Methods, systems and apparatus for approximation of peak summed fundamental and third harmonic voltages in a multi-phase machine

1. A method for approximating a peak summed magnitude of a phase voltage waveform comprising voltage vectors including a fundamental voltage vector and a third harmonic voltage vector in a multi-phase system, the method comprising:

computing an endpoint value based on a magnitude of a fundamental voltage vector and a magnitude of a third harmonic voltage vector, wherein the endpoint value is the lowest value of the peak summed magnitude for a particular ratio of the magnitude of the fundamental voltage vector to the magnitude of the third harmonic voltage vector;

generating a first scaling factor based on a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector;

determining, based on the endpoint value and the first scaling factor, an unscaled approximation of the peak summed magnitude of the phase voltage waveform; and

scaling the unscaled approximation based on a second scaling factor to generate an approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform.

2. A method according to claim 1 , wherein the step of determining, based on the endpoint value and the first scaling factor, an unscaled approximation of the peak summed magnitude of the phase voltage waveform, further comprises:

applying a normalization function to the endpoint value and the first scaling factor to determine an unscaled approximation of the peak summed magnitude of the fundamental voltage vector and the third harmonic voltage vector.

3. A method according to claim 1 , wherein the step scaling the unscaled approximation, further comprises:

scaling the unscaled approximation based on the second scaling factor so that the unscaled approximation has the same magnitude as the fundamental and third harmonic voltage vectors to generate the approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform.

4. A method according to claim 3 , wherein the second scaling factor represents the largest possible magnitude value of the phase voltage waveform, and is defined as the sum of the fundamental voltage vector magnitude and the third harmonic voltage vector magnitude, and wherein the step of scaling the unscaled approximation, further comprises:

multiplying the unscaled approximation by the second scaling factor so that the unscaled approximation has the same magnitude as the fundamental and third harmonic voltage vectors to generate an approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform.

5. A method according to claim 1 , wherein the phase voltage waveform is a sinusoidal waveform that drives any one phase of a five-phase machine, and wherein the peak summed magnitude of the phase voltage waveform is determined based on the sum of the fundamental voltage vector and the third harmonic voltage vector, wherein the phase voltage waveform is defined based on the fundamental voltage vector magnitude, the third harmonic voltage vector magnitude, an electrical angular position that is a function of rotor position, and the voltage angle difference.

6. A method according to claim 1 , wherein the step of generating a first scaling factor, comprises:

generating a first scaling factor by:

determining a value of a first function that depends on a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector when the voltage angle difference is between 0 and 180 degrees; and

determining a value of a second function that depends on 360 degrees minus the voltage angle difference when the voltage angle difference is between 180 and 360 degrees.

7. A method according to claim 1 , wherein the step of generating a first scaling factor, comprises:

generating a first scaling factor by determining the absolute value of a function that depends on a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector.

8. A method according to claim 7 , wherein the step of generating a first scaling factor, comprises:

generating a first scaling factor by determining the absolute value of a cosine function or a sine function that depends on the voltage angle difference.

9. A method executable by an approximation module in a multi-phase system, the method comprising:

approximating, at the approximation module, a peak summed magnitude of a phase voltage waveform, to generate an approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform, wherein the phase voltage waveform comprises a fundamental voltage vector and a third harmonic voltage vector.

10. A method according to claim 9 , further comprising:

determining, at the approximation module, an endpoint value based on a magnitude of a fundamental voltage vector and a magnitude of a third harmonic voltage vector, wherein the endpoint value is the lowest value of the peak summed magnitude for a particular ratio of the magnitude of the fundamental voltage vector to the magnitude of the third harmonic voltage vector;

generating, at the approximation module, a first scaling factor based on a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector;

determining, at the approximation module, based on the endpoint value and the first scaling factor, an unscaled approximation of the peak summed magnitude of the phase voltage waveform; and

generating, at the approximation module, based on the unscaled approximation, the approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform.

11. A method according to claim 10 , wherein the step of determining, at the approximation module, based on the endpoint value and the first scaling factor, an unscaled approximation of the peak summed magnitude of the phase voltage waveform, comprises:

normalizing, at the approximation module, the endpoint value via a normalization function that computes the unscaled approximation based on the endpoint value and the first scaling factor.

12. A method according to claim 10 , wherein the step of generating, at the approximation module, based on the unscaled approximation (Ã), the approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform, comprises:

scaling, at the approximation module, the unscaled approximation based on the second scaling factor so that the unscaled approximation has the same magnitude as the fundamental and third harmonic voltage vectors to generate the approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform.

13. A method according to claim 12 , wherein the second scaling factor represents the largest possible magnitude value of the phase voltage waveform and is defined as the sum of the fundamental voltage vector magnitude and the third harmonic voltage vector magnitude, and wherein the step of scaling, comprises:

scaling the unscaled approximation by multiplying the unscaled approximation by the second scaling factor so that the unscaled approximation has the same magnitude as the fundamental and third harmonic voltage vectors to generate an approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform.

14. A method according to claim 9 , wherein the phase voltage waveform is a sinusoidal waveform that drives any one phase of a five-phase machine, wherein the peak summed magnitude of the phase voltage waveform is determined based on the sum of the fundamental voltage vector and the third harmonic voltage vector, wherein the phase voltage waveform is defined based on the fundamental voltage vector magnitude, the third harmonic voltage vector magnitude, an electrical angular position (θ) that is a function of rotor position, and the voltage angle difference (Δφ).

15. A method according to claim 9 , wherein generating, at the approximation module, a first scaling factor, comprises:

generating a first scaling factor by determining the absolute value of a function that depends on a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector.

16. A method according to claim 15 , further comprising:

generating, at the approximation module, the first scaling factor by determining the absolute value of a cosine function or a sine function that depends on the voltage angle difference.

17. A method according to claim 9 , generating, at the approximation module, a first scaling factor, comprises:

generating a first scaling factor by:

determining a value of a first function that depends on a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector when the voltage angle difference is between 0 and 180 degrees; and

determining a value of a second function that depends on 360 degrees minus the voltage angle difference when the voltage angle difference is between 180 and 360 degrees.

18. A multi-phase system, comprising:

a fundamental voltage vector computation module that generates a fundamental voltage vector based on a fundamental d-axis voltage command signal and a fundamental q-axis voltage command signal;

a third harmonic voltage vector computation module that generates a third harmonic voltage vector based on a third harmonic d-axis voltage command signal and a third harmonic q-axis voltage command signal, wherein the phase voltage waveform is defined based on a fundamental voltage vector magnitude, a third harmonic voltage vector magnitude, an electrical angular position that is a function of rotor position, and a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector; and

an approximation module designed to approximate a peak summed magnitude of a phase voltage waveform and to generate an approximated peak summed magnitude of the phase voltage waveform that corresponds approximately to the peak summed magnitude of the phase voltage waveform.

19. A multi-phase system according to claim 18 , wherein the approximation module is designed to:

determine an endpoint value based on the magnitude of a fundamental voltage vector and the magnitude of a third harmonic voltage vector, wherein the endpoint value is the lowest value of the peak summed magnitude for a particular ratio of the magnitude of the fundamental voltage vector to the magnitude of the third harmonic voltage vector;

determine a voltage angle difference between an angle of the fundamental voltage vector and an angle of the third harmonic voltage vector, and determine an absolute value of a sine function when one-half of the voltage angle difference is applied to the sine function to generate a first scaling factor;

compute an unscaled approximation of a peak value of the phase voltage waveform that is equal to the peak summed magnitude of the fundamental voltage vector and the third harmonic voltage vector by applying the endpoint value and the first scaling factor to a normalization function that computes the unscaled approximation based on the endpoint value and the first scaling factor; and

scale the unscaled approximation, based on a second scaling factor that represents the largest possible magnitude value of the phase voltage waveform, to generate the approximated peak summed magnitude of the phase voltage waveform that substantially corresponds to the peak summed magnitude of the phase voltage waveform, wherein the second scaling factor is the sum of the fundamental voltage vector magnitude and the third harmonic voltage vector magnitude.

20. A multi-phase system according to claim 18 , further comprising:

a comparator module designed to determine whether the approximated peak summed magnitude of the phase voltage waveform is greater than or equal to a maximum available value of the phase voltage.