Chapter 12
Three Phase Circuits
Chapter Objectives:
 Be familiar with different three-phase configurations and how
to analyze them.
 Know the difference between balanced and unbalanced circuits




Learn about power in a balanced three-phase system
Know how to analyze unbalanced three-phase systems
Be able to use PSpice to analyze three-phase circuits
Apply what is learnt to three-phase measurement and
residential wiring
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Three phase Circuits

An AC generator designed to develop a single sinusoidal voltage for each rotation
of the shaft (rotor) is referred to as a single-phase AC generator.

If the number of coils on the rotor is increased in a specified manner, the result is a
Polyphase AC generator, which develops more than one AC phase voltage per
rotation of the rotor

In general, three-phase systems are preferred over single-phase systems for the
transmission of power for many reasons.
1. Thinner conductors can be used to transmit the same kVA at the same voltage, which
reduces the amount of copper required (typically about 25% less).
2. The lighter lines are easier to install, and the supporting structures can be less
massive and farther apart.
3. Three-phase equipment and motors have preferred running and starting
characteristics compared to single-phase systems because of a more even flow of power
to the transducer than can be delivered with a single-phase supply.
4. In general, most larger motors are three phase because they are essentially selfstarting and do not require a special design or additional starting circuitry.
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Single Phase, Three phase Circuits
a) Single phase systems two-wire type
b) Single phase systems three-wire type.
Allows connection to both 120 V and
240 V.
Two-phase three-wire system. The AC sources
operate at different phases.
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Three-phase Generator
 The three-phase generator has three induction coils placed 120° apart on the stator.
 The three coils have an equal number of turns, the voltage induced across each coil
will have the same peak value, shape and frequency.
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Balanced Three-phase Voltages
Three-phase four-wire system
Neutral Wire
A Three-phase Generator
Voltages having 120 phase difference
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Balanced Three phase Voltages
Neutral Wire
a) Wye Connected Source
b) Delta Connected Source
Van  V p 0
Van  V p 0
Vbn  V p   120
Vbn  V p   120
Vcn  V p   240
Vcn  V p   240
a) abc or positive sequence
b) acb or negative sequence
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Balanced Three phase Loads
 A Balanced load has equal impedances on all the phases
a) Wye-connected load
b) Delta-connected load
Balanced Impedance Conversion:
Conversion of Delta circuit to Wye or Wye to Delta.
ZY  Z1  Z 2  Z 3
Z   Z a  Zb  Zc
Z  3ZY
1
ZY  Z 
3
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General Delta to Wye conversion
Delta to Wye
Wye to Delta
works the same way for complex impedances
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Three phase Connections
 Both the three phase source and the three phase load can be
connected either Wye or DELTA.
 We have 4 possible connection types.
• Y-Y connection
• Y-Δ connection
• Δ-Δ connection
• Δ-Y connection
 Balanced Δ connected load is more common.
 Y connected sources are more common.
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Balanced Wye-wye Connection
 A balanced Y-Y system, showing the source, line and load impedances.
Line Impedance
Source Impedance
Load Impedance
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Balanced Wye-wye Connection
Line current In add up to zero.
Neutral current is zero:
In= -(Ia+ Ib+ Ic)= 0
 Phase voltages are: Van, Vbn and Vcn.
 The three conductors connected from a to A, b to B and c to C are called LINES.
 The voltage from one line to another is called a LINE voltage
 Line voltages are: Vab, Vbc and Vca
 Magnitude of line voltages is √3 times the magnitude of phase voltages. VL= √3 Vp
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Balanced Wye-wye Connection
Line current In add up to zero.
Neutral current is zero:
In= -(Ia+ Ib+ Ic)= 0
 Magnitude of line voltages is √3 times the magnitude of phase voltages. VL= √3 Vp
Van  V p 0, Vbn  V p   120, Vcn  V p   120
Vab  Van  Vnb  Van  Vbn  3Vp 30
Vbc  Vbn  Vcn  3Vp   90
Vca  Vcn  Van  Van  Vbn  3Vp   210
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Balanced Wye-wye Connection
 Phasor diagram of phase and line voltages
VL  Vab  Vbc  Vca
= 3 Van  3 Vbn  3 Vcn = 3Vp
V p  Van  Vbn  Vcn
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Single Phase Equivalent of Balanced Y-Y Connection
 Balanced three phase circuits can be analyzed on “per phase “ basis..
 We look at one phase, say phase a and analyze the single phase equivalent circuit.
 Because the circuıit is balanced, we can easily obtain other phase values using their
phase relationships.
Van
Ia 
ZY
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Balanced Wye-delta Connection
 Three phase sources are usually Wye connected and three phase loads are Delta
connected.
 There is no neutral connection for the Y-∆ system.
I AB
VAB

Z
I BC 
I CA
Line currents are obtained from the phase currents IAB, IBC and ICA
VBC
Z
VCA

Z
I a  I AB  I CA  I AB 3  30
I L  I a  Ib  Ic
I b  I BC  I AB  I BC 3  30
I p  I AB  I BC  I CA
I c  I CA  I BC  I CA 3  30
I L  3I p
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Balanced Wye-delta Connection
 Phasor diagram of phase and line currents
I L  I a  Ib  Ic
I p  I AB  I BC  I CA
I L  3I p
 Single phase equivalent circuit of the balanced Wye-delta connection
Z
3
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Balanced Delta-delta Connection
 Both the source and load are Delta connected and balanced.
I AB
VBC
VCA
VAB

, I BC 
, I CA 
Z
Z
Z
I a  I AB  I CA , I b  I BC  I AB , I c  I CA  I BC
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Balanced Delta-wye Connection
Transforming a Delta connected source
to an equivalent Wye connection
Single phase equivalent of Delta Wye connection
Vp   30
3
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Chapter 12
Three Phase Circuits
Chapter Objectives:
 Be familiar with different three-phase configurations and how
to analyze them.
 Know the difference between balanced and unbalanced circuits




Learn about power in a balanced three-phase system
Know how to analyze unbalanced three-phase systems
Be able to use PSpice to analyze three-phase circuits
Apply what is learnt to three-phase measurement and
residential wiring
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
Eeng 224
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Power in a Balanced System
 The total instantaneous power in a balanced three phase system is constant.
v AN  2V p cos(t ) vBN  2V p cos(t  120) vCN  2V p cos(t  120)
ia  2 I p cos(t   ) ib  2 I p cos(t    120) ib  2 I p cos(t    120)
p  pa  pb  pc  v AN ia  vBN ib  vCN ic
p  2V p I p  cos(t ) cos(t   )  cos(t  120) cos(t    120)  cos(t  120) cos(t    120)
1
cos A cos B  [cos( A  B)  cos( A  B )] Using the identity and simplifying
2
p  3V p I p cos
The instantenous power is not function of time.
The total power behaves similar to DC power.
This result is true whether the load is Y or  connected.
The average power per phase Pp  p .
3
Pp  p
3
 V p I p cos 
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Power in a Balanced System
 The complex power per phase is Sp. The total complex power for all phases is S.
p  3V p I p cos 
1
Pp = p  V p I p cos 
3
Sp  Pp  jQp  Vp I p
1
Qp = p  V p I p sin 
S p  Vp I p
3
Complex power for each phase
P  Pa  Pb  Pc  3Pp  3V p I p cos   3VL I L cos 
Q  3Q p  3V p I p sin   3VL I L sin 
S=3Sp  3Vp I p  3I p 2 Z p 
3Vp 2
Zp

Total complex power
S  P  jQ  3VL I L 
Vp , I p , VL and I L are all rms values,  is the load impedance angle
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Power in a Balanced System
S=3Sp  3Vp I p  3I p 2 Z p 
3Vp 2
Zp

Toal complex power
S  P  jQ  3VL I L 
Vp , I p , VL and I L are all rms values,  is the load impedance angle
 Notice the values of Vp, VL, Ip, IL for different load connections.
VL  3 Vp
VL Vp
IL  I p
IL  3 I p
Ip
Vp
Ip
VL
Vp
VL
VL
Vp
VL
Ip
VL
Y connected load.
Ip
Ip
Vp
Ip
Vp
VL
Vp
Δ connected load.
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Power in a Balanced System
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Single versus Three phase systems
 Three phase systems uses lesser amount of wire than single phase systems for the
same line voltage VL and same power delivered.
a) Single phase system
b) Three phase system
Wire Material for Single phase 2( r 2l ) 2r 2 2

 '2  (2)  1.33
'2
Wire Material for Three phase 3( r l ) 3r
3
 If same power loss is tolerated in both system, three-phase system use
only 75% of materials of a single-phase system
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VL=840 V (Rms)
IL
Capacitors for pf
Correction
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IL 
S
73650

 50.68A
3 VL
3 840
Without Pf Correction
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Unbalanced Three Phase Systems
 An unbalanced system is due to unbalanced voltage sources or unbalanced load.
In a unbalanced system the neutral current is NOT zero.
Unbalanced three phase Y connected load.
Line currents DO NOT add up to zero.
In= -(Ia+ Ib+ Ic) ≠ 0
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Three Phase Power Measurement
 Two-meter method for measuring three-phase power
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Residential Wiring
Single phase three-wire residential wiring
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