**1. Introduction**

464 Induction Motors – Modelling and Control

*2006 Pages: 1496 – 1507.* 

*Wiley & Sons, 1995.*

*Congress, Munich 1987.* 

*Conference on Industrial Electronics.*

*IPEMC 2004, Vol. 3, 14-16 August 2004.*

induction machine drive, *EPE 2003, Toulouse*.

*PhD Thesis, Gdansk University of Technology 2007.* 

*Conference on Power Electr., PEDES '06, 2006.*

*Electronics, Volume 52, Issue 2, April 2005 Pages: 523 – 531*.

*in Proc. the American control conference, California, June 1999, pp. 1-5.* 

Glab (Morawiec) M.; Krzeminski Z.; Lewicki A., Multiscalar control of induction machine

Fuschs F. & Kloenne A. dc-link and Dynamic Performance Features of PWM IGBT Current Source Converter Induction Machine Drives with Respect to Industrial Requirements,

Kwak S.; Toliyat H.A., A Current Source Inverter With Advanced External Circuit and Control Method, *IEEE Transactions on Industry Applications, Volume 42, Issue 6, Nov.-dec.* 

Klonne A. & Fuchs W.F., High dynamic performance of a PWM current source converter

Krstic M.; Kanellakopoulos I.; & Kokotovic P., Nonlinear and Adaptive Control Design, *John* 

Krzeminski Z., A new speed observer for control system of induction motor. IEEE Int. Conference on Power Electronics and Drive Systems, *PESC'99, Hong Kong, 1999.*  Krzeminski Z., Nonlinear control of induction motor, *Proceedings of the 10th IFAC World* 

Morawiec M., Sensorless control of induction machine supplied by current source inverter ,

Nikolic Aleksandar B., Jeftenic Borislav I.: Improvements in Direct Torque Control of Induction Motor Supplied by CSI, *IEEE Industrial Electronics, IECON 2006 - 32nd Annual* 

Payam A. F.; Dehkordi B. M.; Nonlinear sliding-mode controller for sensorless speed control of DC servo motor using adaptive backstepping observer, *International* 

Salo M.; Tuusa H., Vector-controlled PWM current-source-inverter-fed induction motor drive with a new stator current control method, *IEEE Transactions on Industrial* 

Tan H.; & Chang J.; Adaptive Backstepping control of induction motor with uncertainties,

Young Ho Hwang; Ki Kwang Park; Hai Won Yang; Robust adaptive backstepping control

for efficiency optimization of induction motors with uncertainties, *ISIE 2008.* 

supplied by current source inverter, *PCIM 2007, Nuremberg* 2007.

Vector control has been widely used for the high-performance drive of the induction motor. As in DC motor, torque control of the induction motor is achieved by controlling torque and flux components independently. Vector control techniques can be separated into two categories: direct and indirect flux vector orientation control schemes. For direct control methods, the flux vector is obtained by using stator terminal quantities, while indirect methods use the machine slip frequency to achieve field orientation.

The overall performance of field-oriented-controlled induction motor drive systems is directly related to the performance of current control. Therefore, decoupling the control scheme is required by compensation of the coupling effect between q-axis and d-axis current dynamics (Jung et al., 1999; Lin et al., 2000; Suwankawin et al., 2002).

The PWM is the interface between the control block of the electrical drive and its associated electrical motor (fig.1). This function controls the voltage or the current inverter (VSI or CSI) of the drive. The performance of the system is influenced by the PWM that becomes therefore an essential element of the system. A few problems of our days concerning the variable speed system are related to the conventional PWM: inverter switching losses, acoustical noise, and voltages harmonics (fig.2).

Harmonic elimination and control in inverter applications have been researched since the early 1960's (Bouchhida et al., 2007, 2008; Czarkowski et al., 2002; García et al., 2003; Meghriche et al., 2004, 2005; Villarreal-Ortiz et al., 2005; Wells et al., 2004). The majority of these papers consider the harmonic elimination problem in the context of either a balanced connected load or a single phase inverter application. Typically, many papers have focused

on finding solutions and have given little attention to which solution is optimal in an application context.

Minimizing Torque-Ripple in Inverter-Fed Induction Motor Using Harmonic Elimination PWM Technique 467

significantly reduced by the new PWM technique. The three-phase inverter is associated with a passive LC filter. The commutation angles are predetermined off-line and stored in the microcontroller memory in order to speed up the online control of the induction motors. Pre-calculated switching is modelled to cancel the greater part of low-order harmonics and to keep a single-pole DC voltage across the polarized capacitors. A passive LC filter is designed to cancel the high-order harmonics. This approach allows substantial reduction of the harmonic ratio in the AC main output voltage without increasing the number of switches per period. Consequently, the duties of the semiconductor power switches are alleviated. The effectiveness of the new harmonic elimination PWM technique for reducing torque-ripple in inverter-fed induction motors is confirmed by simulation results. To show

Figure 3 shows the new structure of the three-phase inverter, with E being the dc input voltage and U12-out=VC1-VC2, U23-out=VC2-VC3 and U31-out=VC3-VC1 are the ac output voltage obtained via a three LC filter. R represents the internal inductors resistance. Qi and Q'

(i=1,2,3) are the semiconductor switches. It is worth to mention that transistors Qi and Q'

undergo complementary switching states. VC1, VC2 and VC3 are the inverter filtered output

VC2 VC3

U31-out

Va Vb Vc

U12-out U23-out

R

Q3

Q'3

Vo3

L

**.**

C3

i

i

the validity of our approach, DSP-based experimental results are presented.

**2. New three-phase inverter model** 

**Figure 3.** New three-phase inverter model.

E

voltages taken across capacitors C1, C2 and C3 respectively.

V Vo2 o1 Q'1 Q'2

**. .**

C1 C2

L L

R R

Q1 Q2

VC1

A pre-calculated PWM approach has been developed to minimize the harmonic ratio within the inverter output voltage (Bouchhida et al., 2007, 2008; Bouchhida, 2008, 2011). Several other techniques were proposed in order to reduce harmonic currents and voltages. Some benefit of harmonic reduction is a decrease of eddy currents and hysterisis losses. That increase of the life span of the machine winding insulation. The proposed approach is integrated within different control strategies of induction machine.

**Figure 1.** Global scheme of the Induction machine control

**Figure 2.** Harmonic spectrum of the PWM inverter output voltage.

A novel harmonic elimination pulse width modulated (PWM) strategy for three-phase inverter is presented in this chapter. The torque ripple of the induction motor can be significantly reduced by the new PWM technique. The three-phase inverter is associated with a passive LC filter. The commutation angles are predetermined off-line and stored in the microcontroller memory in order to speed up the online control of the induction motors. Pre-calculated switching is modelled to cancel the greater part of low-order harmonics and to keep a single-pole DC voltage across the polarized capacitors. A passive LC filter is designed to cancel the high-order harmonics. This approach allows substantial reduction of the harmonic ratio in the AC main output voltage without increasing the number of switches per period. Consequently, the duties of the semiconductor power switches are alleviated. The effectiveness of the new harmonic elimination PWM technique for reducing torque-ripple in inverter-fed induction motors is confirmed by simulation results. To show the validity of our approach, DSP-based experimental results are presented.

## **2. New three-phase inverter model**

466 Induction Motors – Modelling and Control

application context.

on finding solutions and have given little attention to which solution is optimal in an

A pre-calculated PWM approach has been developed to minimize the harmonic ratio within the inverter output voltage (Bouchhida et al., 2007, 2008; Bouchhida, 2008, 2011). Several other techniques were proposed in order to reduce harmonic currents and voltages. Some benefit of harmonic reduction is a decrease of eddy currents and hysterisis losses. That increase of the life span of the machine winding insulation. The proposed approach is

integrated within different control strategies of induction machine.

**Figure 1.** Global scheme of the Induction machine control

**Figure 2.** Harmonic spectrum of the PWM inverter output voltage.

A novel harmonic elimination pulse width modulated (PWM) strategy for three-phase inverter is presented in this chapter. The torque ripple of the induction motor can be Figure 3 shows the new structure of the three-phase inverter, with E being the dc input voltage and U12-out=VC1-VC2, U23-out=VC2-VC3 and U31-out=VC3-VC1 are the ac output voltage obtained via a three LC filter. R represents the internal inductors resistance. Qi and Q' i (i=1,2,3) are the semiconductor switches. It is worth to mention that transistors Qi and Q' i undergo complementary switching states. VC1, VC2 and VC3 are the inverter filtered output voltages taken across capacitors C1, C2 and C3 respectively.

**Figure 3.** New three-phase inverter model.

### **2.1. Harmonic analysis**

In ideal case, the non filtered three inverter output voltage Vo1, Vo2 and Vo3 is desired to be:

$$\begin{cases} V\_{o1-ideal} = \frac{E}{2} \{ 1 + \cos \alpha \} \\ V\_{o2-ideal} = \frac{E}{2} \{ 1 + \cos(\alpha - \frac{2}{3}\pi) \} \\ V\_{o3-ideal} = \frac{E}{2} \{ 1 + \cos(\alpha - \frac{4}{3}\pi) \} \end{cases} \tag{1}$$

Minimizing Torque-Ripple in Inverter-Fed Induction Motor Using Harmonic Elimination PWM Technique 469

*sink ( ) d <sup>k</sup>* , for *k ,* <sup>0</sup> (4)

*a sink ( ) d <sup>k</sup>* , for *k ,N*<sup>0</sup> (5)

π

α

α

2π

α

0 2π

2 2π

 3 4

In practice, the number of harmonics N that can be identical is limited. Thus, a nonlinear

To solve the nonlinear system (5), we propose to use the genetic algorithms, to determine the switching angles αi (Bäck, 1996; Davis, 1991). The optimal switching angles family are

*i*

*i i k*

<sup>1</sup>

 <sup>1</sup>

*k ik*

α<sup>1</sup> α<sup>2</sup> α<sup>3</sup> α<sup>4</sup> α<sup>5</sup>

 3

*i*

system of N+1 equations having Nα unknowns is obtained as:

0 <sup>2</sup> <sup>1</sup> *N*

*i*

Vo1

E

Vo2

E

**Figure 4.** Inverter direct outputs representation for Nα=5.

E

Vo3

listed in table I.

0 <sup>2</sup> <sup>1</sup> *N*

With: α=ωt, and ω is the angular frequency.

The relative Fourier harmonic coefficients of (1), with respect to E, are given by (2.1) or more explicitly by (2.2).

$$d\_k^i = \frac{1}{E} \frac{1}{\pi} \int\_{-\pi}^{+\pi} V\_{oi-ideal} \cdot \cos(k\alpha) d\alpha \tag{2.1}$$

$$d\_0^i = 1 \; \; \; d\_1^i = \frac{1}{2} \; \; \; \; d\_k^i = 0 \; \; \; \; \; \; \; \; \; k \in \left\lceil 2 \right\rceil \infty \left\lceil \right\rceil \tag{2.2}$$

With index i is the phase number.

However, in practice, the non filtered inverter output voltage Vo1 (Vo2, Vo3) is a series of positive impulses (see Fig. 4): 0 when Q' 1 (Q' 2, Q' 3) is on and E when Q' 1(Q' 2, Q' 3) is off, so that voltage of capacitor C1 (C2, C3) is always a null or a positive value. In this case, the relative Fourier harmonic coefficients, with respect to E, are given by (3).

$$a\_k^1 = \frac{2}{k\pi} \sum\_{i=0}^{N\_a} \sin k\alpha\_i (-1)^{i+1} \tag{3}$$

Where:

k is the harmonic order

αi are the switching angles

Nα is the number of αi per half period

The other inverter outputs Vo2 and Vo3 are obtained by phase shifting Vo1 with 2/3 π, 4/3 π, respectively as illustrated in fig. 4 for Nα=5.

The objective is to determine the switching angles αi so as to obtain the best possible match between the inverter output Vo1 and Vo1-ideal.

For this purpose, we have to compare their respective harmonics. A perfect matching is achieved only when an infinite number of harmonics is considered as given by (4).

$$\frac{2}{k\pi} \sum\_{i=0}^{N\_n} \sin k\alpha\_i (-1)^{i+1} = d\_{k'} \text{ for } k \in \left[0, \infty\right] \tag{4}$$

In practice, the number of harmonics N that can be identical is limited. Thus, a nonlinear system of N+1 equations having Nα unknowns is obtained as:

468 Induction Motors – Modelling and Control

In ideal case, the non filtered three inverter output voltage Vo1, Vo2 and Vo3 is desired

*o ideal*

1

*<sup>E</sup> V [ cos ]*

*<sup>E</sup> V [ cos( )]*

*<sup>E</sup> V [ cos( )]*

The relative Fourier harmonic coefficients of (1), with respect to E, are given by (2.1) or more

However, in practice, the non filtered inverter output voltage Vo1 (Vo2, Vo3) is a series of

voltage of capacitor C1 (C2, C3) is always a null or a positive value. In this case, the relative

The other inverter outputs Vo2 and Vo3 are obtained by phase shifting Vo1 with 2/3 π, 4/3 π,

The objective is to determine the switching angles αi so as to obtain the best possible match

For this purpose, we have to compare their respective harmonics. A perfect matching is

achieved only when an infinite number of harmonics is considered as given by (4).

*<sup>k</sup> oi ideal d V .cos(k )d <sup>E</sup>* (2.1)

3) is on and E when Q'

*a sink ( ) <sup>k</sup>* (3)

*i*

*<sup>k</sup> d* , for *k ,* 2 (2.2)

1(Q' 2, Q'

3) is off, so that

*<sup>i</sup>* 1 1

<sup>0</sup> <sup>1</sup> *<sup>i</sup> <sup>d</sup>* , <sup>1</sup>

Fourier harmonic coefficients, with respect to E, are given by (3).

1 2 *<sup>i</sup> d* , 0 *<sup>i</sup>*

> 1 (Q' 2, Q'

 1 1 0 <sup>2</sup> <sup>1</sup> *N*

*k i i*

<sup>2</sup> <sup>1</sup> 2 3 <sup>4</sup> <sup>1</sup> 2 3

(1)

1 2

 

With: α=ωt, and ω is the angular frequency.

With index i is the phase number.

positive impulses (see Fig. 4): 0 when Q'

*o ideal*

2

*o ideal*

3

**2.1. Harmonic analysis** 

explicitly by (2.2).

Where:

k is the harmonic order

αi are the switching angles

Nα is the number of αi per half period

respectively as illustrated in fig. 4 for Nα=5.

between the inverter output Vo1 and Vo1-ideal.

to be:

$$a\_k = \frac{2}{k\pi} \sum\_{i=0}^{N\_a} \sin k a\_i (-1)^{i+1} = d\_{k'} \text{ for } k \in \left[0, N\right] \tag{5}$$

To solve the nonlinear system (5), we propose to use the genetic algorithms, to determine the switching angles αi (Bäck, 1996; Davis, 1991). The optimal switching angles family are listed in table I.

**Figure 4.** Inverter direct outputs representation for Nα=5.


**Table 1.** Optimal switching angles family with genetic algorithms

### **2.2. Dynamic LC filter behavior**

Considering the inverter direct output fundamental, the LC filter transfer function is given by:

$$\overline{T} = \frac{\overline{V}\_{C1}}{\overline{V}\_{o1}} = \frac{1}{1 - L\text{Co}^2 + j\text{RCo}}\tag{6}$$

Minimizing Torque-Ripple in Inverter-Fed Induction Motor Using Harmonic Elimination PWM Technique 471

2 2

1

*<sup>V</sup> x k jkxy* (8)

(9.1)

(9.2)

(9.3)

*x k* (11)

*RC* (13)

*dx* , can be expressed by:

(10)

(12)

2

4

For a given harmonic component of order k, the LC filter transfer function Tk is obtained by

 1

1

Assuming that the filter L and C components are not saturated, and using the superposition principle, we obtain the inverter filtered output voltages VC1, VC2 and VC3, taken across

*V*

*C*

0

<sup>1</sup> 2 *N C k k k k <sup>a</sup> V E( a T cos(k ))* 

1

1

amplitude equal to akTk, Where ak is the amplitude of the kth order harmonic of Vo1.

1

In which case, (12) corresponds to a maximum angular frequency, this last is given by:

*max <sup>y</sup> <sup>T</sup>*

2 2 1 *<sup>k</sup> kxy arctan*

*k*

*N*

*k*

*N*

2 3

2 3

Tk and φk are the kth order magnitude and phase components of the LC filter transfer

Each k harmonic term of the inverter output voltage has a frequency of kω and an

 222 2 22 1

*( xk) yxk*

1 1 4

<sup>2</sup> <sup>2</sup> <sup>1</sup> 2 *max*

*y*

2

*y*

2

1

*C k k k*

*<sup>a</sup> V E( a T cos[k( ) ])*

*C k k k*

*<sup>a</sup> V E( a T cos[k( ) ])*

*k*

capacitors C1, C2 and C3 as given by (9.1), (9.2) and (9.3) respectively:

1

2

3

0

0

The transfer function magnitude and phase are given, respectively, by:

*k k T T*

The maximum value Tmax of Tk , obtained when <sup>0</sup> *dT*

*T*

1

*o k*

replacing ω with kω as:

function respectively.

From (6), one can notice that for ω=0, *T* =1, meaning that the mean value (dc part) of the input voltage is not altered by the filter. Consequently, the inverter dc output part is entirely transferred to capacitor C1. The same conclusion can be drawn for capacitor C2 and C3.

Letting  $\chi = \arg\overline{\text{LC}}$  and  $\underline{\nu} = R\sqrt{\frac{C}{L}}$ , the filter transfer function can rewritten as: 
$$\overline{T} = \frac{\overline{V}\_{\text{C1}}}{\overline{V}\_{\text{o1}}} = \frac{1}{1 - x^2 + jxy} \tag{7}$$

For a given harmonic component of order k, the LC filter transfer function Tk is obtained by replacing ω with kω as:

470 Induction Motors – Modelling and Control

Nα=3

Nα=5

Nα=7

Nα=9

**2.2. Dynamic LC filter behavior** 

Letting *x LC* and *<sup>C</sup> y R <sup>L</sup>*

by:

α1 α2 α3

α1 α2 α3 α4 α5

α1 α2 α3 α4 α5 α6 α7

α1 α2 α3 α4 α5 α6 α7 α8 α9

**Table 1.** Optimal switching angles family with genetic algorithms

Family symbol Angles (radians) Angles (degrees)

0.817809468 1.009144336 1.911639657

1.051000076 1.346257127 1.689593122 2.374938655 2.47770082

0.52422984 0.57159284 1.14918972 1.41548576 1.66041537 2.16577455 2.29821202

0.43157781 0.45713212 0.70162245 0.77452452 0.96140142 1.09916539 1.21595592 1.45409688 1.64220075

Considering the inverter direct output fundamental, the LC filter transfer function is given

From (6), one can notice that for ω=0, *T* =1, meaning that the mean value (dc part) of the input voltage is not altered by the filter. Consequently, the inverter dc output part is entirely transferred to capacitor C1. The same conclusion can be drawn for capacitor C2 and C3.

> 1

1

*V x jxy*

1

*C o V T*

2

1

*V LC jRC*

1

2

, the filter transfer function can rewritten as:

1

1

*C o V T*

1

46.8570309622392 57.8197113723319 109.528884295936

60.2178686227288 77.1348515165077 96.8065549849324 136.073961533976 141.961799882103

30.0361573268184 32.7498573318965 65.8437208158208 81.1013600088678 95.134792939653 124.089741091845 131.677849172236

24.7275870444989 26.1917411558679 40.2000051966286 44.3769861253959 55.0842437838843 62.9775378338511 69.6691422899472 83.3136142271409 94.0911720882184

(6)

(7)

$$\overline{T}\_k = \left(\frac{\overline{V}\_{C1}}{\overline{V}\_{o1}}\right)\_k = \frac{1}{1 - \mathbf{x}^2 k^2 + j kxy} \tag{8}$$

Assuming that the filter L and C components are not saturated, and using the superposition principle, we obtain the inverter filtered output voltages VC1, VC2 and VC3, taken across capacitors C1, C2 and C3 as given by (9.1), (9.2) and (9.3) respectively:

$$V\_{C1} = E(\frac{a\_0}{2} + \sum\_{k=1}^{N} a\_k T\_k \cos(ka + \mathfrak{q}\_k)) \tag{9.1}$$

$$V\_{C2} = E(\frac{a\_0}{2} + \sum\_{k=1}^{N} a\_k T\_k \cos\{k(a - \frac{2}{3}\pi) + \Phi\_k\})\tag{9.2}$$

$$V\_{C3} = E(\frac{a\_0}{2} + \sum\_{k=1}^{N} a\_k T\_k \cos[k(a - \frac{4}{3}\pi) + \Phi\_k])\tag{9.3}$$

Tk and φk are the kth order magnitude and phase components of the LC filter transfer function respectively.

Each k harmonic term of the inverter output voltage has a frequency of kω and an amplitude equal to akTk, Where ak is the amplitude of the kth order harmonic of Vo1.

The transfer function magnitude and phase are given, respectively, by:

$$\left| \overline{T}\_k \right| = T\_k = \frac{1}{\sqrt{(1 - x^2 k^2)^2 + y^2 x^2 k^2}} \tag{10}$$

$$\text{op}\_k = -\arctan \frac{kxy}{1 - \alpha^2 k^2} \tag{11}$$

The maximum value Tmax of Tk , obtained when <sup>0</sup> *dT dx* , can be expressed by:

$$T\_{\text{max}} = \frac{\frac{1}{y^2}}{\sqrt{\frac{1}{y^2} - \frac{1}{4}}} \tag{12}$$

In which case, (12) corresponds to a maximum angular frequency, this last is given by:

$$
\alpha\_{\text{max}} = \frac{\sqrt{2}}{RC} \sqrt{1 - \frac{y^2}{2}} \tag{13}
$$

The maximum angular frequency ωmax exists if, and only if, *y* 2 . The filter transfer function will exhibit a peak value then decreases towards zero. As consequence, the fundamental, as well as the harmonics, are amplified, this leads to undesirable situation, as illustrated in fig.5.

**Figure 5.** LC filter transfer function magnitude for the fundamental.

If (14) is satisfied, the filter transfer function will exhibit a damped behaviour.

$$y = R\sqrt{\frac{C}{L}} > \sqrt{2} \tag{14}$$

Minimizing Torque-Ripple in Inverter-Fed Induction Motor Using Harmonic Elimination PWM Technique 473

*<sup>E</sup> V [ a .T .cos( ) a .T .cos( . ) a .T .cos( . ) ...* (16)

<sup>1</sup> 1 1 1 88 8 10 10 <sup>13</sup> 1 8 10

*<sup>E</sup> V [ a .T .cos([ ] ) a .T .cos( [ ] ) a .T .cos( [ ] ) ...* (17)

*<sup>E</sup> V [ a .T .cos([ ] ) a .T .cos( [ ] ) a .T .cos( [ ] ) ...* (18)

Using (15), (16), (17) and (18), we get the three inverter filtered output voltages expressions

(19)

(20)

 

*k k n k n*

32 2 23 3

> 

*k k* 8

 

 3 1 3 2 *n k n*

34 4 23 3

 

*out k k k*

*U E a T cos( ) a T cos(k )*

*out k k k*

*U E a T cos( ) a T cos(k )*

From the precedent fig. 3, and for each lever, the equations with the currents and the voltages can be written in the following form (Bouchhida et al., 2007, 2008; Bouchhida, 2008,

1

*dv (i i ) dt C*

1

1

2

*dv (i i ) dt C*

1

3

*dv (i i ) dt C*

1 1

*ch*

*ch*

*ch*

2 2

3 3

1

*C*

2

*C*

3

*C*

 

 

*out k k k k k n k n*

*U E a T cos( ) a T cos(k )*

2 1 1 1 88 8 10 10 10 22 2 <sup>1</sup> <sup>8</sup> <sup>10</sup> 23 3 3 *<sup>C</sup>*

3 1 1 1 88 8 10 10 10 44 4 <sup>1</sup> <sup>8</sup> <sup>10</sup> 23 3 3 *<sup>C</sup>*

2 *<sup>C</sup>*

 

12 1 1 1

3 2

23 1 1 1

31 1 1 1

as:

with:

2011).

6 *k k* .

Currents equations

Voltages equations:

This condition matches both practical convenience and system objectives.

### **2.3. Harmonic rate calculation**

Using (1) to (3), the non filtered inverter output voltages Vo1, Vo2 and Vo3 can be expressed as

$$V\_{o1} = \frac{E}{2}(1 + \cos\alpha t) + a\_5 \cos 5\alpha t + a\_6 \cos 6\alpha t + \dots \tag{15.1}$$

$$V\_{o2} = \frac{E}{2} [1 + \cos(\alpha t - \frac{2}{3}\pi)] + a\_5 \cos \mathfrak{S} (\alpha t - \frac{2}{3}\pi) + a\_6 \cos \mathfrak{G} (\alpha t - \frac{2}{3}\pi) + \dots \tag{15.2}$$

$$V\_{o3} = \frac{E}{2} [1 + \cos(\alpha t - \frac{4}{3}\pi)] + a\_5 \cos \dots \text{(\text{\textdegree} } -\frac{4}{3}\pi \text{)} + a\_6 \cos \theta (\alpha t - \frac{4}{3}\pi) + \dots \tag{15.3}$$

Taking into consideration the filter transfer function, we get the expressions (16), (17) and (18) for VC1, VC2 and VC3 respectively.

$$V\_{C1} = \frac{E}{2} \{ 1 + a\_1 T\_1 . \cos(\alpha + \phi\_1) + a\_8 T\_8 . \cos(8.a + \phi\_8) + a\_{10} T\_{10} . \cos(10.a + \phi\_{13}) + \dots \tag{16}$$

$$W\_{C2} = \frac{E}{2} \{ 1 + a\_1 T\_1 \cos(\{a - \frac{2}{3}\pi\} + \phi\_1) + a\_8 T\_8 \cos(8\{a - \frac{2}{3}\pi\} + \phi\_8) + a\_{10} T\_{10} \cos(10\{a - \frac{2}{3}\pi\} + \phi\_{10}) + \dots \tag{17}$$

$$V\_{C3} = \frac{E}{2} [1 + a\_1 T\_1 \cos(\{\alpha - \frac{4}{3}\pi\} + \mathfrak{q}\_1) + a\_8 T\_8 \cos(8\{\alpha - \frac{4}{3}\pi\} + \mathfrak{q}\_8) + a\_{10} T\_{10} \cos(10\{\alpha - \frac{4}{3}\pi\} + \mathfrak{q}\_{10}) + \dots \tag{18}$$

Using (15), (16), (17) and (18), we get the three inverter filtered output voltages expressions as:

$$\begin{aligned} \left[ \begin{aligned} \mathbf{U}\_{12-\alpha t} &= E \frac{\sqrt{3}}{2} \left[ a\_1 T\_1 \cos(\alpha + \beta\_1) + \sum\_{\substack{k=8\\k=3n+1}}^{\infty} a\_k T\_k \cos(k\alpha + \beta\_k) \right] \\\\ \mathbf{U}\_{23-\alpha t} &= E \frac{\sqrt{3}}{2} \left[ a\_1 T\_1 \cos(\alpha + \beta\_1 - \frac{2}{3}\pi) + \sum\_{\substack{k=8\\k=3n+1\\k=3n+2}}^{\infty} a\_k T\_k \cos(k\alpha + \beta\_k - \frac{2}{3}\pi) \right] \\\\ \mathbf{U}\_{31-\alpha t} &= E \frac{\sqrt{3}}{2} \left[ a\_1 T\_1 \cos(\alpha + \beta\_1 - \frac{4}{3}\pi) + \sum\_{\substack{k=8\\k=3n+1\\k=3n+2}}^{\infty} a\_k T\_k \cos(k\alpha + \beta\_k - \frac{4}{3}\pi) \right] \end{aligned} \tag{19}$$

with: 6 *k k* .

472 Induction Motors – Modelling and Control

illustrated in fig.5.

0.5

1

1.5

magnitude T1

2

2.5

The maximum angular frequency ωmax exists if, and only if, *y* 2 . The filter transfer function will exhibit a peak value then decreases towards zero. As consequence, the fundamental, as well as the harmonics, are amplified, this leads to undesirable situation, as

> y=1.4142 y=0.4142 y=1.7

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> <sup>0</sup>

x

Using (1) to (3), the non filtered inverter output voltages Vo1, Vo2 and Vo3 can be expressed as

1 56 1 56

 2 56 2 22 1 56 23 3 3 *<sup>o</sup>*

 3 56 4 44 1 56 23 3 3 *<sup>o</sup>*

Taking into consideration the filter transfer function, we get the expressions (16), (17) and

<sup>2</sup> *<sup>C</sup> y R <sup>L</sup>* (14)

*<sup>E</sup> V ( cos t) a cos t a cos t ...* (15.1)

*<sup>E</sup> V [ cos( t )] a cos ( t ) a cos ( t ) ...* (15.2)

*<sup>E</sup> V [ cos( t )] a cos ( t ) a cos ( t ) ...* (15.3)

If (14) is satisfied, the filter transfer function will exhibit a damped behaviour.

This condition matches both practical convenience and system objectives.

**Figure 5.** LC filter transfer function magnitude for the fundamental.

2 *<sup>o</sup>*

**2.3. Harmonic rate calculation** 

(18) for VC1, VC2 and VC3 respectively.

From the precedent fig. 3, and for each lever, the equations with the currents and the voltages can be written in the following form (Bouchhida et al., 2007, 2008; Bouchhida, 2008, 2011).

Currents equations

$$\begin{cases} \frac{dv\_{C1}}{dt} = \frac{1}{C\_1}(i\_1 - i\_{1ch})\\ \frac{dv\_{C2}}{dt} = \frac{1}{C\_2}(i\_2 - i\_{2ch})\\ \frac{dv\_{C3}}{dt} = \frac{1}{C\_3}(i\_3 - i\_{3ch}) \end{cases} \tag{20}$$

Voltages equations:

$$\begin{cases} \frac{di\_1}{dt} = (V\_{o1} - R\_1.i\_1 - v\_{C1}).\frac{1}{L\_1} \\ \frac{di\_2}{dt} = (V\_{o2} - R\_2.i\_2 - v\_{C2}).\frac{1}{L\_2} \\ \frac{di\_3}{dt} = (V\_{o3} - R\_3.i\_3 - v\_{C3}).\frac{1}{L\_3} \end{cases} \tag{21}$$

Where:

Vds , Vqs : d-axis and q-axis stator voltages;

ids , iqs : d-axis and q-axis stator currents;

Rs , Rr : stator and rotor resistances;

Ls , Lr : stator and rotor inductances;

Msr : mutual inductance

ω<sup>g</sup> : slip speed ωg=(ωs-ωr)

p : number of pole pairs

σ : leakage coefficient, (

flux vector, (ψdr=ψr and ψqr=0), can be expressed as:

*di*

Ω<sup>r</sup> : mechanical rotor angular speed

ψdr ,ψqr : d-axis and q-axis rotor flux linkages;

ωs ,ωr : electrical stator and rotor angular speed

Cr , Cem : external load torque and motor torque

j , f : inertia constant and motor damping ratio

2 1 *sr s r M ) L L*

Equation (23) represents the dynamic of the motor mechanical side and (26) describes the electromagnetic torque provided on the rotor. The model of a three phase squirrel cage induction motor in the synchronous reference frame, whose axis d is aligned with the rotor

 1 *dr sr*

*d M <sup>i</sup>*

*s ds qs dr ds*

*ds dr r r*

*qs* 1

<sup>1</sup> *ds di <sup>K</sup> ii V dt ds s qs dr ds T L*

*i i pK V dt <sup>L</sup>* (28)

*s*

*dt T T* (29)

(27)

*r s*

These equations are put in following matric form:

$$
\frac{d}{dt} \begin{bmatrix} v\_{\mathbb{C}} \\ I \end{bmatrix} = \begin{bmatrix} \mathbf{0}\_{3 \times 3} & \frac{\mathbf{1}}{\mathbf{C}} \times \mathbf{I}\_{3 \times 3} \\\\ -\frac{\mathbf{1}}{\mathbf{L}} \times \mathbf{I}\_{3 \times 3} & -\mathbf{R} \times \mathbf{I}\_{3 \times 3} \end{bmatrix} \begin{bmatrix} v\_{\mathbb{c}} \\ I \end{bmatrix} \\
+ \begin{bmatrix} -\frac{\mathbf{1}}{\mathbf{C}} \times I\_{3 \times 3} & \mathbf{0}\_{3 \times 3} \\\\ \mathbf{0}\_{3 \times 3} & \frac{\mathbf{1}}{\mathbf{L}} \times I\_{3 \times 3} \end{bmatrix} \begin{bmatrix} i\_{ch} \\ V\_o \end{bmatrix} \tag{22}
$$
  $\text{with } : \quad v\_{\mathbb{C}} = \begin{bmatrix} v\_{\mathbb{C}1} \\ v\_{\mathbb{C}2} \\ v\_{\mathbb{C}3} \end{bmatrix}; I = \begin{bmatrix} i\_1 \\ i\_2 \\ i\_3 \end{bmatrix}; \ v\_o = \begin{bmatrix} v\_{o1} \\ v\_{o2} \\ v\_{o3} \end{bmatrix}$ 

## **3. Indirect field- oriented induction motor drive**

The dynamic electrical equations of the induction machine can be expressed in the d-q synchronous reference frame as:

$$\begin{aligned} \frac{d\dot{l}\_{ds}}{dt} &= -\frac{1}{\sigma L\_s} (R\_s + R\_r \frac{M\_{sr}^2}{L\_r^2}) \dot{i}\_{ds} + \alpha\_s \dot{i}\_{qs} + \frac{M\_{sr} R\_r}{\sigma L\_s L\_r^2} \Psi\_{dr} + \frac{M\_{sr}}{\sigma L\_s L\_r} \Psi\_{qr} \alpha\_r + \frac{1}{\sigma L\_s} V\_{ds} \\ \frac{d\dot{l}\_{qs}}{dt} &= -\alpha\_s \dot{i}\_{ds} - \frac{1}{\sigma L\_s} (R\_s + R\_r \frac{M\_{sr}^2}{L\_r^2}) \dot{i}\_{qs} - \frac{M\_{sr}}{\sigma L\_s L\_r} \Psi\_{dr} \alpha\_r + \frac{M\_{sr} R\_r}{\sigma L\_s L\_r^2} \Psi\_{qr} + \frac{1}{\sigma L\_s} V\_{qs} \\ \frac{d\Psi\_{dr}}{dt} &= \frac{M\_{sr} R\_r}{L\_r} \dot{i}\_{ds} - \frac{R\_r}{L\_r} \Psi\_{dr} + \alpha\_g \Psi\_{qr} \\ \frac{d\Psi\_{qr}}{dt} &= \frac{M\_{sr} R\_r}{L\_r} \dot{i}\_{qs} - \alpha\_g \Psi\_{dr} - \frac{R\_r}{L\_r} \Psi\_{qr} \end{aligned} \tag{23}$$

$$\frac{d\Omega\_r}{dt} = -\frac{f}{\dot{\jmath}}\Omega\_r - \frac{1}{\dot{\jmath}}(\mathcal{C}\_{em} - \mathcal{C}\_r) \tag{24}$$

$$
\Omega\_r = \frac{\text{op}\_r}{p} \tag{25}
$$

$$\mathbf{C}\_{em} = p \frac{M\_{sr}}{L\_r} (\boldsymbol{\Psi}\_{dr} \dot{\mathbf{i}}\_{qs} - \boldsymbol{\Psi}\_{qr} \dot{\mathbf{i}}\_{ds}) \tag{26}$$

### Where:

474 Induction Motors – Modelling and Control

with :

1

2

3

 

 

**3 3 3 3**

**<sup>1</sup> 0 I C <sup>1</sup> I RI**

;

**3. Indirect field- oriented induction motor drive** 

2

*ds dr g qr*

*dr sr r r*

*d MR R <sup>i</sup> dt L L*

*r*

*qr sr r*

*dt L*

*<sup>d</sup> M R <sup>R</sup> <sup>i</sup>*

*qs g dr*

*r r*

**33 33**

 1 2 3

*o o*

*v v*

*o*

*v*

*o*

The dynamic electrical equations of the induction machine can be expressed in the d-q

1 1

 

*di <sup>M</sup> MR M (R R )i i <sup>V</sup> dt L L L L LL L*

2 2 2

*ds sr sr r sr*

*qs sr sr sr r*

*r qr r L*

*di M M MR i (R R )i <sup>V</sup> dt L L L L L L L*

<sup>1</sup> *<sup>r</sup>*

 *<sup>r</sup> <sup>r</sup> p*

 *sr em dr qs qr ds r*

*s ds s r qs dr r qr qs s r s r s r s*

1 1

*r em r*

*<sup>d</sup> <sup>f</sup> (C C ) dt j j* (24)

*<sup>M</sup> C p ( i i) <sup>L</sup>* (26)

2 2

*s r ds s qs dr qr r ds s r s r s r s*

*v*

These equations are put in following matric form:

**L**

 1 2 3

*i I i i*

 

*v v v v*

*C*

synchronous reference frame as:

 

*C C C* 1 2 3

;

1 11 1

*o C*

*di (V R .i v ). dt <sup>L</sup>*

*di (V R .i v ). dt <sup>L</sup>*

*di (V R .i v ). dt <sup>L</sup>*

2 22 2

*o C*

3 33 3

*C c ch*

*<sup>I</sup> <sup>d</sup> v v <sup>C</sup> <sup>i</sup> dt I I <sup>V</sup> <sup>I</sup>*

*o C*

1

1

1

2

(21)

(22)

(23)

(25)

*o*

1

3

33 33

<sup>1</sup> <sup>0</sup>

<sup>1</sup> <sup>0</sup>

3 3 3 3

*L*


Equation (23) represents the dynamic of the motor mechanical side and (26) describes the electromagnetic torque provided on the rotor. The model of a three phase squirrel cage induction motor in the synchronous reference frame, whose axis d is aligned with the rotor flux vector, (ψdr=ψr and ψqr=0), can be expressed as:

$$\frac{d\dot{\mathbf{i}}\_{ds}}{dt} = -\gamma \dot{\mathbf{i}}\_{ds} + \alpha \frac{\dot{\mathbf{e}}}{\mathbf{s}} \frac{\mathbf{i}}{\mathbf{q}\mathbf{s}} + \frac{\mathbf{K}}{T\_r} \boldsymbol{\Psi}\_{dr} + \frac{1}{\sigma L\_s} V\_{ds} \tag{27}$$

$$\frac{di\_{qs}}{dt} = -\
\left\|\phi\_s i\_{ds} - \gamma \right\|\_{qs} - p\Omega K \psi\_{dr} + \frac{1}{\sigma L\_s} V\_{ds} \tag{28}$$

$$\frac{d\Psi\_{dr}}{dt} = \frac{M\_{sr}}{T\_r}\dot{\imath}\_{ds} - \frac{1}{T\_r}\Psi\_{dr} \tag{29}$$

$$\frac{d\boldsymbol{\Psi}\_{qr}}{dt} = \frac{M\_{sr}}{T\_r}\dot{\mathbf{i}}\_{qs} - (\boldsymbol{\alpha}\_s - p\boldsymbol{\Omega})\boldsymbol{\Psi}\_{dr} \tag{30}$$

0.012(s). The harmonic spectrum of the filtered output voltage is shown in (figure 7.b). In order to compare the performance of the new three-phase inverter with the conventional PWM inverter, the output voltage of the latter, where a modulation index of 35 was used, is

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

t(s)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

t(s)

(a)

(b) **Figure 7.** (a) Three filtered inverter outputs voltages.(b) Harmonic spectrum of the filtered output voltage.

t(s)

shown in (figure 8.a) and its harmonic spectrum is presented in (figure 8.b).


500


('---' filtered output voltage) ('—' ideal output voltage)

0

0

500


0

500

$$\frac{d\dot{\Omega}}{dt} = \frac{pM\_{sr}}{fL\_r}(\psi\_{dr}\dot{i}\_{qs}) - \frac{C\_r}{J} - f\,\,\Omega\tag{31}$$

With: 2 2 *r sr s r sr r r s r s s r L M R RM T ,K , . <sup>R</sup> LL L L L*

The bloc diagram of the proposed indirect field-oriented induction motor drive is shown in fig.6. Speed information, obtained by encoder feedback, enables computation of the torque reference using a PI controller. The reference flux is set constant in nominal speed. For higher speeds, rotor flux must be weakened.

**Figure 6.** Block diagram of the proposed indirect field oriented induction motor drive system.

### **4. Simulation results**

To demonstrate the performance of a new tree-phase inverter, we simulated three filtered inverter output voltages. Two frequency values are imposed on the inverter, starting with a frequency of 50 Hz, then at time t=0.06(s) the frequency is changed to 60Hz. The three filtered inverter output voltages are illustrated in (figure 7.a) which clearly shows that the three voltages are perfectly sinusoidal and follow the ideal values with a transient time of 0.012(s). The harmonic spectrum of the filtered output voltage is shown in (figure 7.b). In order to compare the performance of the new three-phase inverter with the conventional PWM inverter, the output voltage of the latter, where a modulation index of 35 was used, is shown in (figure 8.a) and its harmonic spectrum is presented in (figure 8.b).

476 Induction Motors – Modelling and Control

*r*

Ω<sup>r</sup>

Ωre f

**4. Simulation results** 

With:

higher speeds, rotor flux must be weakened.

C\* em

Ψ\* r The bloc diagram of the proposed indirect field-oriented induction motor drive is shown in fig.6. Speed information, obtained by encoder feedback, enables computation of the torque reference using a PI controller. The reference flux is set constant in nominal speed. For

*sr r dr qs*

> 2 2

*qs s dr*

*<sup>d</sup> <sup>M</sup> i ( p) dt T* (30)

*<sup>d</sup> pM C ( i) f dt JL <sup>J</sup>* (31)

New Threephase Inverter

E

Control Signals **generator**  Vo1

Vo2

Vo3

ia ib ic

 

  *K R A P*

1

V\* 2

1

K R A <sup>P</sup>

 

V\* ds

V\* qs

 

V\* 3 **IM** 

Ω<sup>r</sup>

*.*

*.*

*r sr s r sr*

*r s r s s r L M R RM T ,K , . <sup>R</sup> LL L L L*

**Field** 

PI V\*

**Oriented** 

**Control** 

ids Ω iqs <sup>r</sup>

*qr sr*

*r*

*r*

**Figure 6.** Block diagram of the proposed indirect field oriented induction motor drive system.

To demonstrate the performance of a new tree-phase inverter, we simulated three filtered inverter output voltages. Two frequency values are imposed on the inverter, starting with a frequency of 50 Hz, then at time t=0.06(s) the frequency is changed to 60Hz. The three filtered inverter output voltages are illustrated in (figure 7.a) which clearly shows that the three voltages are perfectly sinusoidal and follow the ideal values with a transient time of

**Figure 7.** (a) Three filtered inverter outputs voltages.(b) Harmonic spectrum of the filtered output voltage.

0 20 40

> -1 0 1

> > 0 20 40

> > > -1 0 1

q-axis flux(Wb)

(b)

**Figure 9.** (a) Simulation results of the indirect field-oriented control for proposed inverters (b) Simulation results of the indirect field-oriented control for conventional PWM inverters

torque(N.m)

q-axis current(A)

q-axis flux(Wb)

(a)

torque(N.m)

wr wrref

ids idsref

> wr wrref

ids idsref q-axis currant(A)

0 0.5 1 1.5 2 2.5 3

T Tref

iqs iqsref

iqs iqsref

T Tref

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

In order to illustrate the effectiveness of the proposed inverter, the torque response obtained by using the proposed and the conventional PWM inverters are shown in (figure 10.a) and (figure 10.b), respectively. The obtained results clearly show that the conventional PWM inverter generates more oscillations in the torque than the proposed structure (figure 11). Moreover, the switching frequency of the proposed inverter is dramatically reduced (see (figure 12.a)) when compared to its counterpart in the conventional PWM inverter (see figure 12.b). Therefore, the proposed inverter gives a better dynamic response than the

conventional PWM inverter.

0 50 100

> 0 10 20

> > > 50 100

> > > > 0 10 20

d-axis flux(Wb)

d-axis current(A)

speed(rad/sec)

d-axis flux(Wb)

d-axis current(A)

speed(rad/sec)

0 0.5 1 1.5 2 2.5 3

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

0 0.5 1 1.5 2 2.5 3

t(s)

**Figure 8.** (a) PWM inverter output voltage (b) Harmonic spectrum of thePWM inverter output voltage.

We carried out two simulations of the field-oriented control for induction motor drives with speed regulation using the new structure of the three phase inverter in the first simulation (figure 9.a) and the conventional PWM (figure 9.b) inverter in the second simulation. The instruction speed is set to 100 (rad/sec) for both simulations. During the period between 1.3(s) and 2.3(s), a resistive torque equal to 10 (N.m) (i.e the nominal torque) is applied.

In order to illustrate the effectiveness of the proposed inverter, the torque response obtained by using the proposed and the conventional PWM inverters are shown in (figure 10.a) and (figure 10.b), respectively. The obtained results clearly show that the conventional PWM inverter generates more oscillations in the torque than the proposed structure (figure 11). Moreover, the switching frequency of the proposed inverter is dramatically reduced (see (figure 12.a)) when compared to its counterpart in the conventional PWM inverter (see figure 12.b). Therefore, the proposed inverter gives a better dynamic response than the conventional PWM inverter.

478 Induction Motors – Modelling and Control


0

0.2

0.4

0.6

Harmonic Amplitude

0.8

1



0

Vas

100

200

300

400

<sup>0</sup> 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 -400

t(s)

(a)

0 10 20 30 40 50 60 70 80 90 100

Harmonic Order

(b) **Figure 8.** (a) PWM inverter output voltage (b) Harmonic spectrum of thePWM inverter output voltage.

We carried out two simulations of the field-oriented control for induction motor drives with speed regulation using the new structure of the three phase inverter in the first simulation (figure 9.a) and the conventional PWM (figure 9.b) inverter in the second simulation. The instruction speed is set to 100 (rad/sec) for both simulations. During the period between 1.3(s) and 2.3(s), a resistive torque equal to 10 (N.m) (i.e the nominal torque) is applied.

**Figure 9.** (a) Simulation results of the indirect field-oriented control for proposed inverters (b) Simulation results of the indirect field-oriented control for conventional PWM inverters

<sup>0</sup> 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 <sup>0</sup>

t(s) (a)

<sup>0</sup> 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 <sup>0</sup>

t(s) (b) **Figure 12.** (a) Switching frequency for proposed inverters (b) Switching frequency for conventional

The experimental setup was realized based on the DS1103 TMS320F240 dSPACE kit (dSPACE, 2006a, 2006b, 2006c, 2006d, 2006e). Figure 13 gives the global scheme of the experimental setup. This kit allows real time implementation of inverter and induction motor IM speed drive, it includes several functions such as Analog/Digital converters and digital signal filtering. In order to run the application the control algorithm must be written in C language. Then, we use the RTW and RTI packages to compile and load the algorithm on processor. To visualize and adjust the control parameters in real time we use the software

The novel single phase inverter structure for pre-calculated switching is based on the use of IGBT (1000V/25A) with 10 kHz as switching frequency. The switching angles are predetermined off-line using Genetic Algorithms and stored in the card memory in order to speed up the programme running. The non-filtered inverter output voltages are first

control-desk which allows conducting the process by the computer.

0.2

0.2

PWM inverters

**5. Experimental setup** 

0.4

0.6

0.8

1

0.4

0.6

0.8

1

**Figure 10.** (a) Torque response for proposed inverters (b)Torque response for conventional PWM inverters

**Figure 11.** Torque response for proposed and conventional PWM inverters

**Figure 12.** (a) Switching frequency for proposed inverters (b) Switching frequency for conventional PWM inverters
