electrical power and machines- lec02_single phase circuit
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8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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U
n i t 2
S i n g l e - P h
a s e C i r c u i t s
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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L e c t u
r e O u t l i n e s
I n t r o
d u c t i o n t o :
S i n u s o i d s
P h a s o r s
T i m e & P h a s o
r D o m a i n T r a n s f o r m a t i o n
s
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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S i n u s
o i d s
ω π 2 = T
S i n u s o
i d a l V o l t a g e
v ( t ) = V
m
s i n
ω t
A m p l i t u d e
V m :
− − − − V
m ≤ ≤ ≤ ≤
v ( t ) ≤ ≤ ≤ ≤ V
m
2
T
f
F r e q
u e n c y
C y c l i c
1
,
=
−
A r g u m
e n t
ω t
ω π 2
,
= T
P e r i o d
T
=
=
T
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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S i n u s
o i d s
ω π 2 = T
S i n u s o
i d a l V o l t a g e
v ( t ) = V
m
s i n
ω t
A m p l i t u d e
V m :
− − − − V
m ≤ ≤ ≤ ≤
v ( t ) ≤ ≤ ≤ ≤ V
m
2
v ( t ) = V
m
s i n
ω t + φ φ φ φ
A r g u m
e n t
ω t
ω π 2
,
= T
P e r i o d
T
=
=
T
f
F r e q
u e n c y
C y c l i c
1
,
=
−
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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C o m p
a r i n g S i n u s o i d s
W h e n t w o
s i n u s o i d s o f t h e s a m e
f r e q u e n c y a r e c o m p a r e d , t h e y m i g h t
b e
i n p h a
s e o r o u t - o f - p h a s e
)
(
) (
2
2
ω t
s i n
V
t v
m
=
)
(
) (
1
1
ω
t
s i n
V
t v
m
=
v 1 a n d
v 2 a r e i n p h a
s e
v 1
a n d v 2 m i g h t
h a v e d i f f e r e n t a m p l i t u d e s
&
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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C o m p
a r i n g S i n u s o i d s
O u t - o f - p h a s e S i n u
s o i d s
)
(
) (
2
φ
+
=
ω t
s i n
V
t
v
m
ω t
s i n
V
t v
m
=
) ( 1
&
v 2 l e a d s v 1
b y Φ Φ Φ Φ
v 1 l a g s v 2 b y Φ Φ Φ Φ
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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• F o r p r a c t i c a l r e a s o n s
t h a t w e w i l l f i n d o u t l a t e r , i t w i l l b e
c o n v e n i e n t i f t h e s i n u s o i d s i n m a n y d
i f f e r e n t f o r m s a
r e
e x p r e s
s e d i n p o s i t i v e
c o s i n e t e r m s
• T h e m
o s t u s e f u l e q u a
t i o n s a r e
•
-
=
-
o
S i n u s
o i d T r a
n s f o r m a
t i o n s
•
m s
i n
ω t = c
o s ( ω t ± 9 0 o )
• E x a m p l e s :
•
1 2
s i n
( ω
t –
1 0 o ) =
1 2
c o s ( ω t –
1 0
o
–
9 0 o )
=
1 2
c o s ( ω t –
1 0
0 o )
•
- 4
s i n
( 3 7 7 t + 2
5 o ) = 4
c o s ( 3 7 7 t + 2
5 o
+ 9 0 o )
= 4
c o s ( 3 7 7 t + 1 1
5 o
)
•
- 1 0
c o s ( ω
t + 5
0 o ) =
1 0
c o s ( ω
t + 5
0 o
- 1 8 0 o )
= 1
0
c o s
ω
t - 1 3
0 o
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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• ( a ) c o
s ( ω t - 9 0 o ) =
+ s i n
ω t
• ( b ) s i n
( ω t ± 1 8 0 o )
= -
s i n
ω t
• T h e m
o s t u s e f u l e
q u a t i o n s
S i n u s o
i d T r a n s f
o r m a t i o n
u s i n g
G r a p h i c a l A p p r o a c h
a r e
•
- c
o s ω t = c
o s ( ω t -
1 8 0 o )
•
m
s i n
ω t = c
o s ( ω t ± 9 0 o )
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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C a n r e p
r e s e n t a n e x p r e
s s i o n
c o n t a i n i n g s i n e a n d c o s
i n e o f
s i m i l a r f r e q u e n c y a n d p
h a s e
A c o
s ω ω ω ω
+ B s i n
ω ω ω ω
t
G r a p h i c a l R e p r e s e n t a
t i o n
o f C o
s i n e & S
i n e
C
c o s ( ω ω ω ω t - θ θ θ θ )
c a n b e e x p r e s s e d
a s
A B
B
A
C
1
2
2
t a
n
,
−
=
+
=
θ
w h e r e
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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- 1 2 9
c o s ω ω ω ω t - 1 2 s i n
ω ω ω ω t
1 5 2
2 5
1 4 4
8 1
1 2
9
2
2
= =
+
=
+
= C
G r a p h i c a l R e p r e s e n t a
t i o n
o f C o
s i n e & S
i n e : E x
a m p l e
c o s ω t
9
θ θ θ θ
o 1 . 5 3
3 3 3
. 1
t a n
9 1 2
t a n
1 1
−
=
−
=
−
=
− −
θ
9 c o s ω ω ω ω t
- 1 2 s i n ω ω ω ω t =
1 5 c o s ( ω ω ω ω t + 5 3 . 1
o )
s i n
ω t
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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G i v e n v 1 = 2 0 s i n ( ω t
+ 6 0 ° ) a n d v
2 = 6 0 c o s ( ω t
- 1 0 ° ) ,
d e t e r m
i n e t h e p h a s e
a n g l e b e t w e
e n t h e t w o s i n u s o i d s
a n d w h
i c h o n e l a g s t h e o t h e r .
=
°
E x a m
p l e
=
2 0 c o s ( ω
t +
6 0 °
− 9
0 ° )
=
2 0 c o s ( ω
t −
3 0 ° )
v 2
=
6 0
c o s (
ω t −
1 0 ° )
T h i s i n d i c a t e s
t h a t t h e p h a s e
a n g l e b e t w e e
n
t h e
t w o s i g n a l s i s 2
0 ° ° ° ° a
n d
t h a t v 1
l a g s
v 2 .
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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3 c o
m m o n w a y s
t o e x p r e s s t h e c o m p l e x
n u m
b e r z ,
F o r m s o f C o m
p l e x N u
m b e r s
z = x +
( R e c t a n g u l a
r F o r m )
z = r
≈ Φ
( P o l a r F o r m )
z = r e j Φ
( E x p o n e n t i a
l F o r m )
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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F r o m
t h e t r i a n g l e , w e c a n e a s i l y
f i n d r a n d φ
x y
y
x
r
1
2
2
t a n
,
−
=
+
=
φ
R e c t a
n g u l a r t o
P o l a r
T r a n s f o r m a t i o n
A n d , i f w e k n o w r a n d φ ,
φ
φ
s i n
,
c o s
r y
r x
=
= z = x + j y
= r ∠ φ = r ( c o s
φ + j s i n φ )
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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1
1
1
1
1
φ
∠ =
+
=
r
j y
x
z
2
2
2
2
2
φ ∠ =
+
=
r
j y
x
z
G i v e n 2
c o m p l e x n u m b
e r , z 1 a n d z 2
C o m p l
e x n u m b e
r : A d d i t
i o n &
S u b t r
a c t i o n
E a s i e r t o p e r f o r m a d d i t i o n s a n d
s u b s t r a c t i o n s
o f
c o m p l e x n u m
b e r s i n r e c t a n
g u l a r f o r m
2
1
2
1
2
1
y
x
z z
+
+
+
=
+
)
(
2
1
2
1
2
1
y
y j
x
x
z z
−
+
−
=
−
S u b s t r a c t i o n :
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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1
1
1
1
1
φ
∠ =
+
=
r
j y
x
z
2
2
2
2
2
φ ∠ =
+
=
r
j y
x
z
G i v e n 2
c o m p l e x n u m b
e r , z 1 a n d z 2
C o m p l
e x n u m b e
r : M u l t i
p l i c a t i o n
& D i v
i s i o n
)
(
2
1
2 1
2 1
φ
φ −
∠
= r r
z z
2
1
2
1
2
1
+
∠ ⋅
=
⋅
r
r
z
z
M
u l t i p l i c a t i o n :
D i v i s i o n :
E a s
i e r t o p e r f o r m m u l t i p l i c a t i o n
s a n d d i v i s i o n
s o f
c o m p l e x
n u m b e r s i n p o l a r f o r m
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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φ ∠ =
+
=
r
j y
x
z
G i v e
n a c o m p l e x n u m b e r
O t h e
r M a t h
O p e r a t
i o n s
− ∠
= r
z
) 2 / ( φ ∠
= r
z
R e c i p r o c a l :
S q u a r e r o o t :
C o m
p l e x c o n j u g a t e :
φ
φ
j
r e
r
j y
x
z
−
= − ∠ =
−
= *
j
j − =
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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φ
φ
φ
s i n
c o s
j
e j
±
=
±
E x p o n
e n t i a l F
o r m
I m
s i n
)
R e (
c o s
φ φ
φ
j j
e e
= =
W h e r e
;
)
R e (
) (
t
j
e
t v
ω
V
=
φ
φ
∠ =
=
m
j
m
V
e V
V
)
R
e (
)
c o s
(
) (
)
(
φ
ω
φ
ω
+
=
+
=
t
j
m
m
e V
t
V
t v
W h e r e
;
E . g .
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P h a s o r R e p r e s e n t a t i o n
φ
φ
ω
∠ =
⇔ +
=
m
m
V
t
V
t
v
V
)
c o s (
) (
V i s t h u s p h a s o r r e p r e s e n t a t i o n
o f t h e s i n u s o i d
( t )
)
R e (
)
(
t
j e
t v
ω
V
=
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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φ ∠
=
m
V
V V o l t a g
e i n
P h a s o r
f o r m
P h a s o
r D i a g r a
m
θ −
∠ = m I
I C u r r e n
t i n
P h a s o r f o r m
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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V ω j
d t
d v ⇔
D e r i v
a t i v e &
I n t e g r a l ω j
d t v
V
⇔
∫
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φ ∠
=
m
V
V
)
c o s (
) (
φ
ω
+
=
t
V
t v
m
T i m e D o m a i n
P h a s o r
D o m a i n
°
T i m e
&
P h a s o
r D o m a i n s
−
−
=
) 4 .
1 7
4 .
1 2
c o s (
3 8
. 5
) (
°
+
=
t
t i
4 . 1 7
3 8 . 5
∠
= I
- N o t e t h a t f r e q u e n c y
( ω ) d o e s n o t a
p p e a r i n t h e p
h a s o r s
- ω i s m
u s t b e c o n s t a
n t i n p h a s o r a n a l y s i s
- I f t h e
s i g n a l s a r e o f d i f f e r e n t ω , p h a s o r s c a n n o t b
e u s e d
- T h e t i m e f u n c t i o n m
u s t b e e x p r e s s
e d i n + v e c o s i n e f o r m
b e f o r e
t r a n s f o r m a t i o n
.
8/9/2019 Electrical Power and Machines- Lec02_Single Phase Circuit
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v ( t ) i s t h e i n s t a n t a n e o u s r e p r e s e n t a t i o n w
h i l e
V i s
t h e f r e q u e n c
y r e p r e s e n t a t i o n
v ( t ) i s t i m e d e p e n
d e n t w h i l e V
i s n o t
D i f f e
r e n c e s b
e t w e e n
T i m e
&
P h a s o r s D o m a i n s
v t
s a w a y s r e a
w
n o c o m
p e x e r m s
w h i l e V i s g e n e r a
l l y c o m p l e x
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)
7 5
2
c o s (
5 0
3
8
4
°
+
=
−
∫
+
t
d t
d i
d t i
i
°
∠
=
−
+
7 5
5 0
I
3
I 8
I 4
ω
ω
j
2 = ω
P h a s o r s t o s o l v e D i f f e r e
n t i a l
E q u a t i o n s
°
∠
=
−
+
7 5
5 0
I 6
I 4
I 4
j
j
j
j
− =
1
°
∠
=
−
−
7 5
5 0
) 6
j 4
I ( 4
j
°
∠
= °
− ∠
°
∠
=
−
°
∠
=
2 .
1 4 3
6 4 .
4
2 .
6 8
7 7 .
1 0
7 5
5
0
0 1 j
4
7 5
5 0
I
) 2 .
1 4 3
2
c o s (
6 4 . 4
) (
2 .
1 4 3
6 4 .
4 I
°
+
=
⇒ °
∠
=
t i