The space vector modulation technique (SVM ou SVPWM: Space Vector Pulse Width Modulation) is a variant of the pulse width modulation control which is based on the vector representation of the voltages in the complex plane [4] [20] [21] [22]. SVM control technology is used for conventional single-phase and three-phase inverters [12] [23]. It applies well to neutral point voltage balancing problems, making it flexible and also reducing switching losses in switches [12] [14] [24] [25]. From the three-phase inverter in Figure 2, we will measure the current and voltage in one phase of the inverter. Various PWM techniques have been developed for industrial applications. The most commonly used PWM schemes for three-phase voltage inverters are Space Vector PWM (SVPWM) [5] [6] [20] [22]. The principle of vector modeling consists in reconstructing the voltage vector from eight voltage vectors [5]. Each of these vectors corresponds to a combination of the state of the switches of the three-phase voltage inverter. It does not rely on separate modulation calculations by each inverter arm [20] [22]. The Construction of the output vectors of the three-phase inverter and the different states of the inverter are given in [5] [22]. This technique follows the following principles:

A reference vector is calculated globally and approximated over a modulation period T_s;

All half-bridge switches have an identical state at the centers and at the ends of the period. A combinatorial analysis of all the possible states of the switches makes it possible to calculate the voltage vector ( v α , v β ), according to the System (1). [6] [20]

[ v α v β ] = 2 3 ( 1 − 1 2 − 1 2 0 3 2 − 3 2 ) [ v A N v B N v C N ] (1)

The vector v α is approximated over the modulation period, by the generation of an average vector developed by the application of the vectors. It consists in considering the three-phase system globally, and in applying a Concordia transformation to it to be brought back to the plane ( v α , v β ) [5] [20] [22]. The sampling time can then be represented as a single vector in this plane [6]. The major drawback of SVPWM is the enormous computational time required for its implementation due to the trigonometric functions used to determine the times T_k and T_k+1 [20]. When PWM is used to control solid state switches, a relatively high frequency carrier is required to reduce current and voltage ripple and THD [8]. This increase in frequency leads to switching losses [12] [19] on the one hand, and to thermal stress at the switches on the other hand [8] [20], hence the interest of improving it by using the duty cycle regulator. Recent literature [12] [13] [14] [15] presents the duty cycle modulation (DCM) method as a signal modulation technique with interesting characteristics and the particularity of producing fewer harmonics.

In the past, inverters were controlled only using PWM (pulse width modulation) based drivers, but nowadays it is also done from drivers based on the DCM [2]. The DCM is a modulation in which an input signal x is transformed into a train of switching wave where duty cycle and period of the modulated signal vary simultaneously according to control signal [14] [16] [26]. Initially invented and patented by Biard et al. in 1962, the realization of classical DCM circuit requires at least two integrated circuits and seven passive elements [23] [27]. Initially proposed for instrumentation problems, they are today used in many sectors such as ADC [14] [28], DAC [17], control of electronic power converter [13] [29], digital signal transmission [26], power quality [30] [31] [32] and optical transmission [7]. Figure 3(a) shows the basic electronic circuitry of the DCM while its mathematical model is described by Equation (2).

This structure, although functional and allowing low-frequency operation, has some drawbacks, among which: The non-linearity of the function linking the

duty-cycle to the control signal and the dependence of the duty-cycle on the values of the passive components [11].

{ u ( t ) + = α 1 x m ( t , x ( t ) ) + ( 1 − α 1 ) x ( t )       with     α 1 = R 1 R 1 + R 2 ε ( t ) = u ( t ) + − u ( t ) 4 = u ( t ) + − u C ( t ) x m ( t , x ( t ) ) = { + E ,   if   ε ( t ) ≥ 0 − E ,   otherwise d u C ( t ) d t = − 1 R C ( u C ( t ) + x m ( t , x ( t ) ) ) (2)

The high level time T_on, the carrier period T_m, the duty cycle R_m and the low level time T_off give as shown in Equation (3) [3].

{ T o n ( x ( t ) , α ) = R C ln ( ( 1 − α ) x ( t ) − ( 1 + α ) E ( 1 − α ) x ( t ) + ( α + 1 ) E ) T m ( x ( t ) , α ) = R C ln ( ( ( 1 − α ) x ( t ) ) 2 − ( ( 1 + α ) E ) 2 ( ( 1 − α ) x ( t ) ) 2 + ( ( α + 1 ) E ) 2 ) R m ( x ( t ) , α ) = T o n ( x ( t ) , α ) T m ( x ( t ) , α ) T o f f ( x ( t ) , α ) = T m ( x ( t ) , α ) ( 1 − R m ( x ( t ) , α ) ) (3)

This modulator has the advantage of being simpler than a PWM modulator. Its period T_m and duty cycle α are defined by [29]:

{ T m ( x ( t ) , α ) = R C ln ( ( ( 1 − α ) x ( t ) ) 2 − ( ( 1 + α ) E ) 2 ( ( 1 − α ) x ( t ) ) 2 + ( ( α + 1 ) E ) 2 ) α = ln ( ( ( 1 − α ) x ( t ) ) 2 − ( ( 1 + α ) E ) 2 ( ( 1 − α ) x ( t ) ) 2 + ( ( α + 1 ) E ) 2 ) α 1 = R 1 R 1 + R 2   and   α 2 = R 2 R 1 + R 2 (4)

Its minimum or central period and duty cycle (for x ( t ) = 0 𝑥𝑡=0 ) are given by:

{ T m ( 0 ) = 2 R C ln ( 1 + 2 R 1 R 2 ) α = 1 2 (5)

The DCM method discussed has the particularity of producing harmonics. Another particularity of DCM is its modulation frequency which is a function of the amplitude of the modulating signal. The major drawback is the imperative need to use a low-pass filter to eliminate the high frequencies. Subsequently, a new, simpler and better control strategy is proposed to combine the advantages of SVM and DCM.

The principle of the DCSVM control is based on the advantages of the DCM control and the classical SVM control. The implementation of the DCSVM strategy is carried out in the following steps:

Step 1: Determination of reference voltage ( v α , v β )

The generation of vectors in the Concordia landmark (d, q) from the three-phase signals ( v A N , v B N , v C N ) is given by the Relation (6):

{ v d = 2 3 ( v A N − v B N 2 − v C N 2 ) v q = 3 3 ( v B N − v C N ) (6)

[ v d v q ] = 1 3 ( 1 0 − 1 3 − 1 − 3 ) [ v A N v B N v C N ] (7)

Step 2: Cartesian to polar transformation

The system of equations gives in Equation (8) transforms corresponding elements of the two-dimensional Cartesian coordinate arrays d and q into polar coordinates ( ρ , θ ).

{ ρ = v d 2 + v q 2 θ = tan − 1 ( v q v d ) (8)

Step 3: Generation of control signals and intermediate times

The Simulink model is used to determine the control signals for switches T_ag, T_bg and T_cg, and the intermediate times T₁, T₂ and T₀.

Furthermore, assuming that the reference voltage vector V_ref is synthesized from the two basic vectors, the operating time of two basic vectors can be calculated according to the volt-second equilibrium principle [29].

{ T 1 = 3 ( V r e f V d c T S ) sin ( θ − ( k − 1 ) π 3 ) T 2 = 3 ( V r e f V d c T S ) sin ( k π 3 − θ ) T 0 = T S − T 1 − T 2 (9)

With: T₁ and T₂: times allocated to vectors V 1 and V 2 , T₀: time shared between the two null vectors V 7 and V 9 , and T_S: half-carrier time period.

After determining the times T₁, T₂ and T₀, the time of the closing all keys to the rack at work is determined. In each sector the switching time of each key is different and is expressed in (10):

{ for   θ = ] 0 , π 3 ]   ;   we   get : S = 1 for   θ = ] π 3 , 2 π 3 ]   ;   we   get : S = 2 for   θ = ] 2 π 3 , π ]   ;   we   get : S = 3 for   θ = ] π , 4 π 3 ]   ;   we   get : S = 4 for   θ = ] 4 π 3 , 5 π 3 ]   ;   we   get : S = 5 for   θ = ] 5 π 3 , 2 π ]   ;   we   get : S = 6 (10)

Step 4: Generation of the output voltage V_ref

The formula for determining output voltage (which should be achieved by modulation) V_ref depending on the standard vector of boundary voltages and the closing time of the key:

V r e f = α V 1 + β V 2       where   α = T 1 T S   and   β = T 2 T S (11)

Substitute Equation (10) into Equation (8):

V r e f = 3 ( V r e f V d c ) [ sin ( θ − ( k − 1 ) π 3 ) + sin ( k π 3 − θ ) ] (12)

Where the voltage modulation ratio can be expressed as:

V r e f = 3 m [ sin ( θ − ( k − 1 ) π 3 ) + sin ( k π 3 − θ ) ] (13)

And: m = V r e f V d c

This modulation offers very good performance and generates fewer current harmonics [5]. However, in practical implementation, the determination of the sequences does not involve a calculation of the tangent of the vector angle which causes undesirable singularities. This led us to consider DCSVM as the best duty cycle modulation technique for generating SVPWM signals. In the following, the DCSVM control strategy is applied to the structure shown in Figure 2 and the results obtained from the simulation are discussed.