Three-phase Circuits
Balanced 3-phase systems
Unbalanced 3-phase systems
SKEE 1043 Circuit Theory
Dr. Nik Rumzi Nik Idris
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Balanced 3-phase systems
POWER CALCULATION
Instantaneous power
Phase voltage of Y connected
load with Z  per phase:
Phase current, lagging by :
v AN(t)  2Vp cos(t)
ia (t)  2Ip cos(t  )
vBN(t)  2Vp cos(t  120o )
ib (t)  2Ip cos(t    120o )
vCN(t)  2Vp cos(t  120o )
ic (t)  2Ip cos(t    120o )
Total instantaneous power: p = vANia + vBNib+ vCNic
p(t)  2VpIp [cos(t) cos(t  )  cos(t) cos(t    120o )  cos(t) cos(t    120o )]
Which can be re-written as:
p(t)  VpIp [3 cos  cos(2t  )  cos(2t    240o )  cos(2t    240o )]
 p(t)  3VpIp cos
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Balanced 3-phase systems
Instantaneous power
POWER CALCULATION
p(t)  3VpIp cos
• Instantaneous power is NON PULSATING
• Smoother energy conversion for electrical machine in 3phase system
• True for delta () connected load too
3
Balanced 3-phase systems
POWER CALCULATION
Complex, apparent, active, and reactive
Phase voltage Vp  Vpv (rms)
Phase current
Ip  Ipi
(rms)
The complex power per phase:
Sp  VpIp*  VpIp cos  jVpIp sin
 Pp  jQp
Total complex power:
S  3VpIp*  3VpIp
For Y connection, Vp 
VL
3
and Ip  IL
 S  3VLIL
Apparent power is as before, i.e.
S S
4
Balanced 3-phase systems
POWER CALCULATION
Power measurement using a wattmeter
wattmeter is an instrument used for measuring the average power.
The basic structure
Equivalent Circuit with load
If v(t) = Vmcos (t + v) and i(t) = Imcos (t + i)
then the wattmeter will measure the average power:
Reading on wattmeter  P 
VmIm
cos v  i 
2
5
Balanced 3-phase systems
POWER CALCULATION
Power measurement using a wattmeter
wattmeter is an instrument used for measuring the average power.
6
Balanced 3-phase systems
POWER CALCULATION
3-phase power measurement using wattmeters: two-wattmeter method
In order to measure average power in 3-phase system ONLY 2
wattmeters are needed
Vcb
Z Y  Z Y 
Vca
Ic
Vcn
Vab

30o
30o
Van

Vbn
Ia
30o
Vbc
Reading by wattmeter1,
P1  VabIa cos1   VLIL cos(  30o )
Reading by wattmeter2,
P2  VcbIc cos2   VLIL cos(  30o )
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Balanced 3-phase systems
POWER CALCULATION
Reading by wattmeter1,
P1  VabIa cos1   VLIL cos(  30o )
Reading by wattmeter2,
P2  VcbIc cos2   VLIL cos(  30o )
IT CAN BE SHOWN :
P1  P2  3VLIL cos()
 PTotal  P1  P2
P2  P1  VLIL sin()
 QTotal  3(P2  P1)
 tan  
QTotal
PTotal
• This is also true for a  connected load as well
• This is also true if the load is unbalanced (ONLY for P1+P2)
• Two-wattmeter method CANNOT be used for a 3-phase, 4 wire system
UNLESS the neutral current = 0
• P1=P2  resistive load, P2 > P1  inductive load, P2 < P1  capacitive load
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