Transformation of Fortescue

All system of three-phase sizes unbalanced can be put in the form of the sum of three balanced systems (or symmetrical):

• a system balanced direct noted G_d .
• a system balanced opposite noted G_i .
• a sytème of homopolar tension noted G_o (actually a single-phase size which one divides into 3 for matrix algebra).

Homopolar three-phase systems

As explained previously, it is not really a three-phase system because that corresponds to a system of 3 tensions in phase:

$g\_o\; =\; G\_o\; \backslash \; sin\; (\backslash \; Omega\; t+\; \backslash \; varphi\_o)$

$g\_o\; =\; G\_o\; \backslash \; sin\; (\backslash \; Omega\; t+\; \backslash \; varphi\_o)$

$g\_o\; =\; G\_o\; \backslash \; sin\; (\backslash \; Omega\; t+\; \backslash \; varphi\_o)$

The interest of this false three-phase system is to facilitate the matric writing of the transformation of Fortescue.

Matrix of transformation

The goal is to find the values of G_d , G_i and G_o starting from G₁ , G₂ and G₃ .

Calculation of G_o

As the sum of the three sizes of a balanced system is null, one with inevitably:

$3\; G\_o\; \backslash \; sin\; (\backslash \; Omega\; t+\; \backslash \; varphi\_o)\; =\; G\_1\; \backslash \; sin\; (\backslash \; Omega\; t+\; \backslash \; varphi\_1)\; +G\_2\; \backslash \; sin\; (\backslash \; Omega\; t+\; \backslash \; varphi\_2)\; +G\_3\; \backslash \; sin\; (\backslash \; Omega\; t+\; \backslash \; varphi\_3)$

Operator of rotation: a

Remark : An underlined size represents the complex number associated with the sinusoidal size considered.

It is a Complex number of module 1 and argument $\backslash \; tfrac23\; \backslash \; pi$: $\backslash \; underline\; has\; =\; e^\; \{J\; \backslash \; frac23\; \backslash \; pi\}$

The result of its multiplication to the complex number associated with a size corresponds to another size of the same amplitude and out of phase of $\backslash \; tfrac23\; \backslash \; pi$ compared to the initial size. It corresponds to a rotation of $\backslash \; tfrac23\; \backslash \; pi$ in the plan of Fresnel.

It checks the following properties:

• $\backslash \; underline\; a^3\; =\; 1$
• $1\; +\; \backslash \; underline\; a+\; \backslash \; underline\; a^2\; =\; 0$

Stamp of opposite Fortescue

$\backslash begin\{bmatrix\}\; \backslash underline\; G\_d\backslash \backslash \; \backslash underline\; G\_i\backslash \backslash \; \backslash \; underline\; G\_o\; \backslash end\{bmatrix\}$

\ frac13

\begin{bmatrix} 1 & \ underline has & \ underline a^2 \ \ 1 & \ underline a^2 & \ underline has \ \ 1 & 1 & 1 \end{bmatrix}

\begin{bmatrix} \underline G_1\\ \underline G_2\\ \ underline G_3 \end{bmatrix}

See too

Internal bonds

• Transformation of the three-phase systems

External bonds

• Transformation of the three-phase systems Fortescue, Clarke, Park and Ku

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