Fortescue Decomposition

Definition 1: The number  is defined to be[pic 1]

[pic 2]

Definition 2: Let  be a time-varying sinusoidal quantity with constant amplitude , constant angular frequency , and constant phase offset  (). The complex number[pic 3][pic 4][pic 5][pic 6][pic 7]

[pic 8]

is called the phasor representation of  , or simply a phasor.[pic 9]

Definition 3: A three-phase system is a phasor sequence

[pic 10]

  is said to be a balanced three-phase system if[pic 11]

[pic 12]

in which case  is positive-sequence, or[pic 13]

[pic 14]

in which case  is negative-sequence.[pic 15]

Definition 4: A phasor sequence  is zero-sequence if[pic 16]

[pic 17]

Fortescue Decomposition: Let be a three-phase system. Then there exist a zero-sequence phasor sequence , a positive-sequence phasor sequence  and a negative-sequence phasor sequence  such that[pic 18][pic 19][pic 20][pic 21]

[pic 22]

[pic 23]

[pic 24]

Moreover, this phasor sequence triple is unique.

Proof: For existence, we use the fact that for any complex number ,[pic 25]

[pic 26]

Express each phasor in  as[pic 27]

[pic 28]

[pic 29]

[pic 30]

Let

[pic 31]

Then we can re-express , , and  as[pic 32][pic 33][pic 34]

[pic 35]

[pic 36]

[pic 37]

Take , , and .[pic 38][pic 39][pic 40]

For uniqueness, we have

[pic 41]

[pic 42]

[pic 43]

Since  is positive-sequence and  is negative-sequence, it must be the case that[pic 44][pic 45]

[pic 46]

Thus, we can re-express the equations for , , and  as[pic 47][pic 48][pic 49]

[pic 50]

[pic 51]

[pic 52]

or, in matrix form,

[pic 53]

The determinant of the matrix on the right is

[pic 54]

Equivalently, the phasors , , and  are unique. Consequently, the sequences , , and  are also unique.                                                                                                                           □[pic 55][pic 56][pic 57][pic 58][pic 59][pic 60]