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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 Voltages

1-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 Voltages

Phase 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




Types of Three Phase Systems

Balanced 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 unequalphase 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) connection3-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 andare 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




Balance ThreeBalance Three--Phase VoltagesPhase Voltages

• Two possible configurations:

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




Line and Phase Parameters

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

∆ (Delta) connection

A

B

C

a

b

Zab

Zbcc

ZcaA

B

C

ab

c

n

Za

Zb

Zc

N

Phase parameters usually not easily accessible




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

Phase currents, Iph:

|I an| = |I bn| = |I cn| = Iph


Phase sequence a-b-c

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

From phasor diagram:Vab = 2 |Van| Cos 300 = √3 Vph

i.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|



Balanced ∆ System

Phase 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|


-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 Iph

i.e. |Ia| = √3 |I

ab| and I

alags I

abby 300

Vline = Vphase |Iline| = √3 |Iphase|



Balanced 3-Phase Systems

Y → |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 φ

where, φ is the angle between the phase current and the

phase voltage


3-phase power = 3 x Per phase power

P (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




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



Power System Loads


P F C i



Power Factor Correction¾

The design of any power transmission system is very sensitiveto the magnitude of the current in the lines as determined bythe applied loads.

¾ Increased currents result in increased power losses (by a

squared factor since P = I 2 R) in the transmission lines due tothe resistance of the lines.

¾ Heavier currents also require larger conductors, increasing the

amount of copper needed for the system, and they requireincreased generating capacities by the utility company.

¾ Since the line voltage of a transmission system is fixed, the

apparent power is directly related to the magnitude of current .¾ 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 powertransmission/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



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 loadP.F. Correction Dr. H.H.Hanafy 19



Q1 = S 1 sin φold = P tan φold

Q2 = S 2 sin φnew = P tan φnew

)tan(tan21 newoldC PQQQ ϕ ϕ −=−=

22

CV

X

VQ

C

C ω ==

Power triangle of power factor correction

Value of required shunt capacitance :

22 2 Vf

Q

V

QC CC

π ω ==


For 3 phase systems



For 3-phase systems

)tan(tannewoldphCph

PQ ϕ −=

22

cph

C

cCph VC

X

VQ ω ==

Value of required shunt capacitance per phase :

22

2 c

Cph

c

Cph

ph

Vf

Q

V

QC

π ω

==

For Star connected capacitors bank Vc = VL/√3

For delta connected capacitors bank Vc = VL

Hence: CY = 3 C∆


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