In the FOC algorithm, the PI algorithm for the DQ axis calculates the DQ axis voltage. After the inverse PARK transformation, the α axis and β axis voltages can be obtained. However, these voltages are just values in the computer. To convert them into actual terminal voltages applied to the motor phase lines, a drive bridge composed of power electronic switching devices (MOSFETs or IGBTs) is needed.
Suppose the control program wants a 3V voltage on the motor phase line, but the battery power supply is only a stable 12V. How can this 3V be obtained? This is what SPWM or SVPWM does.
The theoretical basis of PWM: impulse equivalence principle
When narrow pulses of equal impulse but different shapes are applied to an inertial element, their effects are basically the same. As shown below, when the voltages of the four pulses are applied to the RL circuit, the low-frequency part of the Fourier series expansion of the current response in the circuit is basically the same, while the high-frequency part is slightly different.
The principle holds true under two indispensable conditions: a narrow pulse and an inertial element. The concept of "narrow" is relative to the time constant of the RL loop. If the time constant of the inertial element is in the millisecond range, then the pulse must be at least on the order of tens of microseconds; if the time constant of the inertial element is in the hundreds of milliseconds, then a pulse narrowing to a few milliseconds is acceptable. The other condition is the presence of an inertial element. This is relatively easy to understand: if the controlled object is a pure resistor, no matter how narrow the four pulses are, the output current response will not be essentially the same.
This theory was developed by mathematicians through theoretical analysis. However, for current semiconductor-based discrete control systems (quantum computing and biological computers are still unknown), the simplest and most feasible approach is obviously to provide only 0 or 1 switching signals. Therefore, based on the current level of human technology, the square wave method shown in Figure a) is the most ideal choice.
SPWM: Sinusoidal Pulse Width Modulation
The concept of sinusoidal pulse width modulation (PWM) is relatively simple and straightforward. After obtaining the α-axis and β-axis voltages, performing an inverse Clark transform yields the three-phase sinusoidal voltages Va, Vb, and Vc of the motor. A triangular wave is then used for modulation to obtain the chopped signal for each phase. Both phase voltages and line voltages are sinusoidal waves.
Taking a power supply voltage of 12V as an example, the peak-to-peak phase voltage under this modulation method can reach a maximum of 12V. Therefore, the peak-to-peak line voltage can reach a maximum of [value missing], and the bus voltage utilization rate is [value missing].
At any given moment, Va + Vb + Vc = 1.5 * bus voltage. (The sum of three sine waves with equal amplitudes and a 120° difference between them is A * sinωt + A * sin(ωt + 120°) + A * sin(ωt + 240°) = 1.5 * A)
SVPWM: Space Vector Modulation
SVPWM takes the combined voltage vector of the three phases as its starting point. Its basic idea is that, mathematically speaking, the voltage generated by the abc axis or the αβ axis should be equal to the voltage synthesized by the dq axis. Therefore, as long as the voltage vector ultimately generated by the drive bridge is the same as the combined voltage vector of the dq axis, this goal is achieved. As for how to obtain the duty cycle of each phase arm from the αβ axis voltage, I won't go into detail here, as there are plenty of resources online.
Taking a 12V power supply voltage as an example, under SVPWM modulation, the phase voltage becomes a 0~12V saddle-shaped wave, but the motor line voltage remains 80% of a sine wave.
