9/30/2003 Electromechanical Dynamics 1
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