PATENT ABSTRACT
A modular-based, zero-current-switched (ZCS), isolated full-bridge boost converter with multiple inputs is disclosed. Each converter module is used to match the connected input source and control the amount of power drawn from the source. The power from various sources are combined together and delivered to the load through a multiphase transformer. The input inductor of each boost-derived converter module keeps the input current constant and acts as a current source to drive the multiphase transformer through a phase-shifted-controlled full bridge converter. By connecting an auxiliary circuit across the full-bridge input in each module, the transformer leakage inductance and output capacitance of the switching devices are used to create resonant paths for facilitating zero-current-switching of all switching devices.

PATENT DESCRIPTION
FIELD 
       [0001]    The present disclosure relates to multi-input (and single-input) DC/DC converters with zero-current-switching. 
       BACKGROUND 
       [0002]    The statements in this section merely provide background information related to the present disclosure and may not constitute prior art. 
         [0003]    Various DC/DC converters are known in the art for supplying dc power to loads, including multi-input DC/DC converters that can combine energy from multiple dc power input sources. Many of these converters incorporate full bridge converters having switches. Ideally, these switches would be turned on or off under zero-current conditions (so-called zero-current-switching) to minimize switching losses. Many existing converters are, however, incapable of achieving zero-current-switching of the main switches during both switching transitions. As a result, such converters suffer from switching losses that reduce the overall efficiency of the DC/DC converter. 
       SUMMARY 
       [0004]    According to one example of the present disclosure, a multi-input dc/dc converter includes a plurality of input modules, a transformer, at least one output terminal for supplying dc power to a load, and a rectifier circuit. Each input module includes at least one input terminal for coupling to a dc power source and a full bridge converter having a plurality of main switches. The transformer has a plurality of primary windings and at least one secondary winding. Each primary winding is coupled to one of the full bridge converters. The rectifier circuit is coupled between the secondary winding of the transformer and the output terminal. Each input module further includes an auxiliary circuit coupled to the full bridge converter for achieving substantially zero-current-switching of the main switches during turn-on and turn-off of the main switches. 
         [0005]    According to another example of the present disclosure, a multi-input DC/DC converter includes a plurality of input modules, a transformer having a plurality of primary windings and at least one secondary winding, at least one output terminal for supplying dc power to a load, and a rectifier circuit coupled between the secondary winding of the transformer and said output terminal. Each input module includes at least one input terminal for coupling to a dc power source, a full bridge converter having a plurality of main switches, and means for achieving substantially zero-current-switching of the main switches during turn-on and turn-off of the main switches. Each primary winding of the transformer is coupled to one of the full bridge converters. 
         [0006]    According to a further example of the present disclosure, a DC/DC converter includes at least one input terminal for coupling to a dc power source, a full bridge converter having a plurality of main switches, a transformer having at least one primary winding coupled to the full bridge converter and at least one secondary winding, at least one output terminal for supplying dc power to a load, a rectifier circuit coupled between the secondary winding of the transformer and the output terminal, and an auxiliary circuit coupled between the input terminal and the full bridge converter for achieving substantially zero-current-switching of the main switches during turn-on and turn-off of the main switches. 
         [0007]    Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
     
    
     
       DRAWINGS 
         [0008]    The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
           [0009]      FIG. 1  is a block diagram of a multi-input DC/DC converter according to one example of the present disclosure. 
           [0010]      FIG. 2  is a block diagram of each input module shown in  FIG. 1 . 
           [0011]      FIG. 3  is a circuit diagram of the ith input module shown in  FIG. 1  according to one specific embodiment of this disclosure. 
           [0012]      FIG. 4  is a circuit diagram of a two-input DC/DC converter according to another example of the present disclosure. 
           [0013]      FIGS. 5(   a )- 5 ( l ) illustrate the twelve operating stages of each input module shown in  FIG. 1  and constructed according to the circuit diagram of  FIG. 3 . 
           [0014]      FIG. 6  is a timing diagram for each input module shown in  FIG. 1  and constructed according to the circuit diagram of  FIG. 3 . 
           [0015]      FIG. 7  illustrates the waveform of i o,i  for determining a voltage conversion ratio. 
           [0016]      FIG. 8  illustrates a control circuit for the auxiliary circuit shown in  FIG. 3  according to another example of the present disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0017]    The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
         [0018]    A multi-input DC/DC converter according to one example of the present disclosure is illustrated in  FIG. 1  and indicated generally by reference number  100 . As shown in  FIG. 1 , the converter  100  includes N input modules  102 ,  104 ,  106 ,  108 , a transformer  110 , a rectifier circuit  112 , and output terminals  114  for supplying dc power to a load (N represents a positive integer greater than 1). Each input module  102 - 108  includes input terminals for coupling to a dc power source. Additionally, each input module  102 - 108  includes output terminals coupled to a primary winding n 1 -n N  of the transformer  110 . The rectifier circuit  112  is coupled between a secondary winding n 0  of the transformer  110  and the converter output terminals  114 , as shown in  FIG. 1 . In this manner, power from various dc power sources are combined together and delivered to the load through the transformer  110 . 
         [0019]    In the example of  FIG. 1 , each input module has the same general circuit configuration, which is illustrated in  FIG. 2 . Accordingly, the example of  FIG. 1  is a modular-based multi-input DC/DC converter. As shown in  FIG. 2 , each input module includes an input choke L i , an auxiliary circuit  220 , and a full bridge converter  230 . The auxiliary circuit  220  is coupled between the input choke L i  and the full bridge converter  230 . As further explained below, the auxiliary circuit is configured for achieving substantially zero current switching (ZCS) (also referred to as “soft-switching”) of the converter&#39;s main switches during turn-on and turn-off of such switches. The auxiliary circuit essentially converts the hard-switched isolated boost converter (the main switches of the full bridge converter) into a soft-switched isolated boost converter. The auxiliary circuit does this without significantly affecting the macro-operation of the multi-input converter  100 . As a result, the converter of  FIG. 1  has reduced losses and thus improved efficiency (on the order of 6-8%) as compared to prior art multi-input DC/DC converters. 
         [0020]      FIG. 3  illustrates one specific embodiment of the auxiliary circuit and full bridge converter shown in  FIG. 2 . More specifically,  FIG. 3  illustrates the equivalent circuit for a specific embodiment of the ith input module  106  shown in  FIG. 1 . In the embodiment of  FIG. 3 , the auxiliary circuit  220  includes an auxiliary inductor L r,i , an auxiliary capacitor C r,i  and an auxiliary switch S Au,i  coupled in series. The auxiliary switch includes a body diode D Au,i  that inhibits current flow from the input choke L i  when the auxiliary switch is off. Further, the full bridge converter  230  is a phase-shift-controlled full bridge converter having four main MOSFET switches S 1,i -S 4,i  each configured for unidirectional operation. In the specific embodiment of  FIG. 3 , the main switches are configured for unidirectional operation by coupling the switch in series with blocking diodes D 1,i -D 4,i . 
         [0021]    The auxiliary circuit forms a quasi-resonant circuit—with the transformer leakage inductance and the output capacitances of main switches—that facilitates zero-current-switching of the main switches in the full bridge converter. During the stages that power is transferred to the output side of the converter, the resonant time is only three quarters of the resonant period. Therefore, the auxiliary circuit is called a quasi-resonant circuit. 
         [0022]    The low side switches S 3,i -S 4,i  of the full-bridge converter achieve zero-current switching during turn-off due to the resonance of the auxiliary capacitor and the auxiliary inductor. The current starts from zero, resonates to a peak value and returns to a level equal to the input current. The current then remains at such level until the power transferring stage ends. The low side switches also achieve zero-current switching during turn-on due to the auxiliary capacitor, the auxiliary inductor and the leakage inductance of the transformer. The high side switches S 1,i -S 2,i  of the full-bridge converter also turn-on and turn-off with zero current due to the transformer winding current reset. Accordingly, the switching losses in the full-bridge converter are quite low. 
         [0023]    Further, the auxiliary switch achieves zero-current switching during turn-on due to the resonance of the auxiliary capacitor and the auxiliary inductor. The auxiliary switch also achieves zero-voltage switching during turn-off due to the resonance of the auxiliary capacitor, the auxiliary inductor and the leakage inductance of the transformer. Thus, soft-switching of the auxiliary switch can be achieved during both switching transitions (turn-on and turn-off). Accordingly, the switching losses in the auxiliary circuit are also quite low. 
         [0024]    One example of a suitable phase-shift controller for each full-bridge converter is the UC3875 integrated circuit available from Texas Instruments Incorporated. This circuit includes a bi-directional synchronization (CLOCKSYNC) pin. This pin of several ICs (one for each full-bridge converter) can be connected together so that the gate signal will be synchronized by the IC with the fastest local oscillating frequency. In this manner, the gate signals of the high side switches are synchronized for all modules. The phase shift of the low side switches is determined by the feedback circuit of the phase-shift controller. 
         [0025]    Each auxiliary circuit can be controlled by a control circuit that operates independently in each module. In other words, the control circuits for the auxiliary circuits need not be synchronized. Each control circuit can include a current transformer for sensing the current in the auxiliary circuit. The turn-on time of the auxiliary switch is pre-set depending on the auxiliary capacitor and the auxiliary inductor values. The auxiliary switch can be turned-off any time that its body diode is conducting. 
         [0026]    Although  FIG. 1  illustrates a DC/DC converter with four input modules, it should be understood that the DC/DC converter can have more or less input modules without departing from the teachings of this disclosure. For example,  FIG. 4  illustrates a multi-input DC/DC converter having only two input modules. Further, while multi-input DC/DC converters are described above, it should be understood that the teachings of present disclosure also apply to single input DC/DC converters (having only one dc power source). For example, the circuit illustrated in  FIG. 3  can be used by itself as a single input DC/DC converter (having only one input module i). 
         [0027]    The operating principles for one specific embodiment of the converter of FIG.  1 —where each input module includes an auxiliary circuit and a full-bridge converter as shown in FIG.  3 —will now be described. Each module goes through twelve operating stages in one switching cycle. These operating stages are described further below, and are illustrated in  FIGS. 5A-5L .  FIG. 6  illustrates the timing diagram for each module. The switches S 1,i  and S 2,i  are synchronized with the other modules, while the switches S 3,i  and S 4,i  are independently controlled for determining the input-to-output voltage requirement. The auxiliary circuit formed by the switch S Au,i , resonant inductor L r,i , and resonant capacitor C r,i  is used to provide ZCS conditions for all switches. 
         [0028]    To simplify the analysis below, the following assumptions have been made: (i) the transistors and diodes are perfect; they have zero on-state resistance, infinite off-state resistance, and zero junction capacitance; (ii) all the energy storage components are free of loss without parasitic element; and (iii) the operations of the switch pair S 1,i  and S 2,i , and the switch pair S 3,i  and S 4,i , are similar; the magnitude of the output voltage is controlled by adjusting the phase shift between the two switch pairs. 
         [0029]    Stage 1 [t 0 -t 1 ] [ FIG. 5(   a )]: Transformer Current Reset Stage 
         [0030]    A closed path formed by the transformer, transformer leakage inductance L l,i , S 1,i , and S 2,i  is created. The output voltage V out  reflected to the transformer primary side is applied across L l,i  and makes the transformer primary current decrease. S 2,i  is then turned off with zero current at the end of this stage. The input inductor L i  is charged by the supply source v S,i . The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             v 
                             
                               S 
                               , 
                               i 
                             
                           
                           
                             L 
                             i 
                           
                         
                          
                         t 
                       
                       + 
                       
                         I 
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             12 
                           
                           , 
                           i 
                         
                       
                     
                   
                   , 
                   
                     
                       i 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     0 
                   
                   , 
                   
                       
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                             + 
                             
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 i 
                               
                             
                           
                           
                             
                               L 
                               i 
                             
                              
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                         
                         ) 
                       
                        
                       t 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                              
                             t 
                           
                           ) 
                         
                       
                       + 
                       
                         I 
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             12 
                           
                           , 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           3 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             v 
                             
                               S 
                               , 
                               i 
                             
                           
                           
                             L 
                             i 
                           
                         
                          
                         t 
                       
                       + 
                       
                         I 
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             12 
                           
                           , 
                           i 
                         
                       
                     
                   
                   , 
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           4 
                         
                         , 
                         i 
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       V 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           0 
                         
                         , 
                         i 
                       
                     
                   
                   , 
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           v 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               3 
                             
                             , 
                             i 
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L12,i  is the input current at the end of the last cycle, 
         [0000]    
       
         
           
             
               v 
               
                 P 
                 , 
                 i 
               
             
             = 
             
               
                 
                   n 
                   i 
                 
                 
                   n 
                   o 
                 
               
                
               
                 V 
                 out 
               
             
           
         
       
     
         [0000]    is the output voltage reflected to the primary side of the transformer, and V C0,i  is the voltage across C r,i  at the end of stage 12 and is also equal to the value at the end of stage 11. 
         [0031]    The duration of this stage is Δt 1,i =t 1 −t 0 . The gate signals that apply to the two high side switches S 1,i  and S 2,i  are overlapped. The duration of this stage is Δt 1,i =t 1 −t 0 . The gate signals that apply to the two high side switches S 1,i  and S 2,i  are overlapped 
         [0032]    2) Stage 2 [t 1 -t 2 ] [ FIG. 5(   b )]: Inductor Charging Stage 
         [0033]    L i  is continually charged by v S,i  through the switches S 1,i  and S 3,i . The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             v 
                             
                               S 
                               , 
                               i 
                             
                           
                           
                             L 
                             i 
                           
                         
                          
                         t 
                       
                       + 
                       
                         I 
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             12 
                           
                           , 
                           i 
                         
                       
                     
                   
                   , 
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               1 
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     0 
                   
                   , 
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       V 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           0 
                         
                         , 
                         i 
                       
                     
                   
                   , 
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L1,i  is the value of i S,i  at t 1 . 
         [0034]    The duration of this stage is Δt 2,i . It determines the average current through L i  and the magnitude of V out . 
         [0035]    3) Stage 3 [t 2 -t 3 ] [ FIG. 5(   c )]: Resonant Stage I 
         [0036]    S Au,i  and S 4,i  are turned on with zero current. C r,i , L r,i , S 1,i , and S 3,i  form a resonant path. The energy stored in C r,i  will generate a resonant current that will make the current through S 1,i  and S 3,i  zero. The stage ends when S 3,i  is turned off with zero current and zero voltage. i Cr,i  will reach to the value equal to i S,i . The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               , 
                               i 
                             
                           
                            
                           
                             i 
                             
                               
                                 S 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               i 
                             
                           
                         
                       
                       = 
                       
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               3 
                             
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                               
                                 L 
                                 i 
                               
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 V 
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     0 
                                   
                                   , 
                                   i 
                                 
                               
                               
                                 Z 
                                 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                   , 
                                   i 
                                 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               ω 
                               
                                 3 
                                 , 
                                 i 
                               
                             
                              
                             t 
                           
                           + 
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     i 
                     
                       Cr 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           V 
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               0 
                             
                             , 
                             i 
                           
                         
                         
                           Z 
                           
                             
                               o 
                                
                               
                                   
                               
                                
                               3 
                             
                             , 
                             i 
                           
                         
                       
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       ω 
                       
                         3 
                         , 
                         i 
                       
                     
                      
                     t 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         V 
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             0 
                           
                           , 
                           i 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         ω 
                         
                           3 
                           , 
                           i 
                         
                       
                        
                       t 
                     
                   
                   , 
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           v 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               4 
                             
                             , 
                             i 
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     where 
                      
                     
                         
                     
                      
                     
                       I 
                       
                         
                           L 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                      
                     
                         
                     
                      
                     is 
                      
                     
                         
                     
                      
                     the 
                      
                     
                         
                     
                      
                     value 
                      
                     
                         
                     
                      
                     of 
                      
                     
                         
                     
                      
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                      
                     
                         
                     
                      
                     at 
                      
                     
                         
                     
                      
                     
                       t 
                       2 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       ω 
                       
                         3 
                         , 
                         i 
                       
                     
                     = 
                     
                       1 
                       
                         
                           
                             L 
                             
                               r 
                               , 
                               i 
                             
                           
                            
                           
                             C 
                             
                               r 
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
                       and 
                        
                       
                           
                       
                        
                       
                         Z 
                         
                           
                             o 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             L 
                             
                               r 
                               , 
                               i 
                             
                           
                           
                             C 
                             
                               r 
                               , 
                               i 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0037]    The duration of this stage, Δt 3,i =t 3 −t 2 , is determined by solving (5) and (6) that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         
                           V 
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               0 
                             
                             , 
                             i 
                           
                         
                       
                       
                         Z 
                         
                           
                             o 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       ω 
                       
                         3 
                         , 
                         i 
                       
                     
                      
                     Δ 
                      
                     
                         
                     
                      
                     
                       t 
                       
                         3 
                         , 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           v 
                           
                             S 
                             , 
                             i 
                           
                         
                         
                           L 
                           i 
                         
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         
                           3 
                           , 
                           i 
                         
                       
                     
                     + 
                     
                       I 
                       
                         
                           L 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The voltage of v Cr,i  at t 3  is determined by solving (6) and (7), 
         [0000]        v   Cr,i ( t   3 )= V   C1,i   =V   C0,i  cos ω 3,i    Δt   3,i    (9) 
         [0038]    4) Stage 4 [t 3 -t 4 ] [ FIG. 5(   d )]: Capacitor Constant Current Charging Stage 
         [0039]    As the duration of this stage is short and the input current has small variation, L i  is considered as a constant current in this stage. The junction capacitance of S 3,i , C S3,i , and C r,i  undergo constant current charging. This stage ends when the two capacitor voltages increase to the level equal to the output voltage reflected to the primary side of the transformer. The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         i 
                         
                           S 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                               - 
                               
                                 V 
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                             
                             
                               Z 
                               
                                 
                                   o 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                 
                                 , 
                                 i 
                               
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               4 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 3 
                               
                               , 
                               i 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               4 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         
                           
                             
                               i 
                               
                                 Cr 
                                 , 
                                 i 
                               
                             
                             = 
                               
                              
                             
                               i 
                               
                                 SAu 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                 
                                   
                                     C 
                                     
                                       r 
                                       , 
                                       i 
                                     
                                   
                                   + 
                                   
                                     C 
                                     
                                       
                                         S 
                                          
                                         
                                             
                                         
                                          
                                         3 
                                       
                                       , 
                                       i 
                                     
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           v 
                                           
                                             S 
                                             , 
                                             i 
                                           
                                         
                                         - 
                                         
                                           V 
                                           
                                             
                                               C 
                                                
                                               
                                                   
                                               
                                                
                                               1 
                                             
                                             , 
                                             i 
                                           
                                         
                                       
                                       
                                         Z 
                                         
                                           
                                             o 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                           
                                           , 
                                           i 
                                         
                                       
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         4 
                                         , 
                                         i 
                                       
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     
                                       I 
                                       
                                         
                                           L 
                                            
                                           
                                               
                                           
                                            
                                           3 
                                         
                                         , 
                                         i 
                                       
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         4 
                                         , 
                                         i 
                                       
                                     
                                      
                                     t 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ≈ 
                                 
                                
                               
                                 i 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                             
                             , 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           
                             i 
                             
                               
                                 S 
                                  
                                 
                                     
                                 
                                  
                                 3 
                               
                               , 
                               i 
                             
                           
                           = 
                             
                            
                           
                             
                               
                                 C 
                                 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                   , 
                                   i 
                                 
                               
                               
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                 + 
                                 
                                   C 
                                   
                                     
                                       S 
                                        
                                       
                                           
                                       
                                        
                                       3 
                                     
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         v 
                                         
                                           S 
                                           , 
                                           i 
                                         
                                       
                                       - 
                                       
                                         V 
                                         
                                           
                                             C 
                                              
                                             
                                                 
                                             
                                              
                                             1 
                                           
                                           , 
                                           i 
                                         
                                       
                                     
                                     
                                       Z 
                                       
                                         
                                           o 
                                            
                                           
                                               
                                           
                                            
                                           4 
                                         
                                         , 
                                         i 
                                       
                                     
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ω 
                                     
                                       4 
                                       , 
                                       i 
                                     
                                   
                                    
                                   t 
                                 
                                 + 
                                 
                                   
                                     I 
                                     
                                       
                                         L 
                                          
                                         
                                             
                                         
                                          
                                         3 
                                       
                                       , 
                                       i 
                                     
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ω 
                                     
                                       4 
                                       , 
                                       i 
                                     
                                   
                                    
                                   t 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             ≈ 
                               
                              
                             0 
                           
                           , 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           
                             i 
                             
                               
                                 S 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               i 
                             
                           
                           = 
                           
                             i 
                             
                               
                                 S 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               , 
                               i 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             i 
                             
                               
                                 S 
                                  
                                 
                                     
                                 
                                  
                                 4 
                               
                               , 
                               i 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             v 
                             
                               Cr 
                               , 
                               i 
                             
                           
                           = 
                             
                            
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     V 
                                     
                                       
                                         C 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 
                                   4 
                                   , 
                                   i 
                                 
                               
                                
                               t 
                             
                             + 
                             
                               
                                 Z 
                                 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                   
                                   , 
                                   i 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                   , 
                                   i 
                                 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 
                                   4 
                                   , 
                                   i 
                                 
                               
                                
                               t 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             ≈ 
                               
                              
                             
                               
                                 
                                   
                                     i 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   
                                     
                                       C 
                                       
                                         r 
                                         , 
                                         i 
                                       
                                     
                                     + 
                                     
                                       C 
                                       
                                         
                                           S 
                                            
                                           
                                               
                                           
                                            
                                           3 
                                         
                                         , 
                                         i 
                                       
                                     
                                   
                                 
                                  
                                 t 
                               
                               + 
                               
                                 V 
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                             
                           
                           , 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L3,i  is the input current at t 3 , 
         [0000]    
       
         
           
             
               
                 ω 
                 
                   4 
                   , 
                   i 
                 
               
               = 
               
                 1 
                 
                   
                     
                       L 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           C 
                           
                             r 
                             , 
                             i 
                           
                         
                         + 
                         
                           C 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               3 
                             
                             , 
                             i 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
             
               
                 and 
                  
                 
                     
                 
                  
                 
                   Z 
                   
                     
                       o 
                        
                       
                           
                       
                        
                       4 
                     
                     , 
                     i 
                   
                 
               
               = 
               
                 
                   
                     
                       L 
                       i 
                     
                     
                       
                         C 
                         
                           r 
                           , 
                           i 
                         
                       
                       + 
                       
                         C 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0040]    The duration of this stage, Δt 4,i =t 4 −t 3 , is obtained by using (10) that 
         [0000]        v   P,i   −v   S,i =−( v   S,i   −V   C1,i )cos ω 4,i    Δt   4,i   +Z   o4,i    I   L3,i  sin ω 4,i    Δt   4,i    (12) 
         [0041]    5) Stage 5 [t 4 -t 5 ] [ FIG. 5(   e )]: Resonant Stage II 
         [0042]    S 4,i  starts conducting. C r,i , L r,i , S 1,i , S 4,i , the output capacitances of S 2,i  and S 3,i , C S2,i  and C S3,i , and the leakage inductance of the transformer L l,i  form a resonant circuit. S Au,i  will be turned off with zero current and zero voltage, when the diode D Au,i  conducts. This stage ends when D Au,i  turns off with zero current. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         i 
                         
                           S 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     v 
                                     
                                       P 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     L 
                                     i 
                                   
                                   + 
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   
                                     C 
                                     
                                       r 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     
                                       Φ 
                                       i 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           L 
                                           i 
                                         
                                         + 
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               5 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                   
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 L 
                                 i 
                               
                               + 
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               5 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         - 
                         
                           
                             
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                               - 
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 L 
                                 i 
                               
                               + 
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             
                               L 
                               i 
                             
                              
                             
                               I 
                               
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                 
                                 , 
                                 i 
                               
                             
                           
                           
                             
                               L 
                               i 
                             
                             + 
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         i 
                         
                           Cr 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     v 
                                     
                                       P 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                  
                                 
                                   
                                     Φ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         L 
                                         i 
                                       
                                       + 
                                       
                                         L 
                                         
                                           l 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               5 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 4 
                               
                               , 
                               i 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               5 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         
                           
                             
                                 
                             
                              
                             
                               
                                 i 
                                 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                               = 
                                 
                                
                               
                                 i 
                                 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                   
                                   , 
                                   i 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           v 
                                           
                                             P 
                                             , 
                                             i 
                                           
                                         
                                         - 
                                         
                                           v 
                                           
                                             S 
                                             , 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       C 
                                       
                                         r 
                                         , 
                                         i 
                                       
                                     
                                      
                                     
                                       L 
                                       i 
                                     
                                      
                                     
                                       L 
                                       
                                         l 
                                         , 
                                         i 
                                       
                                     
                                   
                                   
                                     
                                       ( 
                                       
                                         
                                           L 
                                           i 
                                         
                                         + 
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       
                                         
                                           C 
                                           
                                             r 
                                             , 
                                             i 
                                           
                                         
                                          
                                         
                                           
                                             Φ 
                                             i 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 L 
                                                 i 
                                               
                                               + 
                                               
                                                 L 
                                                 
                                                   l 
                                                   , 
                                                   i 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   
                                     5 
                                     , 
                                     i 
                                   
                                 
                                  
                                 t 
                               
                               - 
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               
                                 
                                   
                                     
                                       
                                         L 
                                         i 
                                       
                                        
                                       
                                         I 
                                         
                                           
                                             L 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                           
                                           , 
                                           i 
                                         
                                       
                                     
                                     
                                       
                                         L 
                                         i 
                                       
                                       + 
                                       
                                         L 
                                         
                                           l 
                                           , 
                                           i 
                                         
                                       
                                     
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ω 
                                     
                                       5 
                                       , 
                                       i 
                                     
                                   
                                    
                                   t 
                                 
                                 - 
                                 
                                   
                                     
                                       
                                         v 
                                         
                                           P 
                                           , 
                                           i 
                                         
                                       
                                       - 
                                       
                                         v 
                                         
                                           S 
                                           , 
                                           i 
                                         
                                       
                                     
                                     
                                       
                                         L 
                                         i 
                                       
                                       + 
                                       
                                         L 
                                         
                                           l 
                                           , 
                                           i 
                                         
                                       
                                     
                                   
                                    
                                   t 
                                 
                                 + 
                                 
                                   
                                     
                                       L 
                                       i 
                                     
                                      
                                     
                                       I 
                                       
                                         
                                           L 
                                            
                                           
                                               
                                           
                                            
                                           4 
                                         
                                         , 
                                         i 
                                       
                                     
                                   
                                   
                                     
                                       L 
                                       i 
                                     
                                     + 
                                     
                                       L 
                                       
                                         l 
                                         , 
                                         i 
                                       
                                     
                                   
                                 
                               
                               , 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               4 
                             
                             , 
                             i 
                           
                         
                          
                         
                           Z 
                           
                             
                               o 
                                
                               
                                   
                               
                                
                               5 
                             
                             , 
                             i 
                           
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           ω 
                           
                             5 
                             , 
                             i 
                           
                         
                          
                         t 
                       
                       - 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   v 
                                   
                                     S 
                                     , 
                                     i 
                                   
                                 
                                 - 
                                 
                                   v 
                                   
                                     P 
                                     , 
                                     i 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                           
                             
                               L 
                               i 
                             
                             + 
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         
                           ω 
                           
                             5 
                             , 
                             i 
                           
                         
                          
                         t 
                       
                       + 
                       
                         
                           
                             
                               L 
                               i 
                             
                              
                             
                               v 
                               
                                 P 
                                 , 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                              
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                           
                         
                         
                           
                             L 
                             i 
                           
                           + 
                           
                             L 
                             
                               l 
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         
                           
                               
                           
                            
                           
                             
                               v 
                               
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 , 
                                 i 
                               
                             
                             = 
                               
                              
                             
                               v 
                               
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         v 
                                         
                                           P 
                                           , 
                                           i 
                                         
                                       
                                       - 
                                       
                                         v 
                                         
                                           S 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     L 
                                     i 
                                   
                                    
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                     2 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         L 
                                         i 
                                       
                                       + 
                                       
                                         L 
                                         
                                           l 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     Φ 
                                     i 
                                   
                                 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 
                                   5 
                                   , 
                                   i 
                                 
                               
                                
                               t 
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                            
                           
                             
                               
                                 
                                   
                                     L 
                                     i 
                                   
                                    
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     I 
                                     
                                       
                                         L 
                                          
                                         
                                             
                                         
                                          
                                         4 
                                       
                                       , 
                                       i 
                                     
                                   
                                 
                                 
                                   
                                     
                                       C 
                                       
                                         r 
                                         , 
                                         i 
                                       
                                     
                                      
                                     
                                       
                                         Φ 
                                         i 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             L 
                                             i 
                                           
                                           + 
                                           
                                             L 
                                             
                                               l 
                                               , 
                                               i 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 
                                   5 
                                   , 
                                   i 
                                 
                               
                                
                               t 
                             
                             + 
                             
                               
                                 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     v 
                                     
                                       P 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     L 
                                     i 
                                   
                                 
                               
                               
                                 
                                   L 
                                   i 
                                 
                                 + 
                                 
                                   L 
                                   
                                     l 
                                     , 
                                     i 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L4,i  is the input current at t 4 , 
         [0000]    
       
         
           
             
               
                 ω 
                 
                   5 
                   , 
                   i 
                 
               
               = 
               
                 
                   
                     
                       L 
                       i 
                     
                     + 
                     
                       L 
                       
                         l 
                         , 
                         i 
                       
                     
                   
                   
                     
                       C 
                       
                         r 
                         , 
                         i 
                       
                     
                      
                     
                       Φ 
                       i 
                     
                   
                 
               
             
             , 
             
               
                 Z 
                 
                   
                     o 
                      
                     
                         
                     
                      
                     5 
                   
                   , 
                   i 
                 
               
               = 
               
                 
                   
                     Φ 
                     i 
                   
                   
                     
                       C 
                       
                         r 
                         , 
                         i 
                       
                     
                      
                     
                       ( 
                       
                         
                           L 
                           i 
                         
                         + 
                         
                           L 
                           
                             l 
                             , 
                             i 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0000]    and Φ i =L r,i  L i +L r,i  L l,i +L i  L l,j . 
         [0043]    The duration of this stage Δt 5,i  is obtained by solving i Cr,i =0in (13) that 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     
                       t 
                       
                         5 
                         , 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         t 
                         5 
                       
                       - 
                       
                         t 
                         4 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             ω 
                             
                               4 
                               , 
                               i 
                             
                           
                         
                          
                         
                           tan 
                           
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               I 
                               
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                 
                                 , 
                                 i 
                               
                             
                              
                             
                               
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                  
                                 
                                   
                                     Φ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         L 
                                         i 
                                       
                                       + 
                                       
                                         L 
                                         
                                           l 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               ( 
                               
                                 
                                   v 
                                   
                                     P 
                                     , 
                                     i 
                                   
                                 
                                 - 
                                 
                                   v 
                                   
                                     S 
                                     , 
                                     i 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                              
                             
                               C 
                               
                                 r 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                       ≈ 
                       
                         
                           
                             3 
                              
                             π 
                           
                           2 
                         
                          
                         
                           1 
                           
                             ω 
                             
                               5 
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0044]    At the end of this stage, v Cr,i =V C5,i  and is determined by using (14) and (15). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                      
                     
                       ( 
                       
                         t 
                         5 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       V 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           0 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           - 
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 4 
                               
                               , 
                               i 
                             
                           
                         
                          
                         
                           Z 
                           
                             
                               o 
                                
                               
                                   
                               
                                
                               5 
                             
                             , 
                             i 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               L 
                               i 
                             
                              
                             
                               v 
                               
                                 P 
                                 , 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                              
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                           
                         
                         
                           
                             L 
                             i 
                           
                           + 
                           
                             L 
                             
                               l 
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0045]    The maximum voltage and current rating of the main switches, v S2(max),i , v S3(max),i , i S1(max),i , and i S4(max),i , can be obtained by solving (13) and (14), respectively. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                            
                           
                             ( 
                             max 
                             ) 
                           
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                               ( 
                               max 
                               ) 
                             
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               L 
                               i 
                             
                              
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                              
                             
                               I 
                               
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                 
                                 , 
                                 i 
                               
                             
                           
                           
                             
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 
                                   Φ 
                                   i 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                       i 
                                     
                                     + 
                                     
                                       L 
                                       
                                         l 
                                         , 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                             + 
                             
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 i 
                               
                             
                           
                           
                             
                               L 
                               i 
                             
                             + 
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               1 
                                
                               
                                 ( 
                                 max 
                                 ) 
                               
                             
                             , 
                             i 
                           
                         
                         = 
                           
                          
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 ( 
                                 max 
                                 ) 
                               
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     v 
                                     
                                       P 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 i 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     L 
                                     i 
                                   
                                   + 
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   
                                     C 
                                     
                                       r 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     
                                       Φ 
                                       i 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           L 
                                           i 
                                         
                                         + 
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 
                                   v 
                                   
                                     P 
                                     , 
                                     i 
                                   
                                 
                                 - 
                                 
                                   v 
                                   
                                     S 
                                     , 
                                     i 
                                   
                                 
                               
                               
                                 
                                   L 
                                   i 
                                 
                                 + 
                                 
                                   L 
                                   
                                     l 
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             
                               ( 
                               
                                 π 
                                 
                                   2 
                                    
                                   
                                     ω 
                                     
                                       5 
                                       , 
                                       i 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 L 
                                 i 
                               
                               
                                 
                                   L 
                                   i 
                                 
                                 + 
                                 
                                   L 
                                   
                                     l 
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             
                               I 
                               
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0046]    6) Stage 6 [t 5 -t 6 ] [ FIG. 5(   f )]: Inductor Discharging Stage 
         [0047]    The energy stored in L i  releases to the output together with the input at a relatively constant current. This stage ends when S 2,i  is turned on with zero current. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             - 
                             
                               v 
                               
                                 P 
                                 , 
                                 i 
                               
                             
                           
                           
                             L 
                             i 
                           
                         
                          
                         t 
                       
                       + 
                       
                         I 
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             5 
                           
                           , 
                           i 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     0 
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                               - 
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               5 
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       V 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           0 
                         
                         , 
                         i 
                       
                     
                   
                   , 
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L5,i  is the input current at t 5 . 
         [0048]    The duration of this stage Δt 6,i  is equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     
                       t 
                       
                         6 
                         , 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         t 
                         6 
                       
                       - 
                       
                         t 
                         5 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             θ 
                             i 
                           
                            
                           
                             
                               T 
                               S 
                             
                             2 
                           
                         
                         ) 
                       
                       - 
                       
                         ( 
                         
                           
                             
                               3 
                                
                               π 
                             
                             2 
                           
                            
                           
                             1 
                             
                               ω 
                               
                                 5 
                                 , 
                                 i 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where θ i =t 6 −t 4 . 
         [0049]    As shown in  FIG. 6 , θ i  is approximately equal to the phase shift between the switch pairs of S 1,i -S 2,i  and S 3,i -S 4,i , as the duration of stage 4 is short. 
         [0050]    7) Stage 7 [t 6 -t 7 ] [ FIG. 5(   g )]: Transformer Current Reset Stage 
         [0051]    This stage is similar to stage 1. The reflected voltage of V out  on the primary side is applied across L l,I  through S 1,i  and S 2,i . This makes the transformer primary current decrease linearly. S 1,i  is turned off with zero current before the end of this stage. The input inductor L i  is charged by the supply source v S,i . The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         i 
                         
                           S 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               6 
                             
                             , 
                             i 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           Cr 
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             1 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             - 
                             
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               6 
                             
                             , 
                             i 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                             + 
                             
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 i 
                               
                             
                           
                           
                             
                               L 
                               i 
                             
                              
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                         
                          
                         t 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               6 
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       V 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           0 
                         
                         , 
                         i 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           v 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               4 
                             
                             , 
                             i 
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L6,i  is the input current at t 6 . 
         [0052]    The duration of this stage is Δt 7,i =t 7 −t 6 . The gate signals that apply to the two high side switches S 1,i  and S 2,i  are overlapped. 
         [0053]    8) Stage 8 [t 7 -t 8 ] [ FIG. 5(   h )]: Inductor Charging Stage 
         [0054]    L i  is continually charged by v S,i  through the switches S 2,i  and S 4,i . The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             v 
                             
                               S 
                               , 
                               i 
                             
                           
                           
                             L 
                             i 
                           
                         
                          
                         t 
                       
                       + 
                       
                         I 
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             7 
                           
                           , 
                           i 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               7 
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       V 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           0 
                         
                         , 
                         i 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L7,i  is the input current at t 7 . 
         [0055]    The duration of this stage is Δt 8,i . Similar to stage 2, it determines the average current through L i  and the magnitude of V out . 
         [0056]    9) Stage 9 [t 8 -t 9 ] [ FIG. 5(   i )]: Resonant Stage I 
         [0057]    S Au,i  and S 3,i  are turned on with zero current. C r,i , L r,i , S 2,i , and S 4,i  form a resonant path. The energy stored in C r,i  will generate a resonant current that will make the current through S 2,i  and S 4,i  zero. The stage ends when S 4,i  is turned off with zero current and zero voltage. i Cr,i  will reach to the value equal to i S,i . The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         i 
                         
                           S 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               8 
                             
                             , 
                             i 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             1 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               3 
                             
                             , 
                             i 
                           
                         
                         = 
                         0 
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               4 
                             
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                               
                                 L 
                                 i 
                               
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 V 
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     0 
                                   
                                   , 
                                   i 
                                 
                               
                               
                                 Z 
                                 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     9 
                                   
                                   , 
                                   i 
                                 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               ω 
                               
                                 9 
                                 , 
                                 i 
                               
                             
                              
                             t 
                           
                           + 
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 8 
                               
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             V 
                             
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 0 
                               
                               , 
                               i 
                             
                           
                           
                             Z 
                             
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 9 
                               
                               , 
                               i 
                             
                           
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         ω 
                         
                           9 
                           , 
                           i 
                         
                       
                        
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         V 
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             0 
                           
                           , 
                           i 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         ω 
                         
                           9 
                           , 
                           i 
                         
                       
                        
                       t 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           v 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               4 
                             
                             , 
                             i 
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L8,i  is the value of i S,i  at t 1 , ω 9,i =ω 3,i  and Z o9,i =Z o3,i . 
         [0058]    The duration of this stage, Δt 9,i =t 9 −t 8 , is determined by solving (26) and (27) that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         
                           V 
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               0 
                             
                             , 
                             i 
                           
                         
                       
                       
                         Z 
                         
                           
                             o 
                              
                             
                                 
                             
                              
                             9 
                           
                           , 
                           i 
                         
                       
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       ω 
                       
                         9 
                         , 
                         i 
                       
                     
                      
                     Δ 
                      
                     
                         
                     
                      
                     
                       t 
                       
                         9 
                         , 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           v 
                           
                             S 
                             , 
                             i 
                           
                         
                         
                           L 
                           i 
                         
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         
                           9 
                           , 
                           i 
                         
                       
                     
                     + 
                     
                       I 
                       
                         
                           L 
                            
                           
                               
                           
                            
                           8 
                         
                         , 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The value of Δt 9,i  is equal to Δt 3,i . 
         [0059]    10) Stage 10 [t 9 -t 10 ] [ FIG. 5(   j )]: Resonant Capacitor Constant Current Charging Stage 
         [0060]    Similar to Stage 4, the junction capacitance of S 4,i , C S4,i , and C r,i  undergo constant current charging. This stage ends when the capacitor voltages increase to the level equal to the output voltage reflected to the primary side of the transformer. The equations of this stage are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         i 
                         
                           S 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                               - 
                               
                                 V 
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                             
                             
                               Z 
                               
                                 
                                   o 
                                    
                                   
                                       
                                   
                                    
                                   10 
                                 
                                 , 
                                 i 
                               
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               10 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 9 
                               
                               , 
                               i 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               10 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         
                           
                             
                               i 
                               
                                 Cr 
                                 , 
                                 i 
                               
                             
                             = 
                               
                              
                             
                               i 
                               
                                 SAu 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                 
                                   
                                     C 
                                     
                                       r 
                                       , 
                                       i 
                                     
                                   
                                   + 
                                   
                                     C 
                                     
                                       
                                         S 
                                          
                                         
                                             
                                         
                                          
                                         4 
                                       
                                       , 
                                       i 
                                     
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           v 
                                           
                                             S 
                                             , 
                                             i 
                                           
                                         
                                         - 
                                         
                                           V 
                                           
                                             
                                               C 
                                                
                                               
                                                   
                                               
                                                
                                               1 
                                             
                                             , 
                                             i 
                                           
                                         
                                       
                                       
                                         Z 
                                         
                                           
                                             o 
                                              
                                             
                                                 
                                             
                                              
                                             10 
                                           
                                           , 
                                           i 
                                         
                                       
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         10 
                                         , 
                                         i 
                                       
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     
                                       I 
                                       
                                         
                                           L 
                                            
                                           
                                               
                                           
                                            
                                           9 
                                         
                                         , 
                                         i 
                                       
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         10 
                                         , 
                                         i 
                                       
                                     
                                      
                                     t 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ≈ 
                                 
                                
                               
                                 i 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                             
                             , 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             1 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               2 
                             
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             i 
                             
                               
                                 S 
                                  
                                 
                                     
                                 
                                  
                                 3 
                               
                               , 
                               i 
                             
                           
                           = 
                           0 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               C 
                               
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                 
                                 , 
                                 i 
                               
                             
                             
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                               + 
                               
                                 C 
                                 
                                   
                                     S 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                   
                                   , 
                                   i 
                                 
                               
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       v 
                                       
                                         S 
                                         , 
                                         i 
                                       
                                     
                                     - 
                                     
                                       V 
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                         , 
                                         i 
                                       
                                     
                                   
                                   
                                     Z 
                                     
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         10 
                                       
                                       , 
                                       i 
                                     
                                   
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   
                                     10 
                                     , 
                                     i 
                                   
                                 
                                  
                                 t 
                               
                               + 
                               
                                 
                                   I 
                                   
                                     
                                       L 
                                        
                                       
                                           
                                       
                                        
                                       9 
                                     
                                     , 
                                     i 
                                   
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   
                                     10 
                                     , 
                                     i 
                                   
                                 
                                  
                                 t 
                               
                             
                             ] 
                           
                         
                         ≈ 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             v 
                             
                               Cr 
                               , 
                               i 
                             
                           
                           = 
                             
                            
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     V 
                                     
                                       
                                         C 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 
                                   10 
                                   , 
                                   i 
                                 
                               
                                
                               t 
                             
                             + 
                             
                               
                                 Z 
                                 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     10 
                                   
                                   , 
                                   i 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     9 
                                   
                                   , 
                                   i 
                                 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 
                                   10 
                                   , 
                                   i 
                                 
                               
                                
                               t 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             ≈ 
                               
                              
                             
                               
                                 
                                   
                                     i 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   
                                     
                                       C 
                                       
                                         r 
                                         , 
                                         i 
                                       
                                     
                                     + 
                                     
                                       C 
                                       
                                         
                                           S 
                                            
                                           
                                               
                                           
                                            
                                           4 
                                         
                                         , 
                                         i 
                                       
                                     
                                   
                                 
                                  
                                 t 
                               
                               + 
                               
                                 V 
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                             
                           
                           , 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L9,i  is the input current at t 9 , ω 10,i =ω 4,i  and Z o10,i =Z o4,i . 
         [0061]    The duration of this stage, Δt 10,i =t 10 −t 9 , is obtained by solving (30) that 
         [0000]        v   P,i   −v   S,i =−( v   S,i   −V   C1,i )cos ω 10,i    Δt   10,i   +Z   o10,i    I   L9,i  sin ω 10,i  Δt 10,i    (31) 
         [0062]    11) Stage 11 [t 10 -t 11 ] [ FIG. 5(   k )]: Resonant Stage II 
         [0063]    S 3,i  starts conducting. C r,i , L r,i , S 2,i , S 3,i , the output capacitances of S 1,i  and S 4,i , C S1,i  and C S4,i , and the leakage inductance of the transformer L l,i  form a resonant circuit. S Au,i  will be turned off with zero current and zero voltage, when the diode D Au,i  conducts. This stage ends when D Au,i  turns off with zero current. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         i 
                         
                           S 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     v 
                                     
                                       P 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     L 
                                     i 
                                   
                                   + 
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   
                                     C 
                                     
                                       r 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     
                                       Φ 
                                       i 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           L 
                                           i 
                                         
                                         + 
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               11 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 I 
                                 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     10 
                                   
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 L 
                                 i 
                               
                               + 
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               11 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         - 
                         
                           
                             
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                               - 
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 L 
                                 i 
                               
                               + 
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             
                               L 
                               i 
                             
                              
                             
                               I 
                               
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   10 
                                 
                                 , 
                                 i 
                               
                             
                           
                           
                             
                               L 
                               i 
                             
                             + 
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         i 
                         
                           Cr 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     v 
                                     
                                       P 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                  
                                 
                                   
                                     Φ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         L 
                                         i 
                                       
                                       + 
                                       
                                         L 
                                         
                                           l 
                                           , 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               11 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           
                             I 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 10 
                               
                               , 
                               i 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               11 
                               , 
                               i 
                             
                           
                            
                           t 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               3 
                             
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       v 
                                       
                                         P 
                                         , 
                                         i 
                                       
                                     
                                     - 
                                     
                                       v 
                                       
                                         S 
                                         , 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                  
                                 
                                   L 
                                   i 
                                 
                                  
                                 
                                   L 
                                   
                                     l 
                                     , 
                                     i 
                                   
                                 
                               
                               
                                 
                                   ( 
                                   
                                     
                                       L 
                                       i 
                                     
                                     + 
                                     
                                       L 
                                       
                                         l 
                                         , 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   
                                     
                                       C 
                                       
                                         r 
                                         , 
                                         i 
                                       
                                     
                                      
                                     
                                       
                                         Φ 
                                         i 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             L 
                                             i 
                                           
                                           + 
                                           
                                             L 
                                             
                                               l 
                                               , 
                                               i 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               ω 
                               
                                 11 
                                 , 
                                 i 
                               
                             
                              
                             t 
                           
                           - 
                           
                             
                               
                                 
                                   L 
                                   i 
                                 
                                  
                                 
                                   I 
                                   
                                     
                                       L 
                                        
                                       
                                           
                                       
                                        
                                       10 
                                     
                                     , 
                                     i 
                                   
                                 
                               
                               
                                 
                                   L 
                                   i 
                                 
                                 + 
                                 
                                   L 
                                   
                                     l 
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               ω 
                               
                                 11 
                                 , 
                                 i 
                               
                             
                              
                             t 
                           
                           - 
                           
                             
                               
                                 
                                   v 
                                   
                                     P 
                                     , 
                                     i 
                                   
                                 
                                 - 
                                 
                                   v 
                                   
                                     S 
                                     , 
                                     i 
                                   
                                 
                               
                               
                                 
                                   L 
                                   i 
                                 
                                 + 
                                 
                                   L 
                                   
                                     l 
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 L 
                                 i 
                               
                                
                               
                                 I 
                                 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     10 
                                   
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 L 
                                 i 
                               
                               + 
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                             
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             1 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           i 
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               4 
                             
                             , 
                             i 
                           
                         
                         = 
                         0 
                       
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               10 
                             
                             , 
                             i 
                           
                         
                          
                         
                           Z 
                           
                             
                               o 
                                
                               
                                   
                               
                                
                               10 
                             
                             , 
                             i 
                           
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           ω 
                           
                             11 
                             , 
                             i 
                           
                         
                          
                         t 
                       
                       - 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   v 
                                   
                                     S 
                                     , 
                                     i 
                                   
                                 
                                 - 
                                 
                                   v 
                                   
                                     P 
                                     , 
                                     i 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                           
                             
                               L 
                               i 
                             
                             + 
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                           
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         
                           ω 
                           
                             11 
                             , 
                             t 
                           
                         
                          
                         t 
                       
                       + 
                       
                         
                           
                             
                               L 
                               i 
                             
                              
                             
                               v 
                               
                                 P 
                                 , 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                              
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                           
                         
                         
                           
                             L 
                             i 
                           
                           + 
                           
                             L 
                             
                               l 
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L10,i  is the input current at t 10 , ω 11,i =ω 5,j , Z o11,i =Z o5,i , and Φ i =L r,i  L i +L r,i  L l,i +L i  L l,i . 
         [0064]    The duration of this stage Δt 11,i =t 11−t   10  is equal to Δt 5,i . 
         [0065]    12) Stage 12 [t 11 -t 12 ] [ FIG. 5(   l )]: Inductor Discharging Stage 
         [0066]    The energy stored in L i  releases to the output together with the input at a relatively constant current. This stage ends when S 1,i  is turned on with zero current. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               v 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                             - 
                             
                               v 
                               
                                 P 
                                 , 
                                 i 
                               
                             
                           
                           
                             L 
                             i 
                           
                         
                          
                         t 
                       
                       + 
                       
                         I 
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             11 
                           
                           , 
                           i 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     0 
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 v 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                               - 
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                             
                             
                               L 
                               i 
                             
                           
                            
                           t 
                         
                         + 
                         
                           I 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               11 
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       i 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           1 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         i 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             4 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       v 
                       
                         Cr 
                         , 
                         i 
                       
                     
                     = 
                     
                       V 
                       
                         
                           C 
                            
                           
                               
                           
                            
                           0 
                         
                         , 
                         i 
                       
                     
                   
                   , 
                   
                     
                       v 
                       
                         
                           S 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         v 
                         
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                           
                           , 
                           i 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where I L11,i  is the input current at t 11 . 
         [0067]    The duration of this stage Δt 12,i =t 12 −t 11  is equal to Δt 6,i . This completes a cycle of operation. 
         [0068]    For the sake of simplicity in the following analysis of steady-state conditions, the input inductor current i S,i  is assumed to be constant throughout the whole period. This is valid because the ripple current is very small, as compared with the average of i S,i, . 
         [0069]    A. Voltage Conversion Ratio 
         [0070]    The voltage conversion ratio is derived by considering the current i o,i  on the primary side of the transformer.  FIG. 7  shows the waveform of i o,i . The electric energy is transferred from the primary side to the secondary in three main intervals in one-half of the switching period. Consider the first half cycle from t 0  to t 6 . The intervals are as follows:
   1) Stage 5—Based on (13), i o,i  can be expressed as   
 
         [0000]        i   o,i   =i   S,i   −i   S,i  cos ω 5,i   t    (36) 
         [0000]    The duration of this interval Δt 5  is shown in (15).Δ
   2) Stage 6—i o,i =i S,i . The duration of this interval Δt 6,i  is shown in (20).   3) Stage 7—i o,i  decreases linearly, as expressed in (22),   
 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     
                       o 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             v 
                             
                               P 
                               , 
                               i 
                             
                           
                           
                             L 
                             
                               l 
                               , 
                               i 
                             
                           
                         
                       
                        
                       t 
                     
                     + 
                     
                       i 
                       
                         S 
                         , 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The duration of this interval Δt 7,i  is determined by solving (37) for i o,i =0. Thus, 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     
                       t 
                       
                         7 
                         , 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         t 
                         7 
                       
                       - 
                       
                         t 
                         6 
                       
                     
                     = 
                     
                       
                         
                           L 
                           
                             l 
                             , 
                             i 
                           
                         
                          
                         
                           i 
                           
                             S 
                             , 
                             i 
                           
                         
                       
                       
                         v 
                         
                           P 
                           , 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, the area under i o,i  is equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           A 
                           
                             P 
                             , 
                             i 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               ∫ 
                               0 
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   
                                     5 
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             
                               
                                 [ 
                                 
                                   
                                     i 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     
                                       i 
                                       
                                         S 
                                         , 
                                         i 
                                       
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         5 
                                         , 
                                         i 
                                       
                                     
                                      
                                     t 
                                   
                                 
                                 ] 
                               
                                
                               
                                  
                                 t 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∫ 
                               0 
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   
                                     6 
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             
                               
                                 i 
                                 
                                   S 
                                   , 
                                   i 
                                 
                               
                                
                               
                                  
                                 t 
                               
                             
                           
                           + 
                           
                             
                               ∫ 
                               0 
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   
                                     7 
                                     , 
                                     i 
                                   
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       - 
                                       
                                         
                                           v 
                                           
                                             P 
                                             , 
                                             i 
                                           
                                         
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                       
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     i 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                  
                                 t 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               i 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   θ 
                                   i 
                                 
                                  
                                 
                                   
                                     T 
                                     S 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               i 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                              
                             
                               
                                 
                                   C 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                       
                                         r 
                                         , 
                                         i 
                                       
                                     
                                     + 
                                     
                                       L 
                                       
                                         l 
                                         , 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     i 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                     2 
                                   
                                 
                                 
                                   v 
                                   
                                     P 
                                     , 
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The average output current I o,i  of the module is equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           I 
                           
                             o 
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             2 
                             
                               T 
                               S 
                             
                           
                            
                           
                             
                               n 
                               i 
                             
                             
                               n 
                               o 
                             
                           
                            
                           
                             A 
                             P 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             i 
                             
                               S 
                               , 
                               i 
                             
                           
                            
                           
                             
                               
                                 n 
                                 i 
                               
                               
                                 n 
                                 o 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   θ 
                                   i 
                                 
                                 + 
                                 
                                   
                                     2 
                                     
                                       T 
                                       S 
                                     
                                   
                                    
                                   
                                     
                                       
                                         C 
                                         
                                           r 
                                           , 
                                           i 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             L 
                                             
                                               r 
                                               , 
                                               i 
                                             
                                           
                                           + 
                                           
                                             L 
                                             
                                               l 
                                               , 
                                               i 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       T 
                                       S 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                          
                                         
                                           i 
                                           
                                             S 
                                             , 
                                             i 
                                           
                                         
                                       
                                       
                                         
                                           
                                             n 
                                             i 
                                           
                                           
                                             n 
                                             o 
                                           
                                         
                                          
                                         
                                           V 
                                           Out 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    For N modules, the total output current I o  is equal to the sum of all module outputs. Thus, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           I 
                           o 
                         
                         = 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             I 
                             
                               o 
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               i 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                              
                             
                               
                                 
                                   n 
                                   i 
                                 
                                 
                                   n 
                                   o 
                                 
                               
                               [ 
                               
                                 
                                   θ 
                                   i 
                                 
                                 + 
                                 
                                   
                                     2 
                                     
                                       T 
                                       S 
                                     
                                   
                                    
                                   
                                     
                                       
                                         C 
                                         
                                           r 
                                           , 
                                           i 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             L 
                                             
                                               r 
                                               , 
                                               i 
                                             
                                           
                                           + 
                                           
                                             L 
                                             
                                               l 
                                               , 
                                               i 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       T 
                                       S 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                          
                                         
                                           i 
                                           
                                             S 
                                             , 
                                             i 
                                           
                                         
                                       
                                       
                                         
                                           
                                             n 
                                             i 
                                           
                                           
                                             n 
                                             o 
                                           
                                         
                                          
                                         
                                           V 
                                           Out 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By using the conservation of energy, the voltage conversion ratio can be shown to be 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             i 
                             
                               S 
                               , 
                               i 
                             
                           
                            
                           
                             V 
                             
                               S 
                               , 
                               i 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             i 
                             o 
                           
                            
                           
                             V 
                             Out 
                           
                         
                       
                     
                   
                   
                     
                       
                         ⇒ 
                           
                          
                         
                           
                             V 
                             Out 
                           
                           
                             V 
                             
                               S 
                               , 
                               i 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               n 
                               o 
                             
                             
                               n 
                               i 
                             
                           
                            
                           
                             
                               1 
                               - 
                               
                                 
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                    
                                   
                                     i 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                 
                                 
                                   
                                     T 
                                     S 
                                   
                                    
                                   
                                     v 
                                     
                                       S 
                                       , 
                                       i 
                                     
                                   
                                 
                               
                             
                             
                               
                                 θ 
                                 i 
                               
                               + 
                               
                                 
                                   2 
                                   
                                     T 
                                     S 
                                   
                                 
                                  
                                 
                                   
                                     
                                       C 
                                       
                                         r 
                                         , 
                                         i 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           L 
                                           
                                             r 
                                             , 
                                             i 
                                           
                                         
                                         + 
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, the output voltage can be controlled by adjusting the value of θ i , which is the phase shift between the two switch pairs. 
         [0074]    B. Selection of L l,i , C r,i , and L r,i    
         [0075]    1. L l,i    
         [0076]    L l,i  is used to achieve ZCS of the two high side switches. The maximum value of L l,i , L l(max),i , determines the critical ZCS condition in stages 1 or 7 at the maximum power P max,i . By using (2), it can be shown that 
         [0000]    
       
         
           
             
               
                 
                   
                     L 
                     
                       
                         l 
                          
                         
                           ( 
                           max 
                           ) 
                         
                       
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         n 
                         i 
                       
                        
                       
                         V 
                         out 
                       
                        
                       
                         v 
                         
                           s 
                           , 
                           i 
                         
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         
                           1 
                           , 
                           i 
                         
                       
                     
                     
                       
                         n 
                         o 
                       
                        
                       
                         P 
                         
                           max 
                           , 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where i S(max),i  is the maximum value of i S,i . 
         [0077]    The duration of Δt 1,i  is an adjustable variable in the circuit implementation. It is practically determined by the controller integrated circuit [10]. Thus, the output power can be increased by increasing Δt 1,i —the overlap time of the two high side switches. 
         [0078]    2. C r,i  and L r,i    
         [0079]    The minimum value of C r,i  is determined by ensuring the ZCS conditions for the two low side switches in stages 3 or 9 at the minimum power P min,i . The durations Δt 3,i  or Δt 9,i  are equal to π/2ω 3,i . Thus, by using (7), 
         [0000]      Δ t   3,i =π/2ω 3,i =π/2√{square root over ( C   r,i   L   r,i )}  (44) 
         [0000]    By assuming I L4,i =i S(min),i  and using (16) and (6), 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           - 
                           
                             
                               V 
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   0 
                                 
                                 , 
                                 i 
                               
                             
                             
                               Z 
                               
                                 
                                   o 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                                 , 
                                 i 
                               
                             
                           
                         
                         = 
                           
                          
                         
                           i 
                           
                             S 
                             , 
                             
                               ( 
                               min 
                               ) 
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                   
                     
                       
                         ⇒ 
                           
                          
                         
                           C 
                           
                             r 
                             , 
                             i 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           L 
                                           
                                             l 
                                             , 
                                             i 
                                           
                                         
                                         + 
                                         
                                           L 
                                           
                                             r 
                                             , 
                                             i 
                                           
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         L 
                                         
                                           r 
                                           , 
                                           i 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   i 
                                   
                                     
                                       S 
                                        
                                       
                                         ( 
                                         min 
                                         ) 
                                       
                                     
                                     , 
                                     i 
                                   
                                 
                               
                               
                                 v 
                                 
                                   P 
                                   , 
                                   i 
                                 
                               
                             
                             ] 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where i S(min),i  is the value of i S,i  at the minimum input power. By solving (44) and (45) for C r,i  and L r,i , it can be shown that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       C 
                       
                         r 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         
                           
                             n 
                             o 
                           
                            
                           
                             P 
                             min 
                           
                         
                         
                           
                             n 
                             i 
                           
                            
                           
                             V 
                             out 
                           
                            
                           
                             v 
                             
                               s 
                               , 
                               i 
                             
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     o 
                                   
                                    
                                   
                                     P 
                                     min 
                                   
                                 
                                 
                                   
                                     n 
                                     i 
                                   
                                    
                                   
                                     V 
                                     out 
                                   
                                    
                                   
                                     v 
                                     
                                       s 
                                       , 
                                       i 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             2 
                              
                             
                               ( 
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   
                                     3 
                                     , 
                                     i 
                                   
                                 
                                  
                                 
                                   2 
                                   π 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   
                     L 
                     
                       r 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 n 
                                 i 
                               
                                
                               
                                 V 
                                 out 
                               
                                
                               
                                 v 
                                 
                                   s 
                                   , 
                                   i 
                                 
                               
                             
                             
                               
                                 n 
                                 o 
                               
                                
                               
                                 P 
                                 min 
                               
                             
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           
                             t 
                             
                               3 
                               , 
                               i 
                             
                           
                            
                           
                             2 
                             π 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       
                         L 
                         
                           l 
                           , 
                           i 
                         
                       
                       - 
                       
                         2 
                          
                         
                           
                             
                               n 
                               i 
                             
                              
                             
                               V 
                               out 
                             
                              
                             
                               v 
                               
                                 s 
                                 , 
                                 i 
                               
                             
                           
                           
                             
                               n 
                               o 
                             
                              
                             
                               P 
                               min 
                             
                           
                         
                          
                         
                           ( 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               t 
                               
                                 3 
                                 , 
                                 i 
                               
                             
                              
                             
                               2 
                               π 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0000]    According to (45), in order to ensure a positive value of C r,i , the value of L l,i  has to satisfy the following constraint of 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       L 
                       
                         l 
                         , 
                         i 
                       
                     
                     &gt; 
                     
                       L 
                       
                         
                           l 
                            
                           
                             ( 
                             min 
                             ) 
                           
                         
                         , 
                         i 
                       
                     
                   
                   = 
                   
                     2 
                      
                     
                       
                         
                           n 
                           i 
                         
                          
                         
                           V 
                           out 
                         
                          
                         
                           v 
                           
                             s 
                             , 
                             i 
                           
                         
                       
                       
                         
                           n 
                           o 
                         
                          
                         
                           P 
                           min 
                         
                       
                     
                      
                     
                       ( 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           t 
                           
                             3 
                             , 
                             i 
                           
                         
                          
                         
                           2 
                           π 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
         [0000]    As shown in (46), the value of C r,i  increases with the increase in the value of L l,i  for a given power level. 
         [0080]    C. Selection of the Transformer Turns-Ratio n o /n i    
         [0081]    The transformer turns ratio is selected by considering the converter at the minimum power condition P min . By rearranging (45), 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     
                       min 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         v 
                         
                           S 
                           , 
                           i 
                         
                       
                        
                       
                         i 
                         
                           
                             S 
                              
                             
                               ( 
                               min 
                               ) 
                             
                           
                           , 
                           i 
                         
                       
                     
                     = 
                     
                       
                         
                           v 
                           
                             S 
                             , 
                             i 
                           
                         
                          
                         
                           v 
                           
                             P 
                             , 
                             i 
                           
                         
                       
                       
                         
                           Z 
                           
                             
                               o 
                                
                               
                                   
                               
                                
                               5 
                             
                             , 
                             i 
                           
                         
                         - 
                         
                           Z 
                           
                             
                               o 
                                
                               
                                   
                               
                                
                               3 
                             
                             , 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Based on (42), the output power decreases as θ i  increases. By substituting (49) into (42) and eliminating i S(min),i , it can be shown that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         n 
                         i 
                       
                       
                         n 
                         o 
                       
                     
                      
                     
                       [ 
                       
                         
                           θ 
                           
                             i 
                              
                             
                               ( 
                               max 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             2 
                             
                               T 
                               S 
                             
                           
                            
                           
                             
                               
                                 C 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     L 
                                     
                                       r 
                                       , 
                                       i 
                                     
                                   
                                   + 
                                   
                                     L 
                                     
                                       l 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         + 
                         
                           
                             1 
                             
                               T 
                               S 
                             
                           
                            
                           
                             ( 
                             
                               
                                 L 
                                 
                                   l 
                                   , 
                                   i 
                                 
                               
                               
                                 
                                   Z 
                                   
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       5 
                                     
                                     , 
                                     i 
                                   
                                 
                                 - 
                                 
                                   Z 
                                   
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       3 
                                     
                                     , 
                                     i 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       v 
                       
                         S 
                         , 
                         i 
                       
                     
                     
                       V 
                       Out 
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where θ i(max)  is the maximum value of θ i  and is derived as follows. 
         [0082]    θ i(max)  occurs when Δt 4,i  is close to zero. Thus, according to  FIG. 6 , 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       θ 
                       
                         i 
                          
                         
                           ( 
                           max 
                           ) 
                         
                       
                     
                      
                     
                       
                         T 
                         S 
                       
                       2 
                     
                   
                   = 
                   
                     
                       
                         T 
                         S 
                       
                       2 
                     
                     - 
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         
                           1 
                           , 
                           i 
                         
                       
                     
                     - 
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         
                           2 
                           , 
                           i 
                         
                       
                     
                     - 
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         
                           3 
                           , 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Δt 1,i , Δt 2,i , and Δt 3,i  are set by the external components in the phase-shift controller, such as UC3875. By substituting (51) into (50), the maximum turns ratio μ max  is equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           μ 
                           max 
                         
                         = 
                         
                           
                             n 
                             o 
                           
                           
                             n 
                             i 
                           
                         
                       
                        
                     
                     max 
                   
                   = 
                   
                     η 
                      
                     
                         
                     
                      
                     
                       
                         2 
                         
                           T 
                           S 
                         
                       
                        
                       
                         [ 
                         
                           
                             θ 
                             
                               max 
                               , 
                               i 
                             
                           
                           + 
                           
                             1 
                             
                               ω 
                               
                                 5 
                                 , 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               L 
                               
                                 l 
                                 , 
                                 i 
                               
                             
                             
                               2 
                               · 
                               
                                 ( 
                                 
                                   
                                     Z 
                                     
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         5 
                                       
                                       , 
                                       i 
                                     
                                   
                                   - 
                                   
                                     Z 
                                     
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         3 
                                       
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
         [0083]      FIG. 8  illustrates one example of a suitable control circuit  800  for the auxiliary circuit  220  shown in  FIG. 3 . As shown in  FIG. 8 , the control circuit  800  includes a NOR gate U 4,i  for monitoring the overlapping period of switching gate signals v GS3,i  and v GS4,i . An LC filter network R 1,i  and C 1,i , together with a positive DC voltage V DC3,i  and a comparator U 3,i , is used to determine the turn-on time of the auxiliary switch S Au,i  during the overlapping period. 
         [0084]    A current transformer T C,i  with turns-ratio 1:n is connected in series in the auxiliary circuit, where n is a large integer such as 140. The current transformer is used to step down the current flowing through the auxiliary circuit. Resistor R S,i  is a current sense resistor which is connected in series with the secondary winding of the current transformer. 
         [0085]    The turn-on signal provided by the comparator U 3,i  cannot be maintained after the overlapping period. Therefore, comparators U 1,i , U 2,i , an RS latch U 5,i , and an OR gate U 6,i  are provided to maintain the turn-on signal v GSAu,i  for the auxiliary switch S Au,i  throughout Stages 3, 4 and 5. 
         [0086]    The comparator U 1,i , which receives a positive DC voltage V DC1,i , latches the output of the RS latch U 5,i  to Logic “1” when it detects the rising voltage across the current sense resistor R S,i  during Stage 3. The OR gate U 6,i  combines the Logic “1” signals from the comparator U 3,i  and the RS latch U 5,i  to maintain the turn-on signal v GSAu,i  high. When the signal from the comparator U 3,i  changes to a Logic “0” during Stage 4, the output of the RS latch U 5,i  maintains the turn-on signal v GSAu,i  high to keep the auxiliary switch S Au,i  on. 
         [0087]    The comparator U 2,i , which receives a negative DC voltage −V DC2,i , is used to remove the gate signal v GSAu,i  when the body diode of the auxiliary switch S Au,i  conducts at Stage 5. When the comparator U 2,i  detects the negative voltage across the current sense resistor R S,i , it resets the RS-latch U 5,i  to Logic “0,” which causes the output of the OR gate to become low so that the auxiliary switch S Au,i  is turned off when its body diode is conducting.