Introduction

This document presents the formulation implemented on FPGA for PMSM three-phase machine with mechanical model. The three-phase to q-d park transform has the sqrt(2/3) factor. The machine can operate in both motor mode, when the mechanical torque is positive, and generating mode, when the mechanical torque is negative.

Q-d Transformation

The 3-phase to q-d transformation and the inverse used for the model are:

θ required for the q-d transformation depends on the chosen reference frame as follows:

• Rotor reference frame: θ=θr,
• Stationary reference frame: θ=0

Synchronous reference frame: θ=θe.
Since the Induction Machine is modeled in rotor reference frame, the θ required for the q-d transformation is the rotor electrical angle.

PMSM Electrical Model

Electric machine models in state space framework based on magnetic fluxes as the state variables and winding currents as the outputs can be represented as follows:

Standard DQ Motor Characteristics

In normal conditions, the ideal sinusoidal stator voltages of the PMSM, back-EMFs, and inductances all have sinusoidal shapes. One can transform the equation using the Park transformation with a referential locked on the rotor position θ using (2a) and (2b).

The Park transform (also called 'DQ' transform) reduces sinusoidal varying quantities of inductances, flux, current, and voltage to constant values in the D-Q frame thus greatly facilitating the analysis and control of the device under study.

It is important to note that there are many different types of Park transforms and this often leads to confusion when interpreting the motor states inside the D-Q frame. The one used here presents the advantage of being orthonormal (notice the sqrt(3/2) factor). This particular Park orthonormal transform is power-invariant which means that the power computed in the D-Q frame by performing a dot product of currents and voltages will be numerically equal to the one computed in the phase domain. With this transform (and only this transform) the PMSM torque can be expressed by (3), where pp is the number of pole pairs.

One may notice the absence of the 3/2 factor in (3), which is usually present in the PMSM torque equation when using non-orthonormal transforms. This is, again, because this model uses the orthonormal Park transform. Figure 1 explains the principle of the Park transform. Considering fixed ABC referential with all quantities ( Vbemf, motor current I) rotating at the electric frequency ω, if we observe these quantities in a D-Q frame turning at the same speed we can see that the motor quantities will be constant.

This is easy to see for the Back-EMF voltage Vbemf that directly follows the Q-axis (because the magnet flux is on the D-axis by definition). In Figure 3, I leads and the Q-axis by an angle called β (beta). The modulus of the vector I is called Iamp. In the figure below, θ is the rotor angle, aligned with the D-axis.

Mechanical Model

The mechanical model is the same for all machine types to calculate the rotor mechanical speed (ωm) as follows:

Nomenclature

• x: state variables vector
• y: outputs vector
• u: inputs vector
• A,B,C: state space coefficient matrices
• Ts: simulation time step
• n: time step number
• Va: stator phase a voltage
• Vb: stator phase b voltage
• Vc: stator phase c voltage
• Vsq: stator q-axis voltage
• Vsd: stator d-axis voltage
• V'rq: rotor q-axis voltage (referred to the stator)
• V'rd: rotor d-axis voltage (referred to the stator)
• I: current vector
• Isq: stator q-axis current
• Isd: stator d-axis current
• I'rq: rotor q-axis current (referred to the stator)
• I'rd: rotor d-axis current (referred to the stator)
• I'c1q: cage 1 q-axis current (referred to the stator)
• I'c1d: cage 1 d-axis current (referred to the stator)
• I'c2q: cage 2 q-axis current (referred to the stator)
• I'c2d: cage 2 d-axis current (referred to the stator)
• Ψ: magnetic flux vector
• ψsq: stator q-axis flux
• ψsd: stator d-axis flux
• ψ'rq: rotor q-axis flux (referred to the stator)
• ψ'rd: rotor d-axis flux (referred to the stator)
• ψ'c1q: cage 1 q-axis flux (referred to the stator)
• ψ'c1d: cage 1 d-axis flux (referred to the stator)
• ψ'c2q: cage 2 q-axis flux (referred to the stator)
• ψ'c2d: cage 2 d-axis flux (referred to the stator)
• R: resistance matrix
• Rs: stator winding resistance
• R'r: rotor winding resistance (referred to the stator)
• R'c1: cage 1 resistance (referred to the stator)
• R'c2: cage 2 resistance (referred to the stator)
• L: inductance matrix
• Ls: stator self inductance (mutual (Lm) +leakage (Lls) )
• L'r: rotor self inductance (referred to the stator) (mutual (Lm) +leakage (Llr) )
• L'c1: cage 1 self inductance (referred to the stator) (mutual (Lm) +leakage (Llc1) )
• L'c2: cage 2 self inductance (referred to the stator) (mutual (Lm) +leakage (Llc2) )
• Lm: mutual inductance
• Ω: speed matrix
• ω, θ: rotation speed, and position of the reference frame
• ωe, θe: rotation speed, and position of the synchronous frame
• ωr, θr: rotation speed, and position of the rotor (electrical) frame
• ωm, θm: rotation speed, and position of the rotor (mechanical) frame
• Te: electromagnetic torque
• Tm: mechanical torque
• pp: number of pole pairs
• J: rotor inertia
• Fv: viscous friction coefficient
• Fs: static friction torque

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