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1. ECE 8830 - Electric Drives Topic 8: Simulation of Voltage-Fed Converters for AC Drives Spring 2004

2. Converter Models for AC Drives Goals: - To describe voltage-fed power converter in terms of switching functions - To convert switching functions into d-q reference frame variables - To illustrate space vector methods for converter model analysis

3. Basic Three-Phase VSI Inverter The basic circuit of a 3 VSI inverter is shown below. Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

4. Six Connection Possibilities of VSI Inverter There are six possible connections when a VSI inverter is connected to an induction motor as shown below: Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

5. Line-line and Line-neutral Voltages for 6-step VSI Inverter Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

6. VSI-Switching Function Formulation Ref: G. Venkataraman, Dynamics and Control of AC Drives, Univ. of Wisconsin Course Notes April 2002

7. Switching Functions for ABC Pole Voltages and DC bus Current Ref: G. Venkataraman, Dynamics and Control of AC Drives, Univ. of Wisconsin Course Notes April 2002

8. Phase Currents In the absence of a neutral connection, the phase currents in the windings of the induction motor must sum to zero, i.e. ias + ibs + ics = 0 This further implies that for a balanced load (e.g. wye connected induction motor) the line-neutral voltages must sum to zero, i.e. vas + vbs + vcs = 0

9. Neutral Voltages The neutral voltages are the voltages at the neutral of the motor windings w.r.t. the negative side of the dc bus. With a balanced load, the neutral voltages may be expressed as:

10. Phase Voltages • Phase A voltage • Phase B voltage • Phase C voltage

11. Phase Voltages - 612 Connection Example Ref: G. Venkataraman, Dynamics and Control of AC Drives, Univ. of Wisconsin Course Notes April 2002

12. Phase Voltages - 123 Connection Example Ref: G. Venkataraman, Dynamics and Control of AC Drives, Univ. of Wisconsin Course Notes April 2002

13. d-q Modeling of VSI Inverter For analysis of motor drives, it is useful to model the inverter in the same reference frame as the induction motor, i.e. in terms of d,q,0 components. This type of modeling is particularly useful when simulating the combined performance of the motor with the inverter.

14. d-q Modeling of VSI Inverter (cont’d) The d,q model for a VSI inverter is obtained by simply applying the d,q transformation seen earlier to the inverter equations on a mode by mode basis.

15. d-q Modeling of VSI Inverter (cont’d) The transformation from the abc axes to the dq0 axes in the stator reference frame is given by: where f represents voltage, v, current, i or flux linkage, .

16. d-q Modeling of VSI Inverter (cont’d) Let us consider the example of the 612 mode for the 6-step VSI inverter. We just saw that the phase voltages are given by: ; Applying the d,q transformation gives: ;

17. d-q Modeling of VSI Inverter (cont’d) Thus, simulation of the inverter during this connection mode can be achieved by applying 2/3 Vdc to the q-axis equivalent circuit of the induction motor while shorting the d- and 0- axis circuits.

18. d-q Modeling of VSI Inverter (cont’d) In the 612 connection, the a-phase current, ias = ii, the instantaneous current supplied by the inverter from the dc link, and since ias+ibs+ics = 0, the d,q transformation of the currents yields: ; ; Performing this analysis for the other five modes of operation yields the relations shown on the next slide. Note: i0s,v0s=0 in all cases and are not included in the table.

19. d-q Modeling of VSI Inverter (cont’d) Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

20. d-q Modeling of VSI Inverter (cont’d) The d-q voltages for different switch positions can be expressed as space vectors as shown below: Ref: G. Venkataraman, Dynamics and Control of AC Drives, Univ. of Wisconsin Course Notes April 2002

21. d-q Modeling of VSI Inverter (cont’d) The d,q relations for the VSI inverter can be conveniently described by defining two switching functions to express the constraint equations given two slides ago. The switching functions are shown on the next slide and are described by the following equations: ; ; These expressions relate the instantaneous inverter input quantities Vdc and ii to the instantaneous d,q output quantities.

22. d-q Modeling of VSI Inverter (cont’d) Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

23. d-q Modeling of VSI Inverter (cont’d) The choice of and as the amplitudes of and is arbitrary and have been chosen so that the fundamental in the Fourier series for the two functions is unity. The complete Fourier series for these two switching functions are given by:

24. d-q Modeling of VSI Inverter (cont’d) The complex vector forms of these d,q equations take on simple forms and are useful for visualizing and manipulating the equations. The complex d,q equations for the VSI voltages become: From the Fourier series of and the complex function is given by:

25. d-q Modeling of VSI Inverter (cont’d) This represents rotating vectors in alternating directions at speeds of e and multiples of e. The complex vector form of the current equation is given by:

26. d-q Modeling of VSI Inverter (cont’d) An alternative and very useful form of the d,q complex vector voltage equation can be obtained for each of the modes described earlier. In mode 1: In mode 2: Repeating for all six modes yields the result shown on the next slide.

27. d-q Modeling of VSI Inverter (cont’d) Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

28. d-q Modeling in the Synchronous Reference Frame Until now we have considered d-q modeling in stationary reference frame. However, the synchronous reference frame is more useful for induction motor-inverter simulation. The switching functions in the synchronous reference frame can be derived as (see handout):

29. d-q Modeling in the Synchronous Reference Frame (cont’d) In these two equations we have assumed that the rotating d,q axes have been synchronized with the fundamental frequency of the inverter output voltage, i.e. . The below figure shows the time functions that generate these series. Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

30. d,q Model for PWM Operation In a PWM inverter the various modulation techniques will alter the switching functions and resulting in unique switching functions for each modulation type.

31. d,q Model for PWM Operation One important difference between PWM and modulated VSI operation is the existence of two additional zero voltage states (giving a total of 8 states). Ref: D.W. Novotny and T.A. Lipo, “Vector Control and Dynamics of AC Drives”

32. d,q Model for PWM Operation The below table shows the various inverter switching states and the corresponding space vectors.

33. Space-Vector PWM The space vector method is a d,q model PWM approach. Let us first consider the linear or undermodulation region. The modulating command voltages are sinusoidal and correspond to a rotating space vector V*. This vector rotates at a speed e. The figure on the next slide shows the rotating space vector in terms on the complex plane together with the inverter switching state space vectors.

34. Space-Vector PWM (cont’d)

35. Space Vector PWM (cont’d) A convenient way to generate the PWM output is to use the adjacent vectors V1 and V2 of sector 1 on for part of the time to meet the average output required. The V* can be resolved into: i.e. and

36. Space Vector PWM (cont’d) During the period TC where the average output should match the command, vector addition can be used to write: or where , , and

37. Space Vector PWM (cont’d) The below pulse pattern satisfies the equations on the previous slide.

38. Space Vector PWM (cont’d) Two overmodulation modes. Overmodulation mode 1 starts when the reference voltage V* exceeds the hexagon boundary. Where V* exceeds boundary, loss of fundamental voltage. To compensate for this loss, a modified trajectory partly on the circle and partly on the hexagon is selected as shown in the next slide. Circular part of trajectory has larger radius (Vm*) and crosses hexagon at angle  - the crossover angle.

39. Space Vector PWM (cont’d)

40. Space Vector PWM (cont’d) Overmodulation mode 1 ends when the trajectory is fully on the hexagon (trajectory comprises only line segments). Overmodulation mode 2 starts when V* is increased further. The trajectory is again modified so that the output fundamental voltage matches the reference voltage. In this case the voltage is partly held at the hexagon corner for a holding angle h and partly by tracking the sides of the hexagon (see next slide).

41. Space Vector PWM (cont’d)

42. Space Vector PWM (cont’d) Details of how to calculate  and h are given in the textbook. The implementation of the SVM algorithm is shown in the below figure.