Topic 1

PMSM field-oriented control fundamentals

Learn how PMSM field-oriented control uses d- and q-axis currents, rotor electrical angle and inverter voltage commands to regulate flux and electromagnetic torque.

LevelBeginner → engineer
ModelPMSM dq equations
NextIMC PI tuning

1. Why field-oriented control?

A three-phase PMSM produces sinusoidal electrical quantities. FOC transforms those changing phase quantities into a rotating coordinate system attached to the rotor. In that frame, balanced sinusoidal currents become nearly constant d- and q-axis values, which makes feedback control much easier.

The physical picture

The d-axis is aligned with the rotor permanent-magnet flux. The q-axis is 90 electrical degrees away. For a surface PMSM, q-axis current mainly creates torque while d-axis current changes the flux-producing component.

2. The PMSM current equations

\[ \begin{aligned} u_d &= R_s i_d + L_d\frac{d i_d}{dt} - \omega_e L_q i_q,\\[4pt] u_q &= R_s i_q + L_q\frac{d i_q}{dt} + \omega_e L_d i_d + \omega_e\psi_f. \end{aligned} \]

\(R_s\) is the stator phase resistance used in the dq model. \(L_d\) and \(L_q\) are the axis inductances. \(\psi_f\) is the permanent-magnet flux linkage. The speed-dependent terms couple the axes and create back-EMF.

\[ T_e=\frac{3}{2}p\left[\psi_f i_q+(L_d-L_q)i_d i_q\right]. \]

3. The complete control flow

References\(i_d^{\star}\) and \(i_q^{\star}\)
→
Current controllerPI / state feedback
→
PWM + inverteraverage voltage
→
PMSMcurrents and torque
Rotor positionresolver / encoder
→
Transformsabc ↔ dq
→
SamplingADC + software
↺
Feedbackmeasured \(i_d,\,i_q\)

The rotor angle is essential because the Park and inverse-Park transforms must use the correct electrical orientation. An angle offset rotates both the measured currents and commanded voltages into the wrong frame.

4. Mechanical speed, electrical speed and timing

Electrical speed is pole-pair count multiplied by mechanical speed. Sampling frequency controls how often measurements and controller calculations update. Switching frequency controls the PWM carrier. They may be equal, but they are not the same concept.

\[ \omega_m=\frac{2\pi n_m}{60},\qquad \omega_e=p\omega_m,\qquad f_e=\frac{\omega_e}{2\pi},\qquad N_{\mathrm{samples/cycle}}=\frac{f_s}{f_e}. \]

Electrical-speed calculator

Mechanical speed—
Electrical speed—
Samples/electrical cycle—
Important timing distinction

\(T_s=1/f_s\) is the control-update period. The effective delay \(T_d\) is the time from sampling and angle capture to the corresponding voltage action. The next topics show why this delay changes the implemented loop.

5. Continue with controller design

Now that the plant and signal flow are clear, start with the continuous-time IMC PI baseline. Then compare it with the exact discrete-time delay-aware design.

Open IMC PI tuning →Jump to resolver debugger