Can It Be Balanced?

Electricity and Magnetism Level pending

In this circuit the voltages are a balanced three phase set.

V A ⃗ = 1 ∠ 0 ∘ V B ⃗ = 1 ∠ − 120 ∘ V C ⃗ = 1 ∠ 12 0 ∘ \vec{V_A} = 1 \angle 0^\circ \\ \vec{V_B} = 1 \angle {-120}^\circ \\ \vec{V_C} = 1 \angle 120^\circ

Is there any value of capacitive reactance ( X C ) (X_C) that can cause the current flowing in the neutral conductor to be zero?

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Steven Chase by Steven Chase , 37, USA

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1 solution

Steven Chase
May 16, 2018

Zero neutral current is equivalent to the following:

V A ⃗ 1 + V B ⃗ 1 + j + V C ⃗ Z C ⃗ = 0 1 ∠ 0 ∘ 1 + 1 ∠ − 120 ∘ 1 + j + 1 ∠ 12 0 ∘ Z C ⃗ = 0 \large{\frac{\vec{V_A}}{1} + \frac{\vec{V_B}}{1 + j} + \frac{\vec{V_C}}{\vec{Z_C}} = 0 \\ \frac{1 \angle 0^\circ}{1} + \frac{1 \angle {-120}^\circ}{1 + j} + \frac{1 \angle 120^\circ}{\vec{Z_C}} = 0 }

Solving for Z C ⃗ \vec{Z_C} yields:

Z C ⃗ ≈ 2.732 ∠ − 3 0 ∘ \large{\vec{Z_C} \approx 2.732 \angle -30^\circ}

Since the resistive portion of the required Z C ⃗ \vec{Z_C} is not equal to 1 1 , no value of capacitive reactance can make the neutral current zero.

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