The Synchronous Machine (SM) is an electrical machine whose rotating speed is proportional to the frequency of the alternating current supply and independent of the load.
It is mostly used as a generator where it is almost without a competitor, but it also serves as a motor in several fields.
The Permanent Magnet Synchronous Motors (PMSM) is a type of SM with no damper winding, and excitation at the rotor is provided by a permanent magnet.
It has high-density power, large torque to inertia ratio and high efficiency making it very useful in many applications, such as home equipment, robotic applications, and electric vehicles [2].

Figure 1: Virtual Test Bench of the PMSM Drive

Model simulation is one of the safest and most efficient ways to tackle real-life problems.
By simulating a dynamic model, a wide range of tests could be done on the simulation to analyze and validate the proposed model before making a functioning prototype.
Real-time simulation gives the user time accurate results of a process, making it a useful tool in stimulating electric machines for a wide range of applications, for instance in the education sector.
Electrical engineer undergraduates have to do electrical machines laboratories in order to apply and further their theoretical knowledge in that domain.
But those labs aren’t as effective as they need to be because of the limitations put on the students to protect them and to preserve the machines tested.
That’s why it is primordial to have virtual test bench laboratories, giving students more flexibility to fully analyze the electrical and mechanical limits of electric machines without the fear of costly damages.
Here, a virtual test bench for variable speed drive of PMSM is implemented (see figure 1 above).
The modeling is such that real-time accurate results are obtained.

The different modules covered in this laboratory are discussed next.

Transformations

Consider a direct three-phase system given by:

The time domain and the spatial representations of the system are shown in figures 2 and 3.

Figure 2: Three-Phase System in Time Domain

Figure 3: Spatial Representation Taken at t = 0

Transformations from (a, b, c) to (α, β)

Knowing that for a three-phase system the summation of signals is zero, it makes sense to get rid of the summation equation and to move to a two-signals system in which those signals are orthogonal, therefore, non-dependent.
For that, two transformations are suggested: Clarke and Concordia.

Clarke Transformation

Clarke transformation applied to the three-phase system leads to:

Therefore,

From which the inverse of Clarke transformation is computed as:

The properties of Clarke transformation are:

• This transformation keeps the amplitude invariant, hence the amplitudes of s_a and s_α are both equal to S. However, the power is variant.
• The determinant of this transformation is 2/(3), hence this is not a rotation.

Concordia Transformation

Therefore,

The properties of Concordia transformation are:

• This transformation changes the amplitude but keeps the power invariant.
• The determinant of this transformation is 1.
  Additionally, columns are orthogonal and have unity-norm.
  Consequently, this is a rotation, and the inverse is nothing but the transpose.

Transformations from (α, β) to (d, q)

Rotation of Angle -Ѳ

This transformation is used to move from (α, β) stationary frame to (d, q) rotating frame.
The goal behind is to remove the ωt component and to end-up with time-invariant signal s_d and s_q.

Knowing that the three-phase system depicted in figure 3 above is undergoing a counterclockwise rotation at ω rd/s, a rotation of an angle Ѳ = -ωt is to be applied in order to get time-invariant signals in the (d, q) frame, leading to:

Obviously, this transformation corresponds to a rotation matrix Q with a determinant of value of 1.
The calculation of the speed matrix goes as follows:

It is noteworthy that the speed matrix Ω is skew-symmetric, which is a necessary condition for the speed term.

Transformations from (a, b, c) to (d, q)

There are four possible combinations to bring the three-phase system (a, b, c) to a (d, q) one, namely:

1. Clarke followed by a rotation of -Ѳ
2. Concordia followed by a rotation of -Ѳ + π/2
3. Clarke followed by a rotation of -Ѳ
4. Concordia followed by a rotation of -Ѳ + π/2.

Notice that these transformations are known as Park transformations in literature.

Combination of Clarke and Rotation of -Ѳ

For the three-phase system defined in (1), s_d and s_q become:

As expected, s_d and s_q are time-invariant since they do not depend on ωt.

Since s_d vanished, it is hard to check whether s_q leads or lags s_d.
For that, consider a three-phase system with a delay δ as follows:

Apply Clarke transformation leads to:

Applying a rotation of -Ѳ leads to:

Figures 4 and 5 show the spatial representation of this system for δ = π/6 and ωt = 0, then ωt = π/3.
It is worth noticing that s_q lags s_d.

Figure 4: Spatial Representation Taken at ωt = 0 and δ = π/6

Figure 5: Spatial Representation Taken at ωt = π/3 and δ = π/6

Combination of Concordia and Rotation of -Ѳ

For the three-phase system defined in (1), s_d and s_q become:

Self-Control

The first drive implemented is the self-control mode of the PMSM, which is a method controlling the three-phase currents (i_a, i_b, and i_c) of the machine.
A synchronous machine is said to be self-controlled if it gets its operating frequency from an inverter whose converters are fired through signals calculated using data obtained from a rotor position sensing unit.
This ensures that the stator and rotor fields move in synchronism for all operating points.
The current references are calculated using the angle coming from the sensor and the amplitude is either obtained directly from a reference or from a higher-level regulation loop, like the speed control loop.

Figure 6: Self-Control Diagram of PMSM

 The self-control diagram of the PMSM is shown in figure 6 above.
Here, the current references (i*_a, i*_b, and i*_c) can be directly imposed or calculated from a torque reference.
Thus, it is essential to establish the relation between torque and current. In order to obtain the maximum torque possible, we must impose through the reference currents the internal angle of the machine δ.

By imposing:

The electromagnetic torque of the synchronous machine is expressed as follows:

The current reference is generated using the following equation: