Chapter 11
AC Power Analysis
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AC Power Analysis
Chapter 11
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
Instantaneous and Average Power
Maximum Average Power Transfer
Effective or RMS Value
Apparent Power and Power Factor
Complex Power
Conservation of AC Power
Power Factor Correction
Power Measurement
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11.1 Instantaneous and
Average Power (1)
• The instantaneously power, p(t)
p(t )  v(t ) i (t )  Vm I m cos (w t   v ) cos (w t   i )
1
1
 Vm I m cos ( v   i )  Vm I m cos (2w t   v   i )
2
2
Constant power
Sinusoidal power at 2wt
p(t) > 0: power is absorbed by the circuit; p(t) < 0: power is absorbed by the source.
3
11.1 Instantaneous and
Average Power (2)
• The average power, P, is the average of the instantaneous
power over one period.
1 T
1
P   p(t ) dt  Vm I m cos ( v  i )
T 0
2
1. P is not time dependent.
2. When θv = θi , it is a purely
resistive load case.
3. When θv– θi = ±90o, it is a
purely reactive load case.
4. P = 0 means that the circuit
absorbs no average power.
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11.1 Instantaneous and
Average Power (3)
Example 1
Calculate the instantaneous power and average
power absorbed by a passive linear network if:
v(t )  80 cos (10 t  20)
i (t )  15 sin (10 t  60)
Answer: 385.7  600cos(20t  10)W, 387.5W
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11.1 Instantaneous and
Average Power (4)
Example 2
A current I  10  30 flows through an
impedance Z  20  22Ω . Find the average
power delivered to the impedance.
Answer: 927.2W
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11.2 Maximum Average Power
Transfer (1)
ZTH  R TH  j XTH
ZL  R L  j X L
The maximum average power
can be transferred to the load if
XL = –XTH and RL = RTH
Pmax 
If the load is purely real, then R L 
VTH
2
8 R TH
2
2
R TH
 X TH
 ZTH
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11.2 Maximum Average Power
Transfer (2)
Example 3
For the circuit shown below, find the load impedance ZL that
absorbs the maximum average power. Calculate that maximum
average power.
Answer: 3.415 – j0.7317W, 1.429W
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11.3 Effective or RMS Value (1)
The total power dissipated by R is given by:
1 T 2
R T 2
2
P   i Rdt   i dt  I rms
R
0
0
T
T
Hence, Ieff is equal to: I eff 
1
T

T
i 2 dt  I rms
0
The rms value is a constant itself which
depending on the shape of the function i(t).
The effective of a periodic current is the dc current that delivers the
same average power to a resistor as the periodic current.
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11.3 Effective or RMS Value (2)
The rms value of a sinusoid i(t) = Imcos(wt)
is given by:
I
2
rms
Im

2
The average power can be written in terms of
the rms values:
Ieff
1
 Vm I m cos (θ v  θi )  Vrms I rms cos (θ v  θi )
2
Note: If you express amplitude of a phasor source(s) in rms, then all the
answer as a result of this phasor source(s) must also be in rms value.
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11.4 Apparent Power and
Power Factor (1)
• Apparent Power, S, is the product of the r.m.s. values of
voltage and current.
• It is measured in volt-amperes or VA to distinguish it from
the average or real power which is measured in watts.
P  Vrms I rms cos (θ v  θi )  S cos (θ v  θi )
Apparent Power, S
Power Factor, pf
• Power factor is the cosine of the phase difference between
the voltage and current. It is also the cosine of the angle
of the load impedance.
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11.4 Apparent Power and
Power Factor (2)
Purely resistive θ – θ = 0, Pf = 1
v
i
load (R)
Purely reactive
θv– θi = ±90o,
load (L or C)
pf = 0
Resistive and
reactive load
(R and L/C)
θv– θi > 0
θv– θi < 0
P/S = 1, all power are
consumed
P = 0, no real power
consumption
• Lagging - inductive
load
• Leading - capacitive
load
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11.5 Complex Power (1)
Complex power S is the product of the voltage and the
complex conjugate of the current:
V  Vmθ v
I  I mθi
1

V I  Vrms I rms  θ v  θ i
2
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11.5 Complex Power (2)
1
S  V I  Vrms I rms  θ v  θ i
2
 S  Vrms I rms cos (θ v  θi )  j Vrms I rms sin (θ v  θi )
S =
P
+ j
Q
P: is the average power in watts delivered to a load and it is
the only useful power.
Q: is the reactive power exchange between the source and
the reactive part of the load. It is measured in VAR.
• Q = 0 for resistive loads (unity pf).
• Q < 0 for capacitive loads (leading pf).
• Q > 0 for inductive loads (lagging pf).
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11.5 Complex Power (3)
 S  Vrms I rms cos (θ v  θi )  j Vrms I rms sin (θ v  θi )
S =
P
+ j
Q
Apparent Power, S = |S| = Vrms*Irms = P 2  Q 2
Real power,
P = Re(S) = S cos(θv – θi)
Reactive Power, Q = Im(S) = S sin(θv – θi)
Power factor,
pf = P/S = cos(θv – θi)
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11.5 Complex Power (4)
 S  Vrms I rms cos (θ v  θi )  j Vrms I rms sin (θ v  θi )
S =
Power Triangle Impedance Triangle
P
+ j
Power Factor
Q
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11.6 Conservation of AC Power (1)
The complex real, and reactive powers of the sources
equal the respective sums of the complex, real, and
reactive powers of the individual loads.
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
The same results can be obtained for a series connection. 17
11.7 Power Factor Correction (1)
Power factor correction is the process of increasing the
power factor without altering the voltage or current to
the original load.
Power factor correction is necessary for economic reason.
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11.7 Power Factor Correction (2)
Qc = Q1 – Q2
= P (tan θ1 - tan θ2)
= ωCV2rms
Q1 = S1 sin θ1
= P tan θ1
P = S1 cos θ1
Q2 = P tan θ2
C 
Qc
P (tan θ1  tan θ 2 )

2
2
ωVrms
ω Vrms
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11.8 Power Measurement (1)
The wattmeter is the instrument for measuring the average
power.
The basic structure
If
Equivalent Circuit with load
v(t )  Vm cos(wt   v ) and i(t )  I m cos(wt   i )
P  Vrms I rms cos (θ v  θi ) 
1
2
Vm I m cos (θ v  θi )
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