Modeling approaches for switching converters by Giorgio Spiazzi - PowerPoint PPT Presentation

Modeling approaches for switching converters by Giorgio Spiazzi University of Padova – ITALY Dept. of Information Engineering – DEI Modeling approaches for switching converters 1/72

Summary of the presentation PWM converters PWM converters • • Switching cell average model in continuous conduction Switching cell average model in continuous conduction mode (CCM) mode (CCM) • Switching cell average model in discontinuous • Switching cell average model in discontinuous conduction mode (DCM): first- -order model order model conduction mode (DCM): first Modeling approaches for switching converters 2/72

Basic DC-DC Converter topologies: 2°order i L Buck Buck i s + S L + D u g i D R o u o - i L i D Boost Boost + L + D u g i S R o S u o - i D i s - S + D Buck- -Boost Boost R o Buck u g i L L u o + Modeling approaches for switching converters 3/72

Basic DC-DC Converter topologies: 4°order u C i 1 i 2 + - - + L 2 L 1 C 1 + Cuk Cuk u g R o S D u D C o u S u o + + - u C i 1 i D D + - + + + L 1 u D + C 1 SEPIC SEPIC u g L 2 R o S C o i 2 u S u o - - Modeling approaches for switching converters 4/72

Commutation Cell for 2°order converters i D i s n a p 2° ° order converters can be order converters can be 2 - + D S described by a unique described by a unique i L L U off U on commutation cell: commutation cell: - + c Buck Boost Buck-boost U on U g -U o U g U g U off U o U o -U g U o U on + U off U g U o U g +U o i g i S i L i S i o i L i D i D Modeling approaches for switching converters 5/72

Averaging t 1 ( ) ( ) ∫ = τ τ x t x d Moving average: T − s t T s Example: instantaneous and average inductor Example: instantaneous and average inductor current in transient condition current in transient condition [A] i L 11 10 i L 9 8 7 6 5 [ms] 2.8 2.9 3 3.1 Modeling approaches for switching converters 6/72

Average model: CCM • Switching frequency ripples are neglected • Only low-frequency dynamic is investigated ( ) di t ( ) = Example: inductors L Example: inductors u t L L dt ( ) ( ) ( ) i t t  − −  1 L L i t i t T ( ) ( ) ∫ ∫ = τ τ = = L L S u t u d di L   L L L   T T T ( ) − − S S S t T i t T S L S Modeling approaches for switching converters 7/72

Average model: CCM   ( ) t d i t d 1 ( ) ∫ = τ τ =   L i d ? L dt dt  T    − S t T S ( ) β t z ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ φ = τ τ = φ = τ τ = α = β t f t , d t , y , z f t , d with y t , z t ( ) α t y ( ) ( ) ( ) ( ) ( ) β t φ τ α β d t df t , d t d t ( ( ) ) ( ( ) ) ∫ = τ − α + β d f t , t f t , t dt dt dt dt ( ) α t ( ) ( ) ( ) ( ) − − d i t d i t i t i t T ( ) = = L L L L S u t L L dt dt T S Modeling approaches for switching converters 8/72

Averaging approximation Non steady- -state state Non steady U ( ) ( ) = + on i dT i 0 dT inductor current inductor current L s L s L waveform: waveform: U ( ) ( ) ( ) s = − − off i T i dT 1 d T u L (t) L s L s U on L u L t -U off U − = − off m i L (t) 2 L U U ( ) ( ) ( ) = + − − on off U i T i 0 dT 1 d T i L (T s ) 1 = on m i L (0) L s L s s L L L t u ( ) = + L dT s (1-d)T s i 0 T L s L Modeling approaches for switching converters 9/72

Averaging • Reactive element voltage Reactive element voltage- -current relations current relations • remain valid also for average quantities; remain valid also for average quantities; • for inductors, the current variation in a switching for inductors, the current variation in a switching • period can be calculated by integrating their period can be calculated by integrating their average voltage; average voltage; • for capacitors, the voltage variation in a for capacitors, the voltage variation in a • switching period can be calculated by integrating switching period can be calculated by integrating their average current. their average current. Modeling approaches for switching converters 10/72

Continuous conduction mode - CCM u L = At steady- -state: state: 0 At steady u L (t) U on − U 1 d ( ) S = t = − on U dT U 1 d T on S off -U off U d off U − = − off m i L (t) 2 L M = d i Buck: L U 1 = on m L 1 = Boost: M t − 1 d i s (t) (1-d)T s S = i i d L d = M Buck-Boost: − t 1 d i d (t) Boundary CCM Boundary CCM- -DCM: DCM: ( ) = − i i 1 d dT s D L ∆ i U U ( ) = = = − Lpp t on off i d 1 d T s L lim 2 2 Lf 2 Lf S S Modeling approaches for switching converters 11/72

Switching cell average model: CCM i D i S i S i D n a p - - S + + D D i L L S u D u off u S u on - + - + c Non linear components Non linear components Average switch and diode voltages and currents: Average switch and diode voltages and currents: ( ) =  ′ = + ′ i d i  d u d u u d S L ⇒ = ⇒ =  S on off i i  u u ( ) ′ ′ = S D = + S D i d i d   u d u u d D L D on off d’=1-d = complement of duty-cycle Modeling approaches for switching converters 12/72

Switching cell average model: CCM ( ) ′ = + ′ =   u d u u i d i d d ⇒ = S on off S L ⇒ =   u u i i ( ) ′ = + S D ′ = S D  u d u u d i d i d  D on off D L i S i D i i S D d’:d i d ′ i - - S D + + i D S - d S u D + u u u S + S D u u - + - + d ′ S D - u + D d d’=1-d = complement of duty-cycle Modeling approaches for switching converters 13/72

Switching cell average model: CCM • The non-linear components (switch and diode) are replaced by controlled voltage and current generators representing the relations between average voltage and currents; • These controlled voltage and current generators can be substituted by an ideal transformer with a suitable equivalent turn ratio. Modeling approaches for switching converters 14/72

Buck average model: CCM i L i S + S L + D u g i D R o u g u o - ′ i d = − = − L u u u u u D g S g D d L + - + u u = d u S u D g D i i R o S D - + C ( ) d d u = = − i i i i o d’:d ′ ′ S D L S + d d u g - i = d i S L Modeling approaches for switching converters 15/72

Buck average model (alternative approach): CCM i L i S + S L + D u g i D R o u o - i g i L Switching Switching Independent variables: u u g , i i L Independent variables: g , + S + L cell cell u g u D Dependent variables: u u D , i i g Dependent variables: D , D g - - Modeling approaches for switching converters 16/72

Buck average model (alternative approach): CCM i g i L i i g L Averaging Averaging d i + + S + + + L u g u u D u g D D d u g - - - - i i g L 1:d + + u u g D - - Modeling approaches for switching converters 17/72

Buck average model: CCM i d i L L + L + + d u R o C u g u g o - -  d i = = − L L u d u u   L g o dt  d u u  C = = − o C i i  C L  dt R o Modeling approaches for switching converters 18/72

Boost average model: CCM i L i D + L + D u g i S R o S u o - i d L = − = − u u u u u ′ S o D o S d L - + ′ = u d u u u + S o S i D u i D g S - + ′ ′ ( ) d d = = − + d’:d i i i i D S L D d d R o C u o - ′ = i d i D L Modeling approaches for switching converters 19/72

Boost average model (alternative approach): CCM i D i L + L + D u g i S R o S u o - i L i D Switching Switching + + Independent variables: u u o , i i L Independent variables: o , D L u S cell cell S u o Dependent variables: u u S , i i D Dependent variables: S , D - - Modeling approaches for switching converters 20/72

Boost average model (alternative approach): CCM i i L i D L d ′ i + + L + + D u u S S u o o d ′ u o - - - i i L D d’:1 + + u u S o - - Modeling approaches for switching converters 21/72

Boost average model: CCM d ′ i i d’:1 L L + + L + d ′ u R o C u o u g o - -  d i ′ = = − L L u u d u   L g o dt  d u u  ′ C = = − o C i d i  C L  dt R o Modeling approaches for switching converters 22/72

Buck-Boost average model: CCM i D i S - S + D u g i L L R o u o + i 1 i S = + − = − d’:d D u u u u u u g S D o D o - - d + u u ( ) ( ) D S ′ = + = + u d u u u d u u + - + D g o S g o u R o ′ ′ g ( ) C d d = = − u i i i i o D S L D d d i L L ′ + = i = i d i d i D L S L Modeling approaches for switching converters 23/72