Ultimate Switching: Toward a Deeper Understanding of Switch Timing Control in Power Electronics and Drives

1. Ultimate Switching: Toward a Deeper Understanding of Switch Timing Control in Power Electronics and Drives P. T. Krein, Director Grainger Center for Electric Machinery and Electromechanics Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign

2. Outline • Fundamentals: power electronics control at its basic level • Motivation • False starts and model-limited control • Small-signal examples • Ultimate formulation • Geometric control examples

3. Fundamentals • In any power electronic circuit or system, control can be expressed in terms of the times at which switches operate. • The fundamental challenge is to find switching times for each device. • Example: • For each switch in a converter, find switching times that best address a set of constraints. • This is an optimal control problem of a sort. • Might represent this with a switching function q(t).

4. Fundamentals • The general problem is daunting, so we simplify and address switch timing indirectly. • Averaging (address duty ratio rather than q) • PWM (use d as the actuation, not just the control) • Sigma-delta (make one decision each period based only on present conditions) • Other approaches • We are researching to try and identify ways to address the timing questions more directly.

5. Motivation • We believe that a new and more fundamental consideration of a switch timing framework has strong potential benefits. • Motivated by our work on switching audio • Showed that sine-triangle PWM, used as a basis for audio amplifiers, provides nearly unlimited fidelity. • Motivated by past work on geometric and nonlinear control • Performance can be achieved in power converters that is unreachable with averaging approaches.

6. False Starts • Many argue that space-vector modulation (SVM) gets more directly at switch timing. • In fact, SVM addresses duty ratios and yields (at best) exactly the same result as a PWM process. It is usually worse because uniform sampling is involved. • Small-signal analysis methods are even less direct. • Sliding-mode controls “confine” the switching without getting to the timing challenge.

7. Space vector modulation Third-harmonic injection sine-triangle PWM Space Vectors in Time Domain

8. Model-Limited Control • Many control methods used in today’s switching power converters are limited by the models of the systems. • “Model-limited control” is an important barrier to improvement of converters.

9. Model-Limited Control • Any type of PWM implies switchingthat takes place much faster thansystem dynamics. • Dc-dc converters use controllersdesigned based on averaging. • We often learn that bandwidths arelimited to a fraction of the switching rate. • We finally have the tools to interpret this rigorously.

10. Model-Limited Control • Distortion in the low-frequency band can be computed as a function of switching frequency ratio. • Distortion must be at least -40 dB (better -60 dB) to justify control loop design. • Based on natural sampling: Frequency ratio In-band distortion5 -9 dB 7 -42 dB 9 -70 dB 11 -110 dB 13 -154 dB 15 -201 dB  10-10 • This is consistent with signal arguments that yield 2 as the minimum ratio and “rules of thumb” about a ratio of 10 for best results.

11. Model-Limited Control • These models are convenient and useful, but do not use the full capability of a conversion circuit. • We gave up a factor of 10 on dynamic performance in exchange for precision. • Consider an example: • Small-signal methods and models are powerful tools for analysis and design. • They can only go so far toward the analysis of large-signals circuits and disturbances.

12. Small-Signal Response Examples • Take a dc-dc converter, with a well-designed feedback control. Explore its response. • In this case, a known sinusoidal disturbance is applied at the line input. • Its frequency is 5% of the switching rate. • Its magnitude is 10%. • The controller is adjusted to cancel line variation completely – the duty ratio tracks and cancels the disturbance based on small-signal analysis.

13. i L I #1 IN OUT + + R LOAD V v V IN OUT OUT - #2 - Buck Converter • In this example, a “feedforward” compensation is used to eliminate changes caused by line variation.

14. Example Dc-Dc Converter Problem • 10% disturbance around 80% reference value. • Frequency is 1/20 of switching (e.g. 5 kHz on 100 kHz).

15. Compensated PWM Output • Filter time constant about 1/10 of switching.

16. Result? • Is the disturbance rejected or not? • Yes and no. • Does this controller achieve the requested bandwidth? • In fact, the controller is completely eliminating linear aspects of the disturbance. • But the output ripple has features that may not be preferred. • Now, ignore small signal limits.

17. Example Dc-Dc Converter Problem • 10% disturbance around 80% reference value. • Frequency is 3/4 of switching.

18. Output Ripple

19. Result? • In several ways, the result is the same, although filtering is less effective because of the higher frequency. • There is an aliasing effect (but there was previously as well). • The disturbance frequency does not appear in the output.

20. Quick Performance Check • Hysteresis control instead, 150 kHz disturbance.

21. Hysteresis Method • Now the ripple is tied only to the switching rate. • The disturbance has no noticeable influence on the output. • This is true even though the disturbance is faster than the switching frequency! • Does this mean the converter has a “bandwidth” greater than its switching frequency?

22. Comments • “Frequency response” and “bandwidth” imply certain converter models. • Physical limits are more fundamental: • When should the active switch operate to provide the best response? • How soon can the next operation take place? • How fast can the converter slew to make a change? • Hysteresis controls respond rapidly. This is an issue of timing flexibility more than of switching frequency.

23. i L I #1 IN OUT + + R LOAD V v V IN OUT OUT - #2 - Consideration of Disturbance Timing • In a buck converter, any line disturbance while the active switch is on will have a direct and immediate effect at the output. • No line disturbance will have any effect if it occurs while the active switch is off. • This means an impulse response cannot be written without a switching function.

24. Consideration of Disturbance Timing • This indicates that the nonlinearity cannot be removed for impulse response. • “Impulse” is not adequate information to determine the response. • Average models cannot capture timing issues. • Notice that similar arguments apply to step responses and others.

25. The Ultimate Formulation • A converter has some number of switches. • For each switch, there arespecific times at which adevice should turn on or off. • The times represent the control action. Selection of the times is the control principle. • For each switch i, find a sequence of times ti,j that produce the desired operation of the converter.

26. The Ultimate Formulation • A converter with ten switches. • Time sequences t1,j through t10,j.

27. The Ultimate Formulation • This is too generic -- there must be constraints and objectives. • Example: for a dc-dc converter with one active switch, find the sequence of times ti that yields an output voltage close to a desired reference value.

28. L + C R V V IN OUT - The Ultimate Formulation • Example: boost dc-dc converter. • Find the best time sequence to correct a step load change and maintain fixed output voltage.

29. The Ultimate Formulation • Still too generic – no unique solution. • Also limited in utility. • The proposed constraint deals with steady-state output and only one specific dynamic disturbance. • There were no constraints on switching rates or other factors.

30. The Ultimate Formulation • More practical: Given an objective that takes into account power loss, output steady-state accuracy, dynamic accuracy, response times, and other desired factors, find a sequence of times that yield an optimum result. • That is, find a set of times tkthat minimizes an objective function.

31. The Ultimate Formulation • This is a general formulation in terms of a hybrid control problem. • Unfortunately, with results framed this way there are very limited results about existence of solutions, uniqueness, stability, and other attributes. • Still very general, but with a well-formed cost function it might even have a solution. • There is a control opportunity every time a switch operates.

32. Implications • For steady-state analysis, this must yield familiar results. • A dc-dc converter with loss constraints must act at a specific switching frequency with readily calculated duty ratio. • For dynamic situations, the implications are deeper. • Should a converter operate for a short time at higher frequency when disturbed? • How do EMI considerations affect times? • Are our models accurate and complete enough?

33. Geometric Control Examples • Dc-dc buck converter, 12 V to 5 V nominal. • L = 200 uH, C = 10 uF, 100 kHz switching.

34. Fixed Duty Ratio • Steady state, fixed duty ratio. • This shows the inductor current and ten times the normalized capacitor voltage. • The “best” solution given fixed 100 kHz switching.

35. Result in State Space • Same data plotted in state space.

36. Hysteresis Control • Alternative: simply switch based on whether the output is above or below 5 V. • No frequency constraint.

37. Hysteresis Control • Same result, in state space. • These controls need timing constraints to prevent chattering.

38. Response to Step Line Input • Line step from 12 V to 15 V at 42 us. • Duty ratio adjusts instantly to the right values. (This would happen in open-loop SCM.) • Transient in voltage occurs.

39. State Space • State space plot shows how much the behavior deviates.

40. Same Step – Different Control • This is a current hysteresis control, with the switch set to turn off at a defined peak and on at a defined valley. Same line step.

41. State Space • The step is cancelled perfectly – essentially in zero time.

42. L I i I IN OUT LOAD + - + i + v C L V v IN in - C R - V OUT Boost Converter – A Harder Test • What about a boost converter step? • Example converter: L = 200 uH, C = 20 uF, 5 V input, 12 V output, 100 kHz switching

43. Steady State Behavior

44. Step Change Behavior • Step input from 5 V to 6 V at 42 us. • Very slow transient – even though the duty ratio values are set to cancel the change.

45. State Space • Suggests a faster transition is possible.

46. Ad Hoc Control • Short-term overshoot can be used to dramatically speed the response.

47. State Space • Rapid move toward final desired result.

48. Augmented Boost • Now alter the boost to achieve timing targets. • This control eliminates the transient.

49. State Space • The response never goes outside ripple limits.

50. More General Result