Pulse-Width Modulation (PWM)

The pulse-width modulation is the most adopted technique for driving a voltage source inverter. It is superior with respect to six-step commutation in terms of precision, efficiency and acoustic noise rejection: the superior harmonics are moved to higher frequency thank to this modulation, so the motor can easily filter them. On the other hand, it requires an increase of hardware and software complexity and extra costs, but the continuous technological evolution has boosted the diffusion of PWM method for all the industrial – and in particular automotive – applications. The principle is to use three control signals and compare them at each instant with a carrier triangular wave with a frequency at least 10 times greater. The control inputs – called reference or modulating signals – are sinusoidal waves out of phase of 120° with respect to each other: if the instantaneous value of the carrier wave is less than that of the sine wave, the PWM output signal is in high level;

if the carrier is higher than the modulating wave, the PWM output is in low level.183 From this comparison between each command voltage and the carrier triangular wave, different duty cycles are obtained for phase a, b and c. «Sinusoidal PWM techniques are characterized by constant amplitude pulses with different duty cycles for each period.

The widths of these pulses are modulated to obtain inverter output voltage control and to reduce its harmonic content».¹⁸⁴ The phase a generates a duty cycle coming from the reference signal a and this will drive the upper switch of the first leg, called S¹ in previous

183 Bird, B., King, K., & Pedder, D. (1993), An Introduction to Power Electronics (Second Edition). Chichester, West Sussex, United Kingdom: Wiley.

184 Raju, N., Islam, M., & Uddin, A. (2013, January), Sinusoidal PWM Signal Generation Technique for Three Phase Voltage Source Inverter with Analog Circuit & Simulation of PWM Inverter for Standalone Load &

Micro-grid System, In International Journal of Renewable Energy Research, 3(3), p. 649.

figure 3.5; the corresponding negated signal will drive the lower switch, called S4. Again, a small delay must be introduced between the commutations to avoid short-circuit situations – two switches of the same branch cannot be switched off simultaneously.

Figure 3.7: Pulse-width modulation command waveforms (modulating sine waves and triangular carrier) and output voltage signals.¹⁸⁵

The desired sinusoidal concatenated voltage output is given by the fundamental harmonic (fig. 3.7) of the PWM output signal and it has the same frequency of the modulating signals; in this way by choosing the desired value for the sinusoidal references, the resulting output will have that imposed frequency. The superior harmonics are shifted to frequencies that are multiple of the one of the carrier signal: considering the fact that the triangular wave has a frequency at least 10 times greater than that of command sinusoidal waves, the distortion caused by higher harmonics is reduced because the motor easily filters

185 Mohan, N., Undeland, T., & Robbins, W. (2003), op. cit., p. 255.

84 the high frequency components – consequently the torque and current ripples are reduced.

An important parameter for this technique is the modulation index 𝑚:

𝑚 = 𝑉_𝑚 𝑉_𝑐

where 𝑉_𝑚 is the peak value of the modulating signal and 𝑉_𝑐 is the peak value of the carrier signal. By changing 𝑚, it is possible to modify the amplitude of the fundamental harmonic of the PWM output signal – without considering the DC bus voltage – in order to obtain the desired amplitude value for driving the motor.¹⁸⁶ In fact, another important advantage of the pulse-width modulation is the possibility of leaving always the 𝑉_𝐷𝐶 constant: there is no need of varying the DC bus voltage of the inverter for modulating the sinusoidal output voltage.

An example of PWM algorithm implementation on LabVIEW is reported here: the modulating 5 Hz sinusoidal signals are three-phase voltages – with a difference of 120° with respect to each other – while the carrier signal is a 1 kHz sawtooth wave – its frequency is the same of the timed loop. By using the sawtooth instead of the triangular carrier, the results are not compromised. The amplitude is equal to 100 V for all the waveforms, so the modulation ratio is unitary. The switching mechanism is obtained through the comparison between the modulating waves and the carrier: if the sawtooth wave is less than the sine, the corresponding switch is on (high level); if the sawtooth wave is higher than the sine, the corresponding switch is off (low level). In this way, different duty cycles are generated for the three switches. Then, for reconstructing the first fundamental the low pass filter of the motor is simulated, using a simple RC structure. Knowing that the frequency of the fundamental is the same of the modulating, while the other harmonics are shifted at multiple frequencies of the carrier, a cut frequency equal to 7 Hz is used for the filter. In order to realize it the following values are chosen:

𝑓_𝑐𝑢𝑡 = 1

2𝜋𝑅𝐶 = 7 Hz 𝐶 = 1 𝜇𝐹

𝑅 = 1

2𝜋𝑓_𝑐𝑢𝑡𝐶 = 22.736 kΩ

186 Nagarajan, R., et al. (2016, September), Implementation of SPWM Technique for Inverter, In International Journal of Advanced Research in Biology, Engineering, Science and Technology (IJARBEST), 2(9), p. 11.

For the simulation, a discrete time interval equal to 1 ms is considered, and the analysis lasts 3000 ms. As shown in figure 3.10, the resultant waveform 𝑉_𝑎 (such as 𝑉_𝑏 and 𝑉_𝑐) is in delay with respect the modulating wave and it is affected by ripple: this degrading effect can be reduced by increasing the carrier frequency, by making a more accurate filter or by using the space vector approach. The used waves (modulating and carrier) are only positive; for a more generic situation, an offset can be inserted for equalizing to zero the mean value of the waveforms. Moreover, the negate waves – for the lower switches in the three legs of the inverter – are not considered in this example.

Figure 3.8: PWM three-phase modulating waves and carrier sawtooth wave.

Figure 3.9: Duty cycle of switch A, with the resultant voltage waveform 𝑉_𝑎.

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Figure 3.10: PWM implementation in LabVIEW.