Three Phase Circuits

Three Phase Systems

Bulk power generation and transmission systems are three-phase (3-Φ) systems. Generation and transmission of electrical power are more economical and efficient in (3-Φ) systems than in (1-Φ) systems.

3-Phase Circuits Dr. H.H.Hanafy 2

Generation of 3-Phase Voltages1-Φ Voltage Generation:

0 90 180 270 360

Vm

0

-Vm

N

S

3-Φ Voltage Generation:

The three phases are called: A-B-C or R-S-T or R-Y-B3-Phase Circuits Dr. H.H.Hanafy 3

Balance ThreeBalance Three--Phase VoltagesPhase VoltagesPhase a →va = Vm sin ωtPhase b →vb = Vm sin (ωt – 1200)Phase c →vc = Vm sin (ωt – 2400) = Vm sin (ωt + 1200)

In Phasor Form

Va

Vb

Vc

1200

-1200

o

o

o

120

120

0

∠=

−∠=

∠=

VV

VV

VV

c

b

a

3-Phase Circuits Dr. H.H.Hanafy 44

Types of Three Phase SystemsBalanced Systems:

- 3 phases (V & I) are equal in magnitude and 1200 out of phase- 3 phase systems are usually balanced in normal operation - Can be analyzed considering only one phase

(per phase equivalent circuit)

Unbalanced Systems: - 3 phases (V or I) are unequal in magnitude and have unequal

phase shift- Relatively difficult to analyze(network theorem, symmetrical components)

Three Phase Systems are connected either in:- Y (wye or star) connection (3 phase 3 wire, or 3 phase 4 wire)- ∆ (delta) connection

3-Phase Circuits Dr. H.H.Hanafy 5

Phase Sequence:The order in which the voltages of the individual phases reach their maximum values.So the phase sequence for the previous voltages is a-b-c

Balanced phase voltages are equal in magnitude and are out of phase with each other by 120°.

A balanced load is one in which the phase impedances are equal in magnitude and in phase

3-Phase Circuits Dr. H.H.Hanafy 6

Balance ThreeBalance Three--Phase VoltagesPhase Voltages

• Two possible configurations:

Three-phase voltage sources: (a) Y-connected ; (b) ∆-connected

3-Phase Circuits Dr. H.H.Hanafy 77

3-Phase Circuits Dr. H.H.Hanafy 8

Line and Phase Parameters

Y (wye or Star) connection (3 phase 3 or 4 wire)

∆ (Delta) connection

A

B

C

a

b

Zab

Zbcc

ZcaAB

C

a b

c

n

Za

Zb

Zc

N

Phase parameters usually not easily accessible

3-Phase Circuits Dr. H.H.Hanafy 9

Phase voltages, Vph:|Van| = |Vbn| = |Vcn| = Vph

Phase currents, Iph:|I an| = |I bn| = |I cn| = Iph

3-Phase Circuits Dr. H.H.Hanafy 10

Phase sequence a-b-c

Vab = Van – Vbn, Vbc = Vbn – Vcn, Vca = Vcn - Van

From phasor diagram: Vab = 2 |Van| Cos 300 = √3 Vphi.e. |Vab| = √3 |Van| and Vab leads Van by 300

-Vbn

300

Vbn

Van

Vcn Vab

Vbc

Vca

Balanced Y System

|Vline| = √3 |Vphase| Iline = Iphase

Line voltages, VL:|Vab| = |Vbc| = |Vca|

Line currents, IL:|I a| = |I b| = |I c|

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Balanced ∆ SystemPhase voltages, Vph:|Vab| = |Vbc| = |Vca| =Vph

Phase currents, Iph:|I ab| = |I bc| = |I ca|

Line voltages, VL:|Vab| = |Vbc| = |Vca|

Line currents, IL:|I a| = |I b| = |I c|

3-Phase Circuits Dr. H.H.Hanafy 12-Ica

300

Ibc

Iab

Ica

Ib

Ic

Ia

Phase sequence a-b-c

I a = Iab - Ica

From phasor diagram:Ia = 2 |Iab| Cos 300 = √3 Iphi.e. |Ia| = √3 |Iab| and Ia lags Iab by 300

Vline = Vphase |Iline| = √3 |Iphase|

Balanced 3-Phase SystemsY → |VL| = √3 |Vp|, IL = Ip, V (line) leads V (ph) by 300

∆ → VL = Vp, |IL| = √3 |Ip|, I (line) lags I (ph) by 300

Power Factor of a 3-φ load → cos φwhere, φ is the angle between the phase current and the phase voltage

3-Phase Circuits Dr. H.H.Hanafy 13

3-phase power = 3 x Per phase powerP (3- Φ) = 3 |Vph|.|Iph|. cos φ = √3.|VL|.|IL|. cos φ

Q (3- Φ) = 3 |Vph|.|Iph|.sin φ = √3.|VL|. |IL|. sin φ

S (3- Φ) = 3 |Vph|.|Iph| = √3.|VL|. |IL|)

Single phase equivalent circuit of the balanced connection

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Equivalent circuit per phase3-Phase Circuits Dr. H.H.Hanafy 15

Power System Loads

3-Phase Circuits Dr. H.H.Hanafy 16

Power Factor CorrectionThe design of any power transmission system is very sensitive to the magnitude of the current in the lines as determined by the applied loads.Increased currents result in increased power losses (by a squared factor since P = I2R) in the transmission lines due to the resistance of the lines.Heavier currents also require larger conductors, increasing the amount of copper needed for the system, and they require increased generating capacities by the utility company.Since the line voltage of a transmission system is fixed, the apparent power is directly related to the magnitude of current .In turn, the smaller the net apparent power, the smaller the current drawn from the supply. Minimum current is therefore drawn from a supply when S = P and QT = 0.

P.F. Correction Dr. H.H.Hanafy 17

Benefits of P.F. Correction :1.Smaller conductor size (less cost) for power

transmission/distribution (or can carry more load, i.e. less transmission congestion)

2. Less transmission losses (I2.R)

3. Less voltage drop (I.Z) in transmission

4. Same transformer can supply more load

5. Less Q generation required.P.F. Correction Dr. H.H.Hanafy 18

Increasing the power factor without changing the voltage or current to the load is called Power Factor Correction.

V

IL

φold V

IL

Ic

IcI

φnew

Original Inductive Load Inductive Load with power factor correction

Phasor diagrams showing the effect of adding a capacitor in parallel with the inductive load

P.F. Correction Dr. H.H.Hanafy 19

Q1 = S1 sin φold = P tan φold

Q2 = S2 sin φnew = P tan φnew

)tan(tan21 newoldC PQQQ ϕϕ −=−=

22

CVXVQ

CC ω==

Power triangle of power factor correction

Value of required shunt capacitance :

22 2 VfQ

VQC CC

πω==

P.F. Correction Dr. H.H.Hanafy 20

For 3-phase systems)tan(tan newoldphCph PQ ϕϕ −=

22

cphC

cCph VC

XVQ ω==

Value of required shunt capacitance per phase :

22 2 c

Cph

c

Cphph Vf

QV

QC

πω==

For Star connected capacitors bank Vc = VL/√3

For delta connected capacitors bank Vc = VL

Hence: CY = 3 C∆P.F. Correction Dr. H.H.Hanafy 21