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Prediction of low-frequency variations in duty cycle to understand their impact on converter voltages and currents, focusing on averaging to remove switching ripples and linearization for simplified analysis. The linearized, averaged circuit model aids in analyzing the converter behavior with AC perturbations and DC operating points considered.
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Lecture 22 EEL 646 POWER ELECTRONICS II Closed-Loop Control
Closed-Loop DC-DC Converter H(s) 2 Based on the textbook by Robert Erickson
Objective of ac converter modeling Predict how low-frequency variations in duty cycle induce low- frequency variations in the converter voltages and currents. Ignore the switching ripple Ignore complicated switching harmonics and sidebands The approach is to Remove switching harmonics by averaging all waveforms over one switching period 4
Example of neglecting Switching Ripples 5 Based on the textbook by Robert Erickson
Output voltage spectrum with sinusoidal modulation of duty cycle 6 Based on the textbook by Robert Erickson
Example Outlining Basics of Averaging and Linearization (Manual Circuit Averaging and Linearization) V V o iL m in m L L iD + vout V DT in vin iL peak i L C R L _ iD DTs Ts i peak Fig. (1.) Buck-Boost Converter DTs Ts From the iDplot we can see the average current is given by: Fig. (2.) Inductor and Diode Current in DCM 2 2 V T D in 2 i D AVG L V o A resulting averaged circuit can be drawn as: + V _ D i AVG C R OUT AVG *Note- In the general case, both D and Vin vary with time and we can see the nonlinear dependence on the circuit parameters. Fig. (3.) Output Portion of Averaged Circuit
Example Outlining Basics of Averaging and Linearization (Manual Circuit Averaging and Linearization) To linearize, we first solve for the DC operating point as: 2 in 2 V T D R T V R I R V V D o D o in Avg 2 L V 2 L o Next, we introduce AC perturbations and neglect second and higher order terms: ~ 2 D 2 d D 2 in 2 in 2 ~ ~ V T V T D ~ ~ I I V I v V d D d i d i d d o d o o L L 2 ~ 2 V v o o Separating the AC and DC quantities in the equation, we have: For DC terms: For AC terms: 2 in 2 2 in V T ~ ~ D V T i~ I I v V d D d d d o o L 2 V L o Substituting the DC expression for Id into the AC expression and solving for we obtain: ~ d 2 in 2 ~ V T D D i~ v d o 2 L V 2 V o o
Example Outlining Basics of Averaging and Linearization (Manual Circuit Averaging and Linearization) Substituting in the DC operation point (expression for Vo) into the expression for we obtain: ~ ~ d 1 2 T ~ v V D i d o in R R L The resulting linearized, averaged circuit is: ~ d i + ~ v ~ d 2 T V D C R R in o R L Fig. (4.) Linearized, Averaged Circuit for DCM *Note- The linearization process introduces an additional resistance, also note LTI elements map to themselves in the model but nonlinear elements are replaced by controlled sources and dampening resistances.
Example Outlining Basics of Averaging and Linearization (Manual Circuit Averaging and Linearization) Another key point here is that the same result can be obtained by the following expression: ~ d 2 in 2 ~ ~ d ~ V T f f D D ~ v v d i o o _ o _ 2 L V 2 V o o v d 2 in 2 _ D V T D f i ) t ( i where D AVG _ ) t ( 2 L v o *Note- Therefore Both approaches are an expression of Taylor Series expansion up to the linear terms. Key Issue
