METHOD FOR DETERMINING THE ELECTRIC POTENTIALS OF THE PHASES OF A POLYPHASE MOTOR

Abstract

A method (100) for determining potentials (V.sub.i) across the terminals of the N phases of a motor, N being an integer greater than or equal to 4, comprising: determining (102) reference potentials (V.sub.REF_i) across the terminals of the N phases for a defined drive voltage V.sub.MAG (101) of phase , with i between 1 and N and

[00001]$V_{\mathrm{REF}\_i}=V_{\mathrm{MAG}}\cos \left(-\frac{2}{N}\left(i-1\right)\right)$ comparing (103) the reference potentials to a first threshold (seuil 1); if the reference potentials are all less than or equal to the first threshold (104), then the potentials across the terminals of the N phases are equal to the reference potentials, or if at least one of the reference potentials is greater than the first threshold, comparing (105) these potentials to a second threshold (seuil 2) greater than the first threshold; if these reference potentials are all less than or equal to the second threshold (106), then the potentials across the terminals of the N phases are V.sub.i=V.sub.REF_i+V.sub.H, with

[00002]$V_H=\frac{\underset{k=1.Math.N}{\max }\left(V_{{\mathrm{REF}}_k}\right)+\underset{k=1.Math.N}{\min }\left(V_{{\mathrm{REF}}_k}\right)}{2}$ or if at least one of the potentials is greater than the second threshold (107), then the potentials across the terminals of the N phases are V.sub.i=V.sub.REF_i+V.sub.S_i+V.sub.HN, with V.sub.HN and V.sub.S_i dependent on N and on the reference potentials.

Claims

1. A method, implemented by an electric controller, for determining electrical potentials (V.sub.i) to be produced across the terminals of the phases of a motor comprising N phases, N being an integer greater than or equal to 4, the method comprising the following steps: defining a drive voltage of amplitude V.sub.MAG and of phase which is less than a limit voltage; determining reference electrical potentials across the terminals of the N phases of the motor for this drive voltage with $V_{\mathrm{REF}\_i}=V_{\mathrm{MAG}}\cos \left(-\frac{2}{N}\left(i-1\right)\right)$ where i is between 1 and N, V.sub.REF_i is the reference electrical potential across the terminals of the i-th phase and is the phase of the drive voltage between 0 and 2; comparing the reference electrical potentials to a first threshold; if the reference electrical potentials are all less than or equal to the first threshold, then the electrical potentials to be produced across the terminals of the N phases of the motor are equal to the reference electrical potentials, or if at least one of the reference electrical potentials is greater than the first threshold, comparing these reference electrical potentials to a second threshold, the second threshold being greater than the first threshold; if these reference electrical potentials are all less than or equal to the second threshold, then the electrical potentials to be produced across the terminals of the N phases of the motor are equal to V.sub.i=V.sub.REF_i+V.sub.H, with $V_H=\frac{\underset{k=1\_N}{\max }\left(V_{{\mathrm{REF}}_k}\right)+\underset{k=1\_N}{\min }\left(V_{{\mathrm{REF}}_k}\right)}{2}$ or if at least one of the reference electrical potentials is greater than the second threshold, then the electrical potentials to be produced across the terminals of the N phases of the motor are equal to V.sub.i=V.sub.REF_i+V.sub.S_i+V.sub.HN, with V.sub.HN a homopolar potential dependent on the number of phases and on the reference electrical potentials and V.sub.S_i a secondary potential dependent on the number of phases and on the reference electrical potentials.

2. The determining method as claimed in claim 1, wherein when at least one of the reference potentials is greater than the second threshold, the secondary and homopolar potentials are expressed by the following functions: $\left(\begin{array}{c}{V}_{S_1}\\ .Math.\\ V_{S_N}\end{array}\right)=T\left(:,3:N-1\right)\left(\begin{array}{c}{V}_{{\mathrm{SAB}}_1}\\ .Math.\\ V_{{\mathrm{SAB}}_{N-1}}\end{array}\right)$ with: if N is odd, with p=(N1)/2: $V_{\mathrm{SABI}}\left(k\right)=\frac{\left[U\left(k+2\right)V\left(2\right)-U\left(2\right)V\left(k+2\right)\right]\left[V\left(k+2\right)U\left(1\right)-V\left(1\right)U\left(k+2\right)\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF}\_N}\end{array}\right)$ with k ranging from 1 to N3, and $V_{\mathrm{HN}}=\frac{1}{\sqrt{2}}\frac{\left[U\left(N\right)V\left(2\right)-U\left(2\right)V\left(N\right)\right]\left[V\left(N\right)U\left(1\right)-V\left(1\right)U\left(N\right)\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF}\_2}\end{array}\right)$ U a vector with N rows defined by U=T.sup.1u and u a vector with N rows defined by a value 1 for its indices iP, a value 0 for its index iM and a value 1 for its indices iN; V a vector with N rows defined by V=T.sup.1v and v a vector with N rows defined by a value 0 for its indices iP and iN and a value 1 for its index iM; iP the p indices of the p greatest values of the reference electrical potentials; iN the p indices of the p smallest values of the reference electrical potentials; iM the index of the median value of the reference electrical potentials; i varying from 1 to N, and T a transformation matrix defined by: the odd columns from 1 to 2 [(N1)/2] written 2k1 (k varying from 1 to [(N1)/2]), correspond to the vector having as i-th component (with i varying from 1 to N): c(i, 2k1)=cos(2k(i1)/N); the even columns from 1 to 2 [(N1)/2 ] written 2k (k varying from 1 to [(N1)/2]) correspond to the vector having as i-th component (with i varying from 1 to N): if N is odd, the column N is defined by a constant vector with the value 1/2; if N is even, the column N1 is defined by a vector of alternate constants of value 1/2 and of value 1/2 and the column N is defined by a constant vector with the value 1/2. If N is even, with p=N/2: V.sub.HN=0 and V.sub.SAB_i(k) is defined by: If k varies from 1 to p2: $V_{\mathrm{SABI}}\left(2+k+2.Math.\left(k-1\right)/2.Math.\right)=Q^{-1}.Math.\left(\begin{array}{c}{V}_M-V_{\mathrm{REF}\mathrm{IP}\left(1\right)}\\ .Math.\\ V_M-V_{\mathrm{REF}\mathrm{IP}\left(2\right)}\end{array}\right)$ For other values of k between 1 and N3: V.sub.SAB_i(k)=0; with $V_M=-\frac{V_{{\mathrm{REF}}_{\mathrm{ip}\left(p-1\right)}}-p_M\left(\begin{array}{c}{V}_{\mathrm{REF}\_\mathrm{iP}\left(1\right)}\\ .Math.\\ V_{\mathrm{REF}\_\mathrm{iP}\left(p-2\right)}\end{array}\right)}{1-p_M\left(\begin{array}{c}1\\ .Math.\\ 1\end{array}\right)}$ $p_M={\mathrm{qQ}}^{-1};$ iP the p1 indices of the p1 greatest values of the reference electrical potentials for i varying from 1 to N; q the row vector formed of the elements of the row of the transformation matrix T corresponding to the last index of iP corresponding to the columns indexed $4+k+2.Math.\frac{k-1}{2}.Math.$ where k varies from 1 to p2; and Q the matrix formed of the elements of the transformation matrix T at the intersection of the rows corresponding to the p2 first indices of the iP and of the columns indexed $4+k+2.Math.\frac{k-1}{2}.Math.$ where k varies from 1 to p2.

3. The determining method according to claim 1, wherein the limit voltage and the second threshold are functions dependent on the number of phases of the motor and on the electrical potential of an electrical power supply of the motor.

4. The determining method according to claim 1, wherein the first threshold is a function dependent on the electrical potential of the electrical power supply of the motor.

5. The determining method according to claim 1, wherein the second threshold is expressed by the following function: $\mathrm{seuil}2=\frac{V_{\mathrm{DC}}}{2}\frac{1}{\sin \left(\frac{}{N}+\frac{}{N}.Math..Math.\frac{N}{2}.Math.\right)}$

6. The determining method according to claim 1, wherein the limit voltage is expressed by the following function: $V_{\mathrm{LIM}}=\frac{V_{\mathrm{DC}}}{2}\frac{2}{N}\cos \left(\right)\underset{j=0}{\overset{N-1}{.Math.}}.Math.\cos \left(\frac{2}{N}j\right).Math.$ with =/(2N) if N is odd or =0 if N is even and a multiple of 4 or =/N if N is even and a non-multiple of 4.

7. The determining method according to claim 1, wherein the first threshold is equal to V.sub.DC/2.

8. The determining method according to claim 1, wherein the motor is a five-phase motor, N is equal to 5, the first threshold is equal to V.sub.DC/2, the second threshold is equal to 1.05V.sub.DC/2 and the limit voltage is equal to 1.23V.sub.DC/2 with V.sub.DC the electrical potential of the electrical power supply of the motor, and if at least one of the reference electrical potentials is greater than the second threshold, then the electrical potentials to be produced across the terminals of the 5 phases of the motor are equal to V.sub.i(i=p)=V.sub.M for p, such that the reference electrical potentials V.sub.REF_p are the two largest reference electrical potentials, V.sub.i(i=l)=V.sub.M for l, such that the reference electrical potentials V.sub.REF_l are the two smallest reference electrical potentials or V.sub.i(i=j)=V.sub.C for j, such that the reference electrical potential V.sub.REF_j is the median reference electrical potential, with: $V_M=\frac{3\sqrt{5}-5}{4}\left(\underset{k=1...5}{\max }\left(V_{\mathrm{REF}k}\right)-\underset{k=1...5}{\min }\left(V_{\mathrm{REF}k}\right)\right)$ $\mathrm{and}$ $V_C=\frac{5}{2}\underset{k=1...5}{\mathrm{median}}\left(V_{\mathrm{REF}k}\right)$ with p, j and l chosen from among {1; 2; 3; 4; 5} and pjl.

9. An electric controller intended to be connected to a motor with N phases and to an electrical power supply source, with N an integer greater than or equal to 4, the controller being configured to define a drive voltage of amplitude V.sub.MAG and of phase which is less than a limit voltage; determine reference electrical potentials across the terminals of the N phases of the motor for the drive voltage with $V_{\mathrm{REF\_t}}=V_{\mathrm{MAG}}\cos \left(-\frac{2}{N}\left(i-1\right)\right)$ where i is between 1 and N, V.sub.REF_i is the reference electrical potential across the terminals of the i-th phase and is the phase of the drive voltage between 0 and 2; compare the reference electrical potentials to a first threshold; if the reference electrical potentials are all less than or equal to the first threshold, then the electrical potentials to be produced across the terminals of the N phases of the motor are equal to the reference electrical potentials, or if at least one of the reference electrical potentials is greater than the first threshold, compare these reference electrical potentials to a second threshold, the second threshold being greater than the first threshold; if these reference electrical potentials are all less than or equal to the second threshold, then the electrical potentials to be produced across the terminals of the N phases of the motor are equal to V.sub.i=V.sub.REF_i+V.sub.H, with $V_H=\frac{\underset{k=1...N}{\max }\left(V_{{\mathrm{REF}}_k}\right)+\underset{k=1...N}{\min }\left(V_{{\mathrm{REF}}_k}\right)}{2}$ or if at least one of the reference potentials is greater than the second threshold, then the potentials to be produced across the terminals of the N phases of the motor are equal to V.sub.i=V.sub.REF_i+V.sub.S_i+V.sub.HN, with V.sub.HN a homopolar potential dependent on the number of phases and on the reference electrical potentials and V.sub.S_i a secondary potential dependent on the number of phases and on the reference electrical potentials.

10. The electric controller as claimed in claim 9, wherein when at least one of the reference potentials is greater than the second threshold, the secondary and homopolar potentials are expressed by the following functions: $\left(\begin{array}{c}{V}_{S_1}\\ .Math.\\ V_{S_q}\end{array}\right)=T\left(:,3:N-1\right)\left(\begin{array}{c}{V}_{{\mathrm{SAB}}_1}\\ .Math.\\ V_{{\mathrm{SAB}}_{q.Math.3}}\end{array}\right)$ with: if N is odd, with p=(N1)/2: $V_{\mathrm{SABi}}\left(k\right)=\frac{\left[U\left(k+2\right)V\left(2\right)-U\left(2\right)V\left(k+2\right)\right]\left[V\left(k+2\right)U\left(1\right)-V\left(1\right)U\left(k+2\right)\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF\_N}}\end{array}\right)$ with k ranging from 1 to N3, and $V_{\mathrm{HN}}=\frac{1}{\sqrt{2}}\frac{\left[U\left(N\right)V\left(2\right)-U\left(2\right)V\left(N\right)\right]\left[V\left(N\right)U\left(1\right)-V\left(1\right)U\left(N\right)\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF\_N}}\end{array}\right)$ U a vector with N rows defined by U=T.sup.1u and u a vector with N rows defined by a value 1 for its indices iP, a value 0 for its index iM and a value 1 for its indices iN; V a vector with N rows defined by V=T.sup.1v and v a vector with N rows defined by a value 0 for its indices iP and iN and a value 1 for its index iM; iP the p indices of the p greatest values of the reference electrical potentials; iN the p indices of the p smallest values of the reference electrical potentials; iM the index of the median value of the reference electrical potentials; i varying from 1 to N, and T a transformation matrix defined by: the odd columns from 1 to 2 [(N1)/2] written 2k1 (k varying from 1 to [(N1)/2]), correspond to the vector having as i-th component (with i varying from 1 to N): c(i, 2k1)=cos(2k(i1)/N); the even columns from 1 to 2[(N1)/2 ] written 2k (k varying from 1 to [(N1)/2]) correspond to the vector having as i-th component (with i varying from 1 to N): if N is odd, the column N is defined by a constant vector with the value if N is even, the column N1 is defined by a vector of alternate constants of value 1/2 and of value 1/2 and the column N is defined by a constant vector with the value 1/2. If N is even, with p=N/2: V.sub.H=0 and V.sub.SAB_i(k) is defined by: If k varies from 1 to p2: $V_{\mathrm{SABi}}\left(2+k+2.Math.\left(k-1\right)/2.Math.\right)=Q^{-1}.Math.\left(\begin{array}{c}{V}_M-V_{\mathrm{REF}\mathrm{iP}\left(1\right)}\\ .Math.\\ V_M-V_{\mathrm{REF}\mathrm{iP}\left(p-2\right)}\end{array}\right)$ For other values of k between 1 and N3: V.sub.SAB_i(k)=0; with $V_M=-\frac{V_{{\mathrm{REF}}_{\mathrm{iP}\left(p-1\right)}}-p_M\left(\begin{array}{c}{V}_{\mathrm{REF\_iP}\left(1\right)}\\ .Math.\\ V_{\mathrm{REF\_iP}\left(p-2\right)}\end{array}\right)}{1-p_M\left(\begin{array}{c}1\\ .Math.\\ 1\end{array}\right)}$ ${}_M={\mathrm{qQ}}^{-1};$ iP the p1 indices of the p1 greatest values of the reference electrical potentials for i varying from 1 to N; q the row vector formed of the elements of the row of the transformation matrix T corresponding to the last index of iP corresponding to the columns indexed $4+k+2.Math.\frac{k-1}{2}.Math.$ where k varies from 1 to p2; and Q the matrix formed of the elements of the transformation matrix T at the intersection of the rows corresponding to the p2 first indices of the iP and of the columns indexed $4+k+2.Math.\frac{k-1}{2}.Math.$ where k varies from 1 to p2.

11. The electric controller according to claim 9, wherein the limit voltage and the second threshold are functions dependent on the number of phases of the motor and on the electrical potential of the electrical power supply intended to be connected to the electric controller.

12. The electric controller according to claim 9, wherein the second threshold is expressed by the following function: $\mathrm{seuil}2=\frac{V_{\mathrm{DC}}}{2}\frac{1}{\sin \left(\frac{}{N}+\frac{}{N}.Math..Math.\frac{N}{2}.Math.\right)}$ with V.sub.DC the voltage of the electrical power supply intended to be connected to the electric controller and N the number of phases of the motor intended to be connected to the electric controller.

13. The electric controller according to claim 9, wherein the limit voltage is expressed by the following function: $V_{\mathrm{LIM}}=\frac{V_{\mathrm{DC}}}{2}\frac{2}{N}\cos \left(\right)\underset{i=0}{\overset{N-1}{.Math.}}.Math.\cos \left(\frac{2}{N}j\right).Math.$ with =/(2N) if N is odd or =0 if N is even and a multiple of 4 or =/N if N is even and a non-multiple of 4, and with V.sub.DC the voltage of the electrical power supply intended to be connected to the electric controller and N the number of phases of the motor intended to be connected to the electric controller.

14. An electric device comprising: an electrical power supply source; an electric controller according to claim 9, an input of which is connected to the electrical power supply source; a power converter, inputs of which are connected to the electric controller; a motor with N phases connected to the power converter, N being an integer greater than or equal to 4, and the power converter being configured to provide electrical potentials as input to the N phases of the motor.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0089] Other features and advantages of this invention will become apparent from the description given below, with reference to the appended drawings which illustrate exemplary embodiments thereof without any limitation.

[0090] FIG. 1 represents, schematically and partially, a flowchart of the method for determining electrical potentials to be produced across the terminals of the phases of a motor with several phases according to an embodiment of the invention.

[0091] FIG. 2A represents the reference electrical potential V.sub.REF_i across the terminals of the five phases of a five-phase motor before the application of the method of the invention for a drive voltage V.sub.MAG of amplitude equal to 90% of V.sub.DC/2.

[0092] FIG. 2B represents the reference electrical potential V.sub.REF_i across the terminals of the five phases of a five-phase motor before the application of the method of the invention, and the homopolar potential V.sub.H as described in the method of the invention for a drive voltage V.sub.MAG of amplitude equal to 100% of V.sub.DC/2.

[0093] FIG. 2C represents the reference electrical potential V.sub.REF_i across the terminals of the five phases of a five-phase motor before the application of the method of the invention, and the electrical potential to be produced V.sub.i across the terminals of the five phases of the motor after the application of the method of the invention, as well as the possible values of the homopolar potential V.sub.H for a drive voltage V.sub.MAG of amplitude equal to 105% of V.sub.DC/2.

[0094] FIG. 3A represents the reference electrical potential V.sub.REF_i across the terminals of the five phases of a five-phase motor before the application of the method of the invention, and the electrical potential to be produced V.sub.i across the terminals of the five phases of the motor after the application of the method of the invention, as well as the possible values of the homopolar potential V.sub.HN and of the components V.sub.SAB_i of the secondary potential V.sub.S_i for a drive voltage V.sub.MAG of amplitude equal to 110% of V.sub.DC/2.

[0095] FIG. 3B represents the reference electrical potential V.sub.REF_i across the terminals of the five phases of a five-phase motor before the application of the method of the invention, and the electrical potential to be produced V.sub.i across the terminals of the five phases of the motor after the application of the method of the invention, as well as the possible values of the homopolar potential V.sub.HN and of the components V.sub.SAB_i of the secondary potential V.sub.S_i for a drive voltage V.sub.MAG of amplitude equal to 123% of V.sub.DC/2, i.e. equal to the limit voltage V.sub.LIM.

[0096] FIG. 4 represents, schematically and partially, an electric device according to an embodiment of the invention.

[0097] FIG. 5A represents the reference electrical potential V.sub.REF_i across the terminals of the six phases of a six-phase motor before the application of the method of the invention for a drive voltage V.sub.MAG of amplitude equal to 90% of V.sub.DC/2.

[0098] FIG. 5B represents the reference electrical potential V.sub.REF_i across the terminals of the six phases of a six-phase motor before the application of the method of the invention, and the electrical potential to be produced V.sub.i across the terminals of the six phases of the motor after the application of the method of the invention, as well as the possible values of the component V.sub.SAB_i (3) of the secondary potential V.sub.S_i for a drive voltage V.sub.MAG of amplitude equal to 105% of V.sub.DC/2.

[0099] FIG. 5C represents the reference electrical potential V.sub.REF_i across the terminals of the six phases of a six-phase motor before the application of the method of the invention, and the electrical potential to be produced V.sub.i across the terminals of the six phases of the motor after the application of the method of the invention, as well as the possible values of the component V.sub.SAB_i(3) of the secondary potential V.sub.S_i for a drive voltage V.sub.MAG of amplitude equal to 115.7% of V.sub.DC/2.

DESCRIPTION OF THE EMBODIMENTS

[0100] In the remainder of the description, in order to simplify the writing, the terms reference electrical potential of a phase or electrical potential to be produced of a phase will be used to refer to the reference electrical potential across the terminals of the phase or the electrical potential to be produced across the terminals of the phase.

[0101] The term potential is also used to refer to an electrical potential.

[0102] In the remainder of the description, V.sub.DC denotes the voltage of the electrical power supply of the motor, the motor being supplied with power by this electrical power supply by way of a power converter.

[0103] FIG. 1 shows a flowchart of the method 100 for determining the potentials to be produced V.sub.i of the phases of a motor with N phases, N being an integer greater than or equal to 4 and k being between 1 and N.

[0104] The method 100 comprises the defining 101 of a drive voltage V.sub.MAG which is less than a limit voltage V.sub.LIM and also the determining 102 of the reference potentials V.sub.REF_i of the N phases of the motor based on the drive voltage V.sub.MAG, i being an integer between 1 and N and V.sub.REF_i representing the reference potential of the i-th phase of the motor.

[0105] The reference potentials V.sub.REF_i of the N phases are expressed by the following formula:

[00027]$V_{\mathrm{REF}\_i}=V_{\mathrm{MAG}}\cos \left(-\frac{2}{N}\left(i-1\right)\right)$

with the phase of the drive voltage between 0 and 2.

[0106] The limit voltage Vum represents the maximum amplitude desired for the drive voltage V.sub.MAG across the terminals of the motor windings, while keeping potentials to be produced between V.sub.DC/2 and +V.sub.DC/2 for the N phases.

[0107] It can be defined by a user and vary between 0 and +V.sub.DC/2.

[0108] It can also be a function dependent on the potential V.sub.DC of the electrical power supply of the motor: for example, it can be equal to a fraction of V.sub.DC, for example 80V.sub.DC/100 or 110V.sub.DC/100. It can also be a function dependent on the potential V.sub.DC and on the number of phases N in the motor.

[0109] Advantageously, the limit voltage V.sub.LIM is expressed by the following function, as a function of the number of phases N of the motor and of the supply potential of the motor V.sub.DC:

[00028]$V_{\mathrm{LIM}}-\frac{V_{\mathrm{DC}}}{2}\frac{2}{N}\cos \left(\right)\underset{j=0}{\overset{N-1}{.Math.}}.Math.\cos \left(\frac{2}{N}j\right).Math.$

with =/(2N) if N is odd or =0 if N is even and a multiple of 4 or =/N if N is even and a non-multiple of 4.

[0110] These functions dependent on N make it possible to determine the limit voltage V.sub.LIM, and therefore the maximum amplitude of the drive voltage V.sub.MAG for a given number of phases N.

[0111] The method 100 then comprises the comparing 103 of the reference potentials V.sub.REF_i determined during the step 102 to a first threshold. The first threshold can be a potential value chosen by the user or a function

[0112] dependent on the potential of the power supply of the motor V.sub.DC. For example, the first threshold may be chosen equal to 0.8V.sub.DC/2 or to V.sub.DC/2. Preferably, the first threshold is equal to V.sub.DC/2.

[0113] If the reference potentials V.sub.REF_i are all less than or equal to the first threshold whatever the value of i (step 104), then the potentials to be produced V.sub.i of the N phases of the motor are equal to the reference potentials V.sub.REF_i determined in the step 102.

[0114] If at least one of the reference potentials V.sub.REF_i determined in the step 102 is greater than the first threshold, then the method comprises the comparing 105 of the reference potentials V.sub.REF_i to a second threshold greater than the first threshold.

[0115] The second threshold can be a potential value chosen by the user or a function dependent on the potential of the power supply of the motor V.sub.DC or else a function dependent on the number of phases N of the motor and on the potential of the power supply of the motor V.sub.DC. For example, the second threshold may be chosen equal to 1.1V.sub.DC/2.

[0116] Advantageously, the second threshold is expressed by the following function, as a function of the number of phases N of the motor and of the power supply potential of the motor V.sub.DC:

[00029]$\mathrm{seuil}2=\frac{V_{\mathrm{DC}}}{2}\frac{1}{\sin \left(\frac{}{N}+\frac{}{N}.Math..Math.\frac{N}{2}.Math.\right)}$

[0117] When the limit voltage V.sub.LIM is chosen by the user between 0 and +V.sub.DC/2, it can more specifically be chosen as being equal to the first threshold or to the second threshold.

[0118] Following this comparison 105, if the reference potentials V.sub.REF_i are all less than or equal to the second threshold whatever the value of i (step 106), then the potentials to be produced V.sub.i of the N phases of the motor are equal to V.sub.i=V.sub.REF_i+V.sub.H, with V.sub.H a homopolar potential expressed by the following formula:

[00030]$V_H=\frac{\underset{k=1.Math.N}{\max }\left(V_{{\mathrm{REF}}_k}\right)+\underset{k=1.Math.N}{\min }\left(V_{{\mathrm{REF}}_k}\right)}{2}$

where

[00031]$\underset{k=1...N}{\max }{V}_{{\mathrm{REF}}_k}$

denotes the maximum potential from among the reference potentials V.sub.REF_i determined in the step 102, and

[00032]$\underset{k=1...N}{\max }{V}_{{\mathrm{REF}}_k}$

refers to the minimum potential from among the reference potentials V.sub.REF_i determined in the step 102.

[0119] If at least one of the reference potentials V.sub.REF_i is greater than the second threshold, then the potentials to be produced V.sub.i of the N phases of the motor are equal to V.sub.i=V.sub.REF_i+V.sub.S_i+V.sub.HN, with V.sub.HN a homopolar potential dependent on the number of phases N and on the reference electrical potentials V.sub.REF_i and V.sub.S_i a secondary potential dependent on the number of phases N and on the reference electrical potentials V.sub.REF_i.

[0120] In this scenario, the secondary V.sub.S_i and homopolar V.sub.HN potentials can be expressed by the following functions:

[00033]$\left(\begin{array}{c}{V}_{S_1}\\ .Math.\\ V_{S_N}\end{array}\right)=T\left(:,3:N-1\right)\left(\begin{array}{c}{V}_{{\mathrm{SAB}}_1}\\ .Math.\\ V_{{\mathrm{SAB}}_{N-2}}\end{array}\right)$

with: [0121] if N is odd, with p=(N1)/2:

[00034]$V_{\mathrm{SABi}}\left(k\right)=\frac{\begin{array}{c}\left[U\left(k+2\right)V\left(2\right)-U\left(2\right)V\left(k+2\right)\right]\\ \left[V\left(k+2\right)U\left(1\right)-V\left(1\right)U\left(k+2\right)\right]\end{array}}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF}\_N}\end{array}\right)$$k\mathrm{ranging}\mathrm{from}1\mathrm{to}N-3,\mathrm{and}$$V_{\mathrm{HN}}=\frac{1}{\sqrt{2}}\frac{\begin{array}{c}\left[U\left(N\right)V\left(2\right)-U\left(2\right)V\left(N\right)\right]\\ \left[V\left(N\right)U\left(1\right)-V\left(1\right)U\left(N\right)\right]\end{array}}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF}\_N}\end{array}\right)$

with: [0122] U a vector with N rows defined by U=T.sup.1u and u a vector with N rows defined by a value 1 for its indices iP, a value 0 for its index iM and a value 1 for its indices iN; [0123] V a vector with N rows defined by V=T.sup.1v and v a vector with N rows defined by a value 0 for its indices iP and iN and a value 1 for its index iM; [0124] iP the p indices of the p greatest values of the reference potentials V.sub.REF_i; [0125] iN the p indices of the p smallest values of the reference potentials V.sub.REF_i; [0126] iM the index of the median value of the reference potentials V.sub.REF_i; [0127] i varying from 1 to N, and [0128] T a transformation matrix defined by: [0129] the odd columns from 1 to 2 [(N1)/2] written 2k1 (k varying from 1 to [(N1)/2]), correspond to the vector having as i-th component (with i varying from 1 to N): c(i, 2k1)=cos(2k(i1)/N); [0130] the even columns from 1 to 2[(N1)/2] written 2k (k varying from 1 to [(N1)/2]) correspond to the vector having as i-th component (with i varying from 1 to N): c(i, 2k)=sin(2k(i1)/N); [0131] if N is odd, the column N is defined by a constant vector with the value 1/2; [0132] if N is even, the column N1 is defined by a vector of alternate constants of value 1/2 and of value 1/2 and the column N is defined by a constant vector with the value 1/2. [0133] If N is even, with p=N/2: V.sub.H=0 and V.sub.SAB_i(k) is defined by: [0134] If k varies from 1 to p2:

[00035]$V_{\mathrm{SABi}}\left(2+k+2.Math.\left(k-1\right)/2.Math.\right)=Q^{-1}.Math.\left(\begin{array}{c}{V}_M-V_{\mathrm{REF}\mathrm{iP}\left(1\right)}\\ .Math.\\ V_M-V_{\mathrm{REF}\mathrm{iP}\left(p-2\right)}\end{array}\right)$ [0135] For other values of k between 1 and N3: V.sub.SAB_i(k)=0;
with

[00036]$V_M=\frac{V_{{\mathrm{REF}}_{\mathrm{iP}\left(p-1\right)}}-p_M\left(\begin{array}{c}{V}_{\mathrm{REF}\_\mathrm{iP}\left(1\right)}\\ .Math.\\ V_{\mathrm{REF}\_\mathrm{iP}\left(p-2\right)}\end{array}\right)}{1-p_M\left(\begin{array}{c}1\\ .Math.\\ 1\end{array}\right)}$$p_M={\mathrm{qQ}}^{-1};$ [0136] iP the p1 indices of the p1 greatest values of the reference potentials V.sub.REF_i for i varying from 1 to N; [0137] q the row vector formed of the elements of the row of the transformation matrix T corresponding to the last index of iP corresponding to the columns indexed

[00037]$4+k+2.Math.\frac{k-1}{2}.Math.$

where k varies from 1 to p2; and [0138] Q the matrix formed of the elements of the transformation matrix T at the intersection of the rows corresponding to the p2 first indices of the iP and of the columns indexed

[00038]$4+k+2.Math.\frac{k-1}{2}.Math.$

where k varies from 1 to p2.

[0139] Thus the electrical potentials to be produced V.sub.i determined by the method 100 may comprise a homopolar potential and/or a secondary potential according to the values of the reference potentials V.sub.REF_i with respect to the first and second thresholds.

[0140] FIGS. 2A, 2B, 2C, 3A and 3B show an implementation of the method of the invention for a five-phase motor, so for N=5.

[0141] Thus, by applying the formulae proposed for the limit voltage V.sub.LIM and for the second threshold, the second threshold is equal to 1.05V.sub.DC/2 and the limit voltage V.sub.LIM is equal to 1.23V.sub.DC/2. In this example, the first threshold is equal to V.sub.DC/2. The drive voltage V.sub.MAG must therefore remain less than or equal to 1.23 x V.sub.DC/2.

[0142] In addition, if at least one of the reference potentials V.sub.REF_i determined in the step 102 is greater than the second threshold, then the potentials to be produced V.sub.i become: [0143] V.sub.i(i=p)=V.sub.M for p such that the reference potentials V.sub.REF_p determined in the step 102 are the two largest potentials; [0144] V.sub.i(i=l)=V.sub.M for l such that the reference potentials V.sub.REF_l determined in the step 102 are the two smallest potentials; and [0145] V.sub.i(i=j)=V.sub.C for j such that the reference potential V.sub.REF_j determined in the step 102 is the median potential,
with plj and p, l and j chosen from among {1; 2; 3; 4; 5}; and V.sub.M and V.sub.C expressed by the following functions:

[00039]$V_M=\frac{3\sqrt{5}-5}{4}\left(\underset{k=1.Math.5}{\max }\left(V_{\mathrm{REFk}}\right)-\underset{k=1.Math.5}{\min }\left(V_{\mathrm{REF}k}\right)\right)$$\mathrm{and}$$V_C=\frac{5}{2}\underset{k=1.Math.5}{\mathrm{median}}\left(V_{\mathrm{REF}k}\right)$

[0146] Specifically, for a five-phase motor, the transformation matrix T is expressed as follows:

[00040]$T=\left(\begin{array}{ccccc}\cos \left(\frac{2}{5}.Math.0\right)& \sin \left(\frac{2}{5}.Math.0\right)& \cos \left(\frac{4}{5}.Math.0\right)& \sin \left(\frac{4}{5}.Math.0\right)& \frac{1}{\sqrt{2}}\\ \cos \left(\frac{2}{5}.Math.1\right)& \sin \left(\frac{2}{5}.Math.1\right)& \cos \left(\frac{4}{5}.Math.1\right)& \sin \left(\frac{4}{5}.Math.1\right)& \frac{1}{\sqrt{2}}\\ \cos \left(\frac{2}{5}.Math.2\right)& \sin \left(\frac{2}{5}.Math.2\right)& \cos \left(\frac{4}{5}.Math.2\right)& \sin \left(\frac{4}{5}.Math.2\right)& \frac{1}{\sqrt{2}}\\ \cos \left(\frac{2}{5}.Math.3\right)& \sin \left(\frac{2}{5}.Math.3\right)& \cos \left(\frac{4}{5}.Math.3\right)& \sin \left(\frac{4}{5}.Math.3\right)& \frac{1}{\sqrt{2}}\\ \cos \left(\frac{2}{5}.Math.4\right)& \sin \left(\frac{2}{5}.Math.4\right)& \cos \left(\frac{4}{5}.Math.4\right)& \sin \left(\frac{4}{5}.Math.4\right)& \frac{1}{\sqrt{2}}\end{array}\right)$

[0147] The homopolar and secondary potentials which are generically expressed as follows:

[00041]$V_{5\_k}=\cos \left(\frac{4}{5}\left(k-1\right)\right)V_{\mathrm{SAB}}\left(1\right)+\sin \left(\frac{4}{5}\left(k-1\right)\right)V_{\mathrm{SAB}}\left(2\right)$$V_{\mathrm{HN}}=\frac{1}{5}{V}_C$$\mathrm{with}:$$V_{\mathrm{SAB}}\left(1\right)=\frac{\left[\begin{array}{cc}\left(U\left(3\right)V\left(2\right)-U\left(2\right)V\left(3\right)\right)& \left(V\left(3\right)U\left(1\right)-V\left(1\right)U\left(3\right)\right)\end{array}\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ V_{\mathrm{REF}\_2}\\ V_{\mathrm{REF}\_3}\\ V_{\mathrm{REF}\_4}\\ V_{\mathrm{REF}\_5}\end{array}\right)$$V_{\mathrm{SAB}}\left(2\right)=\frac{\left[\begin{array}{cc}\left(U\left(4\right)V\left(2\right)-U\left(2\right)V\left(4\right)\right)& \left(V\left(4\right)U\left(1\right)-V\left(1\right)U\left(4\right)\right)\end{array}\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ V_{\mathrm{REF}\_2}\\ V_{\mathrm{REF}\_3}\\ V_{\mathrm{REF}\_4}\\ V_{\mathrm{REF}\_5}\end{array}\right)$$V_{\mathrm{HN}}=\frac{1}{\sqrt{2}}\frac{\left[\left(U\left(5\right)V\left(2\right)-U\left(2\right)V\left(5\right)\right)\left(V\left(5\right)U\left(1\right)-V\left(1\right)U\left(5\right)\right)\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ V_{\mathrm{REF}\_2}\\ V_{\mathrm{REF}\_3}\\ V_{\mathrm{REF}\_4}\\ V_{\mathrm{REF}\_5}\end{array}\right)$

are simplified by setting:

[00042]$W=\frac{\left[\begin{array}{cc}V\left(2\right)& -V\left(1\right)\\ -U\left(2\right)& U\left(1\right)\end{array}\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ V_{\mathrm{REF}\_2}\\ V_{\mathrm{REF}\_3}\\ V_{\mathrm{REF}\_4}\\ V_{\mathrm{REF}\_5}\end{array}\right)$

[0148] The homopolar potential and the secondary components of the secondary potential are then expressed by the following formulae:

[00043]$V_{SAB}\left(1\right)=\left[U\left(3\right)V\left(3\right)\right].Math.W$$V_{SAB}\left(2\right)=\left[U\left(4\right)V\left(4\right)\right].Math.W$$V_{HN}=\frac{1}{\sqrt{2}}\left[U\left(5\right)V\left(5\right)\right].Math.W$

[0149] The main system u.sub.commande is then written:

[00044]$u_{\mathrm{commande}}\left(1\right)=\left[\begin{array}{cc}U\left(1\right)& V\left(1\right)\end{array}\right].Math.W$$u_{\mathrm{commande}}\left(2\right)=\left[\begin{array}{cc}U\left(2\right)& V\left(2\right)\end{array}\right].Math.W$

[0150] The potentials to be produced V.sub.1, V.sub.2, V.sub.3, V.sub.4 and V.sub.5 are therefore written:

[00045]$\left(\begin{array}{c}{V}_1\\ V_2\\ V_3\\ V_4\\ V_5\end{array}\right)=T\left(\begin{array}{c}{u}_{\mathrm{commande}}\left(1\right)\\ u_{\mathrm{commande}}\left(2\right)\\ V_{\mathrm{SAB}}\left(1\right)\\ V_{\mathrm{SAB}}\left(2\right)\\ \sqrt{2}{V}_{\mathrm{HN}}\end{array}\right)=T.Math.T^{-1}.Math.\left[uv\right].Math.W$$i.e.$$\left(\begin{array}{c}{V}_1\\ V_2\\ V_3\\ V_4\\ V_5\end{array}\right)=\{\begin{array}{c}W\left(1\right):\mathrm{if}i,\mathrm{such}\mathrm{that}{V}_{\mathrm{REF}i}\mathrm{is}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{largest}\\ -W\left(1\right):\mathrm{if}i,\mathrm{such}\mathrm{that}{V}_{\mathrm{REF}i}\mathrm{is}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{smallest}\\ W\left(2\right):\mathrm{if}i,\mathrm{such}\mathrm{that}{V}_{\mathrm{REF}i}\mathrm{is}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{median}\mathrm{value}\end{array}$

[0151] With no loss of generality, by considering that the phase of the drive voltage V.sub.MAG varies between 0 and /5, then the values of the reference potentials V.sub.REF_i are ranked from the smallest to the largest in the order 4, 3, 5, 2 and 1. The other scenarios, i.e. for a phase between /5 and 2, are obtained by symmetry and permutation of indices.

[0152] It therefore comes about that the vectors u, v, U and V are defined by:

[00046]$u=\left(\begin{array}{c}1\\ 1\\ -1\\ -1\\ 0\end{array}\right)$$v=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\end{array}\right)$$U=\frac{2}{5}.Math.\left(\begin{array}{c}1+\cos \left(\frac{2}{5}\right)-\cos \left(\frac{4}{5}\right)-\cos \left(\frac{6}{5}\right)\\ \sin \left(\frac{2}{5}\right)-\sin \left(\frac{4}{5}\right)-\sin \left(\frac{6}{5}\right)\\ 1+\cos \left(\frac{4}{5}\right)-\cos \left(\frac{8}{5}\right)-\cos \left(\frac{12}{5}\right)\\ \sin \left(\frac{4}{5}\right)-\sin \left(\frac{8}{5}\right)-\sin \left(\frac{12}{5}\right)\end{array}\right)$$V=\frac{2}{5}.Math.\left(\begin{array}{c}\cos \left(\frac{8}{5}\right)\\ \sin \left(\frac{8}{5}\right)\\ \cos \left(\frac{16}{5}\right)\\ \sin \left(\frac{16}{5}\right)\\ \frac{1}{\sqrt{2}}\end{array}\right)$

[0153] By writing:

[00047]$V_{\mathrm{REF}\_2}-V_{\mathrm{REF}\_3}=\frac{\sin \left(\frac{}{5}\right)}{\sin \left(\frac{2}{5}\right)}.Math.\left(V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}\right)$$\mathrm{and}$$V_{\mathrm{REF}\_2}+V_{\mathrm{REF}\_3}=-\frac{\cos \left(\frac{}{5}\right)}{\cos \left(\frac{2}{5}\right)}.Math.\left(V_{\mathrm{REF}\_1}+V_{\mathrm{REF}\_4}\right)=-2.Math.\cos \left(\frac{}{5}\right).Math.V_{\mathrm{REF}\_5}$

one obtains the following expressions relating the reference potentials V.sub.REF_1, V.sub.REF_2, V.sub.REF_3, V.sub.REF_4 and V.sub.REF_5 to one another:

[00048]$V_{\mathrm{REF}\_1}=\frac{V_{\mathrm{REF}\_1}+V_{\mathrm{REF}\_4}}{2}+\frac{V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}}{2}=\cos \left(\frac{2}{5}\right).Math.V_{\mathrm{REF}\_5}+\frac{V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}}{2}$$V_{\mathrm{REF}\_4}=\frac{V_{\mathrm{REF}\_1}+V_{\mathrm{REF}\_4}}{2}+\frac{V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}}{2}=\cos \left(\frac{2}{5}\right).Math.V_{\mathrm{REF}\_5}+\frac{V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}}{2}$$V_{\mathrm{REF}\_2}=\frac{V_{\mathrm{REF}\_2}+V_{\mathrm{REF}\_3}}{2}+\frac{V_{\mathrm{REF}\_2}-V_{\mathrm{REF}\_3}}{2}=\cos \left(\frac{}{5}\right).Math.V_{\mathrm{REF}\_5}+\frac{\sin \left(\frac{}{5}\right)}{\sin \left(\frac{2}{5}\right)}\frac{V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}}{2}$$V_{\mathrm{REF}\_3}=\frac{V_{\mathrm{REF}\_2}+V_{\mathrm{REF}\_3}}{2}+\frac{V_{\mathrm{REF}\_2}-V_{\mathrm{REF}\_3}}{2}=\cos \left(\frac{}{5}\right).Math.V_{\mathrm{REF}\_5}+\frac{\sin \left(\frac{}{5}\right)}{\sin \left(\frac{2}{5}\right)}\frac{V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}}{2}$

[0154] One then computes the vector W which is written in a simplified manner:

[00049]$W=\left[\begin{array}{c}\frac{3\sqrt{5}-5}{4}.Math.\left(V_{\mathrm{REF}\_1}-V_{\mathrm{REF}\_4}\right)\\ \frac{5}{2}.Math.V_{\mathrm{REF}\_5}\end{array}\right]=\left[\begin{array}{c}{V}_M\\ V_C\end{array}\right]$

[0155] This makes it possible to compute the secondary components of the secondary potential and the homopolar potential:

[00050]$V_{\mathrm{SAB}}\left(1\right)=\frac{2}{5}.Math.\left(1-\cos \left(\frac{}{5}\right)-2.Math.\cos \left(\frac{2}{5}\right)\right).Math.V_M-\frac{2}{5}.Math.\cos \left(\frac{}{5}\right).Math.V_C$$V_{\mathrm{SAB}}\left(2\right)=\frac{2}{5}.Math.\sin \left(\frac{}{5}\right).Math.V_M-\frac{2}{5}.Math.\sin \left(\frac{}{5}\right).Math.V_C$$V_{\mathrm{HN}}=\frac{1}{5}{V}_C$

and one obtains for the potentials to be produced V.sub.k (with k between 1 and 5):

[00051]$V_k=V_{\mathrm{REF}\_k}+V_{S\_k}+V_{\mathrm{HN}}$$i.e.:$$\begin{array}{c}{V}_1=V_M\\ V_2=V_M\\ V_3=-V_M\\ V_4=-V_M\\ V_5=V_C\end{array}$

[0156] FIGS. 2A, 2B and 2C show, more specifically, the implementation of the method for a five-phase motor, in the scenario in which the reference potentials V.sub.REF_i determined at the start of the method are less than or equal to the second threshold and therefore the potentials to be produced V.sub.i comprise only the possible addition of a homopolar potential V.sub.H.

[0157] If the drive voltage V.sub.MAG=90% V.sub.DC/2, the reference potentials of the five phases (curves 201a, 202a, 203a, 204a and 205a of FIG. 2A) are indeed all less than the first threshold. Thus, it is not necessary to add any homopolar potential V.sub.H, or any secondary potential V.sub.S_i in accordance with the method of the invention.

[0158] If the drive voltage V.sub.MAG=V.sub.DC/2, the reference potentials of the five phases (curves 201b, 202b, 203b, 204b and 205b of FIG. 2B) are indeed less than or equal to the first threshold. Thus, it is not necessary to add any homopolar potential V.sub.H, or any secondary potential V.sub.S_i.

[0159] Specifically, the maximum and minimum values of homopolar potential V.sub.H represented by the curves 211b and 212b demonstrate that a homopolar potential equal to 0 does indeed make it possible to keep the potentials of the N phases between V.sub.DC/2 and +V.sub.DC/2. The maximum homopolar potential is expressed by V.sub.DC/2max(V.sub.REF_i) and the minimum homopolar potential is expressed by V.sub.DC/2min(V.sub.REF_i). The curve 213b represents the homopolar potential V.sub.H proposed in the method of the invention, i.e. in the scenario in which the electrical potentials of the five phases are less than or equal to the second threshold but in which at least one of the reference potentials is greater than the first threshold.

[0160] If the drive voltage V.sub.MAG=105% V.sub.DC/2, the reference potentials of the five phases (curves 201c, 202c, 203c, 204c and 205c of FIG. 2C) reach the first threshold, but remain less than or equal to the second threshold. Thus, it is necessary to add a homopolar potential V.sub.H for them to always be between V.sub.DC/2 and +V.sub.DC/2. The curves 211c and 212c represent the possible maximum and minimum values of the homopolar potential V.sub.H for the amplitude of the potentials to be produced of the five phases to remain less than or equal to V.sub.DC/2.

[0161] The curve 213c represents the homopolar potential V.sub.H proposed in the step 106 of the method 100 according to the invention. It can be seen that this curve 213c is indeed contained between the curves 211c and 212c, i.e. between the maximum and minimum values of homopolar potential possible. After application of the method 100 of the invention, it can be seen that the potentials to be produced of the five phases have indeed been corrected (curves 221c, 222c, 223c, 224c and 225c which represent the potential V.sub.i=V.sub.REF_i+V.sub.H) in such a way as to remain between V.sub.DC/2 and +V.sub.DC/2 while being increased for certain values of angle so that the amplitude of the voltage across the terminals of the motor windings can rise all the way to the drive voltage V.sub.MAG, i.e. 105% V.sub.DC/2.

[0162] FIGS. 3A and 3B represent, more specifically, the implementation of the method for a five-phase motor, in the scenario in which at least one of the reference potentials V.sub.REF_i is greater than the second threshold and thus the potentials to be produced V.sub.i comprise the addition of a homopolar potential V.sub.HN dependent on the number of phases and on a secondary potential V.sub.S_i.

[0163] If the drive voltage V.sub.MAG=110% V.sub.DC/2, the reference potentials of the five phases (curves 301a, 302a, 303a, 304a and 305a of the FIG. 3A) are greater than the first threshold and than the second threshold. Thus, in accordance with the method of the invention, it is necessary to add a homopolar potential V.sub.HN and a secondary potential V.sub.S_i. The grey areas around the curves 311a, 312a and 313a respectively represent the possible values of the homopolar potential V.sub.HN and of the two components V.sub.SAB(1) and V.sub.SAB(2) of the secondary potential V.sub.S_i. In particular, the curve 311a represents the first component V.sub.SAB(1) of the secondary potential V.sub.S_i, and the curve 312a represents the second component V.sub.SAB(2) of the secondary potential V.sub.S_i according to the formulae of the invention described previously. The curve 313a represents the value of the homopolar potential V.sub.HN according to the formulae of the invention described previously. After application of the method of the invention, and therefore the addition of the secondary and homopolar potentials V.sub.S_i (i.e. V.sub.SAB(1) and V.sub.SAB(2)) and V.sub.HN represented by the curves 311a, 312a and 313a, the reference potentials of the five phases are corrected and the potentials to be produced are represented by the curves 321a, 322a, 323a, 324a and 325a. It can be seen that the potentials to be produced are now indeed between V.sub.DC/2 and +V.sub.DC/2, and that their amplitude has indeed been modified for certain values of angle to increase the voltage across the terminals of the motor windings.

[0164] If the drive voltage V.sub.MAG is equal to the limit voltage V.sub.LIM, i.e. 123% V.sub.DC/2, the reference potentials V.sub.REF_i of the five phases (curves 301b, 302b, 303b, 304b and 305b of the FIG. 3B) are greater than the first threshold and than the second threshold. Thus, in accordance with the method of the invention, it is necessary to add a homopolar potential V.sub.HN and a secondary potential V.sub.S_i. The gray areas around the curves 311b, 312b and 313b respectively represent the possible values of the homopolar potential V.sub.HN and of the components V.sub.SAB(1) and V.sub.SAB(2) of the secondary potential V.sub.S_i. Unlike the scenario of FIG. 3A, it can be seen that the area around the curves 311b, 312b and 313b has shrunk and that there is now virtually only a single possible value for each component and potential to achieve the drive voltage V.sub.MAG and keep the potentials to be produced of the five phases between V.sub.DC/2 and +V.sub.DC/2. After application of the method of the invention, and therefore the addition of the secondary components V.sub.SAB(1) and V.sub.SAB(2) of the secondary potential V.sub.S_i and of the homopolar potential V.sub.HN represented by the curves 311b, 312b and 313b, the reference potentials V.sub.REF_i of the five phases are corrected and the potentials to be produced V.sub.i are represented by the curves 321b, 322b, 323b, 324b and 325b. It can be seen that these potentials to be produced V.sub.i are indeed between V.sub.DC/2 and +V.sub.DC/2, and that their amplitude has indeed been modified for certain values of angle to increase the voltage across the terminals of the motor windings.

[0165] FIG. 4 represents, schematically and partially, an electric device 400 according to an embodiment of the invention.

[0166] The electric device 400 comprises an electric power supply 401 connected to an input of an electric controller 402. The electric controller 402 is connected to a power converter 403 which is itself connected to an electric motor 404 comprising N phases, N being an integer greater than or equal to 4. The electric controller 402 provides N electrical potentials to the converter 403 so that it can convert them and provide them to the N phases of the electric motor 404. The potentials provided to the motor 404 correspond to the potentials to be produced V.sub.i determined by the method of the invention and converted by the power converter 403.

[0167] The electric controller 402 is particularly configured to implement the method of the invention. Thus, the controller 402 is configured to determine the reference potentials and the potentials to be produced V.sub.i of the N phases of the motor 404 for a drive voltage V.sub.MAG less than a limit voltage V.sub.LIM provided by a user, as shown on FIG. 4, or defined by the controller 402 according to the following formula:

[00052]$V_{\mathrm{LIM}}=\frac{V_{\mathrm{DC}}}{2}\frac{2}{N}\cos \left(\right)\underset{j=0}{\overset{N-1}{.Math.}}.Math.\cos \left(\frac{2}{N}j\right).Math.$

with =/(2N) if N is odd or =0 if N is even and a multiple of 4 or =/N if N is even and a non-multiple of 4.

[0168] The electrical potentials to be produced V.sub.i are determined according to the values of the reference electrical potentials V.sub.REF_i determined by the controller 402, the reference potentials V.sub.REF_i being given by the following formula:

[00053]$V_{\mathrm{REF}\_i}=V_{\mathrm{MAG}}\cos \left(-\frac{2}{N}\left(i-1\right)\right)$

with i between 1 and N, and V.sub.REF_i representing the reference potential of the i-th phase of the motor 404.

[0169] Then the controller 402 compares these reference potentials V.sub.REF_i to the first and second thresholds, which can be defined as indicated with reference to FIG. 1.

[0170] If the reference potentials V.sub.REF_i are all less than or equal to the first threshold, then the potentials V.sub.i provided to the power converter 403 then to the motor 404 will be equal to the reference potentials V.sub.REF_i previously described.

[0171] If the reference potentials V.sub.REF_i are all less than or equal to the second threshold, but at least one of the reference potentials is greater than the first threshold, then the potentials V.sub.i provided to the power converter 403 then to the motor 404 will be equal to V.sub.i=V.sub.REF_i+V.sub.H, with V.sub.H the homopolar potential defined by the following formula:

[00054]$V_H=\frac{\underset{k=1.Math.N}{\max }\left(V_{{\mathrm{REF}}_k}\right)+\underset{k=1.Math.N}{\min }\left(V_{{\mathrm{REF}}_k}\right)}{2}$

where

[00055]$\underset{k=1.Math.N}{\max }{V}_{{\mathrm{REF}}_k}$

represents the maximum reference potential from among the reference potentials V.sub.REF_i determined previously and

[00056]$\underset{k=1.Math.N}{\min }{V}_{{\mathrm{REF}}_k}$

represents the minimum reference potential from among the reference potentials V.sub.REF_i determined previously.

[0172] If the reference potentials V.sub.REF_i are all greater than the second threshold, then the potentials V.sub.i provided to the power converter 403 then to the motor 404 will be equal to V.sub.i=V.sub.REF_i+V.sub.S_i+V.sub.HN, with V.sub.HN a homopolar potential dependent on the number of phases N and on the reference electrical potentials V.sub.REF_i and V.sub.S_i a secondary potential dependent on the number of phases N and on the reference electrical potentials V.sub.REF_i.

[0173] In this scenario, the secondary V.sub.S_i and homopolar V.sub.HN potentials can be expressed by the following functions:

[00057]$\left(\begin{array}{c}{V}_{S_1}\\ .Math.\\ V_{S_N}\end{array}\right)=T\left(:,3:N-1\right)\left(\begin{array}{c}{V}_{{\mathrm{SAB}}_1}\\ .Math.\\ V_{{\mathrm{SAB}}_{N-1}}\end{array}\right)$

with: [0174] if N is odd, with p=(N1)/2:

[00058]$V_{\mathrm{SABi}}\left(k\right)=\frac{\left[U\left(k+2\right)V\left(2\right)-U\left(2\right)V\left(k+2\right)\right]\left[V\left(k+2\right)U\left(1\right)-V\left(1\right)U\left(k+2\right)\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF}\_N}\end{array}\right)$

k ranging from 1 to N3, and

[00059]$V_{\mathrm{HN}}=\frac{1}{\sqrt{2}}\frac{\left[U\left(N\right)V\left(2\right)-U\left(2\right)V\left(N\right)\right]\left[V\left(N\right)U\left(1\right)-V\left(1\right)U\left(N\right)\right]}{U\left(1\right)V\left(2\right)-U\left(2\right)V\left(1\right)}{T}^{-1}\left(1:2,:\right)\left(\begin{array}{c}{V}_{\mathrm{REF}\_1}\\ .Math.\\ V_{\mathrm{REF\_N}}\end{array}\right)$

with: [0175] U a vector with N rows defined by U=T.sup.1u and u a vector with N rows defined by a value 1 for its indices iP, a value 0 for its index iM and a value 1 for its indices iN; [0176] V a vector with N rows defined by V=T.sup.1v and v a vector with N rows defined by a value 0 for its indices iP and iN and a value 1 for its index iM; [0177] iP the p indices of the p greatest values of the reference potentials V.sub.REF_i; [0178] iN the p indices of the p smallest values of the reference potentials V.sub.REF_i; [0179] iM the index of the median value of the reference potentials V.sub.REF_i; [0180] i varying from 1 to N, and [0181] T a transformation matrix defined by: [0182] the odd columns from 1 to 2 [(N1)/2] written 2k1 (k varying from 1 to [(N1)/2]), correspond to the vector having as i-th component (with i varying from 1 to N): c(i, 2k1)=cos(2k(i1)/N); [0183] the even columns from 1 to 2 [(N1)/2] written 2k (k varying from 1 to [(N1)/2]) correspond to the vector having as i-th component (with i varying from 1 to N): c(i, 2k)=sin(2k(i1)/N); [0184] if N is odd, the column N is defined by a constant vector with the value 1/2; [0185] if N is even, the column N1 is defined by a vector of alternate constants of value 1/2 and of value 1/2 and the column N is defined by a constant vector with the value 1/2.
If N is even, with p=N/2: V.sub.H=0 and V.sub.SAB_i(k) is defined by: [0186] If k varies from 1 to p2:

[00060]$V_{\mathrm{SABi}}\left(2+k+2.Math.\left(k-1\right)/2.Math.\right)=Q^{-1}.Math.\left(\begin{array}{c}{V}_M-V_{\mathrm{REF}\mathrm{iP}\left(1\right)}\\ .Math.\\ V_M-V_{\mathrm{REF}\mathrm{iP}\left(p-2\right)}\end{array}\right)$ [0187] For other values of k between 1 and N3: V.sub.SAB_i(k)=0;
with:

[00061]$V_M=-\frac{V_{{\mathrm{REF}}_{\mathrm{iP}\left(p-1\right)}}-p_M\left(\begin{array}{c}{V}_{\mathrm{REF}\_\mathrm{iP}\left(1\right)}\\ .Math.\\ V_{\mathrm{REF}\_\mathrm{iP}\left(p-2\right)}\end{array}\right)}{1-p_M\left(\begin{array}{c}1\\ .Math.\\ 1\end{array}\right)}$$p_M={\mathrm{qQ}}^{-1};$ [0188] iP the p1 indices of the p1 greatest values of the reference potentials V.sub.REF_i for i varying from 1 to N; [0189] q the row vector formed of the elements of the row of the transformation matrix T corresponding to the last index of iP corresponding to the columns indexed

[00062]$4+k+2.Math.\frac{k-1}{2}.Math.$

where k varies from 1 to p2; and [0190] Q the matrix formed of the elements of the transformation matrix T at the intersection of the rows corresponding to the p2 first indices of the iP and of the columns indexed

[00063]$4+k+2.Math.\frac{k-1}{2}.Math.$

where k varies from 1 to p2.

[0191] FIGS. 5A, 5B and 5C show another exemplary implementation of the method according to the invention for N=6, i.e. for a six-phase motor.

[0192] For a six-phase motor, the transformation matrix T is expressed as follows:

[00064]$T=\left(\begin{array}{cccccc}\cos \left(\frac{2}{6}.Math.0\right)& \sin \left(\frac{2}{6}.Math.0\right)& \cos \left(\frac{4}{6}.Math.0\right)& \sin \left(\frac{4}{6}.Math.0\right)& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\\ \cos \left(\frac{2}{6}.Math.1\right)& \sin \left(\frac{2}{6}.Math.1\right)& \cos \left(\frac{4}{6}.Math.1\right)& \sin \left(\frac{4}{6}.Math.1\right)& -\frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\\ \cos \left(\frac{2}{6}.Math.2\right)& \sin \left(\frac{2}{6}.Math.2\right)& \cos \left(\frac{4}{6}.Math.2\right)& \sin \left(\frac{4}{6}.Math.2\right)& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\\ \cos \left(\frac{2}{6}.Math.3\right)& \sin \left(\frac{2}{6}.Math.3\right)& \cos \left(\frac{4}{6}.Math.3\right)& \sin \left(\frac{4}{6}.Math.3\right)& -\frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\\ \cos \left(\frac{2}{6}.Math.4\right)& \sin \left(\frac{2}{6}.Math.4\right)& \cos \left(\frac{4}{6}.Math.4\right)& \sin \left(\frac{4}{6}.Math.4\right)& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\\ \cos \left(\frac{2}{6}.Math.5\right)& \sin \left(\frac{2}{6}.Math.5\right)& \cos \left(\frac{4}{6}.Math.5\right)& \sin \left(\frac{4}{6}.Math.5\right)& -\frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\end{array}\right)$

[0193] As N is even, in accordance with the invention, the reference electrical potentials V.sub.REF_i across the terminals of the six phases for a drive voltage V.sub.MAG are defined by:

[00065]$V_{\mathrm{REF}\_i}=T\left(i,1:2\right).Math.\left(\begin{array}{c}{V}_{\mathrm{MAG}}.Math.\cos \left(\right)\\ V_{\mathrm{MAG}}.Math.\sin \left(\right)\end{array}\right)$

with i between 1 and 6.

[0194] Then, applying the formulae of the invention, as N is even, the homopolar potential V.sub.HN is zero and the secondary potentials are defined by:

[00066]$V_{\mathrm{SAB}\_i}\left(1\right)=V_{\mathrm{SAB}\_i}\left(2\right)=0$$V_{\mathrm{SAB}\_i}\left(3\right)=\frac{V_M-V_{\mathrm{REF}\_\mathrm{iP}\left(1\right)}}{T\left(\mathrm{iP}\left(1\right),5\right)}$$\mathrm{with}$$V_M=\frac{V_{\mathrm{REF}\_\mathrm{iP}\left(1\right)}+V_{\mathrm{REF}\_\mathrm{iP}\left(2\right)}}{2}$

[0195] The electrical potentials to be produced V.sub.i across the terminals of the six phases are then equal to:

[00067]$V_i=V_{\mathrm{REF}\_k}-{\left(-1\right)}^{\mathrm{iP}\left(1\right)+k}\frac{V_{\mathrm{REF}\_\mathrm{iP}\left(1\right)}-V_{\mathrm{REF}\_\mathrm{iP}\left(2\right)}}{2}$

[0196] FIG. 5A represents the reference electrical potential V.sub.REF_i across the terminals of the six phases of the six-phase motor before the application of the method of the invention for a drive voltage V.sub.MAG of amplitude equal to 90% of V.sub.DC/2.

[0197] In this scenario, the reference potentials of the six phases are all indeed less than V.sub.DC/2, which is the first threshold. It is therefore not necessary to add any secondary potential in accordance with the method of the invention. The potentials to be produced are therefore equal to the reference electrical potentials V.sub.REF_i.

[0198] FIG. 5B represents the reference electrical potential V.sub.REF_i across the terminals of the six phases of the six-phase motor before the application of the method of the invention (graph (a)), the secondary potential V.sub.SAB_i(3) as described previously (graph (b)) as well as the potentials to be produced V.sub.i (graph (c)) for a drive voltage V.sub.MAG of amplitude equal to 105% of V.sub.DC/2.

[0199] FIG. 5C represents the reference electrical potential V.sub.REF_i across the terminals of the six phases of the six-phase motor before the application of the method of the invention (graph (a)), the secondary potential V.sub.SAB_i(3) as described previously (graph (b)) as well as the potentials to be produced V.sub.i (graph (c)) for a drive voltage V.sub.MAG of amplitude equal to 115.7% of V.sub.DC/2.