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Chapter 11
AC Power Analysis
Chapter Objectives:
 Know the difference between instantaneous power and average
power
 Learn the AC version of maximum power transfer theorem
 Learn about the concepts of effective or Rms value
 Learn about the complex power, apparent power and power factor
 Understand the principle of conservation of AC power
 Learn about power factor correction
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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An Electical Power Distribution Center
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Apparent Power and Power Factor
 The Average Power depends on the Rms value of voltage and current and the
phase angle between them.
P  12 Vm I m cos(v  i )  VRms I Rms cos(v  i )
 The Apparent Power is the product of the Rms value of voltage and current. It is
measured in Volt amperes (VA).
1
S  Vm I m  VRms I Rms
2
 The Power Factor (pf) is the cosine of the phase difference between voltage and
current. It is also the cosine of the angle of load impedance. The power factor may
also be regarded as the ratio of the real power dissipated to the apparent power of
the load.
P
pf   cos(v  i )
S
P  Apparent Power  Power Factor  S  pf
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Apparent Power and Power Factor
 Not all the apparent power is consumed if the circuit is partly reactive.
Purely resistive
load (R)
θv– θi = 0, Pf = 1
P/S = 1, all power are
consumed
Purely reactive
load (L or C)
θv– θi = ±90o,
pf = 0
P = 0, no real power
consumption
θv– θi > 0
θv– θi < 0
• Lagging - inductive load
• Leading - capacitive load
P/S < 1, Part of the apparent
power is consumed
Resistive and
reactive load
(R and L/C)
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 Power equipment are rated using their appparent power in KVA.
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Apparent Power
and Power Factor
Both have same P
Apparent Powers and pf’s are different
Generator of the second load is
overloaded
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Apparent Power and Power Factor
Overloading of the
generator of the
second load is
avoided by
applying power
factor correction.
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Complex Power
 The COMPLEX Power S contains all the information pertaining to the
power absorbed by a given load.
2
V
1 
S  VI  VRms IRms  I 2 Rms Z  Rms
2
Z
VRms  VRms v
I Rms  I Rms i
S  VRms I Rms (v  i )
 VRms I Rms cos(v  i )  jVRms I Rms sin(v  i )
 P  jQ  Re{S}  j Im{S}  Real Power+Reactive Power
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Complex Power
 The REAL Power is the only useful power delivered to the load.
The REACTIVE Power represents the energy exchange between the
source and reactive part of the load. It is being transferred back and
forth between the load and the source
The unit of Q is volt-ampere reactive (VAR)
S  P  jQ  Re{S}  j Im{S}
=Real Power+Reactive Power
S  I 2 Rms Z  I 2 Rms ( R  jX )  P  jQ
P=VRms I Rms cos(v  i )  Re{S}  I 2 Rms R
Q=VRms I Rms sin(v  i )  Im{S}  I
2
Rms
X
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Resistive Circuit and Real Power
v(t )  Vm sin(t   )
i (t )  I m sin(t )
1
1
p(t )  v(t )i(t )  Vm I m cos( ) 1  cos(2t )   Vm I m sin( ) sin(2t )
2
2
 VRms I Rms cos( ) 1  cos(2t )   VRms I Rms sin( ) sin(2t )
 VRms I Rms  VRms I Rms cos(2t )
p(t ) is always Positive
  0 RESISTIVE
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Inductive Circuit and Reactive Power
v(t )  Vm sin(t   )
i (t )  I m sin(t )
1
1
pL (t )  v(t )i (t )  Vm I m cos( ) 1  cos( 2t )   Vm I m sin( ) sin(2t )
2
2
 VRms I Rms cos( ) 1  cos(2t )   VRms I Rms sin( ) sin(2t )
 VRms I Rms sin(2t )
  90 INDUCTIVE
pL (t ) is equally both positive and negative, power is circulating
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Inductive Circuit and Reactive Power
 If the average power is zero, and the energy supplied is returned
within one cycle, why is a reactive power of any significance?
 At every instant of time along the power curve that the curve is
above the axis (positive), energy must be supplied to the inductor,
even though it will be returned during the negative portion of the
cycle. This power requirement during the positive portion of the
cycle requires that the generating plant provide this energy during
that interval, even though this power is not dissipated but simply
“borrowed.”
 The increased power demand during these intervals is a cost
factor that must that must be passed on to the industrial consumer.
 Most larger users of electrical energy pay for the apparent power
demand rather than the watts dissipated since the volt-amperes
used are sensitive to the reactive power requirement.
 The closer the power factor of an industrial consumer is to 1, the
more efficient is the plant’s operation since it is limiting its use of
“borrowed” power.
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Capacitive Circuit and Reactive Power
v(t )  Vm sin(t   )
i (t )  I m sin(t )
1
1
pC (t )  v(t )i (t )  Vm I m cos( ) 1  cos(2t )   Vm I m sin( ) sin(2t )
2
2
 VRms I Rms cos( ) 1  cos(2t )   VRms I Rms sin( ) sin(2t )
 VRms I Rms sin(2t )
  90 CAPACITIVE
pC (t ) is equally both positive and negative, power is circulating
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Complex Power
 The COMPLEX Power contains all the information pertaining to the power
absorbed by a given load.
1 
Complex Power=S  P  jQ  VI  VRms I Rms ( v  i )
2
Apparent Power=S  S  VRms I Rms  P 2  Q 2
Real Power=P  Re{S}  S cos( v  i )
Reactive Power=Q  Im{S}  S sin( v  i )
P
Power Factor= =cos( v  i )
S
• Real Power is the actual power dissipated by the load.
• Reactive Power is a measure of the energy exchange between source and reactive
part of the load.
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Power Triangle
 The COMPLEX Power is represented by the POWER TRIANGLE similar to
IMPEDANCE TRIANGLE. Power triangle has four items: P, Q, S and θ.
a) Power Triangle
b) Impedance Triangle
Q0
Q0
Resistive Loads (Unity Pf )
Capacitive Loads (Leading Pf )
Q0
Inductive Loads (Lagging Pf )
Power Triangle
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Power Triangle
 Finding the total COMPLEX Power of the three loads.
PT  100  200  300  600 Watt
QT  0  700  1500  800 Var
ST  600  j800  1000  53.13
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Power Triangle
S  P  jQ  S1  S2  ( P1  P2 )  j (Q1  Q2 )
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Real and Reactive Power Formulation
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Real and Reactive Power Formulation
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Real and Reactive Power Formulation
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Real and Reactive Power Formulation
v(t )  Vm cos(t  v )
i (t )  I m cos(t  i )
p(t )  VRms I Rms cos(v  i ) 1  cos 2(t  v )  VRms I Rmssin(v  i ) sin 2(t  v )
=P  1  cos 2(t  v )  Q  sin 2(t  v )
=Real Power  R eactive Power
P is the REAL AVERAGE POWER
Q is the maximum value of the circulating power flowing back and forward
P  Vrms I rms cos
Q  Vrms I rms sin 
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Real and Reactive Powers
REAL POWER
CIRCULATING POWER
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Real and Reactive Powers
• Vrms =100 V Irms =1 A Apparent power = Vrms Irms =100 VA
• From p(t) curve, check that power flows from the supply into the load for the
entire duration of the cycle!
• Also, the average power delivered to the load is 100 W. No Reactive power.
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Real and Reactive Powers
Power Flowing Back
• Vrms =100 V Irms =1 A Apparent power = Vrms Irms =100 VA
• From p(t) curve, power flows from the supply into the load for only a part of
the cycle! For a portion of the cycle, power actually flows back to the source
from the load!
• Also, the average power delivered to the load is 50 W! So, the useful power is
less than in Case 1! There is reactive power in the circuit.
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 Practice Problem 11.13: The 60  resistor absorbs 240 Watt of average power.
Calculate V and the complex power of each branch. What is the total complex power?
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 Practice Problem 11.13: The 60  resistor absorbs 240 Watt of average power.
Calculate V and the complex power of each branch. What is the total complex power?
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 Practice Problem 11.14: Two loads are connected in parallel. Load 1 has 2 kW,
pf=0.75 leading and Load 2 has 4 kW, pf=0.95 lagging. Calculate the pf of two loads
and the complex power supplied by the source.
LOAD 1
2 kW
Pf=0.75
Leading
LOAD 2
4 kW
Pf=0.95
Lagging
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Conservation of AC Power
 The complex, real and reactive power of the sources equal the respective sum of the
complex, real and reactive power of the individual loads.
a) Loads in Parallel
b) Loads in Series
For parallel connection:
S
1
1
1
1
V I* 
V (I1*  I*2 )  V I1* 
V I*2  S1  S2
2
2
2
2
Same results can be obtained for a series connection.
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Complex power is Conserved
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