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SPRUJF4 October 2024

3.1.2.1.1 Mathematical Model and FOC Structure of an IPMSM

The sensorless FOC structure for an IPMSM is illustrated in Figure 3-12. 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.

TIEVM-MTR-HVINV Sensorless FOC Structure of an IPMSM System

Figure 3-12 Sensorless FOC Structure of an IPMSM System

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 20. The dynamic model of a generic PMSM can be written in the d-q rotor reference frame as:

Equation 20.

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

where

$v_{d}$ and $v_{q}$ are the q-axis and d-axis stator terminal voltages, respectively
$i_{d}$ and $i_{q}$ are the d-axis and q-axis stator currents, respectively
$L_{d}$ and $L_{q}$ are the q-axis and d-axis inductances, respectively
p is the derivative operator, a short notation of $\frac{d}{d t}$
$λ_{p m}$ 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

TIEVM-MTR-HVINV Definitions of Coordinate Reference Frames for PMSM Modeling

Figure 3-13 Definitions of Coordinate Reference Frames for PMSM Modeling

By using the inverse Park transformation as shown in Figure 3-13, the dynamics of the PMSM can be modeled in the α-β stationary reference frame as shown in Equation 21:

Equation 21.

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

where

$e_{α}$ and $e_{β}$ are components of extended electromotive force (EEMF) in the α-β axis and can be defined as shown in Equation 22:

Equation 22.

[\begin{matrix} e_{α} \\ e_{β} \end{matrix}] = (λ_{p m} + (L_{d} - L_{q}) i_{d}) ω_{e} [\begin{matrix} - s i n (θ_{e}) \\ c o s (θ_{e}) \end{matrix}]

According to Equation 21 and Equation 22, 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 Equation 21 as a state equation using the stator current as a state variable:

Equation 23.

[\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 24.

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

${\hat{i}}_{α}$ and ${\hat{i}}_{β}$ 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