What is sinusoidal pulse-width modulation

Sinusoidal pulse-width modulation (SPWM) is a method of controlling an inverter in which, instead of equal-width pulses, the output in each half cycle consists of several pulses whose widths vary sinusoidally — narrow near the zero crossings and wide near the peak. The fundamental output is then sinusoidal and low-order harmonics are greatly reduced.

How it is obtained

1. A sinusoidal reference $v_r = A_r\sin\omega_r t$ at the desired output frequency $f_r$ is generated.
2. A triangular carrier $v_c$ of amplitude $A_c$ and higher frequency $f_c$ is generated.
3. The two are compared in a comparator. The pulse widths are the intervals where $v_r > v_c$ (positive half) or $v_r < v_c$ for the complementary switch.
4. Bipolar SPWM (half bridge or full bridge): $v_r > v_c \Rightarrow v_0 = +V_s$, otherwise $v_0 = -V_s$. Unipolar SPWM (full bridge): two legs compared with $+v_r$ and $-v_r$, giving $v_0 = +V_s, 0, -V_s$.
5. The gate pulses (with dead time) drive the inverter switches.

 Ar sin wt --->|+\
               |  >--> gate S1 (S4 = NOT)
 triangle ---->|-/
             comparator

 vc /\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
 vr    .-''-.
 -----'------'------.------.-----
                     '-..-'
 v0 +Vs  _ __ ___ __ _
     0 _| ||  ||   || |_  _ __ _
                       | || ||
    -Vs                |_||_||_|
       narrow-wide-narrow pulses

Key definitions

• Amplitude modulation index $m_a = \dfrac{A_r}{A_c}$. For $m_a \le 1$ the fundamental varies linearly: $\hat V_{1} = m_a V_s$ (full bridge, bipolar) or $m_a V_s/2$ (half bridge).
• Frequency modulation ratio $m_f = \dfrac{f_c}{f_r}$; number of pulses per half cycle $p = m_f/2$ (unipolar counts differ). In three-phase inverters $m_f$ is chosen odd and a multiple of 3.
• Output frequency is changed by $f_r$; output voltage by $m_a$.

Harmonic spectrum

• Low-order harmonics (3rd, 5th, 7th) are nearly eliminated.
• Harmonics appear in groups around $m_f$, $2m_f$, … (e.g. $m_f$, $m_f \pm 2$), which are high and easily filtered by the load inductance or a small LC filter.
• With $m_a > 1$ (overmodulation) the output increases non-linearly and low-order harmonics reappear.

Advantages and uses

• Voltage and frequency control in one stage; near-sinusoidal current; small filters.
• More switching loss than square-wave operation.
• Used in UPS, induction motor drives (V/f control), solar and grid-tie inverters.