotor winding. The currents of the rotor, interacting with a rotating magnetic field, create a torque. The directions of the forces acting on the conductors with current are determined by the rule of the left hand. As can be seen from the drawing, under the action of forces, the rotor will rotate in the same direction as the rotating field itself, i.e., clockwise.
Consider the second case when the inductance of the rotor winding is large. In this case, the phase shift between the rotor current I 2 and e. d. The rotor E 2S will also be large. FIG. 223, b. The magnetic field of the stator of the induction motor is still shown as clockwise rotating poles N and S. Direction induced in the rotor winding e. d. remains the same as in FIG. 223, a, but due to the current lagging in phase, the axis of the magnetic field of the rotor will no longer coincide with the neutral line of the stator field, but will shift by some angle against the rotation of the magnetic field. This will lead to the fact that along with the formation of a torque directed in one direction, some conductors will create a counter torque.
This shows that the total torque of the engine during a phase shift between current and e. d. the rotor is smaller than in the case when I 2 and E 2S coincide in phase. It can be proved that the torque of an induction motor is determined only by the active component of the rotor current, i.e., the current I 2 cos and that it can be calculated by the formula:
Ф m - magnetic stator flux (as well as approximately equal to the resulting magnetic flux of the induction motor);
The phase angle between e. d. and phase winding current
C is a constant factor.
After substitution:
From the last expression it is clear that the torque of the induction motor depends on the slip.
FIG. 224 shows the curve And the dependence of the torque of the engine from the slip. From the curve it can be seen that at the moment of start, when s = l and n = 0, the engine torque is small. This is due to the fact that at the time of start-up, the frequency of the current in the rotor winding is greatest and the inductive resistance of the winding is large. Because of this, cos is of little importance (by
|
row 0.1-0.2). Therefore, despite the large starting current, the starting torque will be small.
With some slip S 1 the engine torque will have the maximum value. With a further decrease in slip or, in other words, with a further, the slightest increase in engine speed, its torque will quickly decrease.
If the s = 0 slip, the torque of the engine will also be zero.
It should be noted that in an asynchronous motor, slip, equal to zero, is practically impossible. This is possible only if the rotor is informed from the outside by a torque in the direction of rotation of the stator field.
The starting torque can be increased if at the moment of starting the phase shift between current and e is reduced. d. rotor. From the formula
it can be seen that if, with a constant inductive resistance of the rotor winding, to increase the active resistance, then the angle itself will decrease, which will lead to the fact that the motor torque will become larger. This is used in practice to increase the starting torque of the engine. At the moment of start-up, active resistance (starting rheostat) is introduced into the rotor circuit, which is then output as soon as the engine increases speed.
An increase in the starting torque causes the maximum torque of the engine to be obtained with a larger slip (point S 2 of the curve B in Fig. 224). By increasing the active resistance of the rotor circuit at start-up, it is possible to ensure that the maximum torque will be at the moment of start-up (s = 1 of curve C).
The torque of the induction motor is proportional to the square of the voltage, so even a small decrease in voltage is accompanied by a sharp decrease in torque.
Power P 1 supplied to the stator winding of the induction motor is equal to:
where m 1 is the number of phases.
The stator has the following energy losses:
1) in the stator winding P es. = m 1 I 1 2 r 1;
2) in the stator steel there is a hysteresis and eddy currents Р C.
The power delivered to the rotor is the power of a rotating magnetic field, also called electromagnetic power P eM.
Electromagnetic power is equal to the difference between the power supplied to the motor and the stator losses of the motor, i.e.
|
The difference between Р эМ and represents the electric losses in the rotor winding Р еP, if we neglect the losses in the steel of the rotor due to their insignificance (the frequency of the magnetization reversal of the rotor is usually very small):
Consequently, the losses in the rotor winding are proportional to the slip of the rotor.
If from the mechanical power developed by the rotor, subtract the mechanical loss P mx due to friction in the rotor bearings, friction against the air, etc., as well as additional losses P D arising under load and caused by the scattering fields of the rotor, and the losses caused by: magnetic ripples field in the teeth of the stator and rotor, then there will be useful power on the motor shaft, which we denote by P 2.
The K. p. D. Induction motor can be determined by the formula:
It is clear from the last expression that the torque of an asynchronous motor is proportional to the product of the magnitude of the rotating magnetic flux, the rotor current and the cosine of the angle between e. d. rotor and its current,
From the equivalent circuit of an asynchronous motor, we obtain the magnitude of the reduced rotor current, which we present without proof.
The magnitude of the torque of an induction motor is greatly influenced by the phase shift between the current I 2 and e. d. E 2S rotor.
Consider the case when the inductance of the rotor winding is small and therefore the phase shift can be neglected (Fig. 223, a).
The rotating magnetic field of the stator is replaced here by the field of the N and S poles rotating, suppose clockwise. Using the rule of the right hand, we determine the direction of e. d. and currents in the rotor winding. The currents of the rotor, interacting with a rotating magnetic field, create a torque. The directions of the forces acting on the conductors with current are determined by the rule of the left hand. As can be seen from the drawing, under the action of forces, the rotor will rotate in the same direction as the rotating field itself, i.e., clockwise.
Consider the second case when the inductance of the rotor winding is large. In this case, the phase shift between the rotor current I 2 and e. d. The rotor E 2S will also be large. FIG. 223, b. The magnetic field of the stator of the induction motor is still shown as clockwise rotating poles N and S. Direction induced in the rotor winding e. d. remains the same as in FIG. 223, a, but due to the current lagging in phase, the axis of the magnetic field of the rotor will no longer coincide with the neutral line of the stator field, but will shift by some angle against the rotation of the magnetic field. This will lead to the fact that along with the formation of a torque directed in one direction, some conductors will create a counter torque.
This shows that the total torque of the engine during a phase shift between current and e. d. the rotor is smaller than in the case when I 2 and E 2S coincide in phase. It can be proved that the torque of an induction motor is determined only by the active component of the rotor current, i.e., the current I 2 cos and that it can be calculated by the formula:
Ф m - magnetic stator flux (as well as approximately equal to the resulting magnetic flux of the induction motor);
The phase angle between e. d. and phase winding current
C is a constant factor.
After substitution:
From the last expression it is clear that the torque of the induction motor depends on the slip.
FIG. 224 shows the curve And the dependence of the torque of the engine from the slip. From the curve it can be seen that at the moment of start, when s = l and n = 0, the engine torque is small. This is due to the fact that at the time of start-up, the frequency of the current in the rotor winding is greatest and the inductive resistance of the winding is large. Because of this, cos is of little importance (by
|
row 0.1-0.2). Therefore, despite the large starting current, the starting torque will be small.
With some slip S 1 the engine torque will have the maximum value. With a further decrease in slip or, in other words, with a further, the slightest increase in engine speed, its torque will quickly decrease.
If the s = 0 slip, the torque of the engine will also be zero.
It should be noted that in an asynchronous motor, slip, equal to zero, is practically impossible. This is possible only if the rotor is informed from the outside by a torque in the direction of rotation of the stator field.
The starting torque can be increased if at the moment of starting the phase shift between current and e is reduced. d. rotor. From the formula
it can be seen that if, with a constant inductive resistance of the rotor winding, to increase the active resistance, then the angle itself will decrease, which will lead to the fact that the motor torque will become larger. This is used in practice to increase the starting torque of the engine. At the moment of start-up, active resistance (starting rheostat) is introduced into the rotor circuit, which is then output as soon as the engine increases speed.
An increase in the starting torque causes the maximum torque of the engine to be obtained with a larger slip (point S 2 of the curve B in Fig. 224). By increasing the active resistance of the rotor circuit at start-up, it is possible to ensure that the maximum torque will be at the moment of start-up (s = 1 of curve C).
The torque of the induction motor is proportional to the square of the voltage, so even a small decrease in voltage is accompanied by a sharp decrease in torque.
Power P 1 supplied to the stator winding of the induction motor is equal to:
where m 1 is the number of phases.
The stator has the following energy losses:
1) in the stator winding P es. = m 1 I 1 2 r 1;
2) in the stator steel there is a hysteresis and eddy currents Р C.
The power delivered to the rotor is the power of a rotating magnetic field, also called electromagnetic power P eM.
Electromagnetic power is equal to the difference between the power supplied to the motor and the stator losses of the motor, i.e.
|
The difference between Р эМ and represents the electric losses in the rotor winding Р еP, if we neglect the losses in the steel of the rotor due to their insignificance (the frequency of the magnetization reversal of the rotor is usually very small):
Consequently, the losses in the rotor winding are proportional to the slip of the rotor.
If from the mechanical power developed by the rotor, subtract the mechanical loss P mx due to friction in the rotor bearings, friction against the air, etc., as well as additional losses P D arising under load and caused by the scattering fields of the rotor, and the losses caused by: magnetic ripples field in the teeth of the stator and rotor, then there will be useful power on the motor shaft, which we denote by P 2.
The K. p. D. Induction motor can be determined by the formula:
It is clear from the last expression that the torque of an asynchronous motor is proportional to the product of the magnitude of the rotating magnetic flux, the rotor current and the cosine of the angle between e. d. rotor and its current,
From the equivalent circuit of an asynchronous motor, we obtain the magnitude of the reduced rotor current, which we present without proof.
The moment developed by the engine is equal to the electromagnetic power divided by the synchronous speed of rotation of the electric drive.
M = P em / ω 0
Electromagnetic power is the power transmitted through the air gap from the stator to the rotor, and it is equal to the losses in the rotor, which are determined by the formula:
P em = m I 2 2 (r 2 ’/ s)
m is the number of phases.
M = M em = (Pm / ω 0) (I 2 ’) 2 (r 2’ / s)
The electromechanical characteristic of an asynchronous motor is the slip dependence of I2 ’. But since the asynchronous machine works only as an electric motor, the main characteristic is the mechanical characteristic.
M = Me m = (Pm / ω 0) (I 2 ’) 2 (r 2’ / s) is a simplified expression of a mechanical characteristic.
Substituting the value of current into this expression, we get: M = / [ω 0 [(r 1 + r 2 ’/ s) 2 + (x 1 + x 2’) 2]]
Instead of ω 0, it is necessary to substitute the mechanical speed, as a result of which the number of pairs of poles is reduced.
M = / [ω 0 [(r 1 + r 2 ’/ s) 2 + (x 1 + x 2’) 2]] is an equation for the mechanical characteristics of an induction motor.
When an asynchronous motor goes into generator mode, the rotational speed is ω\u003e ω 0 and the slip becomes negative (s When the slip changes from 0 to + ∞, the mode is called “electromagnetic brake mode”.
Given the slip values from о to + ∞, we obtain the characteristic:
Full mechanical characteristic of the asynchronous motor.
As can be seen from the mechanical characteristics, it has two extremes: one on the segment of the change in slip in the section from 0 to + ∞, and the other on the interval from 0 to -∞. dM / ds = 0
M max = /] + refers to the motor mode. - refers to the generator mode.
M max = M kr M kr - the critical moment.
The slip, at which the moment reaches a maximum, is called critical slip, and it is determined by the formula: s cr = ±
Critical slip has the same value in both motor and generator modes.
The value of M kr can be obtained by substituting the value of the critical slip in the moment formula.
The moment of sliding equal to 1 is called the starting moment. The expression for the starting torque can be obtained by substituting 1 in the formula:
M n = / [ω 0 [(r 1 + r 2 ’) 2 + (x 1 + x 2’) 2]]
Since the denominator in the formula of maximum moment is several orders of magnitude greater than U f, it is considered to be M cr р U f 2.
The critical slip depends on the magnitude of the active resistance of the rotor winding R 2 ’. The starting torque, as can be seen from the formula, depends on the active resistance of the rotor r 2 ’. This property of the starting torque is used in asynchronous motors with a phase rotor, in which the starting torque is increased by introducing active resistance into the rotor circuit.
7. No-load transformer
The no-load mode of a transformer is called an operating mode when one of the transformer windings is supplied from a source with alternating voltage and with open circuits of other windings. This mode of operation may be in a real transformer when it is connected to the network, and the load supplied from its secondary winding is not yet included. A current I 0 passes through the primary winding of the transformer, at the same time there is no current in the secondary winding, since its circuit is open. The current I 0, passing through the primary winding, creates a sinusoidally varying tray F 0 in the magnetic core, which due to magnetic losses lags in phase from the current by a loss angle δ. 