ee_generateIdealPMSMfluxData

Generate tabulated flux linkage data for ideal PMSM

Syntax

[F,T,dFdA,dFdB,dFdC,dFdX] = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,A,B,C,X)

F = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,A,B,C,X)

[F,T,dFdA,dFdB,dFdC,dFdX] = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,D,Q,X)

F = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,D,Q,X)

Description

[F,T,dFdA,dFdB,dFdC,dFdX] = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,A,B,C,X) generates 4-D flux linkage data, including torque and partial derivatives, for an ideal permanent magnet synchronous motor (PMSM).

Use this function to create test data for the FEM-Parameterized PMSM block, either for validation purposes or to set up a model before the actual flux linkage data is available.

F = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,A,B,C,X) generates 4-D flux linkage matrix F for an ideal PMSM.

[F,T,dFdA,dFdB,dFdC,dFdX] = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,D,Q,X) generates 3-D flux linkage data, including torque and partial derivatives, for an ideal PMSM.

F = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,D,Q,X) generates 3-D flux linkage matrix F for an ideal PMSM.

Examples

Generate 4-D Flux Linkage Data

Specify the motor parameters.

PM = 0.1;     % Permanent magnet flux
N = 6;        % Number of pole pairs
Ld = 0.0002;  % D-axis inductance
Lq = 0.0002;  % Q-axis inductance
L0 = 0.00018; % Zero-sequence inductance
Rs = 0.013;   % Stator resistance

Define the phase current vectors.

iA = linspace(-250,250,5);
iB = iA;
iC = iA;

Specify the rotor angle vector based on the number of pole pairs.

X = pi/180*linspace(0,360/N,180/N+1);

Tabulate flux linkage partial derivatives and torque in terms of A-,B-,C-currents and rotor angle.

[F,T,dFdA,dFdB,dFdC,dFdX] = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,iA,iB,iC,X);

The function returns a 4-D flux linkage matrix F, a 4-D torque matrix T, and four 4-D matrices for the flux linkage partial derivatives. The four partial derivative matrices correspond to the three phase currents and the rotor angle, respectively. The matrix dimensions correspond to the three phase currents and the rotor angle.

Generate 4-D Flux Linkage Matrix F

Specify the motor parameters.

PM = 0.1;     % Permanent magnet flux
N = 6;        % Number of pole pairs
Ld = 0.0002;  % D-axis inductance
Lq = 0.0002;  % Q-axis inductance
L0 = 0.00018; % Zero-sequence inductance
Rs = 0.013;   % Stator resistance

Define the phase current vectors.

iA = linspace(-250,250,5);
iB = iA;
iC = iA;

Specify the rotor angle vector based on the number of pole pairs.

X = pi/180*linspace(0,360/N,180/N+1);

Tabulate flux linkage partial derivatives and torque in terms of A-,B-,C-currents and rotor angle

F = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,iA,iB,iC,X);

The function returns a 4-D flux linkage matrix F. The matrix dimensions correspond to the three phase currents and the rotor angle

Generate 3-D Flux Linkage Data

Specify the motor parameters.

PM = 0.1;     % Permanent magnet flux
N = 6;        % Number of pole pairs
Ld = 0.0002;  % D-axis inductance
Lq = 0.0002;  % Q-axis inductance
L0 = 0.00018; % Zero-sequence inductance
Rs = 0.013;   % Stator resistance

Define the d-axis and q-axis current vectors.

iD = linspace(-250,250,5);
iQ = iD;

Specify the rotor angle vector based on the number of pole pairs.

X = pi/180*linspace(0,360/N,180/N+1);

Tabulate flux linkage partial derivatives and torque in terms of d-axis and q-axis currents and rotor angle.

[F,T,dFdA,dFdB,dFdC,dFdX] = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,iD,iQ,X);

The function returns a 3-D flux linkage matrix F, a 3-D torque matrix T, and four 3-D matrices for the flux linkage partial derivatives. The four partial derivative matrices correspond to the three phase currents and the rotor angle, respectively. The matrix dimensions correspond to the d-axis and q-axis currents and the rotor angle.

Generate 3-D Flux Linkage Matrix F

Specify the motor parameters.

PM = 0.1;     % Permanent magnet flux
N = 6;        % Number of pole pairs
Ld = 0.0002;  % D-axis inductance
Lq = 0.0002;  % Q-axis inductance
L0 = 0.00018; % Zero-sequence inductance
Rs = 0.013;   % Stator resistance

Define the d-axis and q-axis current vectors.

iD = linspace(-250,250,5);
iQ = iD;

Specify the rotor angle vector based on the number of pole pairs.

X = pi/180*linspace(0,360/N,180/N+1);

Tabulate flux linkage partial derivatives and torque in terms of d-axis and q-axis currents and rotor angle.

F = ee_generateIdealPMSMfluxData(PM,Ld,Lq,L0,iD,iQ,X);

The function returns a 3-D flux linkage matrix F. The matrix dimensions correspond to the d-axis and q-axis currents and the rotor angle.

Input Arguments

PM — Peak permanent magnet flux linkage, in weber-turns

scalar

Peak permanent magnet flux linkage, in weber-turns, specified as a scalar.

Data Types: double

Ld — D-axis inductance, in henries

scalar

D-axis inductance, in henries, specified as a scalar.

Data Types: double

Lq — Q-axis inductance, in henries

scalar

Q-axis inductance, in henries, specified as a scalar.

Data Types: double

L0 — Zero-sequence inductance, in henries

scalar

Zero-sequence inductance, in henries, specified as a scalar.

Data Types: double

A — A-phase current, in amperes

vector

A-phase current, in amperes, specified as a vector. The vector must be monotonically increasing and two-sided (contain both positive and negative values). Best practice is to include zero current as one of the points. Use this input argument to generate 4-D flux linkage data.

Data Types: double

B — B-phase current, in amperes

vector

B-phase current, in amperes, specified as a vector. The vector must be monotonically increasing and two-sided (contain both positive and negative values). Best practice is to include zero current as one of the points. Use this input argument to generate 4-D flux linkage data.

Data Types: double

C — C-phase current, in amperes

vector

C-phase current, in amperes, specified as a vector. The vector must be monotonically increasing and two-sided (contain both positive and negative values). Best practice is to include zero current as one of the points. Use this input argument to generate 4-D flux linkage data.

Data Types: double

D — D-axis current, in amperes

vector

D-axis current, in amperes, specified as a vector. The vector must be monotonically increasing and two-sided (contain both positive and negative values). Best practice is to include zero current as one of the points. Use this input argument to generate 3-D flux linkage data.

Data Types: double

Q — Q-axis current, in amperes

vector

Q-axis current, in amperes, specified as a vector. The vector must be monotonically increasing and two-sided (contain both positive and negative values). Best practice is to include zero current as one of the points. Use this input argument to generate 3-D flux linkage data.

Data Types: double

X — Rotor angle, in radians

vector

The rotor angle, in radians, specified as a vector. The values must be in the range from zero to 2π/N, where N is the number of pole pairs.

Data Types: double

Output Arguments

F — Flux linkage

matrix

The flux linkage, in weber-turns, returned as a matrix. The matrix can be four-dimensional or three-dimensional, depending on the syntax used to call the function. In a four-dimensional matrix, the first three dimensions correspond to the phase currents, and the fourth dimension corresponds to the rotor angle. In a three-dimensional matrix, the first two dimensions correspond to the d-axis and q-axis currents, and the third dimension corresponds to the rotor angle.

T — Torque

matrix

Torque, in N*m, returned as a matrix. The matrix can be four-dimensional or three-dimensional, depending on the syntax used to call the function. In a four-dimensional matrix, the first three dimensions correspond to the phase currents, and the fourth dimension corresponds to the rotor angle. In a three-dimensional matrix, the first two dimensions correspond to the d-axis and q-axis currents, and the third dimension corresponds to the rotor angle.

dFdA — Flux linkage partial derivative with respect to the A-phase current

matrix

Flux linkage partial derivative with respect to the A-phase current, returned as a matrix. The matrix can be four-dimensional or three-dimensional, depending on the syntax used to call the function. In a four-dimensional matrix, the first three dimensions correspond to the phase currents, and the fourth dimension corresponds to the rotor angle. In a three-dimensional matrix, the first two dimensions correspond to the d-axis and q-axis currents, and the third dimension corresponds to the rotor angle.

dFdB — Flux linkage partial derivative with respect to the B-phase current

matrix

Flux linkage partial derivative with respect to the B-phase current, returned as a matrix. The matrix can be four-dimensional or three-dimensional, depending on the syntax used to call the function. In a four-dimensional matrix, the first three dimensions correspond to the phase currents, and the fourth dimension corresponds to the rotor angle. In a three-dimensional matrix, the first two dimensions correspond to the d-axis and q-axis currents, and the third dimension corresponds to the rotor angle.

dFdC — Flux linkage partial derivative with respect to the C-phase current

matrix

Flux linkage partial derivative with respect to the C-phase current, returned as a matrix. The matrix can be four-dimensional or three-dimensional, depending on the syntax used to call the function. In a four-dimensional matrix, the first three dimensions correspond to the phase currents, and the fourth dimension corresponds to the rotor angle. In a three-dimensional matrix, the first two dimensions correspond to the d-axis and q-axis currents, and the third dimension corresponds to the rotor angle.

dFdX — Flux linkage partial derivative with respect to the rotor angle

matrix

Flux linkage partial derivative with respect to the rotor angle, returned as a matrix. The matrix can be four-dimensional or three-dimensional, depending on the syntax used to call the function. In a four-dimensional matrix, the first three dimensions correspond to the phase currents, and the fourth dimension corresponds to the rotor angle. In a three-dimensional matrix, the first two dimensions correspond to the d-axis and q-axis currents, and the third dimension corresponds to the rotor angle.

Algorithms

The flux linking each winding has contributions from the permanent magnet plus the three windings. Therefore, the total flux is given by [1]:

$$\left[\begin{array}{c}{\psi }_a\\ \begin{array}{l}{\psi }_b\\ {\psi }_c\end{array}\end{array}\right]=\left[\begin{array}{ccc}{L}_{aa}& L_{ab}& L_{ac}\\ L_{ba}& L_{bb}& L_{bc}\\ L_{ca}& L_{cb}& L_{cc}\end{array}\right]\left[\begin{array}{c}{i}_a\\ \begin{array}{l}{i}_b\\ i_c\end{array}\end{array}\right]+\left[\begin{array}{c}{\psi }_{am}\\ \begin{array}{l}{\psi }_{bm}\\ {\psi }_{cm}\end{array}\end{array}\right]$$

$$\begin{array}{l}{L}_{aa}=L_s+L_m\cos \left(2{\theta }_r\right)\\ L_{bb}=L_s+L_m\cos \left(2\left({\theta }_r-2\pi /3\right)\right)\\ L_{cc}=L_s+L_m\cos \left(2\left({\theta }_r+2\pi /3\right)\right)\\ L_{ab}=L_{ba}=-M_s-L_m\cos \left({\theta }_r+\pi /6\right)\\ L_{bc}=L_{cb}=-M_s-L_m\cos \left({\theta }_r+\pi /6-2\pi /3\right)\\ L_{ca}=L_{ac}=-M_s-L_m\cos \left({\theta }_r+\pi /6+2\pi /3\right)\\ {\psi }_{am}={\psi }_m\cos {\theta }_e\\ {\psi }_{bm}={\psi }_m\cos \left({\theta }_e-2\pi /3\right)\\ {\psi }_{bm}={\psi }_m\cos \left({\theta }_e+2\pi /3\right)\end{array}$$

Here, Θ_e is the electrical angle, which is related to rotor angle Θ_r by Θ_e = N·Θ_r. The function assumes that the permanent magnet flux linking the A-phase winding is at the maximum for Θ_e = 0.

The function output F corresponds to ψ_a tabulated as a function of A-phase current, B-phase current, C-phase current, and rotor angle.

Ls, Lm, and Ms are related to input arguments Ld, Lq, and L0 by:

$$\begin{array}{l}{L}_s=\frac{L_0}{3}+\frac{L_d}{3}+\frac{L_q}{3}\\ M_s=\frac{L_d}{6}-\frac{L_0}{3}+\frac{L_q}{6}\\ L_m=\frac{L_d}{3}-\frac{L_q}{3}\end{array}$$

References

[1] Anderson, P.M. Analysis of Faulted Power Systems. 1st Edition. Wiley-IEEE Press, July 1995, p.187.

Version History

Introduced in R2017a