power electronics chapter02-1
TRANSCRIPT
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Power
Electronics
CHAPTER 2
Power Computat ions
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Power
Electronics
Power & Energy
Instantaneous power
Absorbing
Power
Supplying
Power
)()()( t it vt p Time-varying quantity
• PASSIVE SIGN CONVENTION
0)( t p 0)( t p
Energy
2
1
)(t
t dt t pW
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Power
Electronics
Power
Electronics
Power & Energy
Average power (real power, active power)
T t
t
T t
t
o
o
o
o
dt t it vT
dt t pT
P )()(1
)(1
T
W
P
Periodic voltage and current produce a periodic instantaneous power.
Power generally
means average power.
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Power
Electronics
Inductors & Capacitors Inductors
)(2
1)( 2 t Lit w
t 0 t 0+T t 0+2T
T
i L(t 0)
i L(t 0+T )
i L(t ) In the steady state, the
inductor current is periodic
)()( 00 t iT t i L L
)()(1
)( 000
0
t idt t v L
T t i LT t
t L L
0)(1
)()( 0
000
dt t v
Lt iT t i
T t
t L L L
0)()()(1 0
0
t vt vavg dt t vT
L L
T t
t L
In the steady state,
- the average inductor voltage is zero.
- the net change of the inductor current during one switching period is zero.
- no net energy transfer, i.e. the average power of the inductor is zero (PL = 0)
Inductor volt-second balance
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Power
Electronics
Inductors & Capacitors
• Volt-seconds over T equal zero.
v L
i L
Increasing inductor current
Inductor absorbs power
and energy from externalcircuits
v L
i L
Decreasing inductor current
Inductor supplies power
and energy to external
circuits
dt
di Lv L L
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Power
Electronics
Inductors & Capacitors Capacitors
In the steady state, the
capacitor voltage is periodic
t 0 t 0+T t 0+2T
T
vC (t 0)
vC (t 0+T )
vC (t )
)()( 00 t vT t v C C
)()(1
)( 000
0
t vdt t iC
T t v C T t
t C C
0)(1
)()( 0
000
dt t i
C t vT t v
T t
t C C C 0)()()(
1 0
0
t it iavg dt t iT
C C
T t
t C
In the steady state,
- the average capacitor current is zero.
- the net change of the capacitor voltage during one switching period is zero.
- no net energy transfer, i.e. the average power of the capacitor is zero (PC = 0)2
2
1)( Cvt w
Capacitor ampere-second balance
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Power
Electronics
Inductors & Capacitors
• Amp-seconds over T equal zero.
Increasing capacitor
voltage
Capacitor absorbs
power and energy
from external circuits
(charging)
Decreasing
capacitor voltage
Capacitor
supplies power
and energy to
external circuits
(discharging)
dt
dvC i C C
vC
iC
vC
iC
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Power
Electronics
Energy Recovery
Fuel injector solenoid in automobile
freewheeling pathfor induct current
continuity
Inductors and capacitors must be energized and de-energized by switching
operations
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Power
Electronics
• Reminding
• Without freewheeling path, the Tr is damaged by the overvoltage when it turns off.
• Inductor current continuity must be provided by circuit design
Energy Recovery During Tr on (0 < t < t1)
CC L V v
t
CC
CC L
t
L L L
t V
d V Lid v Lt i 00 0
1
)0()(
1
)( )(
2
1)( 2 t Lit w
Inductor current linearly
increases.
Inductor stores energy inside
itself.
freewheeling path
for induct current
continuity
dt
di Lv L L
)()( t it i L s
(+)
Source supplies
power
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Power
Electronics
Energy Recovery During Tr off (t1 < t < T)
diode turns on due to
the inductor current
continuity
L
t V t i CC L
)()( 11
T t t e L
t V et it i
t t CC t t
L L
1
/)(1/)(
1
11)()(
Inductor current exponentially decreases.
Inductor releases its energy to resistor.
R L
LT
t V dt
T dt
L
t V
T V
dt t iT
V I V P
CC t T
t
CC CC
T
sCC S S S
20
11
)(1
2
1
0
0
1
1
0S i
average power
supplied by
source
LT
t V P P CC S R
2
2
1
average power
absorbed by
resistor
0)( L P
Heat
energy
Power
loss
lossy element
(+)
Source supplies
power
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Power
Electronics
Energy Recovery
Recycle energy
stored in elements
energystorageelement
re overeuse
store
energyenergy
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Power
Electronics
Energy Recovery During Tr on (0 < t < t1)
CC L V v
t
CC CC L
t
L L L
t V d V
Lid v
Lt i
000
1)0()(
1)(
• Inductor current linearly increases.
• Inductor stores energy inside itself.
)()( t it i L s
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Power
Electronics
Energy Recovery
• The inductor current linearly decreases.
• The energy stored in the inductor is recovered to the source &
the source is absorbing power.
During Tr off (t1 < t < T)
CC L V v
1111
1 2211
)(11
t t t t t L
V
L
t V d V
Lt id v
Lt i CC CC
t
t CC L
t
t L L
)()( t it i LS
No power
loss
(+)
Source supplies
power
(-)
Source absorbs
power
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Power
Electronics
Effective value (RMS)
Effective value (Root-Mean-Square value)
The effective value of a periodic voltage (current) delivers the sameaverage power to a resistor as dc voltage (current) does.
R
V P dc
2
R
V P
eff
2
T
T T T
dt t vT R
dt R
t v
T dt t it v
T dt t p
T P
0
2
0
2
00
)(11
)(1)()(
1)(
1
• For a dc voltage • For a ac voltage
V eff 2
T
rmseff dt t vT
V V 0
2 )(1
T
rms dt t iT
I 0
2)(
1
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