BALANCED THREE-PHASE AC CIRCUIT

Transcription

1 BAANCED THREE-PHASE AC CRCUT Balanced Three-Phase oltage Sources Delta Connection Star Connection Balanced 3-hase oad Delta Connection Star Connection Power in a Balanced Phase Circuit

2 ntroduction Three Phase System

3 Balanced Three Phase oltages Three-hase voltage sources a) wye-connected source b) delta-connected source f the voltage source have the same amlitude and frequency ω and are out of hase with each other by 12 o, the voltage are said to be balanced. an bn cn an + bn + cn Balanced hase voltages are equal in magnitude and out of hase with each other by 12 o

4 Balanced Three Phase oltages m a b c abc sequence or ositive sequence: an bn cn acb sequence or negative sequence: an cn bn is the effective or rms value + 12 ωt

5 Balanced Three Phase oads Two ossible three-hase load configurations: a) a Star or Y-connected load b) a delta-connected load For a balanced wye connected load: Y a b c Y 1 3 For a balanced delta connected load: 3 Y

6 Examle 1 Determine the hase sequence of the set of voltages van 2 2 cos(ωt + 1 ) vbn 2 2 cos(ωt 23 ), vcn 2 2 cos(ωt 11 ) Solution: The voltages can be exressed in hasor form as We notice that an leads cn by 12 and cn in turn leads bn by 12. an 2 1 bn 2 23 Hence, we have an acb sequence. cn 2 11 Given that bn 11 3, find an and cn, assuming a ositive (abc) sequence. Answer: an cn 11 9

7 Balanced Y-Y Connection A balanced Y-Y system is a three hase system with a balanced Y connected source and balanced Y connected load. s l Y Source imedance ine imedance oad imedance s + l + Y

8 Balanced Wye-Wye Connection s l Y Y Source imedance ine imedance oad imedance Total imedance er hase s + l + Y

9 Balanced Y-Y Connection ine to line voltages or line voltages given that hase a voltage is reference can be shown to be: cn ab ab 3 3 bc o 3 o an ca 3 21 ab bc an bn ca cn bn 3

10 Balanced Y-Y Connection Given the hase voltages, the line current can be calculated as: Alying K to each hase: an a Y 12 bn an b a 12 Y Y 24 cn an c a 24 Y Y a + b + c n nn nn Thus, the er-hase equivalent circuit can be exressed as: a an Y

11 Y-Y configuration Examle:1 A balanced ositive-sequence Y-connected 6 Hz three-hase source has hase voltage a1. Each hase of the load consists of a.1-h inductance in series with a 5-Ω resistance. Find the line currents, the line voltages, the ower and the reactive ower delivered to the load. Draw a hasor diagram showing line voltages, hase voltages and the line currents. Assuming that the hase angle of an is zero. & an R + jω 5 + j & aa θ 37 & bb , & cc & & & , & ab an bc ca

12 Examle 2 1- Calculate the line currents in the three wire Y-Y system of figure below. 2- A Y-connected balanced three-hase generator with an imedance of.4+j.3 Ω er hase is connected to a Y-connected balanced load with an imedance of 24 + j19 Ω er hase. The line joining the generator and the load has an imedance of.6 + j.7 Ω er hase. Assuming a ositive sequence for the source voltages and that an 12 3 Find: (a) the line voltages (b) the line currents

13 Balanced Y-Delta Connection A balanced Y- system consists of balanced Y connected source feeding a balanced connected load. ine voltages: ab bc ca Phase currents: ine currents: a b c BC CA CA BC BC CA BC ; BC ; CA CA

14 CA a Balanced Y-Delta Connection 24 CA a 3 3 ( ) Magnitude line currents: 3 a b c BC CA A single hase equivalent circuit Y 3 a an Y an / 3

15 Y-Delta configuration: Examle 3 1- A balanced abc sequence Y-connected source with 1 1 is connected to a -connected balanced load (8+j4)Ω er hase. Calculate the hase and line currents. an 2-One line voltage of a balanced Y-connected source is 18 2 f the source is connected to a -connected load of 2 4 Ω, find the hase and line currents. Assume the abc sequence.

16 Balanced Delta-Delta Connection A balanced - system is one in which both balanced source and balanced load are connected.

17 ab bc ca Balanced Delta-Delta Connection A balanced - system is the one in which both balanced source and balanced load are connected. ine voltages: BC CA ine currents: a b c Magnitude line currents: BC CA 3 CA BC Total imedance: Y 3 Phase currents: BC CA BC CA

18 Examle4: A delta-connected source sulies a delta-connected load through wires having imedances of line.3+j.4ω, the load imedance are 3+j6 Ω, the balanced source ab voltage is ab1<3 Find the line current, the line volatge at the load, the current in each hase of the load, the ower delivered to the load, and dissiated in the line.

19 1- A balanced connected load having an imedance of 2-j15 Ω is connected to a connected, ositive sequence generator having ab 33 Calculate the hase currents of the load and the line currents. 2- A ositive-sequence, balanced -connected source sulies a balanced -connected load. f the imedance er hase of the load is 18+j12 Ω and, find and. a A

20 Balanced Delta-Y Connection Relace connected source to equivalent Y connected source. A single hase equivalent circuit a an Y 3 Y 3 Phase voltages: an bn cn

21 Y-Delta configuration: Examle 5 1-A balanced Y connected load with a hase resistance of 4 Ω and a reactance of 25 Ω is sulies by a balanced, ositive sequence connected source with a line voltage of 21. Calculate the hase currents. Use ab as reference. 2-n a balanced -Y circuit, ab and Y (12 + j15) Ω. Calculate the line currents.

22 v AN POWER N A BAANCED SYSTEM For Y connected load, the hase voltage: 2 cosωt, v 2 cos( ωt 12 ), v 2 cos( ωt + 12 BN CN ) f Phase current lag hase voltage by θ. Y θ The hase current: i i i a b c cos( ωt θ) cos( ωt θ cos( ωt θ ) )

23 POWER N A BAANCED SYSTEM Total instantaneous ower: + + v i + v i + a b c AN a BN b v CN i c 3 cos θ Average ower er hase: Reactive ower er hase: P cos θ Q sin θ Aarent ower er hase: S Comlex ower er hase: S P + jq *

24 POWER N A BAANCED SYSTEM Total average ower: P 3P 3 cos θ 3 cos θ Total reactive ower: Q 3Q 3 sin θ 3 sin θ Total comlex ower: S 3S 3 3 * * S P + jq 3 θ

25 Power: Examle 6 1-A three-hase motor can be regarded as a balanced Y-load. A three-hase motor draws 5.6 kw when the line voltage is 22 and the line current is 18.2 A. Determine the ower factor of the motor. 2- Calculate the line current required for a 3-kW three-hase motor having a ower factor of.85 lagging if it is connected to a balanced source with a line voltage of 44.