threephase - Three-Phase AC Circuits Objectives 1 To...

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Three-Phase AC Circuits Objectives : 1. To discuss balanced three-phase voltages. 2. To discuss ABC and ACB phase sequences. 3. To discuss the Δ-to-Y and Y-to-Δ trans- formations. 4. To discuss the relationship between line- to-line and line-to-neutral voltages in a Y connection. 5. To discuss the relationship between line and phase currents in a Δ connection. 1
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6. To discuss the complex power into three- phase four-wire and three-phase three-wire power systems. 7. To discuss the two-wattmeter method for real power measurement in a balanced or unbalanced three-phase three-wire power system. 8. To discuss the two-wattmeter method and the one-wattmeter method for reactive power measurement in a balanced three-phase three- wire system.
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Balanced Three-Phase Voltages an v (t) bn v (t) cn v (t) an v (t) bn V m -V m t v (t) v an ( t ) = V m cos( ωt ) v bn ( t ) = V m cos( ωt - 120 o ) v cn ( t ) = V m cos( ωt + 120 o ) v an ( t ) + v bn ( t ) + v cn ( t ) = 0 at any time t 2
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ABC and ACB Phase Sequences V ~ an V cn ~ V ~ bn + + -120 +120 +120 -120 o o o o ~ V an ~ V V ~ bn cn ˜ V an = V ln 6 0 o ˜ V an = V ln 6 0 o ˜ V bn = V ln 6 - 120 o ˜ V bn = V ln 6 120 o ˜ V cn = V ln 6 120 o ˜ V cn = V ln 6 - 120 o 3
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Line-to-Neutral and Line-to-Line Voltages in a Y Connection n + V ~ V an ~ bn V cn ~ + - V ab ~ + - V bc ~ + - V ca ~ - + - + - - Z AB - Z CA - Z BC ˜ V an = V ln 6 0 o ˜ V ab = V ll 6 30 o ˜ V bn = V ln 6 - 120 o ˜ V bc = V ll 6 - 90 o ˜ V cn = V ln 6 120 o ˜ V ca = V ll 6 150 o V ll = 3 V ln 4
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Line-to-Neutral and Line-to-Line Voltages in a Y Connection V ~ V ~ V ~ V ~ 30 o -120 120 o o bc bn ca ca ab an ~ V ~ V ˜ V ab = ˜ V an - ˜ V bn = 3 ˜ V an e j 30 o = 3 V ln 6 (0 o + 30 o ) = V ll 6 30 o ˜ V bc = ˜ V bn - ˜ V cn = 3 ˜ V bn e j 30 o = 3 V ln 6 ( - 120 o + 30 o ) = V ll 6 - 90 o ˜ V ca = ˜ V cn - ˜ V an = 3 ˜ V cn e j 30 o = 3 V ln 6 (120 o + 30 o ) = V ll 6 150 o 5
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