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Y-Δ transform
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==Δ to Y transformation of a practical generator== During the analysis of balanced [[Three-phase electric power|three-phase]] [[Electric power system|power systems]], usually an equivalent per-phase (or single-phase) circuit is analyzed instead due to its simplicity. For that, equivalent wye connections are used for [[Electric generator|generators]], [[Transformer|transformers]], loads and [[AC motor|motors]]. The stator windings of a practical delta-connected three-phase generator, shown in the following figure, can be converted to an equivalent wye-connected generator, using the six following formulas{{efn|For a demonstration, read the [[Talk:Y-Δ_transform#Derivation_of_the_formulas_for_converting_a_delta_to_wye_practical_generator|Talk page]].}}: [[File:Practical generator connected in delta-triangle (version 2).png|275px|thumb|center|Practical generator connected in delta/triangle/pi. The quantities shown are phasor voltages and complex impedances. Click on image to expand it.]] <math> \begin{align} & Z_\text{s1Y} = \dfrac{Z_\text{s1} \, Z_\text{s3}}{Z_\text{s1} + Z_\text{s2} + Z_\text{s3}} \\[2ex] & Z_\text{s2Y} = \dfrac{Z_\text{s1} \, Z_\text{s2}}{Z_\text{s1} + Z_\text{s2} + Z_\text{s3}} \\[2ex] & Z_\text{s3Y} = \dfrac{Z_\text{s2} \, Z_\text{s3}}{Z_\text{s1} + Z_\text{s2} + Z_\text{s3}} \\[2ex] & V_\text{s1Y} = \left( \dfrac{V_\text{s1}}{Z_\text{s1}} - \dfrac{V_\text{s3}}{Z_\text{s3}} \right) Z_\text{s1Y} \\[2ex] & V_\text{s2Y} = \left( \dfrac{V_\text{s2}}{Z_\text{s2}} - \dfrac{V_\text{s1}}{Z_\text{s1}} \right) Z_\text{s2Y} \\[2ex] & V_\text{s3Y} = \left( \dfrac{V_\text{s3}}{Z_\text{s3}} - \dfrac{V_\text{s2}}{Z_\text{s2}} \right) Z_\text{s3Y} \end{align} </math> The resulting network is the following. The neutral node of the equivalent network is fictitious, and so are the line-to-neutral phasor voltages. During the transformation, the line phasor currents and the line (or line-to-line or phase-to-phase) phasor voltages are not altered. [[File:Equivalent practical generator connected in wye-star (version 2).png|275px|thumb|center|Equivalent practical generator connected in wye/star/tee. Click on image to expand it.]] If the actual delta generator is balanced, meaning that the internal phasor voltages have the same magnitude and are phase-shifted by 120° between each other and the three complex impedances are the same, then the previous formulas reduce to the four following: <math> \begin{align} & Z_\text{sY} = \dfrac{Z_\text{s}}{3}\\ & V_\text{s1Y} = \dfrac{V_\text{s1}}{\sqrt{3} \, \angle \pm 30^\circ} \\[2ex] & V_\text{s2Y} = \dfrac{V_\text{s2}}{\sqrt{3} \, \angle \pm 30^\circ} \\[2ex] & V_\text{s3Y} = \dfrac{V_\text{s3}}{\sqrt{3} \, \angle \pm 30^\circ} \end{align} </math> where for the last three equations, the first sign (+) is used if the phase sequence is positive/''abc'' or the second sign (−) is used if the phase sequence is negative/''acb''.
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