1 Inverter Applications Motor Drives Power back-up systems Others: Example HVDC Transmission systems

2 Single Phase Inverter Square-wave “Modulation” (1) v out (t) -V dc V dc t

3 THD = 0.48 Characteristics: - High harmonic content. - Low switching frequency. - Difficult filtering. - Little control flexibility. Single Phase Inverter Square-wave “Modulation” (2)

4 Single Phase Inverter Square-wave “Modulation” (3) v out (t) -V dc V dc t

5 Characteristics: - High harmonic content. - Low switching frequency. - Difficult filtering. - More control flexibility. THD = 0.3 Example with V out-1 =1.21V dc Single Phase Inverter Square-wave “Modulation” (4)

6 Single Phase Inverter Pulse Width Modulation (1) D is the duty cycle of switch Q 1. D is the portion of the switching cycle during which Q 1 will remain closed. In PWM D is made a function of time D=D(t)

7 Single Phase Inverter Pulse Width Modulation (2) Let’s where Modulation function Fundamental Signal Modulation index

8 Moving average Single Phase Inverter Pulse Width Modulation (3) t

9 Single Phase Inverter Pulse Width Modulation (4) Implementation issue: time variable “t” needs to be sampled. Two basic sampling methods: UPWM NSPWM

10 Single Phase Inverter Pulse Width Modulation (5) t

11 Single Phase Inverter Pulse Width Modulation (6) Considering that

12 Three Phase Inverter Pulse Width Modulation (1) Active States: StateQaQa QbQb QcQc S1S1 001 S2S2 010 S3S3 011 S4S4 100 S5S5 101 S6S6 110 Zero States: StateQaQa QbQb QcQc S0S0 000 S7S7 111

13 Three Phase Inverter Pulse Width Modulation (2) Modulation Functions “Zero-Sequence” Signals

14 Three Phase Inverter Pulse Width Modulation (3) Fundamental Signal Modulation index Triplen Harmonics (3, 9, 15, …) Other Harmonics (5, 7, 11, 13, 17, ….)

15 Three Phase Inverter Pulse Width Modulation (4) t

16 Classic Approaches to PWM (1) Time Domain Use of modulation signal: Duty cycle computation: t Sector limits

17 Disadvantages: Can’t see evolution in time Loss of information about e 0-3 (t) Classic Approaches to PWM (2) Classic SVM - Application Inverter DC Source Motor Switch Control d-PI Controller q-PI Controller Domain Transformer (Modulator) Space Vector Time Park’s Transformation (From Vector Controller) Park’s Transformation

18 SECTOR I SECTOR II SECTOR III SECTOR IV SECTOR V SECTOR VI In O: Space vector domain Classic Approaches to PWM (3) Classic SVM 2 Bases B SVM in sector I B SVM changes in each sector

19 -Track the sector in which is in and based on it select the appropriate set of basis B ij -Calculate the coordinates of in the basis B ij - Change the coordinates of the reference voltage vector S4S4 S6S6 S2S2 S5S5 S1S1 S0S0 S7S7 S3S3 Transitions within sector I Classic Approaches to PWM (4) Classic SVM from basis B dq to basis B ij. The sector dependant transformation yields the period of time T i that the machine remains in each state in a given sampling period. - When the time T i is finished move to the next state following the sequence given by the SVM state machine. SVM Computation T 7 =T 0

20 Then, a 3-D vector can be introduced: Mathematical Framework (1) Control Time Domain Space Vector Domain Output Time Domain Complete representation involves a 3-D space

21 Mathematical Framework (2) Control Time Domain Functions of time are used as basis

22 Mathematical Framework (3) Space Vector Domain When e 0-H (t)=0, describes a circumference with radius equal to m.

23 Mathematical Framework (4) Output Time Domain

24 Mathematical Framework (5) Output Time Domain Length of sides equal to 2 Each corner represents one state State sequence is obtained naturally Sectors: six pyramid-shaped volumes bounded by sides of the cube and |v i |=|v j | planes (i, j = a,b,c; i j). Balanced system plane

25 Mathematical Framework (6) Matrix R When e 0-H (t)=0, describes a circumference with radius equal to m. 1 st Idea: Use Park’s transformation to a synchronous rotating reference frame: Instantaneous values of the voltages

26 Mathematical Framework (7) Matrix R Problem: components in and are constant values I am interested in having a constant value in 2 nd Idea: Freeze the rotational reference frame at t=0

27 Mathematical Framework (8) Matrix R 3 rd Idea: Rearrange the product.

28 Mathematical Framework (9) Matrix R 4 th Idea: Eliminate the dependency on e 0-3 (t) in order to have a constant coordinate in

29 Mathematical Framework (10) Matrix R In order to include e 0-H (t) we need to follow the same steps and apply superposition.

30 Mathematical Framework (11) Matrix W W’

31 Mathematical Framework (12) Matrix W 1)Take the transpose and apply scaling factor 2)Include e 0-3 (t) in order to have it as a component

32 Mathematical Framework (13) Matrix S S=WR

33 3D Analysis and Representation

34 3D Analysis and Representation UPWM

35 3D Analysis and Representation SVM UPWM

36 3D Analysis and Representation 3D Representation: Plot evolution of in output time domain during a complete fundamental period. The resulting curve always lays within the cube defined by the switching states.

37 3D Analysis and Representation Triplen harmonic distortion: Evolution away from the plane v a +v b +v c =0 Other harmonic distortion: non circular projection of the curve over the plane v a +v b +v c =0 Sharp corners indicate the presence of higher order harmonics.

38 Commonly used schemes SVM Square wave

39 3D Analysis and Representation Square wave

40 3D Analysis and Representation Phase and Line Voltages (1)

41 3D Analysis and Representation Phase and Line Voltages (2) Direction of v ab Direction of v oa v oa (t) t v a (t)

42 3D Analysis and Representation Phase and Line Voltages (3) v oa (t) t v a (t)

43 3D Analysis and Representation Maximum non distorting range Radius of circle is m if it is measured in the Space Vector Domain. There is a scaling factor in W

44 3D Analysis and Representation If f switch >>f fund then

45 3D Analysis and Representation t

46 3D Analysis and Representation QaQa QbQb QcQc QbQb QaQa QcQc QbQb QcQc QaQa Sector I Sector IISector III T7T7 T6T6 T4T4 T0T0 T7T7 T6T6 T2T2 T0T0 T7T7 T3T3 T2T2 T0T0

47 Analysis of SVM (1) T 7 =T 0 D min =1-D max v max +v min =0 S is sector dependent (considering e 0-H (t)=0) SVM tends to approximate the trajectory of a square wave, but adds 3 rd harmonic and higher order triplen harmonics No difference in sequence compared to other schemes

48 Analysis of SVM (2) Fundamental and zero-sequence signal Modulating signals tt

49 Analysis of SVM (3) Control Time Domain Space Vector Domain Output Time Domain Digital implementation related to sampling method selected, not to the modulation function used.