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# Power Electronics

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1. Power Electronics Small-signal converter modeling and frequency dependant behavior in controller synthesis by Dr. Carsten Nesgaard

2. Agenda Small-signal approximation Voltage-mode controlled BUCK Converter transfer functions  dynamics of switching networks Controller design (voltage-mode control) Discrete time systems Measurements

3. Small-signal approximation Advantages of small-signal approximation of complex networks: An analytical evaluation of equipment performance An analysis of equipment dynamics Stability Bandwidth A design oriented equipment synthesis The linearization of basic AC equivalent circuit modeling corresponds to the mathematical concept of series expansion.

4. Small-signal approximation Drawbacks of small-signal approximation of complex networks: Limited to rather low frequencies (roughly fS/10) Inability to predict large-signal behavior Transients High frequency load steps Calculation complexity increases quite rapidly

5. Not included in ‘Fundamentals of Power Electronics’ SE. Voltage-mode controlled BUCK Basic BUCK topology with closed feedback loop

6. Voltage-mode controlled BUCK Converter waveforms:

7. AC modeling Converter states: 0 < t < dT: dT < t < (1 – d)T KCL: KVL:

8. DC and 2nd order terms are removed from the equations to the right. AC modeling Averaging and linearization (in terms of input and output variables): Inductor equation: Capacitor equation (same for both intervals): Input current equation:

9. AC modeling Resulting AC equivalent circuit: DC transformer relating input voltage and inductor current, thus behaving ‘almost’ like a real transformer.

10. Canonical AC model Rearranging the AC equivalent circuit found on the previous slide by the use of traditional circuit theory a universal model can be established: A similar model applies to a wide variety of other converter topologies. In Fundamentals of Power Electronics SE a table containing coefficients for the different sources can be found.

11. A : State matrix B : Source matrix E : Control matrix F : Feedback matrix I : Identity matrix Q : Feed forward matrix State equation: M : Output-state-matrix N : Output-source-matrix d : Control variable u : Source variable Control equation: x : State variable y : Output variable Output equation: Converter transfer functions Basic control system

12. Open loop transfer function: Closed loop transfer function: Converter transfer functions Opening the loop and rewriting the system equations the following trans-fer functions can be obtained:

13. Inductor current x1 Capacitor voltage x2 State-space averaging State variables: By definition the following apply: Output variable y (dependent) Source variable u (independent) In order to contain past information all variables are functions of time

14. The use of linearization requires: Insertion into the state equation results in: State-space averaging Averaging the equations previously found results in the following non-linearized matrices:

15. Comparing the above A matrix with the non-linearized A’ matrix found on the previous slide, it can be seen that no changes have occurred. State-space averaging The averaged and linearized matrices can now be identified:

16. Collecting terms in accordance with the control equation, and realizing that multiplication by ‘s’ in the frequency domain is the equivalent to differentiation in the time domain, the matrices F and Q can be identified: and Since feedback is the only means of converter control applied, Q is (as expected) zero. State-space averaging Averaging and linearizing the control variable d (PWM controller) in terms of state variables gives the following relation: is the sawtooth peak voltage is the EA gain, ‘a’ factor and comp.

17. 0 1/L 1/RLOADC -K/VP 1 State-space averaging Summarizing the voltage-mode controlled BUCK matrices:

18. Stability Nyquist stability requirement for closed loop systems: Prh(GCL(s)) = Prh(GOL(s)) +  = 0 Where: Prh = number of right half-plane poles  = number of times the Nyquist contour of the open- loop transfer function circles the point (-1,0) GCL = Closed-loop transfer function GOL = Open-loop transfer function Minimum open-loop transfer function gain margin: 6 - 8 dB Minimum open-loop transfer function phase margin: 30- 60

19. Circuit data: L = 300 H C = 69 F RESR = 0,2  ILoad,m = 1 A U1 = 12 V y2 = 5 V IL = 0,2 A f = 50 kHz Additional data: Vp = 2,45 V a = 0,5 Voltage-mode controlled BUCK

20. Voltage-mode controlled BUCK A plot of the open-loop transfer function is shown below (K = 1):

21. Voltage-mode controlled BUCK A 3D plot of the converter filter transfer function is shown to the right. Note: The zero caused by RESR increases the phase (green curve) as a function of frequency and RESR. Unfortunately due to the same zero the filter attenuation drops (red curve).

22. Compensation PI-comp. Lag-comp. PD comp. Lead-comp. A PI-Lead-comp. (PID) will be used in this presentation

23. Compensation Widely accepted error amplifier configuration: Pole at f = 0 for increased DC gain Pole at fESR for compensation Double zero at resonance peak for increased phase margin

24. -20 dB/dec +20 dB/dec 0 dB/dec fC 4.0 kHz  56,1 0 dB/dec -40 dB/dec -20 dB/dec Red : Converter transfer function Blue : Compensation transfer function Compensation Compensator and converter transfer functions: Amplitude: Phase:

25. Voltage-mode controlled BUCK Using the previously derived matrices an expression for the input impedance Zin can be established:

26. Voltage-mode controlled BUCK A plot of the open-loop transfer function during Discontinuous Conduction Mode (red curve) and EA compensation (blue curve) DCM reduces the converter transfer function to a first order system, since the time derivative of the small-signal inductor current is zero and thus disqualifies the inductor current as a state variable.

27. At the sampling instants: Thus, the dynamics of the two systems are identical at the sampling instants: Discrete time systems Transient response and the relationship between the s-plane and the z-plane: Discrete time: Continuous time: Inserting into the expression to the left, it can be seen that the continuous time stability requirement maps onto the z-plane in form of the unit circle.

28. Discrete time systems Arithmetic and operations: • Integration and differentiation • Plotting the frequency response • Tustin’s rule • Sampling rate

29. Discrete time systems Plot of the discrete compensation transfer function: Cont Continuous time Disc_1 Discrete time with sample frequency = 50 kHz (no prewarping) Disc_2 Discrete time with sample frequency = 100 kHz (no prewarping)

30. Measurements Below is a comparison of the predicted continuous time loop gain, predicted discrete time loop gain and an actual measurement of the loop gain: GH Continuous time GD_2 Discrete time with sample frequency = 50 kHz (no prewarping) Meas Actual measurement

31. Measurements The same transfer function as before, but during Discontinuous Conduction Mode: GH Continuous time Meas Actual measurement