Transcript 2nd Lecture Note_1

DET 309
POWER ELECTRONICS
POWER COMPUTATIONS
3/27/2016
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2.1 INTRODUCTION
o Power computations are essential in analyzing and designing
power electronics circuits.
o Basic power concepts are reviewed in this chapter, with
particular emphasis on power calculations for circuits with nonsinusoidal voltages and currents.
o Extra treatment is given to some special cases that are
encountered frequently in power electronics.
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2.2 POWER AND ENERGY
Instantaneous Power
o The instantaneous power for any device is computed from
the voltage across it and the current in it
o Instantaneous power is
p(t )  v(t )i (t )
(2.1)
This relationship is valid for any device or circuit. Instantaneous
power is generally a time-varying quantity.
o If v(t) is in volts and i(t) is in amperes, power has units of
watts, W
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Observe the passive sign convention illustrated in Fig. 2.1a
o The device is absorbing power if p(t) is positive at a specified time, t.
o The device is supplying power if p(t) is negative.
Sources frequently have an assumed current direction consistent with
supplying power. With the convention of Fig. 2.1b, a positive indicates that
the source is supplying power.
+
i(t)
v(t)
v(t)
-
(a)
Figure 2.1
+
i(t)
(b)
(a) Passive sign convention: p(t) > 0 indicates power is being absorbed.
(b) p(t) > 0 indicates power is being supplied by the source.
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Energy
o Energy or work, is the integral of instantaneous power.
o Observing the passive sign convention, the energy absorbed by
a component in the time interval from t1 to t2 is
W 
t2
t
p (t ) dt
(2.2)
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o If v(t) is in volts and i(t) is in amperes, energy has units of joules, J.
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Average Power
o Periodic voltage and current functions produce a periodic instantaneous power
function.
o Average power is the time average of p(t) over one or more periods.
o Average power, P is computed from
1
P
T
to T
t
o
1
p (t ) dt 
T
to T
t
v(t )i (t ) dt
(2.3)
o
Where T is the period of the power waveform. Combining Eqs. 2.3 and 2.2, power
is also computed from energy per period:
W
P
T
(2.4)
Average power is sometimes called real power or active power, especially in ac
circuits. The term power usually means average power. The total average power
absorbed in a circuit equals the total average power supplied.
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Example 2-1 Power and Energy
Voltage and current (consistent with the passive sign convention) for a device are shown in Figs. 2.2a
and 2.2b.
(a) Determine the instantaneous power absorbed by the device,
(b) Determine the energy absorbed by the device in one period,
(c) Determine the average power absorbed by the device.
v(t)
i(t)
20 V
20 A
(t)
(t)
10 ms
20 ms
6 ms
20 ms
- 15 A
(a)
(b)
p(t)
400 W
(t)
6 ms
10 ms
20 ms
-300 W
(c)
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Solution (a) The instantaneous power is computed from Eq. 2.1.
The voltage and current are expressed as
0  t  10ms
20V
v(t )  
0V 10ms  t  20ms
20 A
i (t )  
 15 A
0  t  6 ms
6 ms  t  20ms
Instantaneous power, shown in Fig. 2.2c, is the product of voltage
and current and is expressed as
0  t  6 ms
400W

p(t )   300W 6 ms  t  10 ms
0
10 ms  t  20 ms

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(b) Energy absorbed by the device in one period is determined
from Eq. 2.2:
W 
t2
6 ms
10 ms
20 ms
t1
0
6 ms
10 ms
 p(t )dt   400 dt    300 dt   0dt
 1.2 J
(c) Average power is determined from Eq. 2.3:
P
1
T
to T
t
o
p (t ) dt 
1
20 ms
6 ms
10 ms
 400 dt  
0
6 ms
20 ms
 300 dt 
 0dt
10 ms
 60W
Average power could also be computed from Eq. 2.4 using the energy
per period from part (b):
W 1.2 J
P 
 60W
T 20ms
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o
A special case that is frequently encountered in power electronics is the power
absorbed or supplied from a dc source. Applications include battery-charging circuits and
dc power supplies.
The average power absorbed by a dc voltage source v(t) which has a periodic current i(t)
is derived from the basic definition of average power in Eq. 2.3:
Pdc
1

T
to T
t
o
1
v(t )i (t ) dt 
T
to T
t
Vdci (t ) dt
o
Bringing the constant Vdc outside of the integral,
Pdc
1
 Vdc 
T
t o T
t
o

i (t ) dt 

The term in brackets is
the average of the
current waveform
Therefore, average power absorbed by a dc voltage source is the product of the voltage
and the average current:
Pdc  Vdc I avg
2.5
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Similarly, average power absorbed by a dc current source i(t) = idc is
Pdc  Vavg I dc
(2.6)
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2.3 INDUCTORS AND CAPACITORS
o Inductors and capacitors have some particular characteristics
that are important in power electronics applications.
o For periodic currents and voltages,
(2.7)
2.3.1 Inductors
o For an inductor, the stored energy is
(2.8)
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2.3.1 Inductors…cont.
If the inductor current is periodic, the stored energy
at the end of one period is the same as at the
beginning.
No net energy transfer indicates that the average
power absorbed by an inductor is zero for steadystate periodic operation:
(2.9)
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2.3.1 Inductors…cont.
Instantaneous power is not necessarily zero because power may
be absorbed during one part of the period and returned to the
circuit during another part of the period. Furthermore, from the
voltage-current relationship for the inductor,
(2.10)
Rearranging and recognizing that the starting and ending values are the
same for periodic currents,
(2.11)
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2.3.1 Inductors…cont.
Multiplying by L/T yields an expression equivalent to the average voltage
across the inductor over one period:
(2.12)
Therefore, for periodic currents, the average voltage across an inductor is zero.
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2.3 INDUCTORS AND CAPACITORS…cont.
2.3.2 Capacitors
For a capacitor, stored energy is
(2.13)
If the capacitor voltage is periodic, the stored energy is the same at the end of a
period as at the beginning.
Therefore, the average power absorbed by the capacitor is zero for steady-state
periodic operation:
(2.14)
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2.3.2 Capacitors…cont.
From the voltage-current relationship for the capacitor,
(2.15)
Rearranging the preceding equation and recognizing that the starting and ending
values are the same for periodic voltages,
(2.16)
Multiplying by C/T yields an expression for average current in the capacitor over
one period:
(2.17)
Therefore, for periodic voltages, the average current in a capacitor is zero.
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Example 2.2 Power and Voltage for an Inductor
The current in the 5-mH inductor of Fig. 2.3a is the periodic triangular wave shown in Fig.
2.3b. Determine the voltage, instantaneous power, and average power for the inductor.
i(t)
+
5 mH
i(t)
4A
v(t)
_
t
1 ms
v(t)
(a)
p(t)
20 V
2 ms
3 ms
4 ms
(b)
80 W
t
- 20 V
t
- 80 W
(c)
(d)
Figure 2.3 (a) Circuit for Example 2.2. (b) Inductor current, (c) Inductor voltage, (d) Inductor instantaneous power.
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Solution
The voltage across the inductor is computed from
in Fig. 2.3c.
and is shown
The average inductor voltage is zero, as can be determined from Fig. 2.3c by
inspection.
The instantaneous power in the inductor is determined from
is shown in Fig. 2.3d.
and
When
is positive, the inductor is absorbing power, and when is negative, the
inductor is supplying power. The average inductor power is zero.
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2.4 ENERGY RECOVERY
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Thank You
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