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Chapter 7: Active, Reactive, and Apparent Power

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AC Power

• The behavior of AC machines and systems are often easier to understand by working with power, rather than working with voltages and currents

• Active, reactive, and apparent power apply to steady-state AC circuits with sinusoidal waveforms only– cannot be used to describe the transient behaviors

– cannot be used to describe dc circuits

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Instantaneous power

• Power is the product of the instantaneous voltage and current )()()( titvtp ⋅=

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• The average value of the instantaneous power over one cycle of the voltage

• The effective power that does real work

where– V = rms voltage [V]

– I = rms current [A]

– φ = phase angle difference between the voltage and current

Active or Real Power

φcosIVP=

( )( ) ( )( ) ( )tdtitvP ωφωωπ −= � sinsin2

0

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Reactive Power

• The circulating power, Q, in the circuit, measured in volt-amperes reactive [VAr]

• Power which does no real work

θsinIVQ =

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Reactive Loads

• Inductive load– consumes reactive power

– current lags the voltage

• Capacitive load– generates reactive power

– current leads the voltage

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Apparent Power

• The complex power, S,associated with the voltage phasor and the conjugate of the current phasor

IVS

IVS

== *

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P, Q, and S Relationships

real or active power, P

reactivepower, Q

apparentpower, S

voltage, V

current, I

angle φ

angle -φ QjPS

SQ

SP

IVS

QPS

+==

=

=

+=

φφ

sin

cos

22

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Power Factor

• Ratio of the active power, P, and the apparent power, S– expressed as a simple number or as a percentage

– can never be greater than unity

SPpf =

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Power Factor

• Cosine of the angle between the voltage and current– phase angle between the voltage and current, φ

• Power factor is said to be leading or lagging– pf is lagging, if the current lags behind the voltage

– pf is leading, if the current leads the voltage

φcos=pf