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ECE 430 – Spring 2009
HW #1 – Solution
( ) 100 cos(377 10 ) 2 cos(377 10 ) 2
v t t t
+ ° V
V
° V
( ) cos(377 55 ) 2 cos(377 55 ) 2
i t t t
° A
I
° A
Average power:
cos( ) cos(10 55 ) 2 2 2
P VI θ v θ i W
power factor:
cos( ) cos(10 55 ) 2
PF = θ v − θ i = ° − ° = leading
(a)
( ) 100 cos(377 15 ) 2 cos(377 15 ) 2
v t t t
+ ° V
V
° V
( ) 10sin(377 45 ) 2 cos(377 135 ) 2
i t t t
A
I
° A
Average power:
cos( ) cos(15 135 ) 250 2 2
P VI θ v θ i W
(b)
( ) 100 cos(377 75 ) 2 cos(377 75 ) 2
v t t t
− ° V
V
° V
( ) 10sin(377 15 ) 2 cos(377 105 ) 2
i t t t
A
I
° A
Average power:
cos( ) cos( 75 105 ) 500 2 2
P VI θ v θ i W
(c)
v t ( ) = 2 100 sin(377 ( ) t − 30 )° = 2 100 cos(377( ) t − 120 )° V ⇒ V = 100 ∠ − 120 ° V
i t ( ) = 2 10 cos(377 ( ) t − 60 )° A ⇒ I = 10 ∠ − 60 ° A
Average power: P = VI cos( θ (^) v − θ i ) = (^) ( 100 )(10 cos( 120 ) − ° − −( 60 ° (^) )) = 500 W
power factor: PF = cos( θ (^) v (^) − θ i ) = cos( 120− ° − −( 60 )° =) 0.5leading
(d)
( ) 100sin(377 30 ) 2 cos(377 60 ) 2
v t t t
− ° V
V
° V
( ) 10sin(377 120 ) 2 cos(377 30 ) 2
i t t t
° A
I
° A
Average power:
cos( ) cos( 60 30 ) 0 2 2
P VI θ v θ i W
power factor: PF = cos( θ v − θ i ) = cos( 60− ° − 30 )° = 0 leading (since θ i > θ v )
(e)
V = 10 + j 10 = 10 2 ∠ 45 ° V
I = 5 + j 10 = 5 5 ∠63.43 ° A
Average power: P = VI cos( θ (^) v − θ i ) = (^) ( 10 2 )( 5 5 cos(45) ° − 63.43 )° = 150 W
power factor: PF = cos( θ v − θ i ) = cos(45 ° − 63.43 )° = 0.9487leading
(f)
V = 100 − j 90 = 18100 ∠ − 41.99 ° = 134.54 ∠ − 41.99° V
I = 10 − j 10 = 10 2 ∠ − 45 ° A
Average power: P = VI cos( θ (^) v − θ i ) = (^) ( 18100 )( 10 2 cos( 41.99) − ° − −( 45 °) ) = 1900 W
power factor: PF = cos( θ (^) v (^) − θ i ) = cos( 41.99− ° − −( 45 ° )) = 0.9986lagging
(g)
V = − 10 − j 10 = 10 2 ∠ − 135 ° V
I = 5 + j 10 = 5 5 ∠63.43 ° A
Average power: P = VI cos( θ (^) v − θ i ) = (^) ( 10 2 )( 5 5 cos( 135) − ° − 63.43 )° = − 150 W
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