ECE 201 Circuit Theory I 1

Instantaneous Power

cos( )

cos( )m v

m i

p vi

v V t

i I t

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

cos

cos( )m

m v i

i I t

v V t

ECE 201 Circuit Theory I 3

Re-writing

cos( ) cosm v i m

p vi

p V t I t

1 1cos cos cos( ) cos( )

2 2

v it

t

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 t

V I V Ip t

V I V I V Ip t t

ECE 201 Circuit Theory I 5

60

0v

i

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

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

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

ECE 201 Circuit Theory I 9

Plot of the instantaneous Real Power

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°.

90

90

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

ECE 201 Circuit Theory I 11

Plot of Reactive Power (vars)

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°.

90

90

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

ECE 201 Circuit Theory I 13

Plot of Reactive Power (vars)

ECE 201 Circuit Theory I 14

Power Factor

• Power factor angle

• Power factor

• Reactive factor

v i

cos( )v ipf

sin( )v irf

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