The Relationship Between Motor Inductance and FOC Frequency in LowInductance PMSM Control
This article explores the challenges of controlling lowinductance permanent magnet synchronous motors (PMSMs) and explains why higher fieldoriented control (FOC) frequencies are needed for motors with lower inductance. It presents mathematical principles and formulas to quantify this relationship.
Permanent magnet synchronous motors (PMSMs) with low inductance present unique control challenges compared to traditional PMSMs. The low inductance leads to rapid changes in current, causing current ripple to increase, especially at high current levels. While speed can help smooth out some ripple due to its integrating effect, speed fluctuations can still be significant.
In traditional pulse width modulation (PWM) control with sixstep commutation, there are conducting and nonconducting intervals within each control period at low current levels. This introduces significant nonlinear disturbances into the control. The voltage equations during the conducting (t1) and nonconducting (t2) intervals of a PWM period T are:
di/dt = (Vdc  E)/L (0 < t <= t1) di/dt = E/L (t1 < t <= T)
Where Vdc is the DC bus voltage, E is the backEMF, and L is the phase inductance. Solving these yields the current ripple ΔI over one period:
ΔI = VdcD(1D)*T/L
Where D is the PWM duty cycle (t1/T). This shows current ripple is inversely proportional to inductance L and control frequency 1/T. Doubling the control frequency halves the ripple.
However, control frequency has practical limits based on hardware and software implementation. With a STM32 microcontroller and threephase shunt current sampling synchronized to the zero vectors of space vector PWM (SVPWM), the control frequency is limited to around 16 kHz. Using dedicated current sense amplifiers like INA210 relaxes this synchronization requirement, allowing control frequencies up to around 50 kHz. But even then, discontinuous current is unavoidable at very low current levels.
Other solutions include:

Adding external series inductance. This effectively increases the motor inductance, reducing di/dt and current ripple. But it does add cost, size and power loss.

Pulse amplitude modulation (PAM). Instead of varying PWM duty cycle, the DC bus voltage amplitude is modulated while using 100% duty cycle, square wave switching. This eliminates the low duty cycle region prone to current discontinuity. But it requires an additional power converter stage.

Advanced control strategies incorporating bus voltage control and current predictive control. By strategically inserting zero voltage vectors, the effective voltagetime product of each active vector is reduced, further suppressing current ripple. But this is computationally intensive.
In summary, low inductance PMSM control remains a fertile area of research. Practical solutions must balance performance, cost and complexity. With continuing advancements in devices, topologies and algorithms, the advantages of low inductance designs can be more fully realized.