Sensorless Field Oriented Control Applied for an Induction Machine by Using the Discontinuous PWM Strategy

The circuit consist two parts, the first contains the power part, represented by: a three-phase diode rectifier, LC filter, a three-phase inverter and the induction motor. While the second contains the control part, represented by the SFOC control and the DPWM strategy. The system description as shown in Figure 1.

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Figure 1. System description

2.1 Modeling of three-phase diode rectifier

A rectifier, also called AC/DC converter or GRAETZ bridge, is a converter intended to supply a load which needs to be supplied by a voltage and a current that are both as continuous as possible, from an AC voltage source. The power supply is, most of the time [7-9], a voltage generator. The average voltage across the load is given by the following equations:

$\mathrm{U}_{\mathrm{AV}}=\frac{3 \sqrt{6}}{\pi} \mathrm{U}_{\mathrm{RMS}, \mathrm{S}}=2.33 \mathrm{U}_{\mathrm{RMS}, \mathrm{S}}$    (1)

with, U_{RMS, S}: is the RMS voltage of the AC source.

2.2 Stabilization LC filter

To correct the DC voltage source, a capacitor C is inserted at the output of the rectifier, this absorbs the difference between the different currents, and suppresses the sudden variations of the switching voltage, on the other hand, to reduce the ripple of the current and protect the inverter against the critical speed of current growth, a smoothing inductance L of internal resistance R is placed in series [10].

The transfer function of the LC filter is represented by the following equation:

$F(s)=\frac{1}{L C  s^2+R C s+1}$    (2)

L: is the inductance in series with the rectifier.

C: is the capacitor in parallel with the rectifier.

R: is an internal resistance.

2.3 Modeling of VSI

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Figure 2. Three-phase VSI

The VSI shown in Figure 2, delivers two levels of voltages or depending on the connection function Sx1 as shown in the Eq. (3) [11-13]:

${{\nu }_{xo}}=\left\{ \begin{align}

  & \frac{{{\nu }_{dc}}}{2}si{{S}_{x1}}=1\quad and{{S}_{x2}}=0 \\ 

 & -\frac{{{\nu }_{dc}}}{2}si{{S}_{x1}}=0\quad and{{S}_{x2}}=1 \\ 

\end{align} \right.$    (3)

where,v_xo is the phase-to-neutral voltage between phase x and midpoint o.

The three phase-midpoint voltages v_ao, v_bo and v_co are expressed by:

$\left[\begin{array}{l}v_{a o} \\ v_{b o} \\ v_{c o}\end{array}\right]=\frac{v_{d c}}{2}\left[\begin{array}{l}2 S_{a 1}-1 \\ 2 S_{b 1}-1 \\ 2 S_{c 1}-1\end{array}\right]$    (4)

And the three phase-to-neutral voltages v_a, v_b and v_c are expressed by the following equation [14]:

$\left[ \begin{align}

  & {{v}_{a}} \\ 

 & {{v}_{b}} \\ 

 & {{v}_{c}} \\ 

\end{align} \right]=\frac{1}{3}{{v}_{dc}}\left[ \begin{matrix}

   2 & -1 & -1  \\

   -1 & 2 & -1  \\

   -1 & -1 & 2  \\

\end{matrix} \right]\left[ \begin{matrix}

   {{S}_{a1}}  \\

   {{S}_{b1}}  \\

   {{S}_{c1}}  \\

\end{matrix} \right]$     (5)

For the generation of logic signals, there are several control techniques such as the PWM strategy.

2.3.1 Discontinuous PWM strategy

The three-phase VSI is a converter with eight switches which requires a good PWM control strategy [15, 16]. In this work, we have chosen the DPWM strategy which offers the possibility of reducing the number of switching in the inverter. The basic principle of this technique is to inject the zero voltage sequence (ZVS) to the sinusoidal reference waves, with the aim of blocking each phase for 60 degrees in two times compared to the modulation period which is 360 degrees [17], to generate a balanced three-phase system. Figure 3 shows the principle of this strategy.

The DPWM strategy uses the obtained modulators from the sinusoidal references by adding the ZVS equal to [18]:

ZVS $=\operatorname{sign}\left(V_m\right) * E / 2-V_m$    (6)

where V_m is the voltage resulting from the maximum amplitude test (absolute maximum of the three reference voltages between v_aref, v_bref and v_cref).

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Figure 3. Principle of the DPWM strategy

2.4 Sensorless field oriented control SFOC

2.4.1 Modeling of induction motor IM

IM depends on the rotational speed of the rotor. If the latter is slightly greater than that of the magnetic field of the stator, it then develops an electromagnetic force similar to that obtained with a synchronous generator [19, 20]. On the other hand, the machine does not generate its own excitation energy. For that, it will be necessary to bring this energy either by a battery of capacitors, or by a static converter control, which will stabilize its output voltage and frequency through capacitors connected across the stator. The model of induction machine IM is composed of an electrical phase windings and a rotor cage squirrel [21].

$\left\{ \begin{align}

  & {{\nu }_{ds}}={{R}_{s}}{{i}_{ds}}\frac{d}{dt}{{\varphi }_{ds}}-\omega {{\varphi }_{qs}} \\ 

 & {{\nu }_{qs}}={{R}_{s}}{{i}_{qs}}\frac{d}{dt}{{\varphi }_{qs}}-\omega {{\varphi }_{ds}} \\ 

 & {{\nu }_{dr}}={{R}_{r}}{{i}_{dr}}\frac{d}{dt}{{\varphi }_{dr}}-\left( {{\omega }_{s}}-{{\omega }_{r}} \right){{\varphi }_{qr}} \\ 

 & {{\nu }_{qr}}={{R}_{r}}{{i}_{qr}}\frac{d}{dt}{{\varphi }_{qr}}+\left( {{\omega }_{s}}-{{\omega }_{r}} \right){{\varphi }_{dr}} \\ 

\end{align} \right.$    (7)

The expression for the stator and rotor flux linkages:

$\left\{ \begin{align}

  & {{\varphi }_{ds}}={{L}_{s}}{{i}_{ds}}+{{L}_{m}}\left( {{i}_{ds}} \right.+\left. {{i}_{dr}} \right) \\ 

 & {{\varphi }_{qs}}={{L}_{s}}{{i}_{qs}}+{{L}_{m}}\left( {{i}_{qs}} \right.+\left. {{i}_{qr}} \right) \\ 

 & {{\varphi }_{dr}}={{L}_{r}}{{i}_{dr}}+{{L}_{m}}\left( {{i}_{ds}} \right.+\left. {{i}_{dr}} \right) \\ 

 & {{\varphi }_{qr}}={{L}_{r}}{{i}_{qr}}+{{L}_{m}}\left( {{i}_{qs}} \right.+\left. {{i}_{qr}} \right) \\ 

\end{align} \right.$    (8)

The electrical model is completed by this mechanical equation [22]:

$\mathop{T}_{em}^{{}}-\mathop{T}_{r}^{{}}=j\frac{d{{\Omega }_{mec}}}{dt}+{{k}_{f}}{{\Omega }_{mec}}$    (9)

The electromagnetic torque expression as a function of stator currents and rotor flux as follows:

$T_{e m}=p \frac{L_m}{L_m+L_r}\left[i_{q s} \varphi_{d r}-i_{d s} \varphi_{q r}\right]$    (10)

2.4.2 Modeling of sensorless field oriented control of IM

The main objective of the vector control induction motors is, as in DC machines, to independently control the torque and the flux. In this order we propose to study the SFOC of the IM [2]. The control strategy used consists to maintain the quadrate component of the flux null (φ_qr=0) and the direct flux equals to the reference (φ_dr= φ_r*) [3, 4]: φ_r* and the torque T^*_em as well as

$\left\{\begin{array}{c}\varphi_{d r}=\varphi_r^* \\ \varphi_{q r}=0 \\ \frac{d}{d t} \varphi_r^*=0\end{array}\right.$    (11)

Substituting (11) into (7) yields:

$\left\{ \begin{align}

  & {{i}_{dr}}=0 \\ 

 & {{i}_{qr}}=-\frac{\mathop{\omega }_{gl}^{*}\ \mathop{\varphi }_{r}^{*}}{\mathop{R}_{r}} \\ 

\end{align} \right.$    (12)

with $\omega_{g l}^*=\omega_s^*-\omega_s$ ($ \omega_g^* l$ is the slip speed).

After calculation and rearrangement of the electromagnetic torque and stator voltages equations, following expressions are obtained [5].

$\left\{ \begin{align}

  & \mathop{i}_{dr}^{{}}=\frac{\mathop{\varphi }_{r}^{*}}{{{L}_{m}}+{{L}_{r}}}-\frac{{{L}_{m}}}{{{L}_{m}}+{{L}_{r}}}\mathop{i}_{ds}^{{}} \\ 

 & \mathop{i}_{qr}^{{}}=\frac{{{L}_{m}}}{{{L}_{m}}+{{L}_{r}}}\mathop{i}_{qs}^{{}} \\ 

\end{align} \right.$    (13)

with:

i_dr and i_qr: are the rotor (d, q) axis current.

i_ds and i_qs: are the stator (d, q) axis current.

φ_r^*: the flux linkage.

L_r: rotor inductance.

L_s: stator inductance.

L_m: mulual inductance.

Substitution (13) into (12), obtain

$\mathop{\omega }_{gl}^{*}=\frac{\mathop{R}_{r}\ {{L}_{m}}\mathop{i}_{qs}^{{}}}{({{L}_{m}}+{{L}_{r}})}\mathop{\varphi }_{r}^{*}$    (14)

The final expression of the torque electromagnetic is

$\mathop{T}_{em}^{*}=p\frac{{{L}_{m}}}{({{L}_{m}}+{{L}_{r}})}\mathop{\mathop{i}_{qs}^{{}}\,\varphi }_{r}^{*}$    (15)

with taking into the rotor field orientation, the stator voltage Eq. (7) can be rewritten as [6] 

$\left\{ \begin{array}{*{35}{l}}

   v_{ds}^{*}={{R}_{s}}{{i}_{ds}}+{{L}_{s}}\frac{d}{dt}{{i}_{ds}}-\omega _{S}^{*}\left( {{L}_{s}}{{i}_{qs}}+{{\tau }_{r}}\varphi _{r}^{*}\left( \omega _{s}^{*}-\omega _{r}^{*} \right) \right)  \\

   v_{qs}^{*}={{R}_{s}}{{i}_{qs}}+{{L}_{s}}\frac{d}{dt}{{i}_{qs}}+\omega _{s}^{*}\left( {{L}_{s}}{{i}_{ds}}+\varphi _{r}^{*} \right)  \\

\end{array} \right.$    (16)

where, $\tau_r=\frac{L_r}{R_r}$.

Consequently, the electrical and mechanical equations for the system after these transformations in the space control may be written as follows [5, 6]:

$\left\{ \begin{align}

  & \frac{d}{dt}{{i}_{ds}}=\frac{1}{{{L}_{s}}}\left( \nu _{ds}^{*}- \right.{{R}_{s}}{{i}_{ds}}+\mathop{\omega }_{s}^{*}\left( {{L}_{s}}{{i}_{qs}}+{{\tau }_{r}}\mathop{\varphi }_{r}^{*} \right.\left( \mathop{\omega }_{s}^{*}-\mathop{\omega }_{r}^{*})) \right. \\ 

 & \frac{d}{dt}{{i}_{qs}}=\frac{1}{\mathop{L}_{s}}\left( \nu _{qs}^{*}- \right.{{R}_{s}}{{i}_{qs}}-\mathop{\omega }_{s}^{*}\left( \mathop{L}_{s}{{i}_{ds}}+\mathop{\varphi }_{r}^{*} \right) \\ 

\end{align} \right.$    (17)

$\frac{d}{dt}{{\varphi }_{r}}=-\frac{{{R}_{r}}}{{{L}_{r}}+{{L}_{m}}}\left( {{\varphi }_{r}}+{{L}_{m}}{{i}_{ds}} \right)$    (18)

$\frac{d}{dt}{{\Omega }_{mec}}=\frac{1}{j}\left( \frac{p{{L}_{m}}{{i}_{qs}}\mathop{\varphi }_{r}^{*}}{({{L}_{r}}+{{L}_{m}})}-{{T}_{g}}-f{{\Omega }_{mec}} \right)$    (19)

image015.png

Figure 4. Current control scheme

For perfect decoupling, stator current control loops (i_ds, i_qs) are added and at their outputs the voltage (v_ds, v_qs) are obtained. The block control diagram is shown in Figure 4 [23]. Dynamic equation for the dc bus voltage, is derived as follows. Equating currents at the dc bus voltage, following equation is obtained:

$\mathop{i}_{dc}=\mathop{C}_{dc}\frac{d}{dt}\mathop{v}_{dc}$    (20)

The current control scheme is given in Figure 4. The references $i_d^*$  and $i_q^*$ are the outputs of the PI regulator [24].

$\left\{ \begin{align}

  & \mathop{i}_{d}^{*}=\left( PI \right)\ \left( \mathop{v}_{d}^{*}-\mathop{v}_{d}^{{}} \right) \\ 

 & \mathop{i}_{q}^{*}=\left( PI \right)\ \left( \mathop{v}_{q}^{*}-\mathop{v}_{q}^{{}} \right) \\ 

\end{align} \right.$     (21)