DSEV: PMSM Dynamic Modelling

A PMSM Dynamic Model needs to be done and implemented if we want to simulate and work with our drive system. For that reason, we will explain all the maths behind the theoretical model of the Simulink/Matlab solution.

It is known that the voltage equation is given in terms of Rs, (average stator resistance), the iabc (phase currents) and the flux linkages of each phase (λabc). The equation is written as:

MotorModelMath

Then, the matrix form of the equation is expressed:

MotorModelMath

A transformation to the dq0 reference frame needs to be done. For that reason, a transformation matrix (K_qd0) is applied to the previous equation:

MotorModelMath

 In order to simplify the final expression, it can be operated to obtain the next one:

MotorModelMath

Where:

MotorModelMath

And operating the last term of the equation, an even easier expression can be obtained:

MotorModelMath

Then, the final equations are obtained:

MotorModelMath

Finally, the expressions for the currents in the frame dq are calculated:

MotorModelMath

Having:

MotorModelMath

Next step:

MotorModelMath

And, after all of this gross math equation, we can obtain the next torque model of the motor:

MotorModelMath

In order to present the mechanical equation of the motor, the J represents the inertia, Bm the friction constant and Tload is the load torque.

Torque equation

Where:

MotorModelMath

Being Npp = number of pair of poles.

We know that all these theoretical explanations and equations can be omitted since we have models in advanced programs such as Matlab, but in this case, as it is a report for a University, or in the case we want to know where they came from, it is interesting to know how a motor works.

Next time we will have a look to the controllers, so stay tuned!

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