Electromechanical control with synchronous machine

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

CHAPTER III

Built an AC drive system withpermanent magnet (PM) synchronous

motor

3.1 PM synchronous motor mathematicalmodel

The general equation of the statorarmature for PM synchronous motor is theusual equation of the AC electrical machines.

(3.1)

All the quantities are expressed ascomplex phasors, in a (d, q) coordinatesystem that rotates with stator voltagefrequency .

Direct and quadrature components of statorflux are:

(3.2)

(3.3)with as PM flux.

Direct and quadrature components ofequation (3.1) are:

1


(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)

2

Fig.3.1 The operational model of equations (3.7)


The transfer functions obtained fromequation (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 PMsynchronous machine.

Input variables: uq, ud, Output variables: id, iq,

;;

;

;

;

;

;

;

;3


;return;

The controls uq, ud, are subjected tothe followings conditions:

The stator voltage frequency and rotorangular frequency are synchronous

(3.15)with p as rotor pair poles and mechanicalrotor angular speed.

The stator voltages U are related withmachine flux by equation (3.1). TakingR=0 and stationary regime, we have:

(3.16)

The best command is obtained by fieldorientation. 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.3.2The C++ program sinc1 simulate the open

loop control schema (fig.3.3), for a PMmachine with the following parameters:R=0.6Ω; Ld=1.4*10-3H; Lq=2.8*10-3H; p=2; U=100V.

4

v1

Φ0

Φ0

Fig.3.2 Simplified control schema of PMM synchronous machine


In figure 3.3 was represented the simulationresults for start up and brake of PM machinewith the parameters mentioned above:

The closed loop control of PM synchronousmachines has three main objectives:

To maintain the id current component nearzero;

To control the torque by the uq voltagecomponent;

To control the machine speed by imposingthe appropriate torque;

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Fig.3.3 The simulation results of open loop control

0.25 0.5 0.75 1

0.5

1

uds uqs V1 w


These three conditions are accomplished

by flux, torque and speed controllers(fig.3.4).

The closed loop control under schemarepresented in fig.3.4 was simulating in C++language. The simulation results obtainedwith C++ program are shown in fig.3.5

These results are very similar with the DCmachine simulations. That is why the PMsynchronous machines is named as brushless DCmachine.

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3.2 Decoupling the torque and flux controlsfor PM synchronous machine

The two control channels for torque andflux are coupled by product nonlinearities ofmachine equations.

From (3.6 a, b) results:

(3.20)

(3.21)

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

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(3.22)

(3.23)

The two controllers, for torque and flux,must to generate the and voltagecomponents in such a way that machine remainsalways flux orientated. That means, if the id

current component is maintained nearly zeroby the flux controller, the torque willdepends mainly of the iq current component(see relation 3.8). Because the inverter thatsupply the motor is voltage controlled, itmust determine the relations between thecurrent controls , and the correspondingvoltage controls , . That is made byinverse model which results from equations(3.20, 3.21).

8


Like in the case of the inductionmachine, based of the inverse machine model,the linear part of the torque and fluxcontrollers are:

(3.24) (3.25)

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

(3.26)

(3.27)

where .

9


The PI torque controllers parametersresults from (2.58):

(3.28)

(3.29)

and for flux controller(3.30)(3.31)(3.32)

3.3 The simulation algorithm of themotion system

10


11

Reference speed and position trajectories

k


12

J

m(k)

(k)

k=k+1


3.4 Experimental results 13


The simplified diagram associated with themotion control algorithm (fig.3.7) of theelectromechanical system with the permanentmagnets synchronous machine, is given in thefig. 3.8.

There are many analogies between the DCmachine algorithms and PM synchronous machinesimulation algorithm. If the PM synchronousmachine is well field orientated (the twocommand channels for torque and fluxdecoupled by fd and fq functions), the controlsystem for speed and position is similar toDC machine.

In figures 3.9 and 3.10 are representedthe dynamic responses of PM synchronous drive

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


system controlled by minimum energy controllaw and respectively minimum time control(bang-bang control).

The controlled system follows very wellthe feed-forward imposed values for torque (mand mm) and speed (om and ω1) in conditions ofstrong variations of the load torque (mres).

In figures 3.9 and 3.10 the mres and mrare the estimated and imposed load torque, m,mm is the machine and imposed electromagnetictorque and ω1, om are the machine speed andrespectively imposed speed.

The PM synchronous machine ratedparameters are: power PN=3Kw, voltage UN=220V,efficiency ηN=0.84, power factor cosφN = 0.81,the time constant Td=0.06s, Tq=0.03s, angularfrequency ωN=314 and electromechanical timeconstant Tem = 0.1s.

The dynamical behavior of PM synchronousmachine represented in figures 3.9 and 3.10has the same performances as DC motor (seefigures 1.23 – 1.28). The simulation programfor PM synchronous machine allows to verifythe efficiency of motion like it was done forDC machine (table 1.1 – 1.4).

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PM-SM

0.5


Conclusions:

In this chapter a PM synchronous machinedrive system simulation was elaborated. Theaim of the simulation program is to confirmthe advantages of minimal energy control forposition drive system. This is important forexample when the mobile robots carry theirenergy sources such as batteries.

16

PM-SM

0.5


First the PM synchronous machine modelwas developed in rotor field orientationtheory. Than we used the minimal energycontrol law presented in section 1 forgeneration of the optimal trajectories fortorque, speed and position. A feed-forwardcontrol system commands the motor actuator(PWM inverter) in order to follow as well apossible the imposed trajectories as well aspossible.

All the techniques used in the simulationprogram (the load and electromagnetic torqueestimators, the tuning methods for torque andflux controllers, the optimal trajectoriesgeneration) are well suited for directimplementation in a real drive system.

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Electromechanical control with synchronous machine

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