Digital simulation of two level inverter based on space

Indian Journal of Science and Technology Vol. 5 No. 4 (Apr 2012) ISSN: 0974- 6846

Research article “Two level inverter” S.Pal & S.Dalapati Indian Society for Education and Environment (iSee) http://www.indjst.org Indian J.Sci.Technol.

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Digital simulation of two level inverter based on space vector pulse width modulation

Snehasish Pal1* and Suvarun Dalapati2

1Department of Electrical Engineering, JIS College of Engineering, Kalyani, West Bengal-741233, India 2Asst. General Manager (R & D), Stesalit Limited, Electronic Systems Division, Kolkata- 700091, India

[email protected]*; [email protected]

Abstract Space Vector Pulse Width Modulation (SVPWM), one of the advanced computation based PWM techniques, has many advantages over conventional carrier-based PWM methodologies. Recently, with the easy availability of Microcontrollers and Digital Signal Processors, this technique is being widely used in industrial inverters. This paper presents a simple model for a three-phase two level SVPWM inverter using MATLAB-Simulink software. The entire model is based on only elementary Simulink blocks, and the use of advanced tool-box functions has been avoided. Hence, the model can be used, understood and modified easily as per the need of the user. The inverter has been divided into several sub-systems. Each such ‘sub-system-block’ is explained individually. Both linear and over-modulation-zone-controls have been included. The model operates successfully for various values of amplitude modulation index. Keywords: Space Vector Pulse Width Modulation, Two-level inverter, MATLAB-Simulink, Linear zone, Over-modulation zone. Introduction

SVPWM technique based inverters have been a widely researched topic in the field of power electronics and machine drives over the last few years (Boost & Ziogas, 1988; Bose, 2006). The technique enjoys several advantages over conventional PWM techniques (e.g. sine-triangle PWM). Some of its advantages include more effective utilization of DC bus voltage, optimum harmonic content for a wide load range, non-requirement of high frequency carriers and the allied synchronization problems etc. (Bose, 2006; Holtz, 1992). Recently, with the easy availability of DSP and Microcontrollers with high computational features, this technique is being applied in various industrial inverters.

MATLAB-Simulink based simulations have been accepted globally both in academic and research institutes, as well as in industry to be a standard tool for simulating various complicated industrial systems (Ayasun & Karbeyaz, 2007). In the recent versions of MATLAB-Simulink, the user has been provided with many additional Simulink-Libraries and Models (in block form) to help the modelling of several complicated systems more easily (Hunt et al., 2006; Shaffer, 2007). However, although the newer libraries / blocks ease the development of models for simulating complicated systems, the price to be paid is in terms of the time and PC-resources (e.g. memory, CPU-speed etc.) required.

This paper presents a model for the three phase two level SVPWM inverter in MATLAB-Simulink environment. The model has been developed by using only basic Simulink Library blocks, thereby reducing simulation time, without compromising on the accuracy of the solution. The model of the inverter comprises of several sub-systems. In the following sections, the function of each sub-system is explained with reference to the SVPWM technique for linear and both the over-modulation zones

(termed as ‘over-modulation zone-1’ and ‘over-modulation zone-2’). Subsequently the mathematical expressions, for deriving the SVPWM control in various zones, are also derived and presented. The blocks are then combined together to synthesize the model of the full SVPWM inverter. This inverter model is simulated in open loop conditions for a three phase star connected balanced inductive load. Some sample results for the inverter operation in open loop conditions for all the three zones of operations are presented to validate the correctness of the model. The basic structure of the SVPWM inverter

The essence of SVPWM technique can be understood easily from any standard text book on power electronics (Bose, 2006). The three phase balanced

Fig.1. Six-switch inverter feeding its output to a three phase star connected inductive load (with floating neutral); the

source neutral point ‘N’ may be tapped by splitting the dc bus into two equal halves by equal capacitors



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windings of an induction motor, when fed with a balanced three-phase sinusoidal voltage-set, will generate a resultant space vector, which has a fixed amplitude and rotates along a circular trajectory in space. The basic aim of the SVPWM inverter is to generate such a ‘rotating space-vector’.

The SVPWM inverter has the conventional six-switch based bridge structure as shown in Fig.1. The resultant space-vector can be readily derived from the load-phase voltages generated from the inverter as follows:

223s an bn cnV v av a v ………………. (1)

Using equation (1), the resultant space vectors for a six-step operation can be easily derived to form the hexagon, as shown in Fig.2. It may be noted that the six ‘active vectors’ connects the centre of the hexagon to the six vertices, while the two ‘null-vectors’ are located at the hexagon centre. This ‘space-vector-hexagon’ defines the area, within which the resultant space vector will always lie.

The basic block diagram of the SVPWM inverter is presented in Fig.3. The voltage and frequency are fed as the reference signals. The ‘Ma & θ calculator block’ calculates the value of Ma and θ. Depending upon the value of Ma, the inverter then must be given a set trajectory, along which the space-vector rotates with respect to the hexagon-centre (Bose, 2006). The ‘Trajectory Selector’ block calculates this trajectory and modifies the value of Ma and θ as Mae and θsect respectively. The ‘time-splitter’ block takes Mae, θsect and

sector number n as inputs and calculates the time durations t0, t7, (for applying the null vectors) and t1,t2 (for applying active vectors) of a given sector. The ‘pulse generator’ block takes these time-values and applies suitable active and null vectors to the load. Description of Constituent Blocks

This section is devoted toward describing each building-block of the SVPWM Inverter model for the various zones of inverter operation (i.e. linear, over-modulation-1 and over-modulation-2 zones). The mathematical deductions for each block are also presented in the corresponding section. The ‘Ma-calculator’ block In this paper the ‘amplitude-modulation index’, termed as Ma, is calculated as follows:

2

3

r r ra

n dDC

V V VM

V VV

…………………….. (2)

(where n = 1, 2, … 6). The structure of the ‘Ma-calculator’ block is shown below

in Fig.4. The ‘theta sampler’ block

Depending upon the chosen switching frequency, the inverter combines the various active and null vectors to synthesize an ‘average-vector’ over one switching cycle to match the given reference space vector. For computing the time of applying the various fundamental vectors, the knowledge of the value of θ is required. The value of θ may be computed from the given value of switching and output frequencies and is to be held constant during one switching cycle. At the beginning of the next switching cycle, the value of θ will have to be ‘refreshed’. The ‘theta-sampler’ block performs the above function. This block is presented in Fig.5. It may be observed that the ‘Zero-Order-Hold’ block, as shown in Fig.5b, must have a sampling frequency which is same as the switching frequency of the inverter (chosen to be 6 kHz in this case). The ‘sector-number-generator’ block

The ‘Sector-Number-Generator’ block computes the sector number from the given value of theta as input. The sector number may be generated by noting the value of θ, as shown in Table 1.

Fig. 2. Six-step operation of the three phase bridge inverter of Fig.1, (a) three load phase waveforms and (b) space vectors forming a hexagon (encircled digits show

sector numbers)

Fig.3. Basic block diagram for the SVPWM inverter model

(a) (b)

Fig. 4. (a) Ma-calculator block and (b) its internal structure



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Trajectory selector block-1 (for linear zone and over-modulation-zone-1)

The trajectory selector block-1 takes in the computed value of Ma and θ as inputs. Based on these quantities, it

defines the trajectory to be followed by the inverter output. Trajectory-selector-block-1 accomplishes this function for the linear zone and the over-modulation zone-1 of the inverter operation. Linear zone

The ‘linear-zone’ for inverter-operation takes place when the reference trajectory is a circle lying completely within the hexagon. This scenario exists when the computed value of Ma lies between 0 and 0.866. The ‘linear-zone-detector-block’ checks whether the computed value of Ma lies within the range 0 – 0.866, or not. Once, Ma lies within this range, the ‘trajectory-selector-block’ passes on the computed value of Ma and θ, which it receives as input, directly to its output (Fig.6). Thus, in this case, Ma = Mae. Over-modulation zone-1

This zone exists when 0.866<Ma<0.9091. In this zone, the ‘original-reference-trajectory’ is a circle, which traverses outside the ‘space-vector-hexagon’. Thus, in every sector of the hexagon, the ‘original-reference-trajectory’, defined by a circle, cuts the hexagon boundary in two distinct points (Fig.7). However, it is not possible for the inverter to follow this trajectory directly. To compensate for this area, which is lost, the ‘trajectory-selector-block-1’ must select a modified trajectory. This new trajectory is shown in Fig. 8. The modified trajectory follows a circular path from angle 0 to α (within a sector)

(a) (b)

Fig. 5. (a) Theta-sampler block and (b) its internal structure

Fig. 6. Original reference trajectory input to ‘trajectory selector block-1’ (circle with light thick line) and

modified reference trajectory output from the same block (circle with dark-narrow line) for linear zone of

operation (Ma is chosen as 0.8 in this case)

Fig. 7. Original reference trajectory traversing outside the space vector hexagon

Fig. 8. Modified reference trajectory to compensate for the area lost



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with a radius (r1) greater than the radius of the original reference trajectory (r). It meets the hexagon boundary at some angle α, follows the hexagon boundary up to the angle 3

, after which it reverts to a circular

trajectory of radius r1 again. This circular trajectory is followed till the end of the sector (at angle

3 ). The angle

α, also called the ‘cross-over angle’ for the modified reference trajectory, may be determined by area-matching principle. Thus, to determine the modified trajectory for a value of Ma, which is received as input, the controller should compute two quantities, namely: (a) the value of α and (b) the value of r1 = Vov. The following mathematical treatment provides the solution. Referring to Fig.8, the actual area determined from the ‘original reference trajectory’, per half-sector, is given by:

2

12ref rA V ……………………………......................... (3)

Area traversed per half-sector by the ‘modified reference trajectory’ is given by:

2 21 tan2 6 6mref ov DCA V V

………….... (4)

Thus, to satisfy the area-matching criterion, the above two areas, given by Equations (3) and (4) must be equated.

2 2 21 tan12 2 6 6r ov DCV V V

…………… (5)

Again, from Fig. 8, it easily follows that:

1 sec63ov DCV V

…………………........... (6)

Using the result of Equation (6) in Equation (5), and after

Fig. 9. Variation of Ma with α

Fig.10. Original reference trajectory input to ‘trajectory selector block-1’(circle with light thick

line) and modified reference trajectory output from the same block (trajectory with dark-narrow line)

for over-modulationzone-1 of operation (Ma is chosen as 0.9 in this case)

(a)

(b)

Fig.11(a) Trajectory selector block-1 (for linear zone and over-modulation zone-1, and (b) its internal structure



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some algebra, it follows that:

2 22 1 tan tan9 6 6aM

……(7)

For every value of α, Equation (7) gives one unique value of Ma (negative value is not considered). Thus, a look-up table may be formed, which will give us a value of α for a given value of Ma. Using this look-up table, for each value of Ma, the corresponding value of the cross-over angle may be obtained. The nature of variation of Ma for various value of α is shown in Fig.9. A typical result from the ‘trajectory-selector-block-1’ is presented next in Fig.10, corresponding to the operation in ‘over-modulation-zone-1’. It may be noted that by following the modified trajectory, the inverter-operation now moves into a non-linear zone. The block diagram and the internal structure for the ‘trajectory-selector-block-1’ is presented next in Fig.11(a) and Fig.11(b) respectively. Trajectory selector block-2 (for over-modulation-zone-2)

Just as in the case of ‘trajectory-selector-block-1’, the ‘trajectory-selector-block-2’ also takes in the computed value of Ma and θ as input and checks whether the value of Ma lies within the range for over-modulation zone-2, which is 0.9091<Ma<1. If the value lies within this range, then the ‘trajectory-selector-block-2’ modifies the space-vector-reference trajectory suitably. If the value of Ma lies below this range, then the value of Ma is either in the linear zone or over-modulation-zone-1. In this case, the computed trajectory of ‘trajectory-selector-block-1’ is allowed to pass through. The principle of operation of this block is explained in the next section. Principle of operation The control algorithm is modified as follows. For an output frequency required, the time by which

one sector is to be traversed, is determined. In every sector, for a certain time (determined by the

value of Ma), the modified reference space vector will lie

fixed at the starting edge of the sector (i.e. at 1V for

sector-1, at 2V for sector-2 etc.).

For the remainder of the time (for one sector) the modified-reference-trajectory will move along the boundary of the hexagon (in that sector) and will have to reach the other edge (i.e. the finishing edge) of the

sector. The finishing edge of this sector (i.e. at 2V for

sector-1, at 3V for sector-2 etc.) will act as the starting

edge for the next sector. Thus, in this zone (over-modulation-zone-2), the value

of θ (which is input to the trajectory selector block) is to be modified into eff (which is to be output from the

trajectory selector block). The value of eff will be used

to compute the value of θsect in the successive stages. The case may be illustrated with the help of Fig.12,

where sector-1 has been taken as an example. Here, as shown in Fig.13, if Ts be the time in which a sector is to be traversed (this is determined by the choice of the inverter output frequency, e.g. 50Hz / 60Hz), then for a particular time Th, the modified-reference-trajectory must

stay at 1V (which forms the starting edge of the sector),

then move along the hexagon-boundary (straight line)

from 1V to 2V , within a time of s hT T . Fig.13 and

Fig.14 show the time plot for the original reference and modified reference vector magnitudes and angles respectively within one sector.

Table 1. Sector number generated by noting the value of θ Range of θ Sector Number 0 / 3 1 / 3 2 / 3 2 2 / 3 3

4 / 3 4 4 / 3 5 / 3 5

5 / 3 2 6

Fig.14. Angles of original reference and Modified reference signals, plotted with respect to time

Fig.13. Magnitudes of original reference and modified reference signals, plotted with respect to time

Fig. 12. Original and modified reference trajectories for over-modulation-zone-2 (in case of sector 1)



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Fig.17. Symmetrical pulse distribution (centre-aligned mode)

Fig.16. Combining vectors to achieve

a resultant vector; 1

1 1

sw

ta V

T

and

Fig. 15 (a) Trajectory selector block for over-modulation-zone-2 and (b) internal structure

(a)

(b)



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Thus, as seen from the above discussion, to control the SVPWM inverter in the over-modulation-zone-2, the time Th must be first determined. The following mathematical treatment will derive a formula for determining Th (or the corresponding angle αh, also called the ‘holding-angle’).

As the ‘modified-reference-vector’ stays at one active vector location for a certain finite time in every sector (of the space-vector-hexagon), the rate, at which it will traverse to the other end of the same sector will be higher than the output frequency. Let this modified value of angular frequency be ω1 (ω1>2πfo). Thus, from Fig.13 and Fig.14, the modified reference space vector can be defined as follows:

.. (8)

Here, 1 3s hT T

Alternatively, in the range h sT t T , F(t) may be defined

as:

1

3 1

2 sin3

d

h

F t Vt T

……….................... (9)

Hence, by substituting z = t-Th, the actual area traversed (in one sector) by the modified reference vector may be calculated as follows:

01

3

2 sin3

3 3ln 3

2

s hT T

mref d h d

d h d s h

dzA V T V

z

V T V T T

………………........ (10)

Again, area traversed in one sector by the original reference is given by:

6o

ref r

TA V ………………….............................................. (11)

Equating the two areas, given by (10) and (11), the following relationship between Ma and Th may be obtained:

3 3ln 3

6 2 6o o

a h h

T TM T T

…………............. (12)

The MATLAB-Simulink model for the ‘trajectory-selector-block-2’, which works in conjunction with ‘trajectory-selector-block-1’, is presented next in Fig.15. Time-splitter block

The time splitter block ‘splits’ one switching period into sub-intervals for applying the active and null vectors corresponding to that sector. Such vectors must be applied in proper sequence to achieve centre-aligned pulse positioning (Fig.16). The time durations, for which the active vectors are to be applied, are given by:

21 sectcos

2 a sw

tt M T ……………….. (13)

and 2 sect

2sin

3a swt M T ……………… (14)

The time-durations for which the null vectors are to be applied can be easily derived by using the following relationship:

1 20 7 2

swT t tt t

……………….. (15)

Thus, for sector-1, the sequence for each vector (with time-duration) may be as follows: V0 (t0/2) V1 (t1/2) V2 (t2/2) V7 (t7) V2 (t2/2) V1 (t1/2) V0 (t0/2)………….. (16)

This gives rise to a pulse distribution, as shown in Fig.17 (assuming that the state ‘1’ implies that the upper switch of any leg is ON, while the state ‘0’ implies that the lower switch of the same leg is ON). The above procedure (of time distribution & symmetrical pulse positioning) is to be followed for each sector individually. The various sectors and the corresponding vector sequence are given in Table 2.

Thus, it may be observed that the vector sequence is of the ‘forward’ type in odd numbered sectors, while the sequence is of the ‘reversed’ type for the even-numbered sectors. Thus, the effective value of angle in a sector for computing time distribution may be obtained as follows:

1

effective sect

1 1. 1

2 3

nn

…………... (17)

Thus, the steps to be followed, for computing time in any sector (for linear zone of operation), are as follows:

1. From the value of θ and the sector number, compute θsect and θeffective using (17).

2. Using the value of Ma and θeffective, compute t1, t2, t7 and t0 using (13) – (15).

3. Split up these times symmetrically among the corresponding active and null vectors to get centre-aligned pulses

For over-modulation zones 1 or 2, (13) and (14) reduces to the following form:

Table 2. various sectors and the corresponding vector sequence Sector

No. Vector Sequence

1 V0 (t0/2) V1 (t1/2) V2 (t2/2) V7 (t7) V2 (t2/2) V1 (t1/2) V0 (t0/2) 2 V0 (t0/2) V3 (t1/2) V2 (t2/2) V7 (t7) V2 (t2/2) V3 (t1/2) V0 (t0/2) 3 V0 (t0/2) V3 (t1/2) V4 (t2/2) V7 (t7) V4 (t2/2) V3 (t1/2) V0 (t0/2) 4 V0 (t0/2) V5 (t1/2) V4 (t2/2) V7 (t7) V4 (t2/2) V5 (t1/2) V0 (t0/2) 5 V0 (t0/2) V5 (t1/2) V6 (t2/2) V7 (t7) V6 (t2/2) V5 (t1/2) V0 (t0/2) 6 V0 (t0/2) V1 (t1/2) V6 (t2/2) V7 (t7) V6 (t2/2) V1 (t1/2) V0 (t0/2)

F t dV ,

0 ht T

1

3sec

2 6d hV t T

,

h sT t T



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effective effective1

effective effective

3 cos sin

3 cos sinswt T

…………..... (18)

effective

2

effective effective

2 sin

3 cos sinswt T

………......

(19)

and 0 70t t ………………............ (20)

Pulse generator block

Various ‘switching states’ represent the inverter output for different sectors. The computed values of time-durations for each state is compared with a fixed frequency saw-tooth waveform and the state-table (shown in Table-3) is utilised to output the pulses to the load. The sector number input determines the row to be selected. Thus, in each sector, the cells of one particular row are selected in sequence, while the row changes whenever the space vector is to move from one sector to another. As the saw-tooth waveform has a negative gradient for a very short duration in each switching cycle, this information (sign of gradient) is used for re-setting the states to ‘000’ at the end of every switching cycle. Re-setting duration being very short as compared to other durations, its effect on the overall pulse pattern is negligible. From Table 3, three separate look-up tables have been generated for three different inverter-legs, by taking one state (out of three) at a time from each cell of Table 2. The pulse-generator block details are presented in Fig.18.

Direction reverser block All the computations for the previous sections have

been completed by using positive values of θ only.

However, the reference frequency command may be positive / negative. However, since a negative reference-frequency implies a change in phase-sequence of the output PWM waveform, the same is achieved by using the ‘direction-reverser-block’, as shown in Fig.19.

The overall block diagram of the three phase SVPWM inverter with all the constituent blocks, as described above, is presented in Fig.20. The ‘space vector display block’ is not a part of the standard inverter, but is kept for display purpose only. Results

To test the effectiveness of this model, it has been simulated in MATLAB-Simulink environment for various

conditions. In this section, the various results from this SVPWM Inverter-Model will be displayed and explained. The various parameter values, used for simulation is presented in Table-4. Initially, a reference is set to yield a load phase voltage of 220V (amplitude) from this inverter. Thus, the inverter operates in the linear zone. Hence, in this case, the Ma-trajectory (as input by the user) matches the computed Ma-trajectory, as shown in Fig.21. As happens in case of linear

zone of operation, with V/f start, the trajectories are having circular shape, gradually expanding in radii to reach a steady state value. The time-plot of the inverter line voltage and load phase voltage are presented in Fig.22 and Fig.23 respectively. The amplitude of load current, for this set value of load voltage, can be calculated to be 11.81A at 50Hz. The load-current-space-vector trajectory and the actual load currents are presented in Fig.24 and Fig.25 respectively. Fig.25a shows three load phase currents together (w.r.t the same set of axes), while Fig.25b shows one load phase voltage and the corresponding load phase current. As in the case for R-L loads, the load-current clearly lags the corresponding load-voltage.

The same model is now used again and the reference value (of phase voltage amplitude) is set at 330V (over-modulation zone-1) and 350V (over-modulation zone-2), keeping the DC bus voltage fixed. The corresponding results for reference and actual space vector trajectories and load current are presented in Fig.26 and 27 respectively. Fig.28 shows the comparison between the FFT plots for a standard Sine-PWM inverter and a space vector PWM inverter. As seen, the Space Vector PWM inverter shows a more evenly distributed harmonic pattern. Conclusion

In this paper, a MATLAB-Simulink model for a three phase SVPWM Inverter has been presented. The model is based on elementary MATLAB-Simulink functional blocks only, and thus provides a medium for quick and easy simulation for several cycles of the output frequency. It can be easily incorporated in larger ‘MATLAB-Simulink’ models for power-electronics and machine drives simulations.

Table 3. Three different inverter-legs, by taking one state (out of three) at a time from each cell of Table 2

Sector No.

Vector States for corresponding time durations t0/2 t1/2 t2/2 t7/2 t7/2 t2/2 t1/2 t0/2 -

1 000 100 110 111 111 110 100 000 000 2 000 010 110 111 111 110 010 000 000 3 000 010 011 111 111 011 010 000 000 4 000 001 011 111 111 011 001 000 000 5 000 001 101 111 111 101 001 000 000 6 000 100 101 111 111 101 100 000 000

Table 4. Values for simulation Sl. No. Parameter Value / Type

1. DC bus voltage 560V

2. Output phase voltage (set as reference) 220V

3. Switching strategy SVPWM 4. Switching / Sampling freq. 6 kHz 5. O/P frequency (rated) 50Hz 6. Type of load Resistive-inductive 7. Load value R =10Ω, L=50mH



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Fig. 20. Detailed internal structure (left) and overall block diagram (right) of the SVPWM inverter

(a) (b)

Fig. 19. (a) Direction-reverser block (left) and (b) internal structure (right)

(a) (b)

(a)

Fig. 18. (a) Pulse generator block (left) and (b) internal structure (right)

(b)



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Fig. 21. User-defined Ma-trajectory (thin dark line) and inverter-computed Ma-trajectory (thick grey line) for the

inverter, operating in the linear zone with V/f start; the steady-state circular

Fig. 22. Inverter output line voltage (VRY) for the inverter operating in the linear-zone

Fig. 24. Load current space vector plot for the inverter delivering an output phase voltage

of 220V (amplitude), by operating in the linear zone

Fig. 25a. Load phase currents for the inverter operating in the linear zone

Fig. 23. Load phase voltages for the inverter operating in the linear zone



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Fig. 27a. User-defined Ma-trajectory (thin dark line) and inverter-computed Ma-trajectory (thick grey line) for the inverter, operating in over-modulation zone-2

Fig. 25b. One load phase voltage and corresponding load phase current for the inverter operating in linear zone

Fig. 26b. Load phase currents for the inverter operating in over-modulation zone-1

Fig. 27b. Load phase currents for the inverter operating in over-modulation zone-2

Fig. 26a. User-defined Ma-trajectory (thin dark line) and inverter-computed Ma-trajectory(thick grey line) for the inverter, operating in over-modulation zone-1



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The model can operate in all the zones for an SVPWM inverter, namely linear-zone, over-modulation-zone-1, and over-modulation-zone-2. The performance of this inverter-model is checked upon a balanced three phase resistive-inductive load for the various zones in open loop mode. The results follow the well-known trends of SVPWM inverters. References 1. Bose BK (2006) Power electronics and motor drives:

Advances and trends. Academic Press, Elsevier Publ.

2. Holtz J (1992) Pulse width modulation- A survey, IEEE Transact. Indus. Elect. 39(5), 410-420.

3. Mondal SK, Bose BK, Oleschuk V and Pinto JOP (2003) Space vector pulse width modulation of three-level inverter extending operation into over modulation region. IEEE Transact. Power Elect. 18(2), 604–611.

4. Zhou KZ and Wang D (2002) Relationship between space-vector modulation and three-phase carrier-based PWM: A comprehensive analysis. IEEE Transact. Indus. Elect. 49(1), 186–196.

5. Kang DW, Lee YH, Suh BS, Choi CH, and Hyun DS (2003) An improved carrier-based SVPWM method using leg-voltage redundancies in generalized cascaded multilevel inverter topology. IEEE Transact. Power Elect. 18(1), 180-187.

6. Boost MA and Ziogas PD (1988) State-of-the-art carrier PWM techniques: A critical evaluation. IEEE Transact. on Indus. Appl. 24(2) March/April.

7. Ayasun S and Karbeyaz G (2007) DC-Motor speed control methods using MATLAB/Simulink and their integration into undergraduate courses. Wiley Periodicals. pp: 347–354.

8. Hunt BR, Lipsman RL, Rosenberg JM, Coombes KR, Osborn JE and Stuck JG (2006) A guide to MATLAB for beginners and experienced users. Cambridge University Press.

9. Shaffer R (2007) Fundamentals of power electronics with MATLAB. Thomson Learning Inc.

Fig. 28a. Fourier spectrum for Sine-PWM inverter

Fig. 28b. Fourier spectrum for SVPWM inverter

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