Transcript Reactive Power

JAMES W. NILSSON
&
SUSAN A. RIEDEL
ELECTRIC
CIRCUITS
EIGHTH EDITION
CHAPTER 10
SINUSOIDAL
STEADY –
STATE POWER
CALCULATIONS
© 2008 Pearson Education
1. Understand the following ac power concepts & how to
calculate them in a circuit.
 instanteneous power
 Average(real) power
 Reactive power
 Complex power
 Power factor
2. Understand the condition to deliver max. average power
to
a load & then how to calculate the load impedance.
3.
In ac circuits, Understand how to calculate ac power with
linear transformer and ideal transformer.
CONTENTS
10.1 Instantaneous Power
10.2 Average and Reactive Power
10.3 The rms Value and Power Calculations
10.4 Complex Power
10.5 Power Calculations
10.6 Maximum Power Transfer
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Generally,
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10.1 Instantaneous Power
 The
positive sign is used when the
reference direction for the current is
from the positive to the negative
reference polarity of the voltage.
 The
frequency of the instantaneous
power is twice the frequency of the
voltage (or current).
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10.1 Instantaneous Power
Instantaneous power, voltage, and current versus ωt for
steady-state sinusoidal operation
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10.2 Average and Reactive Power
Average Power
 Average power is the average value of
the instantaneous power over one period.
 It
is the power converted from electric to
non-electric form and vice versa.
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Note:
10.2 Average and Reactive Power
 This
conversion is the reason that
average power is also referred to as real
power.
 Average power, with the passive sign
convention, is expressed as
Vm I m
P
cos( v   i )
2
 Veff I eff cos( v   i )
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10.2 Average and Reactive Power
Reactive Power
 Reactive power is the electric power
exchanged between the magnetic field of an
inductor and the source that drives it or
between the electric field of a capacitor and
the source that drives it.
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10.2 Average and Reactive Power
Reactive Power
 Reactive
power is never converted to
nonelectric power. Reactive power, with the
passive sign convention, is expressed as
Vm I m
Q
sin(  v   i )
2
 Veff I eff sin(  v   i )
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Note:
Note:
Note:
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10.2 Average and Reactive Power
Power Factor
Power factor is the cosine of the phase angle
between the voltage and the current:
pf  cos( v  i )
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10.2 Average and Reactive Power
The reactive factor is the sine of the phase
angle between the voltage and the current:
rf  sin(  v  i )
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10.3 The rms Value and Power
Calculations
A sinusoidal voltage
applied to the terminals
of a resistor
Average power
delivered to
the resistor
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10.3 The rms Value and Power
Calculations
2
rms
The average power
delivered to R is simply the
rms value of the voltage
squared divided by R.
V
P
R
If the resistor is carrying a sinusoidal
current, the average power delivered
to the resistor is:
PI
2
rms
.R
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rms value = effective value.
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a) max. amplitude 625 V vtg. Is applied to
50 Ω . Average power at this resistor?
b) Repeat (a) by 1st finding IR.
Sol.:
a)rms value of vtg. = 625/√2 ≈ 441.94 V
P = V2rms/R
b) max. amplitude of ct. = 625/50 = 12.5 A
rms value of ct. = 12.5/ /√2 ≈ 8.84 A
Hence,
10.4 Complex Power
 Complex
power is the
complex sum of real
power and reactive power.
S  P  jQ
| S | = apparent power
Q = reactive power
θ
P = average power
A power triangle
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10.4 Complex Power
Quantity
Units
Complex power
volt-amps
Average power
watts
Reactive power
var
Three power quantities and their units
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10.4 Complex Power
Apparent Power is the magnitude of
complex power.
S  P Q
2
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Remind:
>0
Lagging pf: ct. lags vtg. Hence inductive load
<0
Leading pf: ct. leads vtg. Hence capacitive load
PF lagging → inductive → reactive power +
Electric facilities( that is, refrigerators, fans, air
conditioners, fluorescent lighting fixtures, &
washing machines) & most industrial loads
operate at a lagging power factor.
Correction of power factor of these loads
using capacitor.
This method is often used for large industrial
load.
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See:
10.5 Power Calculations(abbreviation.)
The phasor voltage and current
associated with a pair of terminals
Complex power  V I
*
eff eff
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Load impedance: 39 + j26
Line impedance: 1 + j4
Effective(or rms) value of S. : 250 V
load1: lead pf 0.8, 8 kW av. Power absorbed
load2: lag pf 0.6, 20 kVA absorbed
c) S. freq.: 60 Hz .
value of capacitor with pf = 1 if placed in
parallel with 2 loads?
appearance power supplied to load, magnitude of current,
av. power :
Place capacitor with 10 KVAR in parallel with existing load.
Then pf = 1(그림10.15c)
So capacitive reactance is
a) Power supplied to each impedance?
b) Power of each source?
c) Delivered av. power = absorbed av.
power
Delivered reactive power = absorbed
reactive power
Reactive power delivered to (12 – j16)Ω
Reactive power delivered to(1 + j3)Ω
b) Complex
power of independent S.
see: absorbed av. power:1950 W ,
delivered reactive power3900 VAR
Complex power of dependent S.
주: 5850 W 평균전력 공급, 5070 VAR
무효전력 공급
see: delivered av. Power:5850 W ,
delivered reactive power 5070 VAR
10.6 Maximum Power Transfer
A circuit describing
maximum power transfer
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10.6 Maximum Power Transfer
ZL  Z
*
Th
Condition for maximum
average power transfer
The circuit with the
network replaced by its
Thévenin equivalent
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Proof:
Let
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a) ZL value to deliver max. power to ZL?
b) av. power at a)?
Sol.:
a) Thevenin eqvalent cit. seen to a, b
terminal
b) in Fig.
a) ZL to be able to max. power & then max.
power(mW)
b) load: 0 ~ 4000 Ω
reactance: 0 ~ - 2000 Ω
RL & XL to be able to transfer most av.
power to load & then max. av. power?
See:
See: P
See: P(no
restriction) > P(restriction)
= I2eff RL
ZL phase is -36.87o.
Magnitude of ZL changes until av. power
becomes biggest under given restriction.
a) Specfy ZL in rectangular form.
b) Av. Power to delivered to ZL ?
a) In case magnitude of ZL be varied but
its phase not,
The greatest power is transferred to load when magnitude of ZL
is set equal to magnitude of ZTH ; that is,
Therefore,
See:
No restrictions on ZL
Previous knowledge
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Adjust RL until max. av. power is be
delivered to RL.
a) RL?
b) max. av. power to be delivered to RL?
sol.:
a) Thevenin equivalent cit.
so V20
Note that Vth = - V2
Fig. is used to determine the short ckt current.
Using
Rearranging,
Therefore,
RTh = VTh/I2 =
Pmax = I2R =
Home work
Prob. 10.1 10.4 10.5 10.6 10.14 10.16 10.18 10.25 10.29
10.36 10.37 10.41 10.44 10.47 10.52 10.57 10.61
10.63 10.65
제출기한:
- 다음 요일 수업시간 까지
- 제출기일을 지키지않는 레포트는 사정에서 제외함
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THE END
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