ECE 1311 Ch11
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Transcript of ECE 1311 Ch11
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ECE 1311
Chapter 11AC PowerAnalysis
1
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Outlines
11.1 Instantaneous and Average Power11.2 Maximum Average Power Transfer
11.3 Effective or RMS Value11.4 Apparent Power and Power Factor11.5 Complex Power
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Sinusoidal power at 2tConstant power
)2(cos2
1
)(cos2
1
)(cos)(cos)()()(
ivmmivmm
ivmm
tIVIV
ttIVtitvtp
w
ww
11.1 Instantaneous andAverage Power (1)
The instantaneous power (p(t))
p(t) > 0: power is absorbed by the circuit; p(t) < 0: power is supplied by the circuit.
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4
)(cos2
1)(
1
0ivmm
T
IVdttpT
P
11.1 Instantaneous andAverage Power (2)
The average power (P) is the average of the instantaneouspower over one period.
1. P is not time dependent.
2. When v= i , it is a purely
resistiveload case.
3. When vi= 90o, it is a
purely reactiveload case.
4. P = 0 means that the circuit
absorbs no average power.
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5
)6010(sin15)(
)2010(cos80)(
tti
ttv
11.1 Instantaneous andAverage Power (3)
Example 1
Calculate the instantaneous power and average
power absorbed by a passive linear network if:
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Example 2
A current flows through an
impedance . Find the average
power delivered to the impedance.
11.1 Instantaneous andAverage Power (4)
3010I
2220 Z
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Example 3
Calculate the average power absorbed by the resistor and
inductor. Find the average power supplied by the voltage
source.
11.1 Instantaneous andAverage Power (5)
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11.2 Maximum Average PowerTransfer (1)
The maximum average power
can be transferred to the load if:
TH
2
TH
maxR8
VP
If the load is purely real, then:TH
2
TH
2
THL ZXRR
*ThL ZZ
L
2
max RI
2
1P
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11.2 Maximum Average PowerTransfer (2)
Example 4
For the circuit shown below, find the load impedance ZLthatabsorbs the maximum average power. Calculate that maximum
average power.
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11.3 Effective or RMS Value (1)
2
II mrms
The average power can be written in terms of
the rms values:
)(cosIV)(cosIV2
1
ivrmsrmsivmm P
Note: If you express amplitude of a phasor source(s) in rms, then all the
answers as a result of this phasor source(s) must also be in rms value.
The rms value of a sinusoid i(t) = Imcos(wt)is given by:
rmseff II--Note
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11.4 Apparent Power andPower Factor (1)
Apparent Power (S)is the product of the rms values ofvoltage and current.
It is measured in volt-amperesor VA to distinguish it fromthe average or real power which is measured in watts.
Power factor is: the cosine of the phase difference between the voltage and current
or
The cosine of the angle of the load impedance.
)(cosS)(cosIVP ivivrmsrms
Apparent Power, S Power Factor, pf
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11.4 Apparent Power andPower Factor (2)
Purely resistive
load (R)vi = 0, pf = 1 P/S = 1, all power are
consumed
Purely reactive
load (L or C)vi = 90
o,
pf = 0
P = 0, no real power
consumption
Resistive and
reactive load
(R and L/C)
vi > 0
vi < 0
Lagging- inductive
load
Leading- capacitive
load
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11.4 Apparent Power andPower Factor (3)
Example 5
For the circuit shown below, calculate the power factor as seenby the voltage source. What is the average power supplied by
the voltage source?
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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:
*IVIV2
1S
rmsrms
*Z
VZIS
2
rms2
rms
imvm IV IV
irmsvrms IV IV
Note:
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11.5 Complex Power (2)
ivrmsrms IVIV2
1S
)(sinIVj)(cosIVS ivrmsrmsivrmsrms
P: is the average power in wattsdelivered to a load and it is the only useful
power.
Q: is the reactive power exchangebetween the source and
the reactive part of the load. It is measured in volt-ampere-reactive (VAR).
Q = 0 for resistive loads(unity pf).
Q < 0 for capacitive loads(leading pf).
Q > 0 for inductive loads(lagging pf).
S = P + j Q
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11.5 Complex Power (3)
)(sinIVj)(cosIVS ivrmsrmsivrmsrms
Apparent Power:S = |S| = Vrms*Irms=
Real power: P = Re(S) = S cos(vi)
Reactive Power: Q = Im(S) = S sin(vi)
Power factor: pf = P/S = cos(vi)
S = P + j Q
22 QP
S
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11.5 Complex Power (4)
)(sinIVj)(cosIVS ivrmsrmsivrmsrms
S = P + j Q
Impedance TrianglePower Factor
Power Triangle
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11.5 Complex Power (5)
Example 6:
Given:
Determine:
The complex and apparent powers.
The real and reactive powers.
The power factor and the load impedance.
AI
VV
rms
rms
154.0
85110