chapter 03 cont

Prévia do material em texto

Dynamic modelling of the induction motor
Vector control purpose 
Mechanical motion
Linear motion
For linear motion, the forces acting on a body may usually be simplified to a driving force, Fe, acting on the mass, and an opposing force (or load), Fl, as shown on Figure 1.
Figure 1: A body acted on by two forces.
For translational motion the following may be written:
In any speed and position control of linear motion, force is the fundamental variable which needs to be controlled.
Rotary motion
If the motion is rotary about an axis instead of translational, a situation as shown in Figure 2 arises.
Figure 2: A body acted on by two torques.
For rotary motion the following may be written:
In any speed and position control of rotary motion, torque is the fundamental variable which needs to be controlled.
Torque in an electric drive
Electromagnetic torque produced by a motor is opposed by load torque. The difference, 
, will accelerate the system.
Figure 3: A load acted on by a motor
For motor-load motion the following may be written:
Torque is the fundamental variable which needs to be controlled. Note that under steady state conditions angular speed is constant and 
.
DC-motor drive performance
One of the most essential qualities of a motor is the ability to generate torque. The total torque may be described by
where Ia is the current flowing in the armature and ka becomes a factor describing the physical shape of the winding. DC machine equivalent circuit is shown in Figure 4.
Figure 4: DC machine equivalent circuit
To change 
 as a step, the armature current ia is changed as a step by the power-processing unit as shown in Figure 5.
Figure 5: DC-motor drive performance
Emulation of DC-motor drive performance
In vector control of induction-motor drives, the stator phase currents 
, 
 and 
 are controlled in such a manner that 
 delivers the desired electromagnetic torque while 
 maintains the peak rotor-flux density at its rated value. The references values 
 and 
 are generated by the torque, speed, and position control loops. The total torque may be described by
Simulation of induction machine using Matlab and Simulink
Traditionally in analysis and design of 3-phase induction motors, the “per-phase equivalent circuit” is shown in Figure 6 has been widely used. In the circuit, Rs (Rr) is the stator (rotor) resistance and Lm is called the magnetizing inductance of the motor. Note that stator (rotor) inductance Ls (Lr) is defined by
Ls = Lls + Lm, 		Lr = Llr + Lm 						(1.1)
where Lls (Lrs) is the stator (rotor) leakage inductance. Also note that in this equivalent circuit, all rotor parameters and variables are not actual quantities but are quantities referred to the stator.
Figure 6: Conventional Per-phase Equivalent Circuit
It is also known that induction motors do not rotate synchronously to the excitation frequency. At rated load, the speed of induction motors are slightly (about 2 - 7% slip in many cases) less than the synchronous speed. If the excitation frequency injected into the stator is wsyn and the actual speed converted into electrical frequency unit is wm, slip s is defined by
s = (wsyn - wm ) / wsyn = wslip / wsyn, 						(1.2)
and wslip is called the slip frequency which is the frequency of the actual rotor current.
Although the per-phase equivalent circuit is useful in analyzing and predicting steady-state performance, it is not appropriate to explain dynamic performance of the induction motor.
Dynamic model in space vector form
In an induction motor, the 3-phase stator windings are designed to produce sinusoidally distributed mmf in space along the airgap periphery. Assuming uniform airgap and neglecting the effects of slot harmonics, distribution of magnetic flux will also be sinusoidal. It is also assumed that the neutral connection of the machine is open so that phase voltages, currents and flux linkages are always balanced and there are no zero phase sequence component in the system.
For such machines, the notation in terms of the space vector is very useful. For a sinusoidal 3-phase quantity of constant rms value, the space vector of the stator voltage, current and flux linkage are constant-magnitude vectors rotating at the frequency of the sinusoid with respect to the fixed (stationary) reference frame. With space vector notation, voltage equations on the stator and rotor circuits of induction motors are,
			
where the voltages v and currents i are vectors, and where the resistance R and inductance L are matrices. Eq. 34 describes the electromagnetic system by a set of 6x6 matrices of differential equations. The coupling between stator and rotor is dependent on the rotor-position.
Phase transformation
In many cases, analysis of induction motors with space vector model is complicated due to the the fact that we have to deal with variables of complex numbers. When induction motors are controlled by a vector drive, control computation is often done in the synchronous frame. Since actual stator variables either to be generated or to be measured are all in stationary a-b-c frame, frame transform should be executed in the control. The most popular transform is between stationary a-b-c frame quantities to synchronously rotating d-q quantities.
If the goal is to create a rotating space vector describing a circle, three phases with sinusoidal currents are not necessary. From analytic geometry it is known that the circle may be described by two coordinates in space (x and y). This may be used in this case, by placing two coils at 90º and by supplying them with sinusoidal current displaced by 90º (or π/2). These two coils are usually named the d-coil and the q-coil. In the rotating frame of reference the frame of reference in regard to the phase A is named the d-axis (for direct axis) and the other axis is named the q-axis (for quadrature axis).
This method reduces the three-phase system to a two-phase system.
Doing this, it is possible to model the cross-couplings between the individual coils. A further advantage is that in steady-state, the currents flowing in the coils are a DC current. Using the rotating frame of reference, the differentials of any state value (d/dt) are zero in steady-state and when the differentials are different from zero, they give the change from steady-state only. The rotating frame of reference also has the advantage that the rotor-angle is known (it is a state).
Transformation of currents, voltages, flux-linkage, etc.
What remains is to define a method for performing the phase transformations to the rotating frame of reference. The transformation is done by defining a transformation-matrix for the systems as
where f denote currents, voltages, flux-linkage, etc. For current case, this is shown in Figure 7.
Figure 7. Transformation of phase quantities into dq winding quantities (current case).
The electromagnetic torque
Properly the most important task for the induction motor is to produce a torque on the shaft. The developed torque may be written on the flowing form,
d-q equivalent circuit
The result from the above is a set of equations describing the electromagnetic system in the rotating frame of reference. The equations describing the system may be interpreted as equivalent circuits, which may help in understanding the dynamics of the system.
Using this set of equations, it is possible to construct an equivalent-diagram of the d-, and q-axis individually. For the rotating frame of reference the resulting equivalent diagram for each of the axis is shown in Figure 105 .
a) d-axis
b) q-axis
Figure 8. dq-winding equivalent circuits.
Solving the system: Voltages as Inputs
What remains is to find a strategy for solving the differential equations givenin Eq. 5555. One possibility exist: solving for the flux linkages and then calculating the currents. The flux linkage associated with the d-, q-axis are calculated as
Computer simulation
In order to carry out computer simulations, it is necessary to calculate intial values of the state variables, that is, of the flux linkages of the dq windings. These can be calculated in terms of the initial values of the dq windings currents. These currents allow us to compute the electromagnetic torque in steady state, thus the initial loading of the induction machine. Initial conditions are computed in Example 3-1 and in the matlab file EXE_1.m (or EXE_2.m).
Finally, the Simulink model is shown in Figure 9.
Electromagnetic Torque on the Rotor d-Axis
		weber-vueltas
La figura 2.11 muestra las relaciones de causalidad entre i, H, B, Φ, y λ.
El flujo enlazado por el devanado q del rotor es
El devanado q tiene una inductancia constante 
. Por tanto, la fuerza magnetomotriz que genera este flujo enlazado es
El campo magnetico H (aplicando 
)
La densidad de campo en el entrehierro debida a 
 es
Del mismo modo la densidad de campo en el entrehierro debida a 
 es
Por lo tanto:
Analisis dinamico en terminos de los devanados dq
The concept of vector control has opened up a new possibility that induction motors can be controlled to achieve dynamic performance as good as that of DC or brushless DC motors. In order to understand and analyze vector control, the dynamic model of the induction motor is necessary. It has been found that the dynamic model equations developed on a rotating reference frame is easier to describes the characteristics of induction motors.
It is the objective of the article to derive and explain induction motor model in relatively simple terms by using the concept of space vectors and d-q variables. It will be shown that when we choose a synchronous reference frame in which rotor flux lies on the d-axis, dynamic equations of the induction motor is simplified and analogous to a DC motor.
	representacion en los devanados dq
	relaciones matematicas de los devanados dq
	torque electromagnetico
PRINCIPLES OF VECTOR CONTROL
So far, we have not paid attention to the alignment of the rotating reference frame with respect to the physical coordinate. Noting in Eq. 3.28 that torque is directly proportional to Iqs if λqr = 0, one can choose the rotating d-axis to be the angle of the rotor flux linkage. In fact, this choice offers a lot of advantages of simplifying control and analysis of the motor. Other choices frequently used in direct vector control are stator flux linkage frame (d-axis is aligned to the stator flux linkage) and airgap flux linkage frame, which will be discussed briefly at the end of the section.
_1172444234.unknown
_1172472774.unknown
_1172480692.unknown
_1173686816.unknown
_1173693406.unknown
_1173693527.unknown
_1173686911.unknown
_1173687252.unknown
_1173687378.unknown
_1173687025.unknown
_1173686841.unknown
_1173686493.unknown
_1173686512.unknown
_1172847664.unknown
_1172472866.unknown
_1172476731.unknown
_1172472819.unknown
_1172444372.unknown
_1172444441.unknown
_1172444678.unknown
_1172444420.unknown
_1172444293.unknown
_1172444340.unknown
_1172444268.unknown
_1172433924.unknown
_1172436800.unknown
_1172437275.unknown
_1172435761.unknown
_1172433533.unknown
_1172433833.unknown
_1172431844.unknown