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- The Field Orientation Principle in Control of Induction Motors | Andrzej Trzynadlowski | Springer
- Chapter 7. Modelling Induction Motors
- Working Principle of Field Oriented Control

The resulted scalar model-equations are:. The rotor time constant is denoted by. The stator time constant is denoted by. The kinematic equations of the motor in stator coordinates are as follows:. Choosing the rotor flux as orientation quantity, that means all quantities are viewed from the rotor flux point of view, which are usually called field coordinates. The advantage of using this reference frame is that all quantities are constant in the steady state.

The flux is now eliminated by introducing a new quantity called magnetizing current rotor current cannot be measured see eq. The stator current in field coordinates and the magnetizing current are now chosen to be the state variables. Let us now rewrite these equations in a state space form where state variables i Sd, i Sq and i mR are used. Notice that by introducing the magnetizing current we need only three state variables.

Where , that is, the flux speed equals to the rotor speed plus the slip angular velocity of the rotor flux. The kinematic equations of the motor in field coordinates are:. The forms of equations remains unchanged in this coordinate system too. Equations presents the transformation matrices between the three-phase and the Descartes stationary coordinate systems, describes the connection between the stator and the field coordinates. Figure Note that the rotor based variables are completely eliminated. Using the field oriented model of the motor, it has theoretically become possible to realize a control in field orientation, which makes it extremely easy to control speed by controlling.

However, our system of equations is nonlinear.

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We will eliminate this non-linearity by decoupling the nonlinear terms. This method has been described in [ 14 ] , and detailed treatment is also given in [ 17 ] :. We can see from 7. Let us now examine how nonlinearity is eliminated by decoupling the undesirable nonlinear and coupling terms.

We would like to control the speed. Let us recall. The torque is directly proportional to i mr and to i sq. Because T r is large, i mr cannot be used for a quick control. The leakage factor of i Sq is small, that is why i Sq can be used to control the torque and the speed.

We get the best dynamic behaviour when i mr is kept constant, therefore can be assumed. Under this assumption, using , , , and - 7. Assuming that is kept constant, K J is also constant. In this case the system described by can be split into two independent parts, the direct denoted by d and the quadrature denoted by q subsystem :. After this it is possible to control the motor, provided that we are able to convert all values to field co-ordinates. This is made by the flux model see the description later in the Simulink realization in Figure After realizing the flux model control becomes possible.

The first loop of the control is a controller for i mr, which is in fact a controller for the rotor flux. There is a controller for i sd, which will force the correct i mr.

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The second loop begins with a controller for , followed by a controller for i sq, which will force the correct torque, and implicitly will realize the correct velocity. The overall control structure is shown in Figure The actual model of the ASM can be seen in Figure This model is based on the mathematical model described in [ 16 ]. However, it has been adapted to the requirements of Simulink. For the sake of simplicity, all vector variables have been substituted by scalar variables.

The model is in stator and rotor co-ordinates. These equations have the advantage of an easy implementation, and great symmetry.

The stator co-ordinates are indexed with S, the rotor co-ordinates with R, as before, and the indices and a denote the two axes of the co-ordinate systems. The realized equations of the model presented in Figure The basic functionality of the above model was tested with sine voltage signals excitation in open loop.

The speed settled near the speed of the signals, that is, the rotor moved almost synchronously with the generated voltage vector. Then a torque was applied, which introduced a slip into the system. Since the voltages did not change, the speed of the rotor sank to a lower level. This chapter also describes the adaptation of the ASM Asynchronous Motor model for different kinds of algorithms, including sensorless control, too.

The first step to design a Kalman filter, for example, is to have an appropriate model, which is used by the filter. For this purpose, we followed the method presented in [ 19 ] and derived the model, which was compared with the results of [ 14 ] and the results were the same. The model taken as a basis was the field oriented model of the motor based on [ 19 ] , described by the equations presented in -.

These equations can be formulated in such a form which is non-linear, but can be converted into a time-dependent matrix form. Let us see how to convert these equations:. This can be further modified by replacing based on 7.

## The Field Orientation Principle in Control of Induction Motors | Andrzej Trzynadlowski | Springer

We get the following:. These equations can now be written in matrix form. Since we do not have a sensor for the load torque, it will be simply omitted, and regarded as a disturbance. Also note that the transformation matrix between the field and the stator is introduced to the equations, and so the input and output variables are in stator co-ordinates.

This form is easier to handle in the case of the observing problem. Note that in this model, and in all the following models, is part of the output vector. This does not mean that we measure it, but it must be estimated roughly, and this estimated value must be substituted into the Kalman filter where this output vector is needed.

## Chapter 7. Modelling Induction Motors

The substitution is made using the following formula:. For practical implementation we will need the model in discrete time, so let us consider the conversion to discrete time. Let us denote the system matrices of the continuous system with A, B and C , and those of the discrete system with A d , B d , and C d. If we suppose that our sampling time is very short, as compared to the dynamics of the system which is the case in most applications , we can generate the system with the following approximate equations first order Taylor series, see [ 19 ] :.

This system is dependent on the values of and , which were regarded as known values until now. In [ 19 ] , however, the Kalman filter is presented also as a substitute for the flux model, and the flux is observed as well. The first implementation does not follow this method, but it will be presented because further improvements may be based on it.

The reason why a smaller model was chosen is that the dimension of the model is one of the most important factors determining the speed of calculation, and the reduction of a 6th order model into a 4th order one, which is relatively simple, can be done at a greater speed than the extraneous calculation of the flux model. To observe the flux, we must observe and. This will be done by adding these to the system as new state variables. This increases the order of our system to 6.

### Working Principle of Field Oriented Control

To add the new variables, we will assume that the sampling time is short enough for the speed of the flux to remain constant within the sampling period. That is, we can assume that. With these assumptions we can establish the new 6th order system, which can be the base of a Kalman filter, which observes the flux as well. For convenience the discrete-time model of the induction motor in a stationary reference frame is also presented which will serve again for observer application 7.