Instantaneous Power
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Transcript of Instantaneous Power
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ECE 201 Circuit Theory I 1
Instantaneous Power
cos( )cos( )m v
m i
p viv V ti I t
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ECE 201 Circuit Theory I 2
Reference for t = 0
• Call t = 0 when the current is passing through a positive maximum.
• Re-write the equations as
coscos( )
m
m v i
i I tv V t
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ECE 201 Circuit Theory I 3
Re-writing
cos( ) cosm v i m
p vip V t I t
1 1cos cos cos( ) cos( )2 2
v itt
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ECE 201 Circuit Theory I 4
cos( )cos
cos( ) cos(2 )2 2
cos( ) cos cos sin sin
cos( ) cos( )cos 2 sin( )sin 22 2 2
m m v i
m m m mv i v i
m m m m m mv i v i v i
p V I t tV I V I
p t
V I V I V Ip t t
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ECE 201 Circuit Theory I 5
600
v
i
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ECE 201 Circuit Theory I 6
p is at twice the frequency of v and i
p is negative over part of the cycle, even though the elements are passive
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ECE 201 Circuit Theory I 7
Average and Reactive Power
cos( ) cos( ) cos 2 sin( )sin 22 2 2
cos( )2
sin( )2
cos 2 sin 2
m m m m m mv i v i v i
m mv i
m mv i
V I V I V Ip t t
V IP
V IQ
p P P t Q t
Average, or Real Power
Reactive Power
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ECE 201 Circuit Theory I 8
Power for Purely Resistive Circuits
• For a purely resistive circuit, the voltage and current are in phase.
cos( )2 2
sin( ) 02cos 2 sin 2 cos 2
v i
m m m mv i
m mv i
V I V IP
V IQ
p P P t Q t P P t
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ECE 201 Circuit Theory I 9
Plot of the instantaneous Real Power
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ECE 201 Circuit Theory I 10
Power for Purely Inductive Circuits
• Voltage and Current are out of phase by 90°.• The current lags the voltage by 90°.
9090
cos(90 ) 02
sin(90 )2 2cos 2 sin 2 sin 2
i v
v i
m m
m m m m
V IP
V I V IQ
p P P t Q t Q t
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ECE 201 Circuit Theory I 11
Plot of Reactive Power (vars)
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ECE 201 Circuit Theory I 12
Power for Purely Capacitive Circuits
• Voltage and Current out of phase by 90°• The current leads the voltage by 90°.
9090
cos( 90 ) 02
sin( 90 )2 2cos 2 sin 2 sin 2
i v
v i
m m
m m m m
V IP
V I V IQ
p P P t Q t Q t
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ECE 201 Circuit Theory I 13
Plot of Reactive Power (vars)
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ECE 201 Circuit Theory I 14
Power Factor
• Power factor angle
• Power factor
• Reactive factor
v i
cos( )v ipf
sin( )v irf
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ECE 201 Circuit Theory I 15
Power Factor (continued)
• Lagging power factor– Current Lags the Voltage– Inductive Load
• Leading power factor– Current Leads the Voltage– Capacitive Load