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  1. 1. Generalized Averaging Method for Power Conversion Circuits IEEE PAPER PRESENTATION In the subject of APE (Advanced Power Electronics) Prepared by UTSAV YAGNIK (150430707017) M.E. Electrical SSEC, BHAVNAGAR
  2. 2. About the Paper • IEEE Transactions on Power Electronics. • Dated 2nd April 1991 • Innovator(s) 1. Seth R. Sanders 2. J. Mark Noworolski 3. Xiaojun Z. liu 4. George C. Verghese 2
  3. 3. INDEX Sr. No. Topic Slide No. 1 About the paper 2 2 Brief Abstract 4 3 Introduction 5 4 Previously Developed Methods 8 5 The Generalized Averaging Technique 10 6 Key Properties of Fourier Coefficients 13 7 The Utility of the Method 20 8 Example 23 9 Conclusion 32 3
  4. 4. Brief Abstract • The method of state space averaging has its limitations with switched circuits that do not satisfy a small ripple condition. • Here, a more general averaging procedure has been put forward based on state space averaging which can be applied to broader class of circuits and systems including resonant type converters. 4
  5. 5. Introduction • State space averaging has been effective method for analysis and control design in PWM. • But it can be applied to a limited class of converters due to 1. Small ripple condition 2. A small parameter which is related to the switching period and the system time constants. 5
  6. 6. Introduction • Here, the small parameter is typically small for fast switching but not for resonant type converters. • The resonant converters have state variables that exhibit predominantly oscillatory behaviour. • This paper represents a method that can accommodate arbitrary types of waveforms. 6
  7. 7. Introduction • The method is based on a time-dependent Fourier series representation for a sliding window of a given waveform. • For instance, to recover the traditional state averaged model, one would retain only the dc coefficient in this averaging scheme. 7
  8. 8. Previously developed methods Sampled-data modelling: • Creates a small signal model for the underlying resonant converter with the perturbation in switching frequency as the input. Difficulty with this approach: • Requirement of obtaining a nominal periodic solution as a first step in the analysis. 8
  9. 9. Previously developed methods Phase plane techniques: • A basic approach to obtaining a steady state solution for a resonant converter. Limitation: • Its restricted to second order systems. • So, one can not incorporate additional state variables that are associated with load or source dynamics. 9
  10. 10. The Generalized Averaging technique • It is based on the fact that the waveform 𝑥 ∙ can be approximated in the interval (𝑡 − 𝑇, 𝑡) to arbitrary accuracy with a Fourier series representation of the form • 𝑥 𝑡 − 𝑇 + 𝑠 = 𝑘 𝑥 𝑘(𝑡)𝑒 𝑗𝑘𝜔 𝑠(𝑡−𝑇+𝑠) • k = all integers, 𝜔𝑠 = 2𝜋 𝑇 , 𝑠 𝜖 (0, 𝑇] , and 𝑥 𝑘(𝑡) are complex Fourier coefficients which are functions of time since considered interval slides as a function of time. 10
  11. 11. The Generalized Averaging technique • The analysis computes the time evaluation of these Fourier series coefficients as the window of length T slides over the actual waveform. • The 𝑘 𝑡ℎ coefficient, which is index-k coefficient, is determined by • 𝑥 𝑘 𝑡 = 1 𝑇 0 𝑇 𝑥(𝑡 − 𝑇 + 𝑠)𝑒−𝑗𝜔(𝑡−𝑇+𝑠) 𝑑𝑥 … (2) 11
  12. 12. The Generalized Averaging technique • Here the approach is to determine an appropriate state space model in which the coefficients in earlier equation are the state variables. 12
  13. 13. Key properties of Fourier coefficients • 1. Differentiation with respect to time: • 𝑑 𝑑𝑡 𝑥 𝑘 𝑡 = 𝑑 𝑑𝑡 𝑥 𝑘 𝑡 − 𝑗𝑘𝜔𝑠 𝑥 𝑘 𝑡 • When𝜔𝑠(𝑡) is slowly varying the above equation will be a good approximation. • This approximation is useful in analysis of system where drive frequency is not constant. 13
  14. 14. Key properties of Fourier coefficients • 2. Transforms of Functions of Variables: • 𝑓(𝑥1, 𝑥2, … , 𝑥 𝑛) 𝑘 is the general scalar function of arguments which will be computed. • A procedure for exactly computing the above function is available where function is polynomial which is based on following relationship… (contd.) 14
  15. 15. Key properties of Fourier coefficients • 𝑥𝑦 𝑘 = 𝑖 𝑥 𝑘−𝑖 𝑦 𝑖 • Here sum is taken over all integers i. • The computation is done by considering each homogeneous term separately. The constant and linear term are trivial to transform. The transforms of quadratic terms can be computed using above equation. 15
  16. 16. Key properties of Fourier coefficients • Higher order homogeneous terms are dealt with factoring such term into product of two lower order terms. Then the procedure can be applied to each term separately. • This process is guaranteed to terminate since factors with only linear term will eventually arise. 16
  17. 17. Key properties of Fourier coefficients • 3.Application to State-Space Model of Power electronics circuits: • Here the method is applied to a state-space model that has some periodic time dependence. • Model equation: • 𝑑 𝑑𝑡 𝑥 𝑡 = 𝑓{𝑥 𝑡 , 𝑢(𝑡)} 17
  18. 18. Key properties of Fourier coefficients • Where u(t) = some periodic function with time period T. • To apply the generalized averaging scheme to above equation, one simply needs to compute the relevant Fourier coefficients of both sides of that equation. i.e. • 𝑑 𝑑𝑡 𝑥 𝑘 = 𝑓(𝑥, 𝑢) 𝑘 18
  19. 19. Key properties of Fourier coefficients • The first step is to compute derivative of the kth coefficient as below… • 𝑑 𝑑𝑥 𝑥 𝑘 = −𝑗𝑘𝜔𝑠 𝑥 𝑘 + 𝑓(𝑥, 𝑢) 𝑘 • As previously stated, only dc coefficients(index zero) will be retained for a fast switching PWM circuit to capture the low frequency behaviour. • The result would be exact average model. 19
  20. 20. The utility of the method • It is straight forward to obtain a steady state solution for a model by setting its variables derivative to zero. • This approach goes one step further in extending the analysis to transient behaviour as well. • After obtaining a steady state solution, the model may be linearized about the steady state to obtain small signal transfer functions from inputs such as switching frequency or source voltage to variables such as v or i. 20
  21. 21. The utility of the method • The essence in modelling is to retain only the relatively large Fourier coefficients to capture the interesting behaviour of the system. • So, only the index-zero(dc) coefficients for a fast switching PWM circuit to capture the low frequency behaviour is retained. • The result would be a precise state-space model. 21
  22. 22. The utility of the method • For a resonant dc-dc converter which has some of its states exhibiting predominantly dc(or slowly varying) waveforms, the index-one(and minus one) coefficients for those states exhibiting sinusoidal-type behaviour and the index-zero coefficients for those states exhibiting slowly varying behaviour. 22
  23. 23. Example • 1. Series resonant converter with voltage source load 23
  24. 24. Example • The state-space model for this circuit is of the form below: • The tank resonant frequency is 36KHz. 24
  25. 25. Example • And the waveforms are as below: • 25
  26. 26. Example • These waveforms were generated by stepping the drive frequency between 38KHz and 40KHz. • Following each step change in driving frequency, the waveforms appear to be amplitude modulated sinusoids. • The waveforms settle down to an approximately sinusoid steady state. 26
  27. 27. Example • So, the waveforms can be approximated with the fundamental frequency terms in the Fourier series coefficients as below… 27
  28. 28. Example • The term 𝑠𝑔𝑛(𝑖) 𝑙 can be evaluated using the describing function approach by assuming i(t) is approximated as a sinusoid over each interval of length T. 28
  29. 29. Example • So, the model approximated with time invariant model as below: 29
  30. 30. Example • Waveforms obtained by this approach: 30
  31. 31. Example • From the comparison of waveforms it is evident that both waveforms are quite the same and the given analysis technique is a step ahead than steady state computation where slow transient behaviour is also included. 31
  32. 32. Conclusion • A new approach to averaging power electronics circuits is introduced. • It has been effectively tested with the resonant type converters analysis. • This approach refines the state space averaging technique of analysis providing framework for design and study of small ripple conditions. 32
  33. 33. THANK YOU 33