Frequency-Dependent, Time-Invariant DC-DC Converter Modeling without Averaging

1. Frequency-Dependent, Time-Invariant DC-DC Converter Modeling without Averaging Patrick Chapman Asst. Prof. UIUC April 10, 2006 Grainger Center for Electric Machines and Electromechanics

2. Summary • Overview time-invariant (TI) converter modeling • Propose alternative method of TI modeling, avoiding formal averaging • Explore simulation results • Probe small-signal analysis improvements Grainger Center for Electric Machines and Electromechanics

3. Converter Modeling • DC-DC converter difficulties • Nonlinear • Time-varying • Switched • Longstanding modeling desires: • Easy control design • Rapid simulation • Accuracy • No “hand-waving” Grainger Center for Electric Machines and Electromechanics

4. Conventional Approach: Averaging • Method dates from 70’s • Needed way to apply linear control design • Needed time-invariant model for this • Stability was main priority (fast response, less so) • Can be done on schematic (circuit-based) • Can be done on equations (“state-space”) • Methods produce equivalent models Grainger Center for Electric Machines and Electromechanics

5. Conventional State-Space Avg. • Often cast in the “time-window” approach • T is the switching period • x is any time-domain variable • Sometimes cast as weighted average of state matrices • d is duty cycle in period T • n is integer • Methods produce same model Grainger Center for Electric Machines and Electromechanics

6. Synopsis: Conventional Avg. • Removes time-varying state matrices • Removes switching ripple • Models often linearized • Enables small-signal, linear analysis • Very widely used • Calls into question bandwidth limitation • What disturbance frequency is valid? • Aliasing-like effects neglected • Can improperly track the average! Grainger Center for Electric Machines and Electromechanics

7. Tracking Issues in Avg. Models • Occurs in certain converters • Boost and buck-boost in particular • Occurs even under ideal circuits with modest ripple • Two main causes • Neglecting ripple, ESR in steady-state derivations • Sampling effects in feedback control Grainger Center for Electric Machines and Electromechanics

8. Regarding Moving Time-Window • Averaging integral • Relies on continuous time history • Can’t be implemented directly in hardware • Can complicate rigorous analysis • Imposing is questionable • q is switching signal • d is duty cycle command Grainger Center for Electric Machines and Electromechanics

9. Comparing q, d, and Ratio – 20:1 Natural 2-sided PWM should preserve phase Averaging integral known to produce a stair-step, delayed signal Grainger Center for Electric Machines and Electromechanics

10. Improvements to Conv. Avg. • Higher order abstractions, ripple estimates • Krein, Bass, et al (“KBM” formality) • Sanders, Verghese, Caliskan, Stankovic (“generalized” or “multi-frequency”) • Modification of output equation to catch ripple • Lehman (2004) • Stability and convergence analysis (Tadmor) • These approaches, among others, did much to strengthen and extend averaging • Many other improvements and adaptations made • Discontinuous mode, other converters, correction factors, etc. Grainger Center for Electric Machines and Electromechanics

11. Multi-Frequency Averaging (MFA) • Accounts for ripple and switching frequency • Uses higher-order averages • Has implicit assumption of slowly-varying states (at least during time window) ***Also “Generalized” averaging Grainger Center for Electric Machines and Electromechanics

12. Perhaps Redefine MFA *Make these definitions Then… Grainger Center for Electric Machines and Electromechanics

13. Multiple Reference Frames (MRF) • Quite similar to MFA • In context of motor drives (Sudhoff) • Multi-phase systems • E.g., PM motor with nonsinusoidal back EMF • Involves a time-window average w.r.t. rotor position, not time • Makes a time-invariant model  linear analysis of motor drives Grainger Center for Electric Machines and Electromechanics

14. PM Motor MRF Analysis *MRF theory, then, has similar properties as MFA Grainger Center for Electric Machines and Electromechanics

15. Floquet Theory • A linear transformation of states • Applies to models: • Linear • Periodic coefficients • Produces exact time-invariant model • Applied by Visser to open-loop buck, boost • Difficult, maybe infeasible, for closed-loop • Closed-loop not periodic • Nonlinear theory recent, but doesn’t address aperiodicity Grainger Center for Electric Machines and Electromechanics

16. Questions: • Can averaging be avoided? • If so, • Are resulting models “better”? • Is rigorous analysis (error bounds) easier? • Can they be extended? • Aliasing-like effects • Digital control Grainger Center for Electric Machines and Electromechanics

17. Alternative Approach • Decompose signal along the lines of MFA • Quasi-Fourier-Series (QFS) components • 2N+1 variables, a cosine, b  sine • New variables not defined yet – just notational Grainger Center for Electric Machines and Electromechanics

18. Signal Derivatives (2nd order, e.g.) Grainger Center for Electric Machines and Electromechanics

19. Switching Function Model • d is the duty cycle • f is the initial phase shift (usually arbitrary) Can let d be a function of time (so long as between 0, 1) Grainger Center for Electric Machines and Electromechanics

20. Duty Cycle Ripple • Usually neglected • Usually small • Ripple in voltage or current may be fed back • Important in proportional control • Some filtering may be applied • Current-mode control known to exhibit ripple-dependent behavior • Usually suffices to model one harmonic • Analysis here is general, but specific N = 1 example given Grainger Center for Electric Machines and Electromechanics

21. Obtaining d and f • Duty cycle command (d) and d not the same Sawtooth: Triangle: Grainger Center for Electric Machines and Electromechanics

22. Obtaining d and f • Times th and tlwould occur if QFS variables remained the same until intersection • Variables change between switching edges • At first seems suspect • Cancels out in the infinite sum • Necessary step to put in terms of new states Grainger Center for Electric Machines and Electromechanics

23. Switching Function Summary • QFS coefficients: • Solution to transcendental equation • Inconvenient, but actually quite easy • Can use noniterative approximations Grainger Center for Electric Machines and Electromechanics

24. Product Terms • Commonly we have qx – product of switching function and continuous variable • Generates higher order harmonics • Approximation is the neglect of these • Though, can include in “extra” state eqns if desired Grainger Center for Electric Machines and Electromechanics

25. Product Terms, cont’d • Brute-force calculation (preferred here) • Discrete convolution • Use complex exponential QFS for concise statement 2nd order example Grainger Center for Electric Machines and Electromechanics

26. Converter Model • Boost converter example • Tools in place  apply to state equations Grainger Center for Electric Machines and Electromechanics

27. Synthesizing Model • Thus far, we’ve not completely defined variables • After making substitutions • Can equate trigonometric coefficients to satisfy • Only one of infinite choices, but yields TI model • “My states, I make them up” Grainger Center for Electric Machines and Electromechanics

28. Equating Coefficients • Example: • Arbitrary choice: a(t) = 1, b(t) = 0 • Other choices don’t satisfy “constant-in-steady-state” desire, e.g.: Grainger Center for Electric Machines and Electromechanics

29. Extracting the Equations • Model is synthesized as: • vb is constant input, matrices are constants • Will yield constant state variables in steady state Grainger Center for Electric Machines and Electromechanics

30. Closed-loop Simulation (sawtooth) • Current-mode control (parameters from lit.) Grainger Center for Electric Machines and Electromechanics

31. Closed-loop Simulation (triangle) Grainger Center for Electric Machines and Electromechanics

32. Toward Error Bounds • Was tacitly assumed N was sufficient order • Error will occur due to truncation of higher order terms • Usually, error will be small given • Low-pass filters • Low levels of excitation at higher frequencies • Can bound the steady-state error, if not transient Grainger Center for Electric Machines and Electromechanics

33. Comparison with MFA • Generates same state-space equations • Slightly improved consideration of switching function here • Avoided averaging integral • Avoided “slow-varying” assumption • Both neglect aliasing effects Grainger Center for Electric Machines and Electromechanics

34. More Comparison • Both generally capture the average value more precisely • Both rely on truncations of actual signals • Allowed for sawtooth vs. triangular carrier • Proposed method was not a unique formulation (infinite # of zero terms) • Linearization (below) is analytical with proposed technique Grainger Center for Electric Machines and Electromechanics

35. Small-Signal Analysis • Linearize the Qxterms (op = operating point) • Calculate the partials • Messy derivation, but not bad end result (Sawtooh) (Triangle) Grainger Center for Electric Machines and Electromechanics

36. Linearized Boost • Includes 1 harmonic in duty cycle command, otherwise retains generality Grainger Center for Electric Machines and Electromechanics

37. Closed-loop Frequency Response • 0th order component only (sawtooth) Closed-loop gain differs by about 6% Higher frequency analysis questionable due to neglect of aliasing effects Grainger Center for Electric Machines and Electromechanics

38. “Disturbing” Issues • Switching and sampling not exactly the same • Can’t apply directly sampling theorems • But… • Higher frequency disturbances are effectively aliased • Signals aliased to low frequencies can pass through easily • Absolute phase shift of input disturbance matters • (in typical small-signal analysis, only relative phase matters) • How can we account for this in a TI model? Grainger Center for Electric Machines and Electromechanics

39. Sinusoidal Disturbances • Can use ideas from sinusoidal PWM • Once linearized and perturbed, we introduce unmodeled harmonics • Choose important components from spectrum • Let: • Use Bessel-function approach to get specific components, e.g.: Grainger Center for Electric Machines and Electromechanics

40. Example Disturbances • 1 kHz switching, 400 Hz disturbance, dm = 0.1 Actual freq: 200 Hz Grainger Center for Electric Machines and Electromechanics

41. Example Disturbances • 1 kHz switching, 500 Hz disturbance, dm = 0.1 Actual freq: 500 Hz Grainger Center for Electric Machines and Electromechanics

42. Example Disturbances • 1 kHz switching, 900 Hz disturbance, dm = 0.1 Actual freq: 90 Hz Grainger Center for Electric Machines and Electromechanics

43. Example Disturbances • 1 kHz switching, 1 kHz disturbance, dm = 0.1 Actual freq: 1 kHz Grainger Center for Electric Machines and Electromechanics

44. Other Work in Progress • In most applications, the 2nd order effects don’t matter that much • Sometimes, we design so they won’t matter – but how can we be sure? How can we push the limit? • Aliasing effects • Digital sampling techniques • Variable switching frequency • Can we use MFA modeling techniques to investigate advanced controllers? Grainger Center for Electric Machines and Electromechanics

45. Dominant Aliased Harmonic • Looking between 0 and wsw only • Harmonics of wsw reduced by factor of • Model with an additional harmonic for fixed frequency disturbance analysis Relative sizes depend on fd Grainger Center for Electric Machines and Electromechanics

46. Digital Sampling Effects • Voltage or current may be sampled • Analog to digital converter • Working with continuous time model • Sampling can be modeled as 0th or 1st order hold • Quantization error • These create additional harmonics in small-signal analysis • Effects of (intentional?) over- or undersampling Grainger Center for Electric Machines and Electromechanics

47. Variable Switching Frequency • Useful during transients • Speed up switching to increase dynamic performance • Slow down during steady-state for lower switch losses • Let wsw vary with time – must actually work in terms of qsw – the switching angle Grainger Center for Electric Machines and Electromechanics

48. Conclusion • Multi-frequency techniques can capture sometimes-important 2nd order effects • Showed one technique that avoids formal averaging • Seems to have easy instantiation (suggests “tool”) • Perhaps easier analysis • Do other problems “crop up”? • These methods have potential to tackle subtleties near and above fsw/2 Grainger Center for Electric Machines and Electromechanics