Inverters
With the same fundamental, unipolar modulation moves the harmonics to twice the carrier frequency and lowers the distortion from 146 % to 77 %: an inductive load filters them much better.
01The bridge inverter
An inverter generates AC voltage from a DC source. The single-phase H-bridge has two legs of two switches; with the load connected diagonally, between the midpoints, it can apply \( +V_{dc} \), \( -V_{dc} \) or zero. The three-phase one has three legs, one per phase. Each switch carries an antiparallel diode, because with an inductive load the current keeps flowing when the switch opens. Between the turn-off of one switch and the turn-on of the other in the same leg, a dead time of a fraction of a microsecond is left so as not to short-circuit the supply.
Switching once per half-cycle gives a square wave, with the maximum fundamental, \( \frac{4}{\pi} V_{dc} \), but with all the odd harmonics, of amplitude \( 1/h \), and with no way to vary the voltage.
02Sinusoidal pulse-width modulation
In sinusoidal PWM a sinusoidal reference at the desired frequency is compared with a much faster triangular carrier. The amplitude modulation index \( m_a \) is the ratio of their amplitudes and the frequency modulation index \( m_f \), the ratio of their frequencies. In the linear region, the fundamental of the output is proportional to \( m_a \):
An H-bridge fed with 400 V that generates 50 Hz. Top, the output voltage over one cycle with the scaled reference (yellow); bottom, the odd harmonics as a percentage of the fundamental, up to a little beyond \( 2 m_f \).
The voltage distortion looks enormous, but what matters is the current: an inductive load, such as a motor, attenuates each harmonic in proportion to its order, and high-order ones hardly produce any current. That is why a fast carrier is desirable, up to the limit set by the switching losses seen in power devices.
03Three-phase inverter and other techniques
In the three-phase inverter, three references shifted by 120° are used. The harmonics that are multiples of three cancel in the line voltages, and with \( m_f \) a multiple of three the ones around the carrier also disappear. Space-vector modulation (SVPWM) chooses in each period the bridge states that synthesize the desired voltage vector, and makes use of 15 % more of the DC bus than sinusoidal modulation. For high voltages, multilevel inverters are used —neutral-point-clamped or cascaded H-bridge— whose output has more steps, fewer harmonics and switches that withstand only a fraction of the voltage.
04In the laboratory
Generate sinusoidal PWM with a microcontroller on a low-voltage H-bridge and observe the current in an RL load.
With the oscilloscope FFT, measure the voltage spectrum with bipolar and unipolar modulation.
Increase the dead time and observe the distortion of the current near its zero crossings.
05Common mistakes
- Judging the inverter by the voltage distortion. What heats the motor is the harmonic current.
- Forgetting the dead time. Without it, the two switches of a leg overlap; with too much, the current is distorted.
- Expecting linear gain in overmodulation. The fundamental saturates toward that of the square wave.
- Choosing an even mf. Even harmonics appear and the waveform loses its symmetry.
06Self-assessment
What fundamental does a 400 V bridge give with ma = 0.8?
320 V peak, about 226 V RMS.
And with a square wave?
\( \frac{4}{\pi} \cdot 400 \approx 509 \) V peak.
With mf = 21, where are the first important harmonics in unipolar modulation?
Around 2 mf: at 41 and 43.
What are the antiparallel diodes across the switches for?
To give a path to the inductive load current when the switch opens.
07Further reading
- Ned Mohan, Tore M. Undeland and William P. Robbins. Power Electronics: Converters, Applications, and Design. 3rd ed., Wiley, 2003. Inverters with sinusoidal modulation, including their harmonic tables.
- D. Grahame Holmes and Thomas A. Lipo. Pulse Width Modulation for Power Converters: Principles and Practice. Wiley-IEEE Press, 2003. The reference on all the modulation techniques.
- Bin Wu and Mehdi Narimani. High-Power Converters and AC Drives. 2nd ed., Wiley-IEEE Press, 2017. Multilevel inverters and high-power drives.