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					Ch8 Inverters (converting DC to AC)

8-1 Introduction

․Converting DC to AC
․Applications: adjustable-speed AC motor drives.
             Uninterruptible power supply (UPS).
             AC appliances run from an automobile battery.
  8-2 The full-bridge converter               Fig 8-1 :

S1 and S4 should not be closed at the same time, nor should S2 and S3, otherwise,
a short circuit would exist across the dc source
8-3 The square-wave Inverter                              Fig 8-2. (Waveform)
 An inductive load presents some considerations in designing the switches in
full – bridge circuit because the switch current must be bidirectional.
        io t   i f t   in t 
            Vdc     t           T
                Ae  , 0  t 
             R                   2
                           T
                           t 
          Vdc            2             T
                Be                
                                       ,     t T
           R                               2

    i f t  : forced current                          LR
   i n t  : natural current
                               io o  
                                       Vdc
In steady – state.                          Aeo  Im in
                                        R
                                   T   Vdc
                               io           Beo  Im ax
                                  2     R
                                              Vdc
                                A  Im in 
                                               R
                                  B  Im ax  Vdc
                                                    R
                                Vdc            Vdc   t 
                                 R     I min     e , 0  t  T 2
                                                 R 
                      io t                                 T
                                                              t  
                                  Vdc            Vdc   2         T
                                 R       Im ax      e            ,   t T
                                                   R                 2
                                                                                   T      
                               T   Vdc           Vdc                              2 
                            io   
   By symmetry,Imax=-Imin =  2          Im in      e                                
                                      R             R 

                                                   T    
                                         Vdc 1  e 2  
                      Im ax   Im in 
                                          R        T
                                                       2 
                                             1  e
                                                         
                                                          
rms load current :
                                                                               2
                                         2 Vdc 
                                            T
                                                           Vdc               
                           io t dt 
               1       T                                             t
      Irms                               R 
                                                 Im in      e         
                            2               2
                                         T                 R                 dt
               T   0                       0
                                                                              
      PL  I 2 rms R                            R  load   resis tan ce
      Pdc  Vdc Is .


  If the switches are ideal , then Pdc  PL
                   .
Fig 8-3   Full – bridge inverter using BJTs
8-4 Fourier Series Analysis

 With no dc component in the output
              
 v0 t    Vn sinnwo t   n 
             n 1
             
 io t    I n sinnwo t   n                                                Vn
             n 1                                                           In 
                                        
                                                                                 Zn
                                                In
 Irms            In 1
                           2
                           n , rms      (
                                         n 1    2
                                                     )2                     Z n : load impedance at
                                                                               harmonic n.
 P   Pn   I n ,rms R
                2

      n 1                 n 1


                                                                   
                                                     vo t                    sin nwot 
                                                                           4Vdc
   In square wave output                                          
                                                                 n  odd    n
8-5 Total harmonic distortion (THD)

     A quality of the AC output voltage or current.
     Assuming no dc component in the output


                  V                
                  
                                     2

                 n2
                           n , rms
                                             Vrms  V12rms
                                               2
                                                      ,
     THDV                                                  .
                      V1, rms                  V1, rms
     THDI  ?
8-7 Amplitude and harmonic control

By adjusting the interval  , the output voltage can be controlled.
 Fig.8-4
                1                         2
                      Vdc2 d wt   Vdc 1 
                 
Vrms 
                                              
              
vo t      V       n   sinnwo t 
            n  odd




∵Half – wave symmetry
                                                 
                                                       Vdc sin nwo t d wo t  
                                        2                                            4Vdc
                                 Vn 
                                                                                   n
                                                                                          cos n

                                        4Vdc
                                 V1              cos .
                                            


    Harmonic content can also be controlled by adjusting α
    Harmonic n is eliminated if

                      n  90                             90 n
 Amplitude control and harmonic reduction may not be compatible. To control
both amplitude and harmonic, it is necessary to have control over the dc input
voltage.


    Fig 8-5
    A graphical representation of the integration in the Fourier series coefficient.
8-8 The half – bridge Inverter   Fig 8-7
A square-wave output or bipolar pulse-width-modulated (PWM) output.
             Vdc      Vdc
      vo        or 
              2        2


The voltage across an open switch is twice the load voltage

       Vdc 
      2      Vdc
       2 
8-9 pulse-width-modulated (PWM) output

Advantage:Reduced filter requirements to decrease harmonics and the
            control of the output voltage amplitude can be realized
Disadvantage:more complex control circuits for switches and increased losses
               due to more frequent switching.


Sinusoidal PWM requires:
(1). a reference (modulating or control) signal-sinusoidal.
(2). a carrier (triangular wave) signal that controls the switching fre.


Bipolar switching: Fig 8-8           vo    Vdc 

                              vo  Vdc, v sin e  vtri
                              vo  Vdc, v sin e  vtri

S1 and S2 are on when Vsine>Vtri
S3 and S4 are on when Vsine < Vtri
Unipolar Switching: One (First)          Fig 8-9

 vo    Vdc , 0,  Vdc 

        S1 is on when Vsine>Vtri
        S2 is on when -Vsine<Vtri
        S3 is on when -Vsine>Vtri
        S4 is on when Vsine<Vtri



Another (second)                    Fig 8-10

               S1 is on when Vsine>Vtri (high fre)
               S4 is on when Vsine<Vtri (high fre)
                S2 is on when Vsine> 0 (low fre)
                S3 is on when Vsine< 0 (low fre)
8-10 PWM definitions and considerations.

                                              f carrier    f
 (1) Frequency modulation ratio: m f                    tri
                                            f reference f sin e
    The Fourier series of the PWM output voltage has a fundamental fre.
    which is the same as that of the reference signal. Harmonic frequencies
    exist at and around multiple of the switching fre. A simple low-pass filter
    can be effective in removing those (harmonics). Higher losses in switches
                                          Vm , reference Vm , sin e
(2) Amplitude modulation ratio: ma                     
                                          Vm , carrior    Vm , tri
   If ma  1 , then V1  maVdc , (linearly).

   If ma  1 , the amplitude of the output increases with m a     ,
   but not linearly.
(3). Swithes:carrying current in either direction. Feedback.diode allowing
               for switching times in the control.
(4). Reference voltage:sinusoidal, non-sinusoidal
8-11 PWM harmonics
Bipolar switching:(Fig 8-8)

 If mf =odd integer, the PWM output then exhibits odd symmetry
           
  v o t    Vn sinnwo t 
          n 1



   For the k-th pulse of the PWM output. (Fig 8-11)

          2  k k                             k 1
                     Vdc sin nwot d wo t           Vdcsin nwot d wot 
           k
   Vnk 
                                                k k                           
                                                                                  
     2Vdc
          [cosn k  cos n k 1  2 cos n( k   k )]
       n
         4 T
   Vn   2 vo t sin nwot d wo t ,         ,       T  2
        T   0

           p
         Vnk                  p :p     pulse     over     T/2.
          k 1
Normalized frequency spectrum:                 Fig 8-12

Normalized Fourier coefficients :  n V 
                                    V
                                               Table 8-3
                                     dc   
  Unipolar switching:(Fig 8-9)
    If mf =even integer, some harmonics that were in the spectrum for the
bipolar scheme are absent.
    (seeing Fig 8-13 and Table 8-5)
8-13 Three-phase inverters
Six-step Inverter:Fig 8-17
   v AB , v BC , vCA   為  Vdc, 0, or  Vdc

   v AB  v A0  v B 0 ,
   v BC  v B 0  vC 0 ,
   vCA  vC 0  v A0

 Because of the six steps in the output waveform for the line-to-neutral
voltage resulting from the six switching transitions per period.
   v A0  v AN  vN 0  (1)
   vB 0  vBN  vN 0  (2)
                                            v A0 , v B 0 , vC 0   為+Vdc or 0
   vC 0  vCN  v N 0  (3)

                            v Ao  v Bo  vCo 
                            1
   (1)+(2)+(3)     v No
                            3
  ( v AN  v BN  vCN  0          )
                       v Ao  vBo  vCo 
                       2      1
 v AN  v Ao  v No
                       3      3
                       vBo  vCo  v Ao 
                       2      1
 vBN  vBo  v No
                       3      3
                       vCo  v Ao  vBo 
                       2      1
 vCN  vCo  v No
                       3      3
              4Vdc   
Vn ,L  L        cos n 
               n     6
              2Vdc                  2  
Vn ,L  N           2  cos n   cos n 
               3n 
                            3        3 


Where n=1, 6k  1 ,            k=1、2…
                     
(fundamental)         (harmonics)
   THD V  31 %       for line-to-line and line-to-neutral voltages.

   THD I   is load dependent and is smaller for a R-L load.

Output fre. can be controlled by changing the switching fre..
Output voltage can be controlled by adjusting the DC input voltage.



PWM three-phase Inverters            Fig 8-18

 S1 is on when    v A ,ref  vtri      S2 is on when     vC ,ref  vtri

S3 is on when     v B ,ref  vtri      S4 is on when     v A ,ref  vtri

S5 is on when     vC ,ref  vtri        S6 is on when     v B ,ref  vtri

   Harmonics will be minimized if the carrier fre. is chosed to be
 an odd triple multiple of the reference fre. (that is 3,9,15,…times the reference).
For line-to-line voltage,     Vn 3    An 3  Bn 3
                                        2      2


                                              n      n    
                               An 3  Vn sin         sin    
          ….. 三相                              2       3     
                                              n      n    
                               Bn 3  Vn sin         sin    
                                              2       3     

      P
Vn   Vnk            (參考P.313) …… 單相
     k 1




Table 8-8 Significant amplitude coefficients
8-15 Induction motor speed control

                                       2
      Synchronous speed: S 
                                        P
                                 S  r
               S1ip:       S
                                   S

            ω:electric fre.
            P:number of poles
            ωr:rotor speed

  If the applied electrical fre. is changed, the motor speed will change
proportionally.


  To avoid the magnetic flux in the air gap saturated, V  constant should be
held . Fig 8-19                                        f
The six-step inverter can be used for this application if the dc input is adjustable
      8-20
      Fig




 If the DC source is not controllable, a DC-DC converter may be inserted
between the DC source and the inverter. The PWM inverter is also another
selection for this application.

				
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