8. ac power
DESCRIPTION
8. ac Power. CIRCUITS by Ulaby & Maharbiz. Overview. Linear Circuits at ac. Average power. Instantaneous power. Power at any instant of time. Average of instantaneous power over one period. Note: Power is not a linear function, cannot apply superposition. - PowerPoint PPT PresentationTRANSCRIPT
8. AC POWER
CIRCUITS by Ulaby & Maharbiz
Overview
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
Instantaneous Power for Sinusoids
Power depends on phases of voltage and current
i
i
ttIVtptittptItitVt
coscos)()()()(
cos)(cos)(
mm
m
m
BABABA coscos21coscos
ii tIVtp 2coscos21)( mm
Trig. Identity:
Constant in time (dc term)
ac at 2
Average Value
Sine wave
Truncated sawtooth
Average Value for
These properties hold true for any values of φ1 and φ2
Effective or RMS ValueEquivalent 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
TRdtRi
TP
0
2
0
21
RIP 2eff
Tdti
TI
0
2eff
1 T
dtT
V0
2eff
1
tIti cosm 2
cos1 m0
22mrms
IdttIT
IT
Hence:
Similarly,
Average Power
i
i
ttIVtptittptItitVt
coscos)()()()(
cos)(cos)(
mm
m
m
ii tIVtp 2coscos21)( mm
BABABA coscos21coscos
Note dependence on phase difference
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
ac Power Capacitors
2 i
dtdCi C
C
CC CVjI
2CC CVI
Capacitors (ideal) dissipate zero average power
222cos21
2coscos21
mm
mm
tIVtp
tIVtp ii
= 0
ac Power Inductors
2 i
dtdiL L
L
LL LIjV
2LL iLIV
Inductors (ideal) dissipate zero average power
222cos21
2coscos21
mm
mm
i
ii
tIVtp
tIVtp
= 0
Complex PowerPhasor form defining “real” and “reactive” power
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
Power Factor
Power Factor CompensationIntroduces reactive elements to increase Power Factor
Example 8-6: pf Compensation
Maximum Power TransferMax power is delivered to load if load is equal to Thévenin equivalent
*sssLLL ZjXRjXRZ Max power
transfer when
Set derivatives equal to zero 0L
XP 0
L
RP
s
2Th
max 8RV
P
Example 8-7: Maximum Power
Cont.
Example 8-7: Maximum Power
Three Phase
Y & Delta
Y-Source Connected to a Y-Load
Multisim Measurement of Power
Multisim Measurement of Complex Power
Complex Power S
Summary