l3. power in ac circuits
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Power in AC Circuits
IntroductionIntroductionIntroductionIntroduction
Power in Resistive ComponentsPower in Resistive ComponentsPower in Resistive ComponentsPower in Resistive Components
Power in CapacitorsPower in CapacitorsPower in CapacitorsPower in Capacitors
Power in InductorsPower in InductorsPower in InductorsPower in Inductors
Circuits with Resistance and ReactanceCircuits with Resistance and ReactanceCircuits with Resistance and ReactanceCircuits with Resistance and Reactance
Active and Reactive PowerActive and Reactive PowerActive and Reactive PowerActive and Reactive Power
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The instantaneous power dissipated in a component is a
product of the instantaneous voltage and the instantaneous
current
In a resistive circuit the volta e and current are in hase .
Introduction
p = vi
calculation ofp is straightforward.
In reactive circuits, there will normally be some phase shift
between v and i and calculating the power becomes more
complicated.
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Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in a resistorin a resistorin a resistorin a resistor
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Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in ain ain ain a resistor...resistor...resistor...resistor...
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POWER IN CAPACITORS
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Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in a capacitorin a capacitorin a capacitorin a capacitor
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Power in Capacitors
vp=
From our discussion of capacitors we know that the current
leads the voltage by 90. Therefore, if a voltage v = Vp sin t is
applied across a capacitance C, the current will be given by i =
Ip cost
Then
)2
2sin
(
)cos(sin
cossin
t
IV
ttIV
tItV
PP
PP
PP
=
=
=
The average power is zero
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Power in Inductors
Phasor Diagram
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Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in an inductorin an inductorin an inductorin an inductor
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Power in Inductors
vip=
From our discussion of inductors we know that the current lags
the voltage by 90. Therefore, if a voltage v = Vp sin t is
applied across an inductanceL, the current will be given by i =-Ip cos t
Therefore
)2
2sin
(
)cos(sin
cossin
t
IV
ttIV
tItV
PP
PP
PP
=
=
=
Again the average power is zero
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Circuit with Resistance and Reactance
When a sinusoidal voltage v = Vp sin tis applied across a
circuit with resistance andreactance, the current will be of the
general form i =Ip sin (t-)
Therefore, the instantaneous power,p is given by
)2cos(2
1cos
2
1
)}2cos({cos2
1
)sin(sin
=
=
=
tIVIVp
tIV
tItV
PPPP
PP
PP
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)2cos(2
1cos
2
1 = tIVIVp PPPP
The expression forp has two components
The second part oscillates at 2and has an average value of
zero over a com lete c cle
Circuit with Resistance and Reactance
cos)(cos22
)(cos2
1VI
IIVP PPPP ===
this is the power that is stored in the reactive elements and
then returned to the circuit within each cycle
The first part represents the power dissipated in resistive
components. Average power dissipation is
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cos)(cos2
1VIIVP PP ==
The average power dissipation given by
Circuit with Resistance and Reactance
is termed theactive power(or) real power in the circuit and is
measured in watts (W).This is due to the power consumption of
circuit resistance.
The product of the r.m.s. voltage and current VI is termed the
apparent power(or)complex power, S. To avoid confusion
this is given the units of volt amperes (VA)
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cos
cos
S
VP
=
=
From the above discussion it is clear that
In other words the active ower is the a arent ower times the
Circuit with Resistance and Reactance
factorPoweramperes)volt(inpowerApparent
watts)(inpowerActive=
cosfactorPower ==P
cosine of the phase angle.
This cosine is referred to as the power factor.power factor.power factor.power factor.
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When a circuit has resistive and reactive parts, the resultant
power has 2 parts: The first is dissipatedin the resistive element. This is the
active power,P
Active and Reactive Power
.
This is the reactive power, Q , which has units of volt
amperes reactive or varvarvarvar
While reactive power is not dissipated it does have an effect on
the system for example, it increases the current that must be supplied
and increases losses with cables
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Therefore
Active Power P = VIcos watts
Active and Reactive Power
eac ve ower = s n var
Apparent Power S = VI VA
S2 = P2 + Q2
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