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DC to DC CONVERTER CHOPPER

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DC to DC CONVERTER CHOPPER Powered By Docstoc
					              DC to DC CONVERTER
                    (CHOPPER)
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                    Chapter 3

     DC to DC CONVERTER
           (CHOPPER)
•   General
•   Buck converter
•   Boost converter
•   Buck-Boost converter
•   Switched-mode power supply
•   Bridge converter
•   Notes on electromagnetic compatibility
    (EMC) and solutions.




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  DC-DC Converter (Chopper)
• DEFINITION: Converting the unregulated
  DC input to a controlled DC output with a
  desired voltage level.

• General block diagram:



        DC supply
     (from rectifier-
      filter, battery,           DC output LOAD
      fuel cell etc.)


                             Vcontrol
                          (derived from
                         feedback circuit)




• APPLICATIONS:
    – Switched-mode power supply (SMPS), DC
      motor control, battery chargers
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                 Linear regulator
• Transistor is
  operated in linear
  (active) mode.                                              IL
                                                  + VCE −

                                                                       +

• Output voltage                     Vs
                                                              RL       Vo

                                                                       −

      Vo = I L RT

                                               MODEL OF LINEAR
• The transistor can                             REGULATOR
  be conveniently
  modelled by an                                  + VCE −
                                                              IL
  equivalent                                       RT                  +
  variable resistor,                 Vs
                                                              RL       Vo
  as shown.                                                            −



                                                EQUIVALENT
• Power loss is high                              CIRCUIT
  at high current due
  to:
    Po = I L 2 RT
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           Switching Regulator
• Power loss is zero
  (for ideal switch):                               + VCE −       IL


    – when switch is                                              RL
      open, no current            Vs                                    +

      flow in it,                                                       Vo

                                                                        −
    – when switch is                         MODEL OF LINEAR
      closed no voltage                        REGULATOR
      drop across it.                                              IL

    – Since power is a                              SWITCH
                                                                        +
      product of voltage           Vs
                                                                   RL   Vo
                                                                        −
      and current, no
      losses occurs in the
      switch.                               EQUIVALENT CIRCUIT
    – Power is 100%                    Vo

      transferred from
      source to load.                   (ON) (OFF) (ON)
                                        closed open closed

                                               DT    T

• Switching regulator                        OUTPUT VOLTAGE
  is the basis of all
  DC-DC converters
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   Buck (step-down) converter

              S                        L

                                                              +
   Vd                          D            C      RL
                                                           Vo

                                                           −

                     CIRCUIT OF BUCK CONVERTER
                          iL
             S                     + vL −
                                                          +
   Vd                     D                        RL     Vo

                                                          −


             CIRCUIT WHEN SWITCH IS CLOSED

                 S         iL

                                   +   vL −
                                                                  +
    Vd                                              RL            Vo
                           D

                                                                  −

            CIRCUIT WHEN SWITCH IS OPENED


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  Circuit operation when switch
      is turned on (closed)
• Diode is reversed                                          + vL -
  biased. Switch                              S              iL                      +
  conducts inductor                                     +
                                                                      C              Vo
                             Vd                         VD                    RL
  current                                               −                            −



• This results in                   vL

  positive inductor        Vd−Vo
  voltage, i.e:                          closed
                                                       opened
                                                                  closed
                                                                           opened



   vL = Vd − Vo                                                                     t



• It causes linear           −Vo
  increase in the               iL

  inductor current
                            iLmax
          diL
   vL = L                     IL
                            iLmin
           dt
           1
   ⇒ iL = ∫ vL dt                                 DT         T
                                                                                         t
           L
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  Operation when switch turned
          off (opened)
                                                              + vL -

                                               S
• Because of                                                  iL                         +
                                                                       C                 Vo
  inductive energy            Vd                          D                    RL
                                                                                         −
  storage, iL
  continues to flow.
                                     vL

                            Vd−Vo
• Diode is forward                                      opened              opened
  biased                                  closed                   closed

                                                                                      t


• Current now flows
  through the diode           −Vo
                                 iL
  and
                             iLmax
    vL = −Vo                   IL
                             iLmin

                                                        (1-D)T
                                                                                             t
                                                   DT         T


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    Analysis for switch closed
The inductor voltage,
vL = Vd − Vo                          vL
      diL
  =L                           Vd− Vo
       dt                                     closed
                                                                           t
  diL Vd − Vo
⇒    =
  dt     L
                                      iL
Note : since the deri -
vative of iL is a posi -                   iL max

tive constant.          I                                                  ∆iL
Therefore iL must                 L



 increase linearly.                        iL min

                                                                               t
From Figure                                            DT    T



diL ∆iL ∆iL Vd − Vo
   =   =   =
dt   ∆t DT     L
                 V −V
(∆iL )closed =  d o  ⋅ DT
                     
                         L       
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      Analysis for switch opened
For switch opened,
                                  vL

vL = −Vo                                                opened
                             Vd− Vo
      diL
  =L
       dt                                                                 t

   di     −V
⇒ L= o
   dt      L
                                   iL
Note : since the deri -
vative of iL is a nega -                iL max

tive constant, iL must                  IL
                                                                          ∆iL
decrease linearly.
                                         iL min
From Figure                                                                   t
                                                  DT        T

diL ∆iL    ∆iL     − Vo
   =    =        =
                                                       (1− D)T

dt   ∆t (1 − D)T    L

                 −V
(∆iL )opened =  o  ⋅ (1 − D)T
                   
                  L 

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        Steady-state operation
   iL
                                                      Unstable current

                                              t

   iL                                                 Decaying current


                                                  t

   iL                                                 Steady-state current


                                              t
  Steady - state operation requires that i L at the
  end of switching cycle is the same at the
  begining of the next cycle. That is the change
   of i L over one period is zero, i.e :

   (∆iL )closed + (∆iL )opened = 0
    Vd − Vo           − Vo 
             ⋅ DTs −        ⋅ (1 − D)Ts = 0
    L                 L 
   ⇒ Vo = DVd
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       Average, Maximum and
      Minimum inductor current
      iL


  Imax
 IL                                                     ∆iL

  Imin

                                                          t

Average inductor current = Average current in R L
            V
⇒ IL = IR = o
            R
Maximum current :
             ∆i  V 1 V            
I max = I L + L = o +  o (1 − D)T 
              2   R 2 L           
                 1 (1 − D) 
           = Vo  +         
                R    2 Lf 

Minimum current :
             ∆i       1 (1 − D ) 
I min = I L − L = Vo  −          
              2      R    2 Lf 
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Continuous current operation
              iL

       Imax




        Imin                                                        t
               0




                   From previous analysis,
                                ∆i       1 (1 − D) 
                   I min = I L − L = Vo  −         
                                 2       R   2 Lf 

                   For continuous operation, I min ≥ 0,

                         1 (1 − D) 
                   ⇒ Vo  −         ≥0
                         R   2 Lf 

                                  (1 − D)
                   ⇒ L ≥ Lmin =           ⋅R
                                    2f
                   This is the minimum inductor current to
                   ensure continous mode of operation.
                   Normally L is chosen be be >> Lmin



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            Output voltage ripple
                                   iL                                  imax
               iL        iR
                                                                      iL=IR
        L                                           Vo / R
                    iC        +                                        imin
                              Vo   0
                              −


                                   0



  ic = iL + iR
                                     ∆Q
  Q = CVo ⇒ ∆Q = C∆V ⇒ o ∆Vo =
                                      C
  From figure, use triangle area formula :
        1  T  ∆iL  T∆iL
  ∆Q =            =
        2  2  2      8
            T∆iL (1 − D )
  ∴ ∆Vo =         =
             8C 8 LCf 2
  So, the ripple factor,
        ∆Vo (1 − D)
  r=       =
        Vo   8 LCf 2
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  Design procedures for Buck
                 SWITCH                       L

                                           Lmin= ?
                                                                      RL
                                           L = 10Lmin
     Vd                                                               Po = ?
                f=?                 D
  (input                                              C               Io = ?
                D=?                                ripple ?
  spec.)
                TYPE ?




• Calculate D to obtain required output
  voltage.

• Select a particular switching frequency:
    – preferably >20KHz for negligible acoustic
      noise
    – higher fs results in smaller L, but higher device
      losses. Thus lowering efficiency and larger heat
      sink. Also C is reduced.
    – Possible devices: MOSFET, IGBT and BJT.
      Low power MOSFET can reach MHz range.



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  Design procedures for Buck
• Determine Lmin. Increase Lmin by about 10
  times to ensure full continuos mode.

• Calculate C for ripple factor requirement.

• Capacitor ratings:
    – must withstand peak output voltage
    – must carry required RMS current. Note RMS
      current for triangular w/f is Ip/3, where Ip is the
      peak capacitor current given by ∆iL/2


• Wire size consideration:
    – Normally rated in RMS. But iL is known as
      peak. RMS value for iL is given as:

                                              2
                         2     ∆iL 2 
       I L , RMS = I L       +       
                               3 
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    Examples of Buck converter
•   A buck converter is supplied from a 50V battery
    source. Given L=400uH, C=100uF, R=20 Ohm,
    f=20KHz and D=0.4. Calculate: (a) output voltage
    (b) maximum and minimum inductor current, (c)
    output voltage ripple.

•   A buck converter has an input voltage of 50V and
    output of 25V. The switching frequency is 10KHz.
    The power output is 125W. (a) Determine the duty
    cycle, (b) value of L to limit the peak inductor
    current to 6.25A, (c) value of capacitance to limit
    the output voltage ripple factor to 0.5%.

•   Design a buck converter such that the output
    voltage is 28V when the input is 48V. The load is
    8Ohm. Design the converter such that it will be in
    continuous current mode. The output voltage ripple
    must not be more than 0.5%. Specify the frequency
    and the values of each component. Suggest the
    power switch also.


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        Boost (step-up) converter
                    L                 D




    Vd                                         C             +
                         S
                                                   RL        Vo

                                                             −
                 CIRCUIT OF BOOST CONVERTER
         iL         L                      D
                + vL −

    Vd                                                       +
                             S                 C
                                                   RL       Vo
                                                            −


               CIRCUIT WHEN SWITCH IS CLOSED
                    L
                                       D
                + vL -
                                                             +
   Vd                                          C   RL
                         S                                   Vo

                                                             −

              CIRCUIT WHEN SWITCH IS OPENED



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  Boost analysis:switch closed
       iL
               L                         D
             + vL −
                                                              +
  Vd                                         C                vo
                      S
                                                              −




vL = Vd                            vL


    diL                      Vd

=L
    dt                                  CLOSED

   diL Vd
⇒       =
                                                                          t

   dt     L               Vd − V o


diL ∆iL ∆iL                   iL

   =      =                                                         ∆iL
dt     ∆t DT
   diL     Vd
⇒        =
   dt       L                                                        t
                                                 DT   T

                 Vd DT
(∆iL )closed   =
                   L
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                 Switch opened
                iL
                                        D
            + vL -
                                                                    +
Vd                                          C                       vo
                     S
                                                                    -




vL = Vd − Vo                          vL
    di
=L L                            Vd

    dt
         V − Vo
                                                   OPENED
   di
⇒ L= d                                                                               t
   dt        L
                             Vd − V o
diL ∆iL
   =
                                 iL

dt    ∆t                                                                     ∆iL

        ∆iL
   =
     (1 − D)T                                      ( 1-D )T

                                                DT            T                  t

  diL Vd − Vo
⇒     =
   dt       L
                (V − Vo )(1 − DT )
⇒ (∆iL )opened = d
                       L
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         Steady-state operation
         (∆iL )closed + (∆iL )opened = 0
         Vd DT (Vd − Vo )(1 − D)T
              −                   =0
           L           L
                     Vd
         ⇒ Vo =
                    1− D
• Boost converter produces output voltage
  that is greater or equal to the input
  voltage.
• Alternative explanation:
     – when switch is closed, diode is reversed. Thus
       output is isolated. The input supplies energy to
       inductor.
     – When switch is opened, the output stage
       receives energy from the input as well as from
       the inductor. Hence output is large.
     – Output voltage is maintained constant by
       virtue of large C.

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    Average, Maximum,
  Minimum inductor current
      Input power = Output power
               Vo 2
      Vd I d =
                 R
                            2
                Vd 
                         
                 (1 − D )          Vd 2
      Vd I L =               =
                     R          (1 − D) 2 R
      Average inductor current
               Vd
      IL =
           (1 − D) 2 R
      Max, min inductor current
                    ∆iL       Vd      Vd DT
      I max = I L +     =        2
                                    +
                     2    (1 − D) R    2L

                     ∆iL       Vd      Vd DT
      I min   = IL −     =        2
                                     −
                      2    (1 − D) R    2L

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                     L and C values
For continous operation,
                                          Vd
I min ≥ 0                         vL


   Vd         Vd DT
        2
            −        ≥0
(1 − D) R       2L
                                               Vd−Vo
         D(1 − D )2 TR
Lmin =                                                                imax
               2                  iL
   D(1 − D ) R
            2
=                                                                      imin
       2f
                                                                       imax

Ripple factor                     iD                                   imin


      V 
∆Q =  o  DT = C∆Vo                     Io=Vo / R

      R
      Vo DT Vo D
∆Vo =       =
       RCf    RCf                 ic

   ∆V      D
r= o =                                     ∆Q
    Vo    RCf
                                                     DT   T


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                     Examples

• The boost converter has the following
  parameters: Vd=20V, D=0.6, R=12.5ohm,
  L=65uH, C=200uF, fs=40KHz. Determine
  (a) output voltage, (b) average, maximum
  and minimum inductor current, (c) output
  voltage ripple.




• Design a boost converter to provide an
  output voltage of 36V from a 24V source.
  The load is 50W. The voltage ripple factor
  must be less than 0.5%. Specify the duty
  cycle ratio, switching frequency, inductor
  and capacitor size, and power device.


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           Buck-Boost converter

                    S
                                             D
                                                              +

      Vd                                         C
                                  L                    RL    Vo
                                                              −


                CIRCUIT OF BUCK-BOOST CONVERTER


                S                       D
                                  +                           +

      Vd                iL       vL                           Vo
                                 −                            −



               CIRCUIT WHEN SWITCH IS CLOSED

                S                       D
                                  +                           +
      Vd                iL       vL
                                                              Vo
                                 −
                                                              −

               CIRCUIT WHEN SWITCH IS OPENED



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          Buck-boost analysis
Switch closed                             Vd
            di
vL = Vd = L L                     vL

            dt
   di   V
⇒ L= d
   dt    L                                     Vd−Vo

∆iL ∆iL Vd                                                            imax
     =       =                    iL
 ∆t DT L
               V DT                                                    imin
(∆iL ) closed = d
                  L                                                    imax
                                  iD
Switch opened                                                          imin

            di
vL = Vo = L L                            Io=Vo / R
            dt
   di    V
⇒ L= o
   dt     L                       ic

∆iL   ∆iL   V
    =      = o                             ∆Q
∆t (1 − D)T L
                                                     DT   T

                      Vo (1 − D)T
(∆iL ) opened       =
                            L
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                     Output voltage
            Steady state operation :
            Vd DT Vo (1 − D )T
                 +             =0
              L         L
                        D 
            ⇒ Vo = −Vs       
                        1− D 


• NOTE: Output of a buck-boost converter
  either be higher or lower than the source
  voltage.
   – If D>0.5, output is higher
   – If D<0.5, output is lower


• Output voltage is always negative
• Note that output is never directly
• connected to load. Energy is stored in
  inductor when switch is closed and
  transferred to load when switch is opened.
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     Average inductor current
     Assuming no power loss in the converter,
     power absorbed by the load must equal
     power supplied the by source, i.e.

     Po = Ps

     Vo2
         = Vd I s
      R
     But average source current is related to
     average inductor current as :

     Is = ILD

       Vo2
     ⇒     = Vd I L D
        R
     Substituting for Vo ,

             Vo2   Po     Vd D
     ⇒ IL =      =    =
            Vd RD Vd D R (1 − D ) 2
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                L and C values
       Max and min inductor current,
                     ∆iL      Vd D        Vd DT
       I max = I L +     =            2
                                        +
                      2    R (1 − D )       2L
                     ∆iL      Vd D        Vd DT
       I min = I L −     =           2
                                        −
                      2    R (1 − D)       2L

       For continuous current,
          Vd D        Vd DT
                  2
                    +          =0
       R (1 − D )       2L
                    (1 − D ) 2 R
       ⇒ Lmin =
                        2f
       Output voltage ripple,
              Vo 
       ∆Q =   DT = C∆Vo
             R
             Vo DT Vo D
       ∆Vo =         =
               RC      RCf
          ∆Vo       D
       r=       =
           Vo     RCf
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 Control of DC-DC converter
using pulse width modulation-
            PWM

Vo (desired)
                   +      Vcontrol                   Switch control
                                                         signal
Vo (actual)                            Comparator
                   -
                       Sawtooth
                       Waveform                          Sawtooth
                                                         Waveform


                                                             Vcontrol 1
                                                             Vcontrol 2




                                                            Switch
                                                            control
       ton 2                                                signal


       ton 1
               T




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  Switch-mode power supply
          (SMPS)
• Advantages over linear power
  -Efficient (70-95%)
  -Weight and size reduction

• Disadvantages
  -Complex design
  -EMI problems

• However above certain ratings,
  SMPS is the only feasible choice

• Types of SMPS
  -Flyback
  -forward
  -Push-pull
  -Bridge (half and full)

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 Linear and switched mode
power supplies block diagram
  Basic Block diagram of linear power supply
                                           C             E
                                      Vce=Vd-Vo                    +Vo

                  Rectifier
                                                     B                     +
                                 +
                                            Base/gate                 RL
                                 Vd           Drive                        Vo
    Line
   Input                                                                   -
   1φ / 3φ 50/60 Hz               -
                                                 Error           Vo
            Isolation
                                                 Amp.
          Transformer

                                                                 Vref




  Basic Block diagram of SMPS
                                      DC-DC CONVERSITION + ISOLATION
                                                                                   DC
                                                                    High        Regulated
               RECTIFIER                                                               Vo
     EMI                                                         Frequency
                 AND
   FILTER                                                         rectifier
                FILTER
                                                                    and
                            DC                                      filter
                        Unregulated                                                  Vref


                                         Base/
                                                              PWM                  error
                                         gate
                                                             Controller            Amp
                                         drive


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   High frequency transformer
   Basic function :

   i) Input - output electrical isolation

   ii) step up/down time - varying voltage
   Basic input - output relationship
   v1 N1                               i1 N 2
     =    ;                              =
   v2 N 2                              i2 N1

   Models :
                i1   N1    N2     i2
           +                             +
          V1                            V2           Ideal model
          −                             −

               i1    N1   N2     i2
          +                             +
               Lm                                Model used for
          V1                            V2
          −
                                                most PE application
                                        −




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                 Flyback Converter
                                                                                       +
                         LM                                           C          R     Vo
     Vs                                                                                −

                                                            Flyback converter circuit
                                                                 iD
                   iS                        N1        N2
                                        i1
                                                        - +v -                    +
                                             +              D
                                  iLM                   v2                iR      Vo
                                             v1            iC
     Vs                                                 +                         −
                                             -
                + vSW −                                       i2
                                                            Model with magnetising
                                                                  inductance
                                        0         N1   N2
             is=iLM
                                                                                  +
                                             +
                                 iLM         v1                                   Vo
     Vs
                                              -                                   −
                                            v1=Vs
                                                                 0 Switch closed
                                                                  iD
                                             N1        N2
                                            +           −                        +
                                 iLM        v1         v2= -VS                   Vo
     Vs                                     −           +                        −
                + vSW −                         N 
                                        v1 − Vo  1 
                                                N              Voltage and current
                                                 2
                                                                 conditions when switch
                        N 
          Vsw = Vs + Vo  1 
                        N                                      opened
                         2

                                Power Electronics and                                       33
                                Drives (Version 2): Dr.
                                    Zainal Salam
www.sayedsaad.com                                                    www.tkne.net


                 Flyback waveforms
                 v1
        Vs




                      DT      T
                                          -V(N1/N2)       t
            is

                                                              ∆iLM


                      DT      T                           t

        iD


                      DT      T                           t


        iLm



                                                      t
       iC             DT      T




                      DT      T
      Vo/ R                                           t




                           Power Electronics and                               34
                           Drives (Version 2): Dr.
                               Zainal Salam
www.sayedsaad.com                                www.tkne.net


    Analysis: switched closed
                  diLm
     v1 = Vd = Lm
                   dt
     diLm ∆iLm ∆iLm Vd
         =    =    =
      dt   dt   DT   Lm

         (
     ⇒ ∆iLm         )closed = VdLDT
                                    m

     On the load side of the transformer,

              N2        N2 
     v2 = v1      = Vd     
              N1        N1 

                   N 
     vD = −Vo − Vd  2  < 0
                    N1 
     Therefore,

     i1 = 0
     i2 = 0
                       Power Electronics and               35
                       Drives (Version 2): Dr.
                           Zainal Salam
www.sayedsaad.com                                              www.tkne.net


     Analysis: switch opened
                   N 
          v1 = −V0  1 ;
                   N                     v2 = −V0
                    2
                    N         N 
          ⇒ v1 = v2  1  = −V0  1 
                    N         N 
                     2         2
                 diL m             N 
           Lm           = v1 = −V0  1 
                                   N 
                     dt             2

           diL m         ∆iL m        ∆iL m          − V0 N1
                     =           =               =
            dt            dt         (1 − D )T       Lm N 2

                              V (1 − D)T  N1 
          ⇒ (∆iL m )open = − 0           
                                         N  
                                  Lm      2
          For steady - state operation,
           (∆iL )closed + (∆iL )opened = 0
                 m                     m


           V DT V0 (1 − D )T  N1 
          ⇒ d  +             
                             N =0
                                  
            Lm       Lm       2

                               D  N1 
          ⇒ V0 = Vd                      
                            (1 − D)  N 2 
                                         


                           Power Electronics and                         36
                           Drives (Version 2): Dr.
                               Zainal Salam
www.sayedsaad.com                             www.tkne.net


                Output voltage
 • Input output relationship is similar to buck-
   boost converter.

 • Output can be greater of less than
   input,depending upon D.

 • Additional term, i.e. transformer ratio is
   present.




                    Power Electronics and               37
                    Drives (Version 2): Dr.
                        Zainal Salam
www.sayedsaad.com                                www.tkne.net


       Average inductor current
Ps = P0
         V0 2
Vd I s =
          R

I s is related to I Lm as :

       I Lm DT
Is =
          T
                     ( )
                    = I Lm D

Substitute and solving for I Lm
            V02
   ( )
Vd I Lm D =
              R
           V0 2
⇒ I Lm =
         Vd DR

The average inductor current is also written as :
                        2
          Vd D  N 2         V0  N 2 
I Lm =                =               
              2
       (1 − D) R  N1    (1 − D) R  N1 

                       Power Electronics and               38
                       Drives (Version 2): Dr.
                           Zainal Salam
  www.sayedsaad.com                                        www.tkne.net



       Max, Min inductor current,
            Lmin, C values
                                                       2
                         ∆iLm            N 2  V d DT
                                         Vd D
I Lm          = I Lm   +   =         2
                                              +
       ,max              2   (1 − D ) R  N1     2 Lm
                                                       2
                        ∆iLm             N 2  Vd DT
                                         Vd D
I Lm ,min = I Lm       −   =         2
                                              −
                         2   (1 − D ) R  N1     2 Lm

For continuos operation, I Lm , min = 0

                         2
   Vd D     N 2  Vd DT Vd D
        2
                 =      =
(1 − D ) R  N1     2 Lm   2 Lm f

                                2              2
                Vd (1 − D) R  N1 
(Lm )min =                       
                      2f      N2 
The ripple calculation is similar to boost converter,

   ∆V0    D
r=     =
   V0    RCf
                             Power Electronics and                   39
                             Drives (Version 2): Dr.
                                 Zainal Salam
www.sayedsaad.com                                               www.tkne.net


             Full-bridge converter

                SW1       SW3
                                                   Lx
                           +                  +                 +
                                        NS   vx                 Vo
                                                        C   R
                                               −                −
                          vp
VS
                                        NS
                           −

                SW4       SW2


SW1,SW2


                DT                  T
SW3,SW4


                      T    T
                             + DT
          VP          2    2
          VS




         -VS
        Vx
        N 
     VS  S 
        N 
         P


                DT    T    T        T
                             + DT
                      2    2
                           Power Electronics and                          40
                           Drives (Version 2): Dr.
                               Zainal Salam
www.sayedsaad.com                             www.tkne.net


    Full bridge: basic operation
• Switch “pair”: [S1 & S2];[S3 & S4].

• Each switch pair turn on at a time as
  shown. The other pair is off.

• “AC voltage” is developed across the
  primary. Then transferred to secondary via
  high frequency transformers.

• On secondary side, diode pair is “high
  frequency full wave rectification”.

• The choke (L) and © acts like the “buck
  converter” circuit.
                           Ns 
• Output Voltage Vo = 2Vs 
                          N 
                               ⋅ D
                           p

                    Power Electronics and               41
                    Drives (Version 2): Dr.
                        Zainal Salam

				
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