Download Three Phase Power Calculations File

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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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)
ic  2 I p cos(t    120)
p  pa  pb  pc  v AN ia  vBN ib  vCN ic
cos(t ) cos(t   )  cos(t  120) cos(t    120)  
p  2V p I p 

cos(

t

120

)
co
s(

t



120

)


1
cos A cos B  [cos( A  B)  cos( A  B)]
2
Using the above identity and simplifying,  =2 t-  we obtain that:


1
p  V p I p 3cos  cos   2    cos    3V p I p cos

 2

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Power in a Balanced System
 The important consequences of the instantenous power equation of a balanced three
phase system are:
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 is obtained as 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 
(Total Instantenous Power)
1
Pp = p  V p I p cos  (Average Power per phase)
3
1
Qp = p  V p I p sin  (Reactive Power per phase)
3
(Apparent Power per phase)
S p  Vp I p
Sp  Pp  jQp  Vp I p
Complex power for each phase
V p and I p refer to magnitude values whereas
Vp and I p refer to phasor values (Both magnitude and phase)
Eeng 224
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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.
Sp  Pp  jQp  Vp Ip
Complex power for each phase
S=P  jQ  3Sp  3Vp I p

Total Complex power for three phase
P  Pa  Pb  Pc  3Pp  3V p I p cos   3VL I L cos 
Q  Qa  Qb  Qc  3Q p  3V p I p sin   3VL I L sin 

S=3Sp  3Vp I p  3I p Z p 
2
S  P  jQ  3VL I L 
3Vp 2
Zp

Total complex power
Total complex power using line values
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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