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DC to AC Conversion inverter

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					                    DC to AC Conversion (inverter)


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             Chapter 4
        DC to AC Conversion
           (INVERTER)
• General concept
• Basic principles/concepts
• Single-phase inverter
   – Square wave
   – Notching
   – PWM
• Harmonics
• Modulation
• Three-phase inverter




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           DC to AC Converter
                (Inverter)
• DEFINITION: Converts DC to AC power
  by switching the DC input voltage (or
  current) in a pre-determined sequence so as
  to generate AC voltage (or current) output.

• TYPICAL APPLICATIONS:
    – Un-interruptible power supply (UPS), Industrial
      (induction motor) drives, Traction, HVDC


• General block diagram

            IDC                               Iac


      +                                              +

      VDC                                           Vac
                                                    −
       −




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                   Types of inverter
• Voltage Source Inverter (VSI)
• Current Source Inverter (CSI)



            "DC LINK"                          Iac

           +                                                +
                     C
       VDC                                             Load Voltage
           −                                                −




               L                               ILOAD

       +       IDC                           Load Current
      VDC
       −




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    Voltage source inverter (VSI)
       with variable DC link
                                DC LINK

    +                                                                +
                                       +
    Vs                          C      Vin                           Vo
                                        -
    -                                                                -


              CHOPPER                          INVERTER
         (Variable DC output)           (Switch are turned ON/OFF
                                        with square-wave patterns)




•    DC link voltage is varied by a DC-to DC converter
     or controlled rectifier.

•    Generate “square wave” output voltage.

•    Output voltage amplitude is varied as DC link is
     varied.

•     Frequency of output voltage is varied by changing
     the frequency of the square wave pulses.
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  Variable DC link inverter (2)
• Advantages:
   – simple waveform generation
   – Reliable

• Disadvantages:
   – Extra conversion stage
   – Poor harmonics

        Vdc2                                   Higher input voltage
                                               Higher frequency

       Vdc1                                         Lower input voltage
                                                    Lower frequency
                            T1            T2
                                                                      t




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      VSI with fixed DC link
                             INVERTER


        +
                                                    +
        Vin
                     C                              Vo
       (fixed)
         −                                          −


                         Switch turned ON and OFF
                             with PWM pattern




• DC voltage is held constant.

• Output voltage amplitude and frequency
  are varied simultaneously using PWM
  technique.

• Good harmonic control, but at the expense
  of complex waveform generation


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 Operation of simple square-
     wave inverter (1)
• To illustrate the concept of AC waveform
  generation
                              SQUARE-WAVE
                                INVERTERS

                     T1                             T3
                                   D1                        D3

                                         + VO -
    VDC
                                    IO
                     T4                             T2
                                   D2                        D4




                            S1               S3


                                                    EQUAVALENT
                                                      CIRCUIT
                            S4               S2




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      Operation of simple square-
          wave inverter (2)
              S1,S2 ON; S3,S4 OFF              for t1 < t < t2
                                        vO



               S1            S3     VDC


VDC
                                                                                t
                    + vO −                      t1         t2
               S4                  S2




             S3,S4 ON ; S1,S2 OFF              for t2 < t < t3
                                          vO



               S1                  S3


VDC                                                        t2       t3
                    + vO −                                                      t

               S4            S2
                                    -VDC




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Waveforms and harmonics of
   square-wave inverter
                                    INVERTER
                                     OUTPUT
     Vdc



    -Vdc

                                FUNDAMENTAL
     V1
                    4VDC
                     π




                                3RD HARMONIC
     V1
     3




                                5RD HARMONIC
     V1
     5




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                               Filtering
• Output of the inverter is “chopped AC
  voltage with zero DC component”.In some
  applications such as UPS, “high purity” sine
  wave output is required.

• An LC section low-pass filter is normally
  fitted at the inverter output to reduce the
  high frequency harmonics.

• In some applications such as AC motor
  drive, filtering is not required.

                  (LOW PASS) FILTER
                                L
                         +                          +
                                      C
                        vO 1                   vO 2      LOAD
                         −
                                                    −

                 vO 1                        vO 2




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    Notes on low-pass filters
•   In square wave inverters, maximum output voltage
    is achievable. However there in NO control in
    harmonics and output voltage magnitude.

•   The harmonics are always at three, five, seven etc
    times the fundamental frequency.

•   Hence the cut-off frequency of the low pass filter is
    somewhat fixed. The filter size is dictated by the
    VA ratings of the inverter.

•   To reduce filter size, the PWM switching scheme
    can be utilised.

•    In this technique, the harmonics are “pushed” to
    higher frequencies. Thus the cut-off frequency of
    the filter is increased. Hence the filter components
    (I.e. L and C) sizes are reduced.

•   The trade off for this flexibility is complexity in
    the switching waveforms.
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    “Notching”of square wave
                     Notched Square Wave
          Vdc




         −Vdc


                     Fundamental Component



          Vdc




         −Vdc




• Notching results in controllable output
  voltage magnitude (compare Figures
  above).

• Limited degree of harmonics control is
  possible
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      Pulse-width modulation
             (PWM)
• A better square wave notching is shown
  below - this is known as PWM technique.

• Both amplitude and frequency can be
  controlled independently. Very flexible.




 1
                                                pwm waveform




                                                  desired
 1                                                sinusoid

         SINUSOIDAL PULSE-WITDH MODULATED
             APPROXIMATION TO SINE WAVE




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     PWM- output voltage and
       frequency control




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      Output voltage harmonics
•     Why need to consider harmonics?
       – Waveform quality must match TNB supply.
          “Power Quality” issue.
       – Harmonics may cause degradation of
          equipment. Equipment need to be “de-rated”.
•     Total Harmonic Distortion (THD) is a measure to
      determine the “quality” of a given waveform.

•     DEFINITION of THD (voltage)
                   ∞                    ∞
                   ∑ (Vn, RMS )2
                                       ∑ (VRMS )2 − (V1, RMS )2
       THDv = n = 2                = n=2
                 V1, RMS                           V1, RMS

       where n is the harmonics number.
       Current THD can be obtained by replacing
       the harmonic voltage with harmonic current :
                   ∞
                  ∑ (I n, RMS )2
       THDi = n = 2
                     I1, RMS
             V
       In = n
             Zn
       Z n is the impedance at harmonic frequency.
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                Fourier Series
• Study of harmonics requires understanding
  of wave shapes. Fourier Series is a tool to
  analyse wave shapes.

   Fourier Series

       1 2π
   ao = ∫ f (v)dθ
       π 0
       1 2π
   an = ∫ f (v) cos(nθ )dθ
       π 0
       1 2π
   bn = ∫ f (v) sin (nθ )dθ
       π 0

   Inverse Fourier
                 ∞
          1
   f (v) = ao + ∑ (an cos nθ + bn sin nθ )
          2     n =1
   where θ = ωt
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Harmonics of square-wave (1)

  Vdc




                                                  θ=ωt
                    π                  2π



  -Vdc




     1 π            2π        
ao =  ∫ Vdc dθ + ∫ − Vdc dθ  = 0
     π 0            π         
     Vdc π                2π             
an =      ∫ cos(nθ )dθ − ∫ cos(nθ )dθ  = 0
      π 0                  π             
     Vdc π               2π             
bn =      ∫ sin (nθ )dθ − ∫ sin (nθ )dθ 
      π 0                 π             


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Harmonics of square wave (2)

 Solving,

 bn =
      Vdc
      nπ
              [        π
            − cos(nθ ) 0 + cos(nθ ) π
                                     2π
                                              ]
   Vdc
 =     [(cos 0 − cos nπ ) + (cos 2nπ − cos nπ )]
   nπ
   Vdc
 =     [(1 − cos nπ ) + (1 − cos nπ )]
   nπ
   2Vdc
 =       [(1 − cos nπ )]
    nπ
 when n is even, cos nπ = 1
 bn= 0

 when n is odd, cos nπ = −1
     4Vdc
 bn=
      nπ



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        Spectra of square wave
                 Normalised
                Fundamental
                1st




                    3rd (0.33)

                            5th (0.2)
                                     7th (0.14)
                                           9th (0.11)
                                                  11th (0.09)
                1       3        5        7    9     11
                                      n

• Spectra (harmonics) characteristics:
     – Harmonic decreases as n increases. It decreases
       with a factor of (1/n).
     – Even harmonics are absent
     – Nearest harmonics is the 3rd. If fundamental is
       50Hz, then nearest harmonic is 150Hz.
     – Due to the small separation between the
       fundamental an harmonics, output low-pass
       filter design can be quite difficult.
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  Quasi-square wave (QSW)
          Vdc
                α               α        α



                                     π                  2π


         -Vdc



 Note that an = 0.

 Due to half - wave symmetry,

        1 π −α              2Vdc
bn = 2  ∫ Vdc sin (nθ )dθ  =     − cos nθ α [
                                            π −α
                                                             ]
       π α                  nπ
  2Vdc
=       [cos(nα ) − cos n(π − α )]
   nπ
 Expanding,

cos n(π − α ) = cos(nπ − nα )
= cos nπ cos nα + sin nπ sin nα
= cos nπ cos nα


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          Harmonics control

        2Vdc
  ⇒ bn =     [cos(nα ) − cos nπ cos nα ]
         nπ
   2V
  = dc cos(nα )[1 − cos nπ ]
    nπ
  If n is even, ⇒ bn = 0,

                         4Vdc
  If n is odd, ⇒ bn =         cos(nα )
                          nπ
  In particular, amplitude of the fundamental is :
       4V
  b1 = dc cos(α )
        π
  The fundamental , b1, is controlled by varying α

  Harmonics can also be controlled by adjusting α ,
  For example if α = 30o , then b3 = 0, or the third
  harmonic is eliminated from the waveform. In
  general, harmonic n will be eliminated if :
      90o
  α=
       n

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                       Example
A full - bridge single phase inverter is fed by square wave
 signals. The DC link voltage is 100V. The load is R = 10R
 and L = 10mH in series. Calculate :
a) the THDv using the " exact" formula.
b) the THDv by using the first three non - zero harmonics
c) the THDi by using the first three non - zero harmonics

Repeat (b) and (c) for quasi - square wave case with α = 30
degrees




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       Half-bridge inverter (1)

                                           S1 ON
                                       Vdc S2 OFF
          +
                      S1                2
        VC1
          -
                   − V +
 Vdc                  o
               G                        0
                                                                      t
                     RL
           +
        VC2               S2
           -                           Vdc
                                   −
                                        2           S1 OFF
                                                    S2 ON




• Also known as the “inverter leg”.
• Basic building block for full bridge, three
  phase and higher order inverters.
• G is the “centre point”.
• Both capacitors have the same value.
  Thus the DC link is equally “spilt”into
  two.


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           Half-bridge inverter (2)

 • The top and bottom switch has to be
   “complementary”, i.e. If the top switch is
   closed (on), the bottom must be off, and
   vice-versa.

 • In practical, a dead time as shown below is
   required to avoid “shoot-through” faults.

                                     S1
                                   signal
 + S1                              (gate)
                     Ishort

Vdc                            G
                                     S2
                          RL
                                   signal
 −                                 (gate)
      S2



"Shoot through fault" .                              td              td
  Ishort is very large
                                                  "Dead time' = td


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       Single-phase, full-bridge (1)
      • Full bridge (single phase) is built from two
        half-bridge leg.
      • The switching in the second leg is “delayed
        by 180 degrees” from the first leg.

                                                              VRG
                                                              Vdc
                                                               2
                        LEG R                    LEG R'                       π       2π          ωt
             +
            Vdc                                                         Vdc
                                S1          S3                      −
             2                                             VR 'G         2
+             -                                           Vdc
                                       + Vo -              2
Vdc                            R                  R'
                  G                                                           π       2π          ωt
-

             +                                            Vdc
                                                          −
                                                           2
             Vdc                S4                 S2      Vo
              2
                                                          Vdc
             -
                                                                              π       2π          ωt
                   Vo = V RG − VR 'G
              G is " virtual groumd"

                                                          − Vdc


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           Three-phase inverter
• Each leg (Red, Yellow, Blue) is delayed by
  120 degrees.

• A three-phase inverter with star connected
  load is shown below


+Vdc
            +
           Vdc/2    S1                    S3                      S5

            −
       G                      R                         Y                      B
                              iR                        iY                    iB
           +
                    S4                    S6                      S2
           Vdc/2
            −




                                     ia                      ib
                         ZR                    ZY                        ZB


                                                    N

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            Square-wave inverter
                waveforms
                                                                        VDC/2
                  VAD                                                    t
                                                                        -VDC/2



                  VB0                                                    t




                  VC0
                                                                         t

                          (a) Three phase pole switching waveforms
                                                                        VDC
                                        600       1200
                  VAB                                                    t

                                                                        -VDC
                          (b) Line voltage waveform
                                                                        2VDC/3
                                                                        VDC/3
                 VAPH                                                    t
                                                                        -VDC/3
                                                                        -2VDC/3
                          (c) Phase voltage waveform (six-step)


               Interval     1      2     3      4      5     6
 Positive device(s) on      3     3,5    5     1,5     1    1,3
Negative devise(s) on      2,4     4    4,6     6     2,6    2
                        Quasi-square wave operation voltage waveforms

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         Three-phase inverter
        waveform relationship

• VRG, VYG, VBG are known as “pole
  switching waveform” or “inverter phase
  voltage”.

• VRY, VRB, VYB are known as “line to line
  voltage” or simply “line voltage”.

• For a three-phase star-connected load, the
  load phase voltage with respect to the “N”
  (star-point) potential is known as VRN ,VYN,
  VBN. It is also popularly termed as “six-
  step” waveform




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 MODULATION: Pulse Width
    Modulation (PWM)
                                     Modulating Waveform    Carrier waveform
      +1
      M1



        0




      −1


      Vdc
       2

        0   t0 t1 t 2   t 3 t 4 t5

      Vdc
  −
       2


• Triangulation method (Natural sampling)
       – Amplitudes of the triangular wave (carrier) and
         sine wave (modulating) are compared to obtain
         PWM waveform. Simple analogue comparator
         can be used.
       – Basically an analogue method. Its digital
         version, known as REGULAR sampling is
         widely used in industry.

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                     PWM types
• Natural (sinusoidal) sampling (as shown
  on previous slide)
    – Problems with analogue circuitry, e.g. Drift,
      sensitivity etc.
• Regular sampling
    – simplified version of natural sampling that
      results in simple digital implementation
• Optimised PWM
    – PWM waveform are constructed based on
      certain performance criteria, e.g. THD.
• Harmonic elimination/minimisation PWM
    – PWM waveforms are constructed to eliminate
      some undesirable harmonics from the output
      waveform spectra.
    – Highly mathematical in nature
• Space-vector modulation (SVM)
    – A simple technique based on volt-second that is
      normally used with three-phase inverter motor-
      drive
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    Natural/Regular sampling
 MODULATION INDEX = M I :
      Amplitude of the modulating waveform
 MI =
       Amplitude of the carrier waveform
 M I is related to the fundamental (sine wave)
 output voltage magnitude. If M Iis high, then
 the sine wave output is high and vice versa.
 If 0 < M I < 1, the linear relationship holds :

V1 = M I Vin

 where V1, Vin are fundamental of the output
 voltage and input (DC) voltage, respectively.
 −−−−−−−−−−−−−−−−−−−−−−−−−−−−
 MODULATION RATIO = M R (= p )

                Frequency of the carrier waveform
 MR = p =
              Frequency of the modulating waveform
 M R is related to the " harmonic frequency". The
 harmonics are normally located at :
 f = kM R ( f m )

 where f m is the frequency of the modulating signal
 and k is an integer (1,2,3...)
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Asymmetric and symmetric
    regular sampling
                                     T
            +1                                         M1 sin ω mt
                            sample
                             point




                                                                     t
                       T             3T           5T          π
                       4              4            4          4




            −1




            Vdc
             2                                               asymmetric
                                                              sampling

                                                                         t
                  t0       t1             t2     t3
                                                               symmetric
                                                                sampling
            V
           − dc
             2

             Generating of PWM waveform regular sampling




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      Bipolar and unipolar PWM
          switching scheme
• In many books, the term “bipolar” and
  “unipolar” PWM switching are often
  mentioned.

•      The difference is in the way the sinusoidal
      (modulating) waveform is compared with
      the triangular.

• In general, unipolar switching scheme
  produces better harmonics. But it is more
  difficult to implement.

• In this class only bipolar PWM is
  considered.



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     Bipolar PWM switching


                               ∆    modulating
               ∆          δ=                                 carrier
                               4    waveform                waveform




                                      π                             2π




            kth
            pulse




                                      π                             2π
            δ 1k
                   δ 2k


      αk




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   Pulse width relationships

                              ∆       modulating
              ∆          δ=                                    carrier
                              4       waveform                waveform




                                        π                          2π




           kth
           pulse




                                        π                          2π
           δ 1k
                  δ 2k


     αk




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    Characterisation of PWM
   pulses for bipolar switching

                                   ∆
                       δ0     δ0        δ0      δ0
               + VS
                 2
                                       δ 2k
                             δ1k




                 V
                − S
                  2
                              αk




                    The kth PWM pulse




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 Determination of switching
angles for kth PWM pulse (1)
                              AS2
                                      v   Vmsin( θ )
       AS1




      + Vdc
        2


     Ap1                            Ap2




      V
     − dc
       2




     Equating the volt - second,
     As1 = Ap1
     As 2 = Ap 2

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    PWM Switching angles (2)
The average voltage during each half cycle
of the PWM pulse is given as :

       Vdc  δ1k − (2δ o − δ1k ) 
V1k =      
                                  
                                   
       2           2δ o         
   V  δ − δ o           V 
=  dc  1k
                    = β1k  s 
                    
   2  δ o               2

                  δ1k − δ o 
where β1k       =
                  δ         
                             
                      o     
Similarly,
              Vdc                   δ 2k − δ o 
V2 k = β 2 k       ; where β 2 k = 
                                      δ          
                                                  
              2                          o     
The volt - second supplied by the sinusoid,
           αk
As1 =       ∫ Vm sin θdθ = Vm [cos(α k − 2δ o ) − cos α k ]
        α k − 2δ o

    = 2Vm sin δ o sin(α k − δ o )
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            Switching angles (3)
Since,
sin δ o → δ o for small δ o ,
As1 = 2δ oVm sin(α k − δ o )

Similarly,
As 2 = 2δ oVm sin(α k + δ o )

The volt - seconds of the PWM waveforms,
            Vdc                       Vdc 
Ap1 = β1k       2δ o ;  Ap 2 = β 21k      2δ o
            2                         2 
To derive the modulation strategy,
Ap1 = As1; Ap 2 = As 2

Hence, for the leading edge
     Vdc 
β1k      2δ o = 2δ oVm sin(α k − δ o )
     2 
             Vm
⇒ β1k =            sin(α k − δ o )
           (Vdc 2)

                     Power Electronics and                39
                     Drives (Version 2): Dr.
                      Zainal Salam, 2002
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   PWM switching angles (4)
The voltage ratio,
        Vm
MI =             is known as modulation
      (Vdc 2 )
index or depth. It varies from 0 to 1.

 Thus,

 β1k = M I sin(α k − δ o )

 Using similar method, the trailing edge
 can be derived :

 β 2 k = M I sin(α k − δ o )

Substituting to solve for the pulse - width,
      δ1k − δ o
β1k =
         δo
⇒ δ1k = δ o [1 + M I sin(α k − δ o )]

 and
   δ 2k = δ o [1 + M I sin(α k + δ o )]

                    Power Electronics and                40
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           PWM Pulse width

 Thus the switching angles of the kth pulse is :

     Leading edge : α k − δ1k

    Trailing edge : α k + δ1k

 The above equation is valid for Asymmetric
 Modulation, i.eδ1k and δ 2k are different.
 For Symmetric Modulation, δ1k = δ 2k = δ k

 ⇒ δ k = δ o [1 + M I sin α k ]




                    Power Electronics and                41
                    Drives (Version 2): Dr.
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                                Example
•     For the PWM shown below, calculate the switching
      angles for all the pulses.




                                                                             carrier
        2V
                                                                            waveform
      1.5V



                                          π                                 2π
                                                                             modulating
                                                                             waveform




                  1     2      3      4       5     6     7      8      9



                                          π              t13   t15   t17
             t1   t2   t3 t4 t5 t6   t7 t8 t9 t10 t11 t12    t14   t16 t18 2π




             α1




                                Power Electronics and                                  42
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                                 Zainal Salam, 2002
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 Harmonics of bipolar PWM
Assuming the PWM waveform is half - wave
symmetry,harmonic content of each (kth)
PWM pulse can be computed as :

        1T                    
bnk = 2 ∫ f (v) sin nθdθ 
        π                     
         0                    
  2  k 1k  Vdc                 
      α −δ
                                 
=  ∫ −                sin nθdθ 
  π α −2δ  2                   
     k o                         
  2α k +δ 2 k  V            
+  ∫             dc sin nθdθ 
                               
  π α −δ  2                  
     k 1k                      
  2α k + 2δ o  V              
                                  
+  ∫  − dc  sin nθdθ 
  π α +δ  2                    
         k   2k                  
 Which can be reduced to :
         Vdc
 bnk = −     {cos n(α k − 2δ o ) − cos n(α k − δ1k )
         nπ
          + cos n(α k + δ 2 k ) − cos n(α k − δ1k )
          + cos n(α k + δ 2 k ) − cos n(α k + 2δ o )}
                    Power Electronics and                43
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          Harmonics of PWM
  Yeilding,
       2V
  bnk = dc [cos n(α k − δ1k ) − cos n(α k − 21k )
        nπ
            + 2 cos nα k cos n 2δ o ]

  This equation cannot be simplified
  productively.The Fourier coefficent for the
  PWM waveform isthe sum of bnk for the p
  pulses over one period, i.e. :
          p
  bn =   ∑ bnk
         k =1

  The slide on the next page shows the
  computation of this equation.




                    Power Electronics and                44
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                     Zainal Salam, 2002
   www.sayedsaad.com                                                www.tkne.net


                   PWM Spectra




                                                                                               M = 0.2

Amplitude

                                                                                     M = 0.4

1.0

0.8                                                                       M = 0. 6

0.6
                                                                                  Depth of
                                                                                 Modulation
0.4                                                             M = 0.8

0.2

  0                                                   M = 1.0
              p        2p          3p         4p

Fundamental
                       NORMALISED HARMONIC AMPLITUDES FOR
                        SINUSOIDAL PULSE-WITDH MODULATION




                            Power Electronics and                                       45
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                             Zainal Salam, 2002
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      PWM spectra observations
•     The amplitude of the fundamental decreases or
      increases linearly in proportion to the depth of
      modulation (modulation index). The relation ship is
      given as: V1= MIVin

•     The harmonics appear in “clusters” with main
      components at frequencies of :
      f = kp (fm);    k=1,2,3....
      where fm is the frequency of the modulation (sine)
      waveform. This also equal to the multiple of the
      carrier frequencies. There also exist “side-bands”
      around the main harmonic frequencies.

•     The amplitude of the harmonic changes with MI. Its
      incidence (location on spectra) is not.

•     When p>10, or so, the harmonics can be normalised
      as shown in the Figure. For lower values of p, the
      side-bands clusters overlap, and the normalised
      results no longer apply.



                        Power Electronics and                46
                        Drives (Version 2): Dr.
                         Zainal Salam, 2002
www.sayedsaad.com                                   www.tkne.net


    Bipolar PWM Harmonics
           0.2      0.4        0.6         0.8     1.0
 h
      MI
1          0.2      0.4        0.6         0.8     1.0
MR         1.242    1.15       1.006       0.818   0.601
MR +2      0.016    0.061      0.131       0.220   0.318
MR +4                                              0.018

2MR +1     0.190    0.326      0.370       0.314   0.181
2MR +3              0.024      0.071       0.139   0.212
2MR +5                                     0.013   0.033


3MR        0.335    0.123      0.083       0.171   0.113
3MR +2     0.044    0.139      0.203       0.716   0.062
3MR +4              0.012      0.047       0.104   0.157

3MR +6                                     0.016   0.044


4MR +1     0.163    0.157      0.008       0.105   0.068
4MR +3     0.012    0.070      0.132       0.115   0.009

4MR+5                          0.034       0.084   0.119
4MR +7                                     0.017   0.050



                     Power Electronics and                     47
                     Drives (Version 2): Dr.
                      Zainal Salam, 2002
www.sayedsaad.com                             www.tkne.net

     Bipolar PWM harmonics
       calculation example
  Note : for full bridge single - phase bipolar PWM,
  vo = vRR, = vRG − vR 'G
     = 2vRG

  The harmonics are computed from :
  (VˆRG )n  VDC 
                 
            2 
  as a function of M I

  Example :

  In the full - bridge single phase PWM inverter,
  VDC = 100V, M I = 0.8, M R = 39. The
  fundamentalfrequency is 47Hz. Calculate
  the values of the fundamental - frequency
  voltage and some of the dominant harmonics.




                    Power Electronics and                48
                    Drives (Version 2): Dr.
                     Zainal Salam, 2002
www.sayedsaad.com                             www.tkne.net

     Three-phase harmonics:
     “Effect of odd triplens”
• For three-phase inverters, there is
  significant advantage if p is chosen to be:

    – odd and multiple of three (triplens) (e.g.
      3,9,15,21, 27..)
    – the waveform and harmonics and shown on the
      next two slides. Notice the difference?


• By observing the waveform, it can be seen
  that with odd p, the line voltage shape
  looks more “sinusoidal”.

• The even harmonics are all absent in the
  phase voltage (pole switching waveform).
  This is due to the p chosen to be odd.



                    Power Electronics and                49
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 www.sayedsaad.com                             www.tkne.net


         Spectra observations
• Note the absence of harmonics no. 21, 63
  in the inverter line voltage. This is due to p
  which is multiple of three.

• In overall, the spectra of the line voltage is
  more “clean”. This implies that the THD is
  less and the line voltage is more sinusoidal.

• It is important to recall that it is the line
  voltage that is of the most interest.

• Also can be noted from the spectra that the
  phase voltage amplitude is 0.8
  (normalised). This is because the
  modulation index is 0.8. The line voltage
  amplitude is square root three of phase
  voltage due to the three-phase relationship.
                     Power Electronics and                50
                     Drives (Version 2): Dr.
                      Zainal Salam, 2002
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     Waveform: effect of “triplens”
    Vdc
                                         π                        2π
     2                                                             V RG


    Vdc
−
     2
    Vdc
     2                                                             VYG


    Vdc
−
     2
    Vdc
                                                                  V RY



− Vdc
                        p = 8, M = 0.6
    Vdc
     2                                                             V RG


    Vdc
−
     2
    Vdc
     2                                                             VYG

    Vdc
−
     2
    Vdc
                                                                   VRY


− Vdc
                        p = 9, M = 0.6
          ILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIO
          THAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER

                               Power Electronics and                      51
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                                Zainal Salam, 2002
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                         Harmonics: effect of
                             “triplens”
Amplitude




                    0.8 3 (Line to line voltage)
1.8


 1.6

 1.4


 1.2

 1.0


 0.8

 0. 6

                                                                                                                                          B
 0. 4
                                                               41         43                 61   65                  83   85
                                  19         23         37                     47       59          67          79              89
 0. 2

   0                                                                                                                                 A
                        21                                                     63
                                                                                                     83   85
                   19        23              41    43                     61        65          81             87
                                        39              45           59               67     79                  89
                                   37                        47 57                      69 77                         91 Harmonic Order
Fundamental
              COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE
                                  (B) HARMONIC (P=21, M=0.8)




                                              Power Electronics and                                                             52
                                              Drives (Version 2): Dr.
                                               Zainal Salam, 2002
www.sayedsaad.com                             www.tkne.net
 Comments on PWM scheme

• It is desirable to push p to as large as
  possible.
• The main impetus for that when p is high,
  then the harmonics will be at higher
  frequencies because frequencies of
  harmonics are related to: f = kp(fm), where
  fm is the frequency of the modulating
  signal.
• Although the voltage THD improvement is
  not significant, but the current THD will
  improve greatly because the load normally
  has some current filtering effect.
• In any case, if a low pass filter is to be
  fitted at the inverter output to improve the
  voltage THD, higher harmonic frequencies
  is desirable because it makes smaller filter
  component.


                    Power Electronics and                53
                    Drives (Version 2): Dr.
                     Zainal Salam, 2002
    www.sayedsaad.com                                                www.tkne.net


                              Example
The amplitudes of the pole switching waveform harmonics of the red
phase of a three-phase inverter is shown in Table below. The inverter
uses a symmetric regular sampling PWM scheme. The carrier frequency
 is 1050Hz and the modulating frequency is 50Hz. The modulation
 index is 0.8. Calculate the harmonic amplitudes of the line-to-voltage
 (i.e. red to blue phase) and complete the table.

        Harmonic        Amplitude (pole switching      Amplitude (line-to
         number                waveform)                 line voltage)
           1                        1
           19                      0.3
           21                      0.8
           23                      0.3
           37                      0.1
           39                      0.2
           41                     0.25
           43                     0.25
           45                      0.2
           47                      0.1
           57                     0.05
           59                      0.1
           61                     0.15
           63                      0.2
           65                     0.15
           67                      0.1
           69                     0.05



                             Power Electronics and                              54
                             Drives (Version 2): Dr.
                              Zainal Salam, 2002

				
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