Linear Model (Electric Motor Simulation Toolkit)
- Updated2023-02-21
- 4 minute(s) read
Linear Model (Electric Motor Simulation Toolkit)
The switched reluctance motor (SRM) shows strong nonlinearity because the flux of a stator winding depends on both the phase current and the rotor position. Therefore, the linear model utilizes the piece-wise linear method to describe the flux of an SRM.
In an SRM, the non-linear characteristics of the magnet affect the operation of the SRM. High levels of saturation occur cyclically as the rotor continuously moves from an unaligned position to an aligned position, with reference to the poles on the stator. The following figure depicts different rotor positions.
The positions are specific to one of the phases. The aligned position means that the pole of the rotor and the pole of the stator are in the same line. The unaligned position means that the rotor of the specific phase is completely detached from the stator. The middle position is when the pole of the rotor has intersection with the pole of the stator. The flux linkage of the motor phase varies at different positions.
The SRM linear model function complies with the following equations.
Phase voltage equation:
Instant torque equation:
| where | λ is the flux linkage |
| θ is the rotor position | |
| R is the resistance | |
| I is the current | |
| V is the voltage | |
| T is the torque |
The following figure shows an analytical expression for the flux linkage of a linear model. The aligned and unaligned curves represent the flux changes at the aligned position and unaligned position. The two curves in the middle represent the flux changes at the middle positions.
In the following figure, the curves with dots show the flux linkage of a real motor at different rotor position. The curves without dots represent the linear model approximation.
The slope of the straight line in the figure approximates the flux curve for the unaligned position, as shown in the following equation.
where Lu is a constant that represents the equivalent inductance of the coil for the unaligned position. The u subscript means unaligned.
For aligned positions, when I is no greater than Is,
where λa is the flux of Is at the aligned position. The a subscript means aligned and the s subscript means saturated.
When I is greater than Is,
The dependency of λ on the rotor position θ is approximated by a sine wave oscillating between the maximum value λa(I) and the minimum value λu(I), as illustrated in the following equation.
where p is the number of poles in pairs.
From the previous equations, a linear model function solves the current and torque in the SRM.
When I is no greater than Is,
where the k subscript indicates the variable calculated at the k step, while the k-1 subscript indicates the variable calculated at the k-1 step. For example, Ik is the phase current at the k step, while Ik-1 is the phase current at the k-1 step.
When I is greater than Is,
