Single pulse width modulation

In single PWM control, there is only one pulse per half cycle and the width of the pulse is varied to control the inverter output voltage. The gating signals are generated by comparing a rectangular reference signal of amplitude Ar with a triangular carrier wave of amplitude Ac. the frequency of the reference signal determines the fundamental frequency of output voltage. The instantaneous output voltage is vo = Vs (g1-g4). The ratio of Ar to Ac is the is the control variable and defined as the amplitude modulation index which is given by

M=Ar/Ac

The rms output voltage can be found from

Vo = [ 22π ∫

(π−δ)/2

(π+δ)/2

V s2]1/2 = Vs√ δπ

By varying Ar from 0 to Ac, the pulse width δ can be modified from 0o to 180o and the rms output voltage Vo, from 0 to Vs.

The Fourier series of output voltage yields

vo(t) = ∑n=1,3,5 , ..

∞ 4Vsnπ

sin nδ2sin nωt

Multiple pulse width modulation

The harmonic content can be reduced by using several pulses in each half cycle of output voltage. The generation of gating signals for turning on and off of transistors can be obtained by comparing a reference signal with a triangular carrier wave. The frequency of reference signal sets the output frequency fo, and the carrier frequency fc determines the number of pulses per half cycle p. the modulation index controls the output voltage. This type of modulation is also known as uniform pulse width modulation (UPWM). The number of pulses per half cycle is found from

p = fc2 fo =

mf2

where mf = fc/fo is defined as the frequency modulation ratio.

The instantaneous output voltage is vo = Vs (g1-g4).

If δ is the width of each pulse, the rms output voltage can be found from

Vo = [2 p2π ∫

(πp−δ)/2

( πp +δ)/2

V s2d (ωt )]1/2 = Vs√ pδπ

The variation of the modulation index M from 0 to 1 varies the pulse width d from 0 to T/2p and the rms output voltage Vo from 0 to Vs. The general form of a Fourier series for the instantaneous output voltage is

v0(t) = ∑n=1,3,5 ,…

∞

Bnsinnωt

The coefficient Bn can be determined by considering a pair of pulses such that the positive pulse of duration δ starts at ωt=α and the negative one of the same width starts at ωt=π+α. The effects of pulses can be combined together to obtain the effective output voltage.