Pulse Width Modulation ( PWM )

Voltage Control in the Inverter

When an inverter is connected to load, the output voltage of inverter is controlled due to following reasons. 

• To adjust output voltage of inverter
• When there is any change in input supply source, the output also changed. 
• When the frequency or speed changed, the output of inverter also changes. This is done particular in the induction motor speed control by inverter.

The output voltage of inverter is controlled by following methods.

( 1 ) External voltage control of AC output voltage

• The AC voltage controller is connected between load and inverter output. 
• The load voltage is controlled by controlling firing angle of AC voltage controller. 
• As the harmonics is produced in the output voltage, this method is used for low power application.

( 2 ) External control of DC input voltage

• The chopper is connected between inverter and DC input source when the input supply is DC. 
• The output voltage of inverter changes due to change in chopper output voltage. 
• When the input supply is AC, the AC to DC conversion is done by controlled rectifier. 
• The DC voltage is adjusted by controlling firing angle of controlled rectifier. 
• This will result in output voltage of inverter is also adjustable.
• When the input voltage is AC, the AC output voltage is controlled by ac voltage controller. 
• The AC to DC conversion is done by uncontrolled rectifier therefore the inverter output voltage is adjustable. 
• When the input voltage is AC, the uncontrolled rectifier converts AC into DC. 
• The chopper converts fixed DC into variable DC supply and the output of chopper is feed to input of the inverter.

There are following advantages and disadvantages of above mentioned methods.

Advantages

• When the inverter output voltage is adjusted by controlling the DC input voltage, there is no change in output waveform and harmonics.
• When the inverter output voltage is adjusted by controlling input supply source, the design of inverter is done for specific voltage limit and its efficiency increases due to small power loss.

Disadvantages

• There is filter requires at the input side of input in order to reduce DC voltage ripple. 
• This filter circuit increases the weight, volume and cost of inverter.
• The inverter efficiency decreases as the power stages increases more than one.
• When it is require to control output voltage for constant current, the DC input voltage control method used because the commutating capacitor voltage decreases as the DC input voltage decreases and this will result in turn off time of SCR decreases.

(3) Internal control of Inverter

• When load is connected at the output of inverter, the output voltage of inverter is controlled by internal control of inverter. 

There are following methods of internal control of inverter voltage.

(A)  Series Inverter control

• This method is also called as multiple inverter control.
• There are two or more than two inverters connected in series in this method. 
• The output of the inverter is connected with primary winding of the transformer whereas the load is connected to the secondary winding of the transformer.

Let the secondary voltage of the transformer is V1 and V2 , therefore the load voltage is

           V = √ ( V1 )² + ( V2 )² + 2V1V2sinα

Where α = firing angle of inverter

If α = 0^o

V = √ ( V1 )² + ( V2 )² + 2V1V2

    = ( V1 + V2 )² 

           OR

       ( V1 + V2 ) 

If α = π

V = √ ( V1 )² + ( V2 )² – 2V1V2

    = ( V1 – V2 )² 

           OR

  V =   ( V1 – V2 ) 

• The load voltage can be changed by changing the firing angle of the inverter.

(B)  Pulse width modulation ( PWM )

• This is most efficient method of inverter output voltage control. 
• The constant DC input voltage is applied at the input of the inverter and output voltage is controlled by switching semiconductor device of the inverter in this method.

Advantages of Pulse Width Modulation techniques

• There are no necessary of any extra components to control output voltage of inverter.
• As the low order harmonics ( 3rd, 5th ) reduces whereas higher order harmonics ( 7^th , 9^th and 11^th ) are filter out, less requirement of filter.

Disadvantages of Pulse Width Modulation techniques

• As the switching semiconductor device requires low turn on and turn off time, cost of semiconductor device increases.

Methods of PWM Control

Single Pulse Width Modulation ( SPWM )

• As the semiconductor device receives only one pulse during one half cycle, one semiconductor device is switched on. 
• The output voltage of the inverter can be controlled by controlling width of pulse. 
• Figure A shows the gate signal and output voltage waveform for single phase full bridge inverter.
• The gate signal is generated by comparing V_R amplitude reference signal and V_C amplitude control signal. 
• The width of gate pulse can be varies from 0^o to 180^o by controlling the reference signal from 0 to V_R. 
• This will control the output voltage of the inverter. 

• The frequency of the output voltage depends upon frequency of reference signal.
• The amplitude modulation M is ratio of reference signal ( V_R ) and carrier signal ( V_C ).

       M = V_R / V_C

• The analysis of waveform shown in the figure A is done by fourier series. 
• The output voltage becomes maximum when the width of pulse becomes π radian.

V_L = 4V_DC / π ............................................(1)

RMS output voltage

V_RMS = V_DC √ d / π....................................(2)

And maximum value of nth harmonic

V_Ln = 4V_DC / nπ ( Sin nd / 2 ) ..................(3)

From equation (1) and (3)

V_Ln / V_L = Sin ( nd / 2) / n ......................(4)

• The graphical representation of pulse width in degree ( x – axis ) and n = 1, 3, 5 and 7 ( y – axis ) is shown in the figure B.

Waveform		Y - axis	X – axis
Fundamental	n = 1	Sin ( d / 2 )	Pulse width in degree
3rd harmonic	n = 3	Sin ( 3d / 2 ) / 3
5th harmonic	n = 5	Sin ( 5d / 2 ) / 5
7th harmonic	n = 7	Sin ( 7d / 2 ) / 7

• When a value of the fundamental component becomes equal to 0.143, the third, fifth and seventh harmonics becomes equal. 
• This will conclude the higher harmonics remains present when the output voltage is low.

Multiple pulse width modulation ( MPWM )

• There are more than one pulse per half cycle in the MPWM. 
• These gate pulses are used to control output voltage of inverter as well as reduce harmonics. 
• The magnitude and width of the pulses are equal in this method.
• The reference signal and higher frequencies carrier signals are compared in this method in order to generate more than one gatting pulses. 
• The number of gate pulses depends upon carrier frequencies whereas the output voltage depend frequencies of reference signal.

From figure J,

• Carrier frequency = f_C in Hz
• Reference frequency = f_R in Hz

       1 / f_C = π / 3..................................................(5)

           OR

        T_C = π / 3

Similarly

        1 / 2f_R = 1 / π .............................................(6)

           OR

        T_R = π / 2

Number of pulses per half cycle ( N_P ) = Length of half cycle reference signal /  Length of one cycle triangular waveform

                                             = ( f_R / 2 ) / ( 1 / f_C )

                                      N_P  =  f_C / 2 f_R

Number of generated pulses N_P = ( 3 / π ) × π    [ from equation (5) and (6) ]

                                                    = 3

The RMS voltage when pulse width is equal to d

V_RMS = V_DC √ ( N_P × d / π )

• As the number of pulses increases in the each half cycle, lower order harmonics reduces but higher order harmonics increases. 
• The higher order harmonics are reduced by using filter. 
• It is to be noted that the switching losses of the semiconductor devices increases as there are more number of pulses per half cycle.
• This modulation technique is also called as symmetrical modulation control.

Sinusoidal Pulse width Modulation ( SINPWM )

• The reference signal is taken as sinusoidal waveform whereas the carrier signal is taken as triangular waveform in this method. 
• The width of pulse in the SINPWM is not equal due to reference signal is taken as sinusoidal waveform. 
• The amplitude of sinusoidal waveform is also not constant. 
• The width of gate pulse is determined by intersect point of the sinusoidal waveform and triangular waveform. 
• The frequency of inverter output voltage depends upon frequency of reference signal f_R and amplitude of reference signal V_R controls the modulation index ( M ).

• The number of pulses per half cycle when the amplitude of triangular waveform becomes maximum and sinusoidal waveform becomes zero.

        N_P  =  f_C / 2 f_R

      Where

       f_C = Carrier wave frequency = 3 / π

       f_R = Reference wave frequency = 1 / 2π

Therefore

       N_P = ( 3 / π ) × ( 2π / 2 )

            = 3

• The number of pulses per half cycle when the amplitude of triangular waveform and sinusoidal becomes zero at same time.

       N_P  = ( f_C / 2 f_R – 1 )

             = 2     

             The modulation index = V_R / V_C

• The analysis of harmonics is done in the sinusoidal PWM control is below.
• When the value of modulation index is less than one, the maximum harmonic number in the output voltage is

       f_C / f_R ± 1

         OR

       2N_P  ± 1

      Where  N_P = Number of pulses per half cycle

• As the number of pulses per half cycle increases, the higher order harmonics also increases. 
• Let N_P = 4, it will generates 7^th harmonic and 9thharmonic but higher order harmonics are easily filtered out. 
• As the number of pulses increases per half cycle, the switching losses also increase and it will affect the efficiency of inverter.
• When the modulation index is greater than one, lower order harmonics induces in the output of the inverter.

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