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Download ELEC 3105 Lecture 22 Three Phase Power
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THREE PHASE POWER Generation Transportation Consumption THREE PHASE GENERATION THREE PHASE GENERATION A B C ππ΄ = πβ²π πππ ππ‘ A ππ΅ = πβ²π πππ ππ‘ β 120π ππΆ = πβ²π πππ ππ‘ β 240π Expressions relative to ground 3G A ππ΄π΅ = ππ πππ ππ‘ B ππ΅πΆ = ππ πππ ππ‘ β 120π C ππΆπ΄ = ππ πππ ππ‘ β 240π Between terminals THREE PHASE GENERATION A 3G B C C ο ο· ππ΄π΅ = ππ πππ ππ‘ A ππ΅πΆ = ππ πππ ππ‘ β 120π ππΆπ΄ = ππ πππ ππ‘ β 240π Between terminals B THREE PHASE GENERATION Advantage of three phase Consider resistive load Smooth flow of energy to load Instantaneous power P(t) 2 2 2 (t ) VCA (t ) VAB (t ) VBC P(t ) ο½ ο« ο« R R R ππ΄π΅ = ππ πππ ππ‘ ππ΅πΆ = ππ πππ ππ‘ β 120π ππΆπ΄ = ππ πππ ππ‘ β 240π P(t ) ο½ 3V p2 2R ο« ο cos(2ο·t ) ο« cos(2(ο·t ο 120 )) ο« cos(2(ο·t ο 240 ))ο 2R V p2 o Add to zero o THREE PHASE GENERATION P(t ) ο½ 3V p2 2R ο« οcos(2ο·t ) ο« cos(2(ο·t ο 120 )) ο« cos(2(ο·t ο 240 ))ο 2R V p2 o o ο₯ο½ 0 P(t ) ο½ 3V p2 2R ο½ constant THREE PHASE GENERATION Smooth transfer of power Smooth operation of motor P(t ) ο½ 3V p2 2R Each phase carries 1/3 of the total power delivered to the load ο½ constant A 3G B C How to connect to transmission lines THREE PHASE GENERATION Three generators (ABC) Aβ A 3G Bβ Cβ B 3T C Three transmission lines Terminal board For the generators chose: Identical generators VAA' ο½ VBB ' ο½ VCC ' and THREE PHASE GENERATION Three generators (ABC) βDELTAβ connected Aβ Bβ A 3G B Cβ Aβ A Bβ B Cβ C C 3T THREE PHASE GENERATION Three generators (ABC) βDELTAβ connected Aβ A Bβ B Cβ C C C Bβ G B Aβ G G A Cβ A B 3T THREE PHASE GENERATION Phase diagram for DELTA connected generators C C Bβ G B Aβ G G A Cβ A B ~ VCA ~ VCA ~ VAB Voltages are across pairs of transmission lines Simple VBC ο ο· VBC ~ VAB THREE PHASE GENERATION Three generators (ABC) βSTARβ connected Aβ Bβ A 3G B Cβ Ground Neutral Aβ A Bβ B Cβ C C 3T THREE PHASE GENERATION Aβ A Bβ B Cβ C Ground Neutral Three generators (ABC) βSTARβ connected C C G Aβ Bβ Cβ G B G 3T A A B THREE PHASE GENERATION Phase diagram for STAR connected generators C G Aβ Bβ Cβ B G ~ VBB ' ~ VCC ' VCB ~ VCC ' ~ VAC ~ VAA' G C A We will start with the voltage between lines A and B ο ο· A ~ VBA B ~ V AA' ~ VBB ' Recall the generator phase diagram THREE PHASE GENERATION Aβ Bβ ~ VBB ' B ~ VAA' G G A A ~ VBA B ~ ~ ~ VBA ο½ VBB ' ο« VA' A ~ ~ ~ VBA ο½ VBB ' ο VAA' Phasor βAdditionβ (- = flit direction by 180o) Change order of indices Change sign in front of supply Back to phasor diagram for generators THREE PHASE GENERATION Original Diagram for AAβ and BBβ ~ ~ ~ VBA ο½ VBB ' ο V AA' ο ο· ~ V AA' ο ο· ~ ο VAA' ~ VBB ' ~ VBB ' ~ VBA 30o ~ ο VAA' 120o ~ VBB ' THREE PHASE GENERATION ~ ~ ~ VAC ο½ VAA' ο VCC ' ~ VCC ' For VCB ~ For V AC ~ ~ ~ VCB ο½ VCC ' ο VBB ' ~ VCC ' ο ο· ο ο· ~ V AA' ο ο· 30o ~ VAC ~ V AA' ~ ο VCC ' ~ VBB ' ~ ο VBB ' VCB ~ VCC ' 30o ο ο· THREE PHASE GENERATION Phase diagram for STAR connected generators C G Aβ Bβ Cβ G B G ο ο· 120o VCB ~ VAC A ~ VAC A ~ VBA ~ VCC ' B ~ VBA 120o Phase diagram Line to line voltages have 120 degree phase difference ο ο· ~ V AA' ~ VBB ' C 120o ~ VCC ' ~ VAA' ~ VBB ' VCB Recall the generator phase diagram What about amplitude of the voltages THREE PHASE GENERATION VAA' ο½ VBB' ο½ VCC ' ~ VBA 30o ~ VBA The length of the red vector related to the length of the two black vectors ~ ο VAA' ~ VBB ' 3 VBB ' 2 3 VAA' 2 ~ ο VAA' 30o ~ VBB ' ~ VBA ο½ 3 VAA' THREE PHASE GENERATION Phase diagram for STAR connected generators C G Aβ Bβ Cβ G B G ο ο· 120o VCB ~ VAC A ~ VAC A ~ VBA B ~ VBA 120o Phase diagram ~ VBA ο½ 3 VAA' ~ VCC ' ο ο· ~ V AA' ~ VBB ' C 120o ~ VCC ' ~ VAA' ~ VBB ' VCB VAA' ο½ VBB' ο½ VCC ' Recall the generator phase diagram ~ VAC ο½ 3 VAA' ~ VCB ο½ 3 VAA' THREE PHASE TERMINOLOGY Phase Current: Current through a single generator G ~ Iο¦ Phase Voltage: Voltage across a single generator G ~ Vο¦ THREE PHASE ROTATION CONVENTION ~ VCA ~ VCC ' ο ο· VBC ο ο· ~ VAB ABC ABC ABC β¦ VBB ' ABC Motor turns clockwise ACB Motor turns counter clockwise ~ VAA' ABC ABC ABC β¦ THREE PHASE TERMINOLOGY Line current: Current in a single line feeding a load A ~ IL B 3T 3G C ~ VL Line voltage: Voltage between the lines carrying the power PHASE LINE RELATIONSHIP For the STAR connected generators C C ~ VCC ' G ~ Iο¦ VCB ~ VAC ~ VAA' G B ~ Vο¦ ~ VBB ' G A ~ VL ο½ 3 Vο¦ A ~ VBA B ~ ~ I L ο½ Iο¦ PHASE LINE RELATIONSHIP For the Delta connected generators ~ Iο¦ B Aβ C C Bβ G G G ~ Vο¦ A Cβ A B ~ VCA ~ VAB ~ VL ο½ Vο¦ ~ ~ I L ο½ 3 Iο¦ THREE PHASE TRANSFORMER Ap As NAp NAs Apβ Asβ Bp Bs NBp NBs Bpβ Bsβ Cp Cs NCp Cpβ NCs Csβ How do you connect the primary coils together and separately the secondary coils together THREE PHASE TRANSFORMER Star ----- Star THREE PHASE TRANSFORMER Star ----- Delta THREE PHASE TRANSFORMER Delta ----- Star THREE PHASE TRANSFORMER Delta ----- Delta THREE PHASE LOADS A 3G B C 3 Transformer 3T 3 Tran sfor mer 3Z Three phase load Step up or step down 3Z What does the three phase load look like? Voltages, currents, power, β¦.. THREE PHASE LOADS Made up of three impedances 3Z Usually three identical values Can be connected in STAR or DELTA Zο ZY Zο Z ο ο½ 3Z y Zο DELTA Transformation expression ZY ZY USE OF TRANSFORMATION EXPRESSION IN THREE PHASE LOADS ZL Zο Zο ZY ο½ 3 Zο ZL ZL ZY Zο ZL DELTA ZY Transmission line impedance ZL Now two impedances are in series ~ ~ ~ Zο Z ο½ ZL ο« 3 ZL ZY THREE PHASE LOADS A 3G B 3Z C Generators can be connected in STAR or DELTA Load can be connected in STAR or DELTA 4 possible configurations Delta --- Delta Delta --- Wye (Star) Wye (Star) --- Wye (Star) Wye (Star) --- Delta THREE PHASE LOADS ~ VAB ~ VBC ~ VCA Line to line voltages ~ IA ~ IB ~ IC A B C ~ IA A ~ I AB Line currents ~ IB B 0 o 120o 240 o ~ ~ ~ VAB ο½ VBC ο½ VCA Delta Connected 3Z ~ IC R R R ~ I CA C Phase currents ~ I BC DELTA Balanced three phase load THREE PHASE LOADS ~ VCA ~ VAB ~ VBC ~ IA ~ IB ~ IC Line to line voltages A B 3Z C ~ IA 120o 240 o ~ ~ ~ VAB ο½ VBC ο½ VCA ~ ~ ~ I A ο½ I AB ο« I AC A ~ I AB B 0o Delta Connected: Consider node βAβ R R ~ I CA ~ ~ ~ I A ο½ I AB ο I CA C Since load is only resistive ~ ~ VAB in phase with I AB Flip THREE PHASE LOADS ~ IA ~ VAB Delta Connected: Consider node βAβ A ~ I AB R R ~ I CA ~ ~ ~ I A ο½ I AB ο I CA C B ~ I CA ~ VAB ~ I AB Result ~ I AB ο½ ~ ~ VAB in phase with I AB ~ IA 3 ~ ~ VAB ~ I AB ο½ R Leads I A by 30 degrees 30o ~ I BC ~ ο I CA ~ IA THREE PHASE LOADS Delta Connected: Consider node βAβ Per phase power ~ IA A ~ I AB ~ IB B ~ IC R Pο¦ ο½ Vο¦ Iο¦ cos(ο¦ ) R ~ I CA C R Power factor (PF) For purely resistive load Power factor = 1.0 ~ I BC Total power: P3ο¦ ο½ 3Pο¦ ο½ 3Vο¦ Iο¦ cos(ο¦ ) Note: ο¦ is the phase angle of the load impedance. It may or may not be the phase angle between the line voltage and line current. ~ For complex load: Z L ο½ R ο« jX X R Phase angle ο¦ ο½ atn( ) THREE PHASE LOADS Delta Connected: Consider node βAβ Per phase power ~ IA ~ I AB ~ IB B ~ IC Pο¦ ο½ Vο¦ Iο¦ cos(ο¦ ) A R R R ~ I BC ~ I CA C Total power: P3ο¦ ο½ 3Pο¦ ο½ 3Vο¦ Iο¦ cos(ο¦ ) Note: Phase voltage and phase current may be difficult to measure. For example inside a motor. However line voltage and line current can always be measured IL P3ο¦ ο½ 3Pο¦ ο½ 3VL cos(ο¦ ) 3 P3ο¦ ο½ 3Pο¦ ο½ 3VL I L cos(ο¦ ) Phase angle of load (Can be measured) THREE PHASE LOADS ~ VCA ~ VAB ~ VBC ~ IA ~ IB ~ IC A B Wye (Star) Connected 3Z C A 0o STAR ~ VAB ~ ~ ~ VAB ο½ VBC ο½ VCA ZL N ~ VCA ~ VCN ~ VBC ~ IA ~ VAN 120o 240 o ~ VBN ~ IB ZL B ZL ~ IC C Z L ο½ R ο« jX Balanced three phase load THREE PHASE LOADS A STAR ~ IA ~ VAN ~ VAB ~ ~ ~ VAB ο½ VAN ο« VNB ZL N ~ VCA ~ VCN ~ VBN ~ IB ZL B ~ VBC ~ ο VBN ~ VAN ~ VAB Analysis ~ ~ ~ VAB ο½ VAN ο VBN ~ IC ZL C flip ~ VCN ~ ο VBN ~ VAN ~ VBN THREE PHASE LOADS ~ VCN ~ VAN ~ ο VBN ~ VAB 30o ~ VAN ~ VBN ~ VAN ο½ ~ ~ VAB 3 Lags V AB by 30 degrees STAR ~ ο VBN ~ VAN ~ VAB THREE PHASE LOADS Star Connected: A ~ IA Per phase power Pο¦ ο½ Vο¦ Iο¦ cos(ο¦ ) ZL N ~ IB ZL B ~ IC ZL C Power factor (PF) For purely resistive load Power factor = 1.0 ~ For complex load: Z L ο½ R ο« jX Total power: X R Phase angle ο¦ ο½ atn( ) P3ο¦ ο½ 3Pο¦ ο½ 3Vο¦ Iο¦ cos(ο¦ ) Note: ο¦ is the phase angle of the load impedance. It may or may not be the phase angle between the line voltage and line current. THREE PHASE LOADS Star Connected: A ~ IA Per phase power Pο¦ ο½ Vο¦ Iο¦ cos(ο¦ ) ZL N ~ IB ZL B ~ IC ZL C Per phase power Total power: Power factor (PF) Consider phase for point A and neutral ~ ~ Vο¦ ο½ VAN ~ ~ I and and ο¦ ο½ IL VL Pο¦ ο½ I L cos(ο¦ ) 3 P3ο¦ ο½ 3Pο¦ ο½ 3VL I L cos(ο¦ ) Same as for Delta load ~ VAN ο½ ~ VAB 3 THREE PHASE LOADS SUMMARY A 3G B 3Z C Generators can be connected in STAR or DELTA Real Power P3ο¦ ο½ 3Vο¦ Iο¦ cos(ο¦ ) P3ο¦ ο½ 3VL I L cos(ο¦ ) Load can be connected in STAR or DELTA Reactive Power Apparent Power Q3ο¦ ο½ 3Vο¦ Iο¦ sin(ο¦ ) S ο½ P2 ο« Q2 Q3ο¦ ο½ 3VL I L sin(ο¦ ) S ο½ 3Vο¦ Iο¦ S ο½ 3VL I L
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