A three-phase wye-connected load is supplied by an abc-sequence balanced three-phase wye-connected source with a phase voltage of 120 V rms. If the line impedance and load impedance per phase are 1 + j1 Ω and 20 + j10 Ω, respectively, we wish to determine the value of the line currents and the load

Question

A three-phase wye-connected load is supplied by an abc-sequence balanced three-phase wye-connected source with a phase voltage of 120 V rms. If the line impedance and load impedance per phase are 1 + j1 Ω and 20 + j10 Ω, respectively, we wish to determine the value of the line currents and the load voltages.

Accepted Answer

The phase voltages are

[latex]\begin{aligned}&\mathbf{V}_{a n}=120 \underline{/ 0^{\circ}} \mathrm{V} \mathrm{rms} \\&\mathbf{V}_{b n}=120 \underline{/-120^{\circ}} \mathrm{V} \mathrm{rms} \\&\mathbf{V}_{c n}=120 \underline{+120^{\circ}} \mathrm{V} \mathrm{rms}\end{aligned}[/latex]

The per-phase circuit diagram is shown in Fig. 11.15. The line current for the a phase is

[latex]\begin{aligned}\mathbf{I}_{a A} &=\frac{120 \underline{/ 0^{\circ}}}{21+j 11} \\&=5.06 \underline{/-27.65^{\circ}} \mathrm{A} \mathrm{rms}\end{aligned}[/latex]

The load voltage for the a phase, which we call [latex]\mathbf{V}_{A N}[/latex], is

[latex]\begin{aligned}\mathbf{V}_{A N} &=\left(5.06\underline{/-27.65^{\circ}} ight)(20+j 10) \\&=113.15\underline{/-1.08^{\circ}} \mathrm{V} \mathrm{rms}\end{aligned}[/latex]

The corresponding line currents and load voltages for the b and c phases are

[latex]\begin{array}{ll}\mathbf{I}_{b B}=5.06 \underline{/-147.65^{\circ}} \mathrm{A} \mathrm{rms} &\qquad \mathbf{V}_{B N}=113.15 \underline{/-121.08^{\circ}} \mathrm{V} \mathrm{rms} \\\mathbf{I}_{c C}=5.06\underline{/-267.65^{\circ}} \mathrm{A} \mathrm{rms} &\qquad \mathbf{V}_{C N}=113.15\underline{/-241.08^{\circ}} \mathrm{V} \mathrm{rms}\end{array}[/latex]

To reemphasize and clarify our terminology, phase voltage, [latex]V_{p}[/latex], is the magnitude of the phasor voltage from the neutral to any line, while line voltage, [latex]V_{L}[/latex], is the magnitude of the phasor voltage between any two lines. Thus, the values of [latex]V_{L}[/latex] and [latex]V_{p}[/latex] will depend on the point at which they are calculated in the system.

[ hint ]

[latex]\begin{aligned}&\underline{/ I_{b B}}=\underline{/ I_{a A}}-120^{\circ} \&\underline{/ I_{c C}}=\underline{/ I_{a A}}+120^{\circ}\end{aligned}[/latex]

Question 11.2: A three-phase wye-connected load is supplied by an abc-seque...

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