8. AC POWER CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013...
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Transcript of 8. AC POWER CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013...
8. AC POWER
CIRCUITS by Ulaby & MaharbizAll rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not copy or distribute. ©
2013 National Technology and Science Press
Linear Circuits at ac
Instantaneous power Average power
)()()( tittp
Power at any instant of time Average of instantaneous power over one period
T
dttpT
P0
)(1
Power delivery (utilities) Electronics (laptops, mobile phones, etc.) Logic circuits
Power is critical for many reasons:
Note: Power is not a linear function, cannot apply superposition
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Instantaneous Power for Sinusoids
Power depends on phases of voltage and current
i
i
ttIVtp
tittp
tIti
tVt
coscos)(
)()()(
cos)(
cos)(
mm
m
m
BABABA coscos2
1coscos
ii tIVtp 2coscos2
1)( mm
Trig. Identity:
Constant in time (dc term)
ac at 2w
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Average Value
Sine wave
Truncated sawtooth
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Average Value for
These properties hold true for any values of φ1 and φ2
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Effective or RMS Value
Equivalent Value That Delivers Same Average Power to Resistor as in dc case
For current given by
Effective value is the (square) Root of the Mean of the Square of the periodic signal, or RMS value
TTdti
T
RdtRi
TP
0
2
0
21
RIP 2eff
Tdti
TI
0
2eff
1 T
dtT
V0
2eff
1
tIti cosm 2
cos1 m
0
22mrms
IdttI
TI
T
Hence:
Similarly,
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Average Power
i
i
ttIVtp
tittp
tIti
tVt
coscos)(
)()()(
cos)(
cos)(
mm
m
m
ii tIVtp 2coscos2
1)( mm
BABABA coscos2
1coscos
Note dependence on phase difference
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Average Power
Since and a similar relationship applies to I,
Power factor angle:
0 for a resistor= 90 degrees for inductor ‒90 degrees for capacitor
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ac Power Capacitors
2
i
dt
dCi C
C
CC CVjI
2CC CVI
Capacitors (ideal) dissipate zero average power
222cos2
1
2coscos2
1
mm
mm
tIVtp
tIVtp ii
= 0
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ac Power Inductors
2
i
dt
diL L
L
LL LIjV
2LL iLIV
Inductors (ideal) dissipate zero average power
222cos2
1
2coscos2
1
mm
mm
i
ii
tIVtp
tIVtp
= 0
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Complex Power
Phasor form defining “real” and “reactive” power
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Power Factor for Complex LoadInductive/capacitive loads will require more from the power supply than the average power being consumed Power supply needs to
supply S in order to deliver Pav to load
Power factor relates S to Pav
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Power Factor
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Power Factor Compensation
Introduces reactive elements to increase Power Factor
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Example 8-6: pf Compensation
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Maximum Power Transfer
Max power is delivered to load if load is equal to Thévenin equivalent
*sssLLL ZjXRjXRZ Max power
transfer when
Set derivatives equal to zero 0L
X
P0
L
R
P
s
2
Thmax 8R
VP
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Example 8-7: Maximum Power
Cont.
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Technology and Science Press
Example 8-7: Maximum Power
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Multisim Measurement of Power
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Multisim Measurement of Complex Power
Complex Power S
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Summary
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