Mathematical Model for PMSM

This document provides a comprehensive synthesis of the mathematical foundations, state-space modeling, transfer functions, and control strategies for permanent magnet synchronous motor (PMSM) speed control.

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1. Introduction to PMSM

Permanent Magnet Synchronous Motors (PMSM) are electrical motors that use permanent magnets to produce the air gap magnetic field rather than electromagnets.

Key Advantages

• High Efficiency: No copper losses in the rotor
• High Power Density: Compact structure with high torque-to-weight ratio
• Fast Dynamic Response: Superior transient performance
• Low Maintenance: No brushes or slip rings
• Noiseless Operation: Smooth torque with minimal noise
• Extended Speed Range: Suitable for high-speed applications

Motor Structure

• Stator: Three-phase windings carrying AC current
• Rotor: Permanent magnets producing constant flux
• Power Electronics: Inverter for frequency conversion and control

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2. Coordinate Transformations

Vector control requires transforming three-phase stationary quantities into a rotating two-phase d-q frame.

2.1 Clarke Transformation

Converts (a, b, c) → (α, β):

$$\begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix}$$

• $i_α$: Real-axis component
• $i_β$: Imaginary-axis component

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2.2 Park Transformation

Converts (α, β) → (d, q) rotating frame:

$$\begin{bmatrix} i_d \\ i_q \end{bmatrix} = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} i_\alpha \\ i_\beta \end{bmatrix}$$

• θ: Rotor position
• d-axis: aligned with flux
• q-axis: perpendicular

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3. PMSM Mathematical Modeling

3.1 State-Space Electrical Model

Voltage equations:

$$v_d = R_s i_d + \frac{d\lambda_d}{dt} - \omega_r \lambda_q$$ $$v_q = R_s i_q + \frac{d\lambda_q}{dt} + \omega_r \lambda_d$$

Flux linkages:

$$\lambda_q = L_q i_q, \qquad \lambda_d = L_d i_d + \lambda_{af}$$

Current dynamics:

$$\frac{di_d}{dt} = \frac{v_d - R_s i_d + \omega_r L_q i_q}{L_d}$$ $$\frac{di_q}{dt} = \frac{v_q - R_s i_q + \omega_r L_d i_d - \omega_r \lambda_{af}}{L_q}$$

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3.2 Electromagnetic Torque

$$T_e = \frac{3}{2}\frac{P}{2}(\lambda_d i_q - \lambda_q i_d)$$

For surface-mounted PMSM ($L_d=L_q$):

$$T_e = \frac{3}{2} P \lambda_{af} i_q$$

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3.3 Mechanical Dynamics

$$T_e = T_l + B\omega_r + J \frac{d\omega_r}{dt}$$

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4. Example Motor Parameters

Parameter	Description	Value	Unit
$R_s$	Stator Resistance	0.2	Ω
$L_d$	Direct axis Inductance	8.5	mH
$L_q$	Quadrature axis Inductance	8.5	mH
$P$	Number of Poles	8	-
$λ_{af}$	Flux	0.175	Wb
$B$	Friction	0.005	N·m·s
$J$	Intertia	0.089	kg·m²
Rated Speed	Rated Speed	1000	RPM

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9. References

• Krishnan, R. Permanent Magnet Synchronous and Brushless DC Motor Drives (2010)
• Pillay, P. & Krishnan, R. Modeling of Permanent Magnet Motor Drives (1989)
• Maamoun, A., et al. Fuzzy Logic Based Speed Controller for PMSM Drive (2013)
• Wang, Z., et al. SVPWM Techniques and Applications in HTS PMSM Machines Control (2010)