3

Practical Exercises in Drives Systems with C++ Simulation Software and Synchronous Motor

CHAPTER III

Built an AC drive system with permanent magnet (PM) synchronous motor

3.1 PM synchronous motor mathematical model

The general equation of the stator armature for PM synchronous motor is the usual equation of the AC electrical machines.

(3.1)

All the quantities are expressed as complex phasors, in a (d, q) coordinate system that rotates with stator voltage frequency.

Direct and quadrature components of stator flux are:

(3.2)

(3.3)

with as PM flux.

Direct and quadrature components of equation (3.1) are:

(3.4)

(3.5)

From equation (3.2.3.5) results:

(3.6a)

(3.6b)

The equations (3.6a, 3.6b) take the form:

(3.7a)

(3.7b)

In figure 3.1 is represented the (operational) model derived from equation (3.6a, 3.6b).

The torque equation is:

(3.8)

The transfer functions obtained from equation (3.7a, 3.7b) are:

(3.9)

(3.10)

where

Td=Ld / R , Tq=Lq / R;

(3.11)

(3.12)

(3.13)

(3.14)

Using z transform we have the following discrete mathematical model for PM synchronous machine.

Input variables: uq, ud,

Output variables: id, iq,

;

;

;

;

;

;

;

;

;

;

return;

The controls uq, ud, are subjected to the followings conditions:

The stator voltage frequency and rotor angular frequency are synchronous

(3.15)

with p as rotor pair poles and mechanical rotor angular speed.

The stator voltages U are related with machine flux by equation (3.1). Taking R=0 and stationary regime, we have:

(3.16) The best command is obtained by field orientation. In the case id= 0 and

(3.17)

We analyze first the open loop control. From the relation (3.13) results:

(3.18)

(3.19)

The open loop command implementation is represented in fig.3.2

The C++ program sinc1 simulate the open loop control schema (fig.3.3), for a PM machine with the following parameters:

R=0.6; Ld=1.4*10-3H; Lq=2.8*10-3H; p=2; U=100V.In figure 3.3 was represented the simulation results for start up and brake of PM machine with the parameters mentioned above:

The closed loop control of PM synchronous machines has three main objectives:

To maintain the id current component near zero;

To control the torque by the uq voltage component;

To control the machine speed by imposing the appropriate torque;

These three conditions are accomplished by flux, torque and speed controllers (fig.3.4).

The closed loop control under schema represented in fig.3.4 was simulating in C++ language. The simulation results obtained with C++ program are shown in fig.3.5

These results are very similar with the DC machine simulations. That is why the PM synchronous machines is named as brushless DC machine.

3.2 Decoupling the torque and flux controls for PM synchronous machine

The two control channels for torque and flux are coupled by product nonlinearities of machine equations.

From (3.6 a, b) results:

(3.20)

(3.21)

The nonlinear parts of equations (3.20, 3.21) are:

(3.22)

(3.23)

The two controllers, for torque and flux, must to generate the and voltage components in such a way that machine remains always flux orientated. That means, if the id current component is maintained nearly zero by the flux controller, the torque will depends mainly of the iq current component (see relation 3.8). Because the inverter that supply the motor is voltage controlled, it must determine the relations between the current controls , and the corresponding voltage controls , . That is made by inverse model which results from equations (3.20, 3.21).

Like in the case of the induction machine, based of the inverse machine model, the linear part of the torque and flux controllers are:

(3.24)

(3.25)

Applying the tuning relations (2.53, 54) we have:

(3.26)

(3.27)

where .

The PI torque controllers parameters results from (2.58):

(3.28)

(3.29)

and for flux controller

(3.30)

(3.31)

(3.32)3.3 The simulation algorithm of the motion system

3.4 Experimental results

The simplified diagram associated with the motion control algorithm (fig.3.7) of the electromechanical system with the permanent magnets synchronous machine, is given in the fig. 3.8.

There are many analogies between the DC machine algorithms and PM synchronous machine simulation algorithm. If the PM synchronous machine is well field orientated (the two command channels for torque and flux decoupled by fd and fq functions), the control system for speed and position is similar to DC machine.

In figures 3.9 and 3.10 are represented the dynamic responses of PM synchronous drive system controlled by minimum energy control law and respectively minimum time control (bang-bang control).

The controlled system follows very well the feed-forward imposed values for torque (m and mm) and speed (om and 1) in conditions of strong variations of the load torque (mres).

In figures 3.9 and 3.10 the mres and mr are the estimated and imposed load torque, m, mm is the machine and imposed electromagnetic torque and 1, om are the machine speed and respectively imposed speed.

The PM synchronous machine rated parameters are: power PN=3Kw, voltage UN=220V, efficiency N=0.84, power factor cosN = 0.81, the time constant Td=0.06s, Tq=0.03s, angular frequency N=314 and electromechanical time constant Tem = 0.1s.

The dynamical behavior of PM synchronous machine represented in figures 3.9 and 3.10 has the same performances as DC motor (see figures 1.23 1.28). The simulation program for PM synchronous machine allows to verify the efficiency of motion like it was done for DC machine (table 1.1 1.4).

Conclusions:

In this chapter a PM synchronous machine drive system simulation was elaborated. The aim of the simulation program is to confirm the advantages of minimal energy control for position drive system. This is important for example when the mobile robots carry their energy sources such as batteries.

First the PM synchronous machine model was developed in rotor field orientation theory. Than we used the minimal energy control law presented in section 1 for generation of the optimal trajectories for torque, speed and position. A feed-forward control system commands the motor actuator (PWM inverter) in order to follow as well a possible the imposed trajectories as well as possible.

All the techniques used in the simulation program (the load and electromagnetic torque estimators, the tuning methods for torque and flux controllers, the optimal trajectories generation) are well suited for direct implementation in a real drive system.

PM-SM

PM-SM

Fig. 3.8 The diagram of the electromechanical system with PM synchronous machine

k=k+1

(k)

m(k)

J

k

Reference speed and position trajectories

EMBED Equation.3

EMBED Equation.3

Fig.3.2 Simplified control schema of PMM synchronous machine

0

0

v1

Fig.3.1 The operational model of equations (3.7)

w

V1

uqs

uds

1

0.5

1

0.75

0.5

0.25

Fig.3.3 The simulation results of open loop control

0.5

0.5

PAGE 12

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