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