Delta Winding - Common Mode Removal

A set of three-phase AC (alternating current) transformer windings is connected in a delta configuration, as shown above. Other windings are not shown. Winding currents \((\vec{I_A},\vec{I_B}, \vec{I_C})\) and line currents \((\vec{I_{AL}},\vec{I_{BL}}, \vec{I_{CL}})\) are also pictured.

The following relationship holds for these currents (to a close numerical approximation):

\[\large{\vec{I_{AL}} = \vec{\alpha} (\vec{I_A} - \vec{\beta}) \\ \vec{I_{BL}} = \vec{\alpha} (\vec{I_B} - \vec{\beta}) \\ \vec{I_{CL}} = \vec{\alpha} (\vec{I_C} - \vec{\beta})}\]

To one decimal place, what is the magnitude (absolute value) of \((\vec{\alpha} + \vec{\beta})\)?

Notes:

  • \(\vec{\alpha}\) and \(\vec{\beta}\) are complex numbers.

  • In general, \(A \angle \theta\) denotes a complex number with a magnitude of \(A\) and an angle \(\theta\) with respect to the positive real axis

  • Angles are given in degrees

  • Vector multiplication here is NOT the dot product. For example: \((A + jB)(C + jD) = AC - BD + j(AD + BC)\).

  • Feel free to use a calculator to do the complex arithmetic.

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