**5.1. Torque and flux control by means of SVM**

The electromagnetic torque of the PMSM can be expressed as:

$$\Gamma\_{\rm em} = \frac{3}{2} \mathcal{P} \frac{\Phi\_{\rm sr}}{\mathcal{L}\_{\rm d} \mathcal{L}\_{\rm q}} \left[ \Phi\_{\rm PM} \mathcal{L}\_{\rm q} \sin \delta + \frac{1}{2} \Phi\_{\rm sr} (\mathcal{L}\_{\rm d} - \mathcal{L}\_{\rm q}) \sin \delta \right] \tag{19}$$

Where δ is the angle between the stator and rotor flux linkage (or the torque angle).

Above equation consist of two terms, the first is the excitation torque, which is produced by permanent magnet flux and the second term is the reluctance torque.

In the case where Ld = Lq=L, the expression of electromagnetic torque becomes:

$$
\Gamma\_{\rm em} = \frac{3}{2} \mathbf{P} \frac{\Phi\_{\rm sr}}{\mathbf{L}} \Phi\_{\rm PM} \sin \delta \tag{20}
$$

Under the condition of constant stator flux amplitude Φsr , by diffentiating equation 20 with respect to time, the rate of increasing of torque at t=0 can be obtained in equation 21.

$$\frac{d\mathbf{r}}{dt} = \left(\frac{3}{2} \mathbf{P} \frac{\Phi\_{\rm sr}}{\rm L} \Phi\_{\rm PM} \cos \delta \right) \frac{d\delta}{dt} \tag{21}$$

From equations 20 and 21, it can be seen that for constant stator flux amplitude Φsr and flux produced by Permanent Magnets (PM) ΦPM, the electromagnetic torque can be changed by control of the torque angle; quick dynamic reponse can be achieved by changing this angle as quickly as possible, this is the basis of DTC for PMSM (Tang et al., 2003). This is the angle between the stator and rotor flux linkage, when the stator resistance is neglected. The torque angle δ, in turn, can be changed by changing position of stator flux vector in respect to PM vector using the actual voltage vector supplied by PWM inverter.

When the PMSM drive, we distinguish between two cases:


**Figure 23.** Vector diagram of illustrating torque and flux control conditions

The change of the angle δ is done by varying the position of the stator flux vector relative to the rotor flux vector with the vector Vsref provided by the predictive controller to the power of the SVM. The figure 23 above shows the evolution of the stator flux vector at the beginning and the end of a period vector modulation. At the beginning, stator flux vector is at the position δ with an amplitude Φs, it's at this moment that the predictive controller calculated the variation Δδ of the stator flux angle, it's also at this same moment that the space vector modulator receives the new position and amplitude of the voltage vector that must be achieved at the end of the modulation period. Of course, this vector will allow the stator flux to transit to the location as defined by the predictive controller to adjust the torque fluctuations, and this by calculating the time of application of the adjacent vectors V1, V2 and V0 as well as their sequence that depends on the symmetry of the modulation vector.

The internal structure of the predictive torque and flux controller is shown in figure 24. So the average change Δδ of the angle δ is expressed as:

$$
\Delta \delta = T\_{\rm s} \frac{d}{dt} \left[ \text{Arcsin}(\frac{\Phi\_{\rm sr}}{\mathcal{L}\_{\rm q} \, \text{l}\_{\rm sq}}) \right] \tag{22}
$$

where: Ts is the sampling time.

184 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

�� <sup>=</sup> � � <sup>P</sup> ��� � 

�Г �� = �� � <sup>P</sup> ��� � 

vector using the actual voltage vector supplied by PWM inverter.

**Figure 23.** Vector diagram of illustrating torque and flux control conditions

The change of the angle δ is done by varying the position of the stator flux vector relative to the rotor flux vector with the vector Vsref provided by the predictive controller to the power

ΦPM

θ<sup>m</sup>

Φ<sup>s</sup>

θsref

Φsr

**θs**

α

d

When the PMSM drive, we distinguish between two cases:

different speeds (see figure 23).

q

Under the condition of constant stator flux amplitude Φsr , by diffentiating equation 20 with

From equations 20 and 21, it can be seen that for constant stator flux amplitude Φsr and flux produced by Permanent Magnets (PM) ΦPM, the electromagnetic torque can be changed by control of the torque angle; quick dynamic reponse can be achieved by changing this angle as quickly as possible, this is the basis of DTC for PMSM (Tang et al., 2003). This is the angle between the stator and rotor flux linkage, when the stator resistance is neglected. The torque angle δ, in turn, can be changed by changing position of stator flux vector in respect to PM


**∆δ**

**δ**

β

�� cos δ � ��

respect to time, the rate of increasing of torque at t=0 can be obtained in equation 21.

�� sin δ (20)

�� (21)

**Figure 24.** Internal structure of predictive controller used in DTC-SVM

From equation 22, the relation between error of torque and increment of load angle Δδ is non linear. In odrder to generate the load angel increment required to minimize the instantaneous error between reference and actual estimated torque; a PI controller has been applied as indicated in figure 24. The step change Δδ that corresponds to the torque error is added to the current position � of the stator flux vector to determine the new position of this vector at the next simple time.

The module and argument of the reference stator voltage vector is calculated by the following equations, based on stator resistance Rs, Δδ signal and actual stator flux argument:

$$|\mathbf{V\_{sref}}| = \sqrt{\mathbf{V\_{s\alpha-ref}}^2 + \mathbf{V\_{s\beta-ref}}^2} \tag{23}$$

$$
\langle \text{V}\_{\text{sref}} \rangle = \arctan(\frac{\text{V}\_{\text{s\%}-\text{ref}}}{\text{V}\_{\text{sa-ref}}}) \neq \tag{24}
$$

Where:

$$\mathbf{V\_{sa-ref}} = \frac{\boldsymbol{\Phi\_{\rm Sr}} \cos(\boldsymbol{\theta\_s} + \boldsymbol{\Lambda}\boldsymbol{\delta}) - \boldsymbol{\Phi\_s} \cos\boldsymbol{\theta\_s}}{\mathbf{T\_s}} + \mathbf{R\_{s^\*}} \mathbf{I\_{sa}} \tag{25}$$

$$\mathbf{V\_{s\beta-ref}} = \frac{\boldsymbol{\Phi\_{sr}}\sin(\boldsymbol{\theta\_s} + \boldsymbol{\Lambda}\boldsymbol{\delta}) - \boldsymbol{\Phi\_s}\sin\boldsymbol{\theta\_s}}{\mathbf{r\_s}} + \mathbf{R\_s} \cdot \mathbf{I\_{s\beta}}\tag{26}$$

$$\mathbf{V}\_{\rm sref} = \frac{1}{T\_s} \left[ \left( \frac{\mathbf{T}\_1}{2} \mathbf{V}\_1 + \frac{\mathbf{T}\_2}{2} \mathbf{V}\_2 + \frac{\mathbf{T}\_0}{2} \mathbf{V}\_0 \right) + \left( \frac{\mathbf{T}\_0}{2} \mathbf{V}\_0 + \frac{\mathbf{T}\_2}{2} \mathbf{V}\_2 \frac{\mathbf{T}\_1}{2} \mathbf{V}\_1 \right) \right] \tag{27}$$

In this proposed technique, the same flux and torque estimators and the predictive torque and flux controller as for the DTC-SVM are still used. Instead of the SVM generator, a SPWM technique is used to determine reference stator flux linkage vector. It is seen that the proposed scheme retains almost all the advantages of the DTC-SVM, such as no current control loop, constant switching frequency, low torque and flux ripple, etc. But, the main advantage of the DTC-SPWM is the simple algorithm of PWM (SPWM) used to control the VSI. Of course, the SVM algorithm needs more calculation time than SPWM and the same advantages of DTC-SVM will be obtained by using DTC-SPWM. Whatever is the load torque and speed variation, SPWM guarantees a constant switching frequency, which greatly improves the flux and torque ripples.

**Figure 29.** Sinusoidal PWM pulses generator scheme

## **6.1. Simulation results**

The sampling period has been chosen equal to 100 µs (10 KHz) for DTC-SVM; in order to compare this strategy with basic DTC; despite the fact that the sampling time used to simulate DTC is less than that used in case of DTC-SVM. Whereas, the sampling frequencies used to simulate FDTC and DTC-SVM are equal; so as to compare these two techniques in the same conditions.

**Figure 30.** Stator current spectrum at 800 rpm with nominal load (on the left) and Stator flux in (α,β) axes under load variations (on the right) in case of DTC-SPWM

**Figure 31.** Stator current waveform at 800 rpm with nominal load under DTC-SPWM

**Figure 32.** Stator current spectrum at 800 rpm with nominal load (on the left) and Stator flux in (α,β) axes under load variations (on the right) in case of DTC-SPWM
