TIDUF67 Design guide

TIDUF67 April 2024 – December 2024

3.1.1 Mathematical Model and FOC Structure of PMSM

The FOC structure for a PMSM is illustrated in Figure 2-1. In this system, the eSMO is used for achieving the sensorless control an IPMSM system, and the eSMO model is designed by utilizing the back EMF model together with a PLL model for estimating the rotor position and speed.

An IPMSM consists of a three-phase stator winding (a, b, c axes), and permanent magnets (PM) rotor for excitation. The motor is controlled by a standard three-phase inverter. An IPMSM can be modeled by using phase a-b-c quantities. Through proper coordinate transformations, the dynamic PMSM models in the d-q rotor reference frame and the α-β stationary reference frame can be obtained. The relationship among these reference frames are illustrated in Equation 1. The dynamic model of a generic PMSM can be written in the d-q rotor reference frame as:

Equation 1.

[\begin{matrix} v_{d} \\ v_{q} \end{matrix}] = [\begin{matrix} R_{s} + {pL}_{d} & {- ω}_{e} L_{q} \\ ω_{e} L_{d} & R_{s} + {pL}_{q} \end{matrix}] [\begin{matrix} i_{d} \\ i_{q} \end{matrix}] + [\begin{matrix} 0 \\ ω_{e} λ_{pm} \end{matrix}]

where

v_d is the q-axis stator terminal voltage
v_q is the d-axis stator terminal voltage
i_d is the d-axis stator current
i_q is the q-axis stator current
L_d is the q-axis inductance
L_q is the d-axis inductance
p is the derivative operator
a short notation of Equation 2 is the flux linkage generated by the permanent magnets
R_S is the resistance of the stator windings
ω_e is the electrical angular velocity of the rotor

Equation 2.

\frac{d}{d t}; λ_{p m}

TIDM-02018 Definitions of Coordinate Reference Frames for PMSM Modeling

Figure 3-2 Definitions of Coordinate Reference Frames for PMSM Modeling

By using the inverse Park transformation as shown in Figure 3-2, the dynamics of the PMSM can be modeled in the α-β stationary reference frame as:

Equation 3.

[\begin{matrix} v_{α} \\ v_{β} \end{matrix}] = [\begin{matrix} R_{s} + {pL}_{d} & ω_{e} (L_{d} - L_{q}) \\ {- ω}_{e} (L_{d} - L_{q}) & R_{s} + {pL}_{q} \end{matrix}] [\begin{matrix} i_{α} \\ i_{β} \end{matrix}] + [\begin{matrix} e_{α} \\ e_{β} \end{matrix}]

Where the e_a and e_β are components of extended electromotive force (EEMF) in the α-β axis and can be defined as:

Equation 4.

[\begin{matrix} e_{α} \\ e_{β} \end{matrix}] = (λ_{pm} + (L_{d} - L_{q}) i_{d}) ω_{e} [\begin{matrix} - \sin (θ_{e}) \\ \cos (θ_{e}) \end{matrix}]

According to Equation 3 and Equation 4, the rotor position information can be decoupled from the inductance matrix by means of the equivalent transformation and the introduction of the EEMF concept, so that the EEMF is the only term that contains the rotor pole position information. And then the EEMF phase information can be directly used to realize the rotor position observation. Rewrite the IPMSM voltage Equation 5 as a state equation using the stator current as a state variable:

Equation 5.

[\begin{matrix} {\dot{i}}_{α} \\ {\dot{i}}_{β} \end{matrix}] = \frac{1}{L_{d}} [\begin{matrix} {- R}_{s} & {- ω}_{e} (L_{d} - L_{q}) \\ ω_{e} (L_{d} - L_{q}) & {- R}_{s} \end{matrix}] [\begin{matrix} i_{α} \\ i_{β} \end{matrix}] + \frac{1}{L_{d}} [\begin{matrix} {V_{α} - e}_{α} \\ {V_{β} - e}_{β} \end{matrix}]

Since the stator current is the only physical quantity that can be directly measured, the sliding surface is selected on the stator current path:

Equation 6.

S (x) = [\begin{matrix} {\hat{i}}_{α} - i_{α} \\ {\hat{i}}_{β} - i_{β} \end{matrix}] = [\begin{matrix} {\tilde{i}}_{α} \\ {\tilde{i}}_{β} \end{matrix}]

where

Equation 7 and Equation 8 are the estimated currents
the superscript ^ indicates the estimated value
the superscript ˜ indicates the variable error which refers to the difference between the observed value and the actual measurement value

Equation 7.

{\hat{i}}_{α}

Equation 8.

{\hat{i}}_{β}