ALB Universitat Bremen Otto-Hahn-Allee NW1 Institut fur elektrische Antriebe 28359 BREMEN eistungselektronik und Bauelemente Te.:0421-2184436 Prof. Dr.-Ing B. Orlik FaX:0421-2184318 Power Converter Technology Version 1.5 April 2006
Universität Bremen Otto-Hahn-Allee, NW1 Institut für elektrische Antriebe, 28359 BREMEN Leistungselektronik und Bauelemente Tel.: 0421-218-4436 Prof. Dr.-Ing. B. Orlik Fax: 0421-218-4318 Power Converter Technology Version 1.5 April 2006
IALB-Institute of Electrical Drives, Power Electronics and Components 1 One-Quadrant Converter As an example of a one-quadrant converter a buck converter(also known as step-down converter)is presented in the picture below. The name of the step-down converter results from the arithmetic mean at the output side which is less than the arithmetic mean at the input de Figure 1-1 shows a typical step-down converter with an idealized freewheeling diode and idealized switch as a replacement for real power-semiconductor devices Is R L Figure 1-1: One-quadrant converter UL: Voltage at the load-side UD: Direct voltage of the sourc S closed u Destination:0≤ul≤U t+0 f (To= Period of oscillation) Note 0 u Figure 1-2 Current and voltage at an idealized switch Pvs=is·us Pvs: Power losses at the switch, at a real switch (IGBT, MOSFET, etc. )the power losses are Pvs>0. This means less efficiency and the need of bigger heat sinks Requirement: Power losses Pvs at the switches have to be as low as possible
IALB – Institute of Electrical Drives, Power Electronics and Components - 1 - 1 One–Quadrant Converter As an example of a one-quadrant converter a buck converter (also known as step-down converter) is presented in the picture below. The name of the step-down converter results from the arithmetic mean at the output side which is less than the arithmetic mean at the input side. Figure 1-1 shows a typical step-down converter with an idealized freewheeling diode and an idealized switch as a replacement for real power-semiconductor devices. UD Figure 1-1: One–quadrant converter UL: Voltage at the load-side UD: Direct voltage of the source S closed: L UD u = : ton S opened: uL = 0: toff Destination: 0 ≤ u L ≤UD 14243 0 0 1 0 f T t t t U t t U t t u on off on D on off D on off L = + = + ⋅ + ⋅ = (T0 = Period of oscillation) Note: Figure 1-2: Current and voltage at an idealized switch PVS = iS ⋅ uS PVS: Power losses at the switch; at a real switch (IGBT, MOSFET, etc.) the power losses are PVS > 0. This means less efficiency and the need of bigger heat sinks. Requirement: Power losses PVS at the switches have to be as low as possible!
IALB-Institute of Electrical Drives, Power Electronics and Components Switching performances Parasitic capacities cause a high current at the moment of turning on the switch by a high voltage change. This causes a current peak at the switch(which might be dangerous for the switch). Parasitic capacities may result from turning a freewheeling diode off, too! Parasitic inductances at the load side(or the inductive load itself) impel the current afte storang the voltage of the source off. This is a result of the magnetic energy which is still stored in the magnetic field of the inductances. Opening the switch without any freewheeling diode in the circuit will lead to an electric arc which can destroy any semiconductor switch this effect can be observed by turning a usual light switch off) Because of this effect a freewheeling diode is used in any circuit containing strong inductive component Schalter Turn on: capacitive parasitic effects On Turn off: inductive parasitic effects Off Figure 1-3: Parasitic effects at the switc geschlossen 100%=UR LO Figure 1-4: Current and voltage at a quasi-static state
IALB – Institute of Electrical Drives, Power Electronics and Components - 2 - Switching Performance: Parasitic capacities cause a high current at the moment of turning on the switch by a high voltage change. This causes a current peak at the switch (which might be dangerous for the switch). Parasitic capacities may result from turning a freewheeling diode off, too! Parasitic inductances at the load side (or the inductive load itself) impel the current after turning the voltage of the source off. This is a result of the magnetic energy which is still stored in the magnetic field of the inductances. Opening the switch without any freewheeling diode in the circuit will lead to an electric arc which can destroy any semiconductor switch (this effect can be observed by turning a usual light switch off). Because of this effect a freewheeling diode is used in any circuit containing strong inductive components. Turn on: capacitive parasitic effects Turn off: inductive parasitic effects Off On Figure 1-3: Parasitic effects at the switch t t UL UZ uSpule 100% = U/RZ iL 63% τ Sgeschlossen Soffen t1 t2 iL1 iL0 Figure 1-4: Current and voltage at a quasi-static state
The displayed characteristics in Figure 1-4 are only for steady-state condition(quasi-static state). The inductance at the load side works as an energy storage after opening the switch The current can only flow across the freewheeling diode. The negative field voltage across the inductance stays at a constant level until the magnetization of the inductance is not longer ble to supply the necessary current IL =ID. Calculation of iL: S closed R+L U R dt R L Time constant T=n Possible highest current level Solution of der homogeneous differential equation R+L 0→1R=-L R t=InIl wItn T= Special solution thru variation of constants: k
IALB – Institute of Electrical Drives, Power Electronics and Components - 3 - The displayed characteristics in Figure 1-4 are only for steady-state condition (quasi-static state). The inductance at the load side works as an energy storage after opening the switch. The current can only flow across the freewheeling diode. The negative field voltage across the inductance stays at a constant level until the magnetization of the inductance is not longer able to supply the necessary current: iL = iD. Calculation of iL: S closed: uR + uind = uL L u L U D dt di i ⋅ R + L = = (I) R u dt di R L i L L + = ; R L Time ; Possible highest current level constant τ = R u i L Lstat = Solution of der homogeneous differential equation: i RL di dt L L ⋅+ = 0 ⇒ ⋅ =− iR L di dt L L ∫ − = ∫ R L dt di i L L ln ln c R L t i 1 − = L i ce L R L t = ⋅ − 1 i ce L ogen t ,hom = ⋅ − 1 τ R L with τ = Special solution thru variation of constants: c kt 1 = ( ) i kt e L t = ⋅ − ( ) τ
IALB-Institute of Electrical Drives, Power Electronics and Components di k(t) tk Insert this expression in I provides Rk. r+L..er-k kLe+kRe-Le=U k L R R General solution iL=INhomogeneous +iL, -Ci'e+D Insert initial values i0=i2(t=0)=0=c1 CL R R Solution(for0≤t≤t1; S closed): e 0z2(1) R
IALB – Institute of Electrical Drives, Power Electronics and Components - 4 - di dt kt e ke L t t = ⋅−⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⋅ +⋅ − • − ( ) 1 τ τ τ Insert this expression in I provides: D t t t R k e L k e k e =U ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − • − − τ τ τ τ 1 D t t t e U L k L e k R e = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ + ⋅ ⋅ −⋅ ⋅ • − − − 14 244 4 344 0 τ τ τ τ τ t D e L U k = • τ τ τ t D t D e R U e L U k = ⋅ = R U e e R U i D t t D ⇒ L special = ⋅ = − τ τ , General solution: R U i i i c e D t L = L ogeneous + L special = ⋅ + − τ ,hom , 1 Insert initial values: R U c R U i i t c e D D = L = = = ⋅ + = + − 1 0 0 0 1 ( 0) τ R U c D ⇒ 1 = − Solution (for 0 ≤ ≤ 1 t t , S closed): ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⇒ = − = − − − τ τ t D t D D L e R U e R U R U i 1 for iL0 ≠ 0 0 0 ( ) 1 L t L D L i e i R U i t +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ = − − τ
B-Institute of Electrical Drives, Power Electronics and Components S opened L find= R+L-=0 dt R R L L Calculation of c respectively if the final valueD is not reached R G=L(D)=iL Solution(S opened) genela. L R Upe respectively for 4 SIS4+t2:i,(0=ine
IALB – Institute of Electrical Drives, Power Electronics and Components - 5 - S opened: UL +Uind = 0 i RL di dt L L ⋅+ = 0 iR L di dt L L ⋅ =− − = ∫ ∫ R L dt di i L L −+ = R L tc iL ln ln 1 i ce L t = − 1 τ Calculation of c1: τ 0 0 1 ( 0) − = = = = c e R U i i t D L R U c D 1 = respectively if the final value R UD is not reached: c it i 1 1 = = L L ( ) 1 Solution (S opened): general: τ t D L e R U i − = respectively for t tt t 1 12 ≤ ≤ + : i t ie L L t t ( ) = − − 1 1 τ
B-Institute of Electrical Drives, Power Electronics and Components Time average of the current i e|+ R i2(t=1+12)=io=ie ILL= ILoe R R V(1- R +ile i dt To=4+t2 R -1, ame e tirol e T-1+tiLe R 2t1 1 + iol e-e ToI R 1U , U To,+t2 R
IALB – Institute of Electrical Drives, Power Electronics and Components - 6 - Time average of the current iL : it t L ( ) = = 1 iL1 = 0 1 L0 t L D i e i R U +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ − − τ it t t L ( ) =+ = 1 2 iL0 = i e L t 1 2 − τ iL1 = i e L t 0 2 τ i e L t 0 2 τ 0 0 1 1 L t L D i e i R U +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ = − − τ τ τ 1 1 1 0 t L t D e i e R U − − +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − − − − τ τ τ 2 1 1 0 1 t D t t L e R U i e e ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − ∫ ∫ − + − − e i e dt i e dt R U T i t t t t t L t t L t D L 1 2 1 1 1 1 0 0 0 1 1 τ τ τ ; Ttt 0 12 = + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + − − − − 1 1 1 1 1 1 2 1 0 1 0 τ τ τ τ τ τ t L t L t D D e i e i e R U t R U T ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + − − − 1 1 1 1 1 1 2 1 0 0 0 τ τ τ τ τ τ t L t L t D D e i e i e R U t R U T ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + − − − τ τ τ τ τ 1 2 1 1 0 0 1 1 t t L t D D e i e e R U t R U T R u R U t t t t R U T i D D L L = + = = 1 2 1 1 0 1
B-Institute of Electrical Drives, Power Electronics and Components 2 Four-Quadrant-DC Chopper with Separated oltage ources The following diagram shows the equivalent circuit diagram of a four-quadrant-DC chopper converter D R D2本 u Figure 2-1: four-quadrant-dC chopper converter Basically there are two different cases Case 1 s1 closed, S2 open→l2 Case 2: s1open,S2 closed→u2 Case 1: S1 closed, S2 open Case 2: S1 open, S2 closed R Figure 2-2 U,+INductor 2 =ll U2+b22
IALB – Institute of Electrical Drives, Power Electronics and Components - 7 - 2 Four-Quadrant-DC Chopper with Separated Voltage Sources The following diagram shows the equivalent circuit diagram of a four-quadrant-DC chopper converter. UD UD 2 Figure 2-1: four-quadrant-d.c. chopper converter Basically there are two different cases: Case 1: S1 closed, S2 open ⇒ 2 D L U u = Case 2: S1 open, S2 closed ⇒ 2 D L U u = − Case 1: S1 closed, S2 open Case 2: S1 open, S2 closed D D Figure 2-2 L D R Inductor u U U +U = = 2 L D R Inductor u U U +U = − = 2
IALB-Institute of Electrical Drives, Power Electronics and Components i,R+L dt 2 l di, U differential equations l di, U r dt 2R r dt 2R C1 C1 2R 2R 2r 20//ex general solution lze 2R-41-e)+o Initial value(V): ILo=0 when t→∞ 2R newⅣV: 2R when t→∞ 2R UpI I When t→∞ U newⅣV: 2R Diagram 2. 3 shows the current and voltage characteristics t,: time in which S, is closed t2 time in which S2 is closed Generally the average voltage ur is as following: (1 l1+l2 where To= t1+ t2
IALB – Institute of Electrical Drives, Power Electronics and Components - 8 - 0 0 1 1 2 2 2 2 L t L D D t L L D L L D L i e i R U R U i C e R U dt di R L i U dt di i R L +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ − = + = + = + = − − τ τ differential equations general solution 0 0 1 1 2 2 2 2 L t L D D t L L D L L D L i e i R U R U i C e R U dt di R L i U dt di i R L +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ − − = − = + = − + = − − − τ τ Initial value (IV): iL0 = 0 ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − − τ t D L e R U i 1 2 new IV: R U i D L 2 0 = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⇒ = − − τ t D L e R U i 2 1 when t → ∞ when t → ∞ when t → ∞ new IV: R U i D L 2 0 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⇒ = − − − τ t D L e R U i 2 1 new IV: R U i D L 2 0 = Diagram 2.3 shows the current and voltage characteristics. t1: time in which S1 is closed t2: time in which S2 is closed Generally the average voltage u L is as following: 2 2 D L D U u U− < ≤ ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − + − = 2 1 2 2 0 1 1 2 1 2 T U t t t U t t u D D L where T0 = t1 + t2
IALB-Institute of Electrical Drives, Power Electronics and Components Figure 2-3: Current and voltage characteristic in quasi-static operations The disadvantage of four-quadrant-d C current converter is that two voltage sources are needed, though only one of them is in operation
IALB – Institute of Electrical Drives, Power Electronics and Components - 9 - D D Figure 2-3: Current and voltage characteristic in quasi-static operations The disadvantage of four-quadrant-d.c. current converter is that two voltage sources are needed, though only one of them is in operation
