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7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
1/23
Lesson 24
AC Power andPower Triangle
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
2/23
Learning Objectives
Define real (active) power, reactive power, average, andapparent power.
Calculate the real, reactive, and apparent power in ACseries parallel networks.
raph the real and reactive power of purel! resistive,inductive, or capacitive loads in AC series parallel networksas a function of ti"e.
Deter"ine when power is dissipated, stored, or released in
purel! resistive, inductive, or capacitive loads in AC seriesparallel networks.
#se the power triangle deter"ine relationships betweenreal, reactive and apparent power.
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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AC $ower
AC %"pedanceis a complex &uantit! "ade upof real resistanceand i"aginar! reactance.
ACApparent$ower is a complex &uantit! "ade
up of real activepower and i"aginar! reactive
power'
R jX= +Z ( )
jQ AP V= +S ( )
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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AC eal (Active) $ower ($)
he Active power is the power that is dissipatedin the resistance of the load.
%t uses the sa"e for"ula used for DC (* + % are
the "agnitudes, not the phasors)'22
[watts, W]P I R V
R= =
A-%- #1 mistake with AC power calculations!
he oltagein the above e&uation is the *oltage drop across the resistor, not
across the entire circuit
CA#%O-
/AL value of resistance () is used in /AL power calculations, not
%0$/DA-C/ (1)
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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AC %"aginar! (eactive) $ower (2)
he reactive power is the power that is e3changedbetween reactive co"ponents (inductors and capacitors)
he for"ulas look si"ilar to those used b! the active
power, but use reactance instead of resistances.
#nits' *olts4A"ps4eactive (*A)
2 is negative for a capacitor b! convention and positive
for inductor. 5ust like 6 is negative for a capacitor (46cj)
2
2 [VAR]Q I XXV= =
A-%- #1 mistake with AC power calculations!
he oltagein the above e&uation is the *oltage drop across the reactance, not
across the entire circuit
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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AC Apparent $ower (7)
he apparent power is the power that is8appears9 to flow to the load.
he "agnitude of apparent power can be
calculated using si"ilar for"ulas to those for
active or reactive power'
#nits' *olts4A"ps (*A)
* + % are the "agnitudes, not the phasors
22
[VA]V
S VI I Z
Z
= = =
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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AC $ower -otice the relationship between 1 and 7'
R j X= +Z ( )
P j Q VA= +S ( )
(ealpow
ercalculated
with(
(eactivep
owercalculate
dwith6
Apparentp
owercalculate
dwith1
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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$ower riangle
he power trianglegraphicall! shows therelationship between real (P), reactive (Q) and
apparent power (S).
2 2
LP jQ
QPS
S
= +
= +
=
S
S
v
v
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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Example Problem 1
Deter"ine the real and reactive power of eachco"ponent.
Deter"ine the apparent power delivered b! the
source.
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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eal and eactive $ower
he power trianglealso shows that we can findreal (P) and reactive (Q) power.
cos
sin
S IV
P S
Q S
=
=
=
(*A)
()
(*A)
-O/' he i"pedance angle and
the 8power factor angle9 are the
sa"e value
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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Example Problem 2
Deter"ine the apparent power, total real andreactive power using the following e&uations'
cos
sin
S VI
P S
Q S
=
=
=
(*A)
()
(*A)
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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otal $ower in AC Circuits
he total power real (PT) and reactive power
(QT) is si"pl! the su" of the real and reactive
power for each individual circuit ele"ents.
How elements are connected does not matterfor computation of total power.
1 2 3 4
1 2 3 4
T
T
P P P P P
Q Q Q Q Q
= + + +
= + + +
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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otal $ower in AC Circuits
7o"eti"es it is useful to redraw the circuit tos!"bolicall! e3press the real and reactive power loads
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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Example Problem 3
a. Deter"ine the unknown real ($:) and reactive powers
(2;) in the circuit below.
b. Deter"ine total apparent power
c. Draw the power triangled. %s the unknown ele"ent in Load
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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Example Problem 4
a. Deter"ine the value of , $and 2
b. Draw the power triangle and deter"ine 7.
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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#se of co"ple3 nu"bers in $ower calculations
AC power can be calculated using co"ple3 e&uations.
Apparent $ower can be represented as a co"ple3 nu"ber
he resultant can be used to deter"ine real and reactive power b!
changing it to rectangular for".
*
2
2
P j
= = +
= =
I
S VI
V
S I ZZ
v uvv
uv
v v uv
uv
is co"ple3 conjugate of %2
-O/
he co"ple3 conjugate of Current is used to "ake the power angle the sa"e as
the i"pedance angle
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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$ower >actor
Power actor(FP) tells us what portion of theapparent power(S) is actuall! realpower(P).
$ower factor is a ratio given b!
FP= P / S
$ower factor is e3pressed as a nu"ber
between ? to @.? (or as a percent fro" ? to
@??)
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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$ower >actor
>ro" the power triangle it can be seen that
FP= P / S = cos
Power actor angleis thus given
= cos-1(P / S)
>or a pure resistance, = 0
>or a pure inductance, = 90 >or a pure capacitance, = -90
-O/' B is the phase angle of 1, not the
current or voltage.
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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"nit power actor(FP @)
%"plies that allof a loads apparent power isreal power (S = P).
%f FP @, then = 0.
%t could also be said that the load looks purel!resistive.
Load current and voltage are in phase.
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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Lagging power actor(> 0)
he load current lags load voltage ELI %"plies that the load looks inductive.
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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Leading power actor(< 0)
he load current leads load voltage ICE %"plies that the load looks capacitive.
7/25/2019 EE301 Lesson 24 AC Power and Pwr Triangle
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Example Problem 5
a. Deter"ine $,2,7 and the power factor for this circuit.
Draw the power triangle.
b. %s it a leading or lagging power factor=
c. %s the circuit inductive or capacitive=
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Example Problem 6
a. Deter"ine total current, apparent power, and the power
factor for this circuit. %s it a leading or lagging power
factor=
b. Deter"ine total current, apparent power, and the power
factor if the capacitor reactance is decreased to E?
oh"s. hat kind of power factor does it have=
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