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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
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Power in a Balanced System The total instantaneous power in a balanced three phase system is constant.
2 cos( ) 2 cos( 120 )
2 cos( 120 )
2 cos( ) 2 cos( 120 )
2 cos( 120 )
cos( )cos( ) cos( 120 )cos( 120 )2
cos( 120 )co
AN p BN p
CN p
a p b p
c p
a b c AN a BN b CN c
p p
v V t v V t
v V t
i I t i I t
i I t
p p p p v i v i v i
t t t tp V I
t
Using the above identity and simplifying, =2 t- we obtain
s( 120 )
1cos cos [cos( ) cos( )]
2
that:
13cos
2
cos 2 oos 3c c sp pp p
t
A B A B A B
VV I Ip
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Power in a Balanced System The important consequences of the instantenous power equation of a balanced three
phase system are:
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.
T
3 cos p pp V I
he per phase is obtained as .3
AVERAGE POWE
R
cos 3
p
p p p
pP V I
pP
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Power in a Balanced System The complex power per phase is Sp. The total complex power for all phases is S.
p
p
p p p p p
3 cos
1= cos
3
1= sin
(Total Instantenous Power)
(Average Power per phase)
(Reactive Pow 3
er per phase)
(Apparent Power per pha
S V I
s
e
)
p p
p p
p p
p p p
p V I
P p V I
Q p V I
S V I
P jQ
p p
and refer to magnitude val
C
ues whereas
V and I refer to phasor
omplex power for e
values (Both magn
ach pha
itude a
se
nd phase)
p pV I
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Power in a Balanced System
The complex power per phase is Sp. The total complex power for all phases is S.
p p p p p
p p p
p p p
Complex power for each phase
Total Complex power for three phas
S V I
S= 3S 3V I
3 3 cos 3 cos
3 3 sin 3 sin
S=3S 3V
e
I 3
a b c p p p L L
a b c p p p L L
P jQ
P jQ
P P P P P V I V I
Q Q Q Q Q V I V I
I
2
p
p p L
2
p Total complex power
Total complex power using
, , and are all rms values, is the load imp
3
line val
edance
3
angl
s u e
e
p
p
L L
LV I V I
VZ
Z
P jQ V I
S
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Power in a Balanced System
Notice the values of Vp, VL, Ip, IL for different load connections.
2
p2
p p p
p
p
p L, , and are all rms values, is the load impe
To
da
3S=3S 3V I 3
nce
al c
an
3
omplex ower
gle
p
L
p
p
L L
V
VI Z
Z
P
V
V I
I
jQ
I
S
VL VL
VL
Vp Vp
Vp Ip
Ip Ip
VL
Vp
Ip
VL
VL Vp
Vp
Ip
Ip
Y connected load. Δ connected load.
3L p L pV V I I 3L p L pV V I I
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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
2 2
'2 '2
Wire Material for Single phase 2( ) 2 2(2) 1.33
Wire Material for Three phase 3( ) 3 3
r l r
r l r
If same power loss is tolerated in both system, three-phase system use
only 75% of materials of a single-phase system Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
VL=840 V (Rms)
Capacitors for pf
Correction
IL
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224
7365050.68A
3 3840
Without Pf Correction
L
L
SI
V
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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
Payam zarbakhsh
EElE301 Circuit Theory II
Department of Electrical and Electronic Engineering
Cyprus International university
‹#› Eeng 224