אאא
Single Phase Transformers
אא
א
F
א
E
אא
א
א
אאא
א
K
אאאאאא
K
אא
אא
אאא
אא
K
אאאא
א
K
א
אאאאאא
א
K
אאאאא
אא
K
אאאא
א
K
אאאא
K
אאאאאאאאא
K
א
אאאאאא
א
אא
K
٤
J
١
א
א
אאא
(core)
א
א
F
٤
J
١
E
אאא،
K
א
אאא
(Magnetic flux)
،
אאאאא
אא
א
K
א א א ، א א א
אא א א
(primary winding)
N
١
א א ،
א א
א
(secondary winding)
N
٢
K
אא
אא
،אאא
א
א א א א
K
א
א א א א
א
،א
؟א
٤
J
١
J
١
אא
א
W
-
א
(Power transformers)
אאאא
אאאאא
K
-
א
(Electronic transformers)
אאא
א א
א א
K
אא
K
-
א
(Instrument transformers
)
א א
א
א
אא،א
K
אא
W
J
א
J
א
F
א
E
J
J
אא
W
J
א
J
א
א،אאאא
א،אא
אאא
אאא
K
٤
J
١
٤
J
١
J
٢
א
א אא א
W
א א
(core)
א ،
(windings)
אאא
אאאא
K
J
אא
(core
)
W
אאא
(legs)
אא
(yoke)
אאא
K
אאא
א
א
F
אאא
E
،
א
א
٥٣
,
K
٥,
K
K
אאא
٢K
,
٣K
,
א
،
אאא
K
אאאא
אאאא
K
،א
،אא
אא
K
٤
J
٢
K
א
א
א
א
،אא
א א אאא
א
K
א
אאא
K
٤
J
٢
(Leg)
א
:
J
א
אא
(Core type)
א א א א
٤
J
١
א
א،אא
L
א
א
٤
J
٣
א א ،
F
E
،
אא
F
E
א،
F
E
א
אא
K
אאאאאא
،אא،
א
K
٤
J
٣
אאא
core type
J
אא
א
F
אא
E
Shell type
אאא
E
I
٤
J
٤
א،
E
I
F
E
،
I
E
F
E
אא
אאאאאא،
א
אא
K
אאא
אא
אאאאאא،
K
٤
J
٤
אא
shell type
J
א
(windings)
W
א א ،א א
א
(cylindrical windings)
אא،
(disc windings)
אא،
א
אאא
א
אאא،
K
٤
J
٢
א
אא
א،
א
(Sandwich)
K
٤
J
٢
J
١
אאא
א א
א
א א א א
core) (type
K
٤
J
٥
،א
א
א
،אאא
אאאא
،
אא
א
٥,
K
א
١,
K
א
K
אא
א
א
אא
،אא
א
א
א
F
E
אא
،א
אא
K
א
F
E
אא
K
٤
J
٥
אא
٤
J
٢
J
٢
אאא
،אאאא،א
אאאא
،
אאאאאא
א
،אא
אאא
،א
٤
J
٦K
٤
J
٦
אא
،א א א א אא
א
א
אאאאאא
K
٤
J
٣
אאא
Ideal Transformer
אאאא
אא
K
אא
אאאאאא
אאא
K
אא
א،אא
א
K
א
אא،אאא
א
٤
J
٧
אאאא،
،
F
E
א א
א
אאא
K
אאאא
אא
א א
K
אא
v
١
א א ،
א
Ф
،
e
١
אא
א
W
١ ٤
dt
d
N
e
v
Φ
=
=
1
1
1
N
١
א א א א ،אא א
W
٢ ٤
dt
d
N
e
v
Φ
=
=
2
2
2
N
٢
،א
e
٢
אאאאא
K
א
٤
J
١
א
٤
J
٢
אא
W
٤-٣
2
1
2
1
2
1
N
N
e
e
v
v
=
=
2
1
N
N
א ، א
N
١
>N
٢
א
א،
N
٢
>N
١
אא
K
אאא
א
،אאאא،א
אאאא
W
٤ ٤
2
2
1
1
i
v
i
v
=
א
٤
J
٣
א
٤
J
٤
אאאאאא
K
٥ ٤
1
2
2
1
N
N
i
i =
٤
J
٧
אא
אאא
אאאא
٤
J
١
٤
J
٥
K
אאא
٤
J
٧
א،א
אאא
K
א
אאאאא
א
א
א א א א א
א
א
K
א
א
v
١
אא
v
٢
K
אאאא
٤
J
٣
א
٤
J
٥
א،
W
٦ ٤
2
2
2
2
1
1
1
I
V
N
N
I
V
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
א
(load impedance)
Z
٢
W
٧ ٤
2
2
2
Z
I
V =
א
٤
J
٧
א
٤
J
٦
٨ ٤
2
2
2
1
1
1
Z
N
N
I
V
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
א
٤
J
٨
א
Z
٢
אאאא
Z
٢’
٩ ٤
2
2
2
1
2
Z
N
N
Z
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
٤
J
٨
א،אאאא
Z’
٢
א
Z
٢
אאא
K
א
K
٤
J
٨
אאאא
٤
J
٤
אאאא
אאאאא
K
٠١ ٤
t
t
m
ω
sin
)
(
Φ
=
Φ
א
א
٤
J
١
אא
W
٤-١١
t
N
t
e
m
ω
ω
cos
)
(
1
1
Φ
=
=
٢πf
ω
،
f
א
א
٤
J
١١
אא
W
٢١ ٤
)
90
sin(
2
)
(
1
1
o
m
t
f
N
t
e
+
Φ
=
ω
π
א
אאאאאא
W
٣١ ٤
)
90
sin(
2
)
(
1
1
o
t
E
t
e
+
=
ω
E
١
אאאאא
e
١
W
f
N
E
m
Φ
=
1
1
2
2
π
٤١ ٤
f
N
E
m
Φ
=
1
1
44
.
4
א،אאאאאאאא
W
٥١ ٤
f
N
E
m
Φ
=
2
2
44
.
4
א
٤
J
٤١
א
٤
J
٥١
אאאא
א
K
א
٤
J
٠١
א
٤
J
٣١
א
E
١
אאא
٠٩
o
K
א،אאא
אאאאאאאאא
K
٤
J
١
א
٠٥
L
א،
אא
א
٠٢
אאאאא
א
٠٠٠٠١
L
٢
–
א אא א א א א
א
٠٢٢
L
٠٠٠٣
K
א
B=
٠٠٠٠١lines/cm
٢
A=
٠٢x٠٢=٤٠٠cm
٢
E
١
=
٠٠٠٣V E
٢
=
٠٢٢V
Φ=BA=
٠٠٠٠١*٤٠٠*١٠
٨
=
٠{٠٤ wb
E=
٤{٤٤fΦN
E
١
=
٤{٤٤fΦN
١
N
١
=
٤(/٠٠٠٣{٤٤*٥٠*٠{٠٤)=٣٣٨ turns
E
٢
=
٤{٤٤fΦN
٢
N
٢
=
٤(/٠٢٢{٤٤*٥٠*٠{٠٤)=٢٥ turns
٤
J
٢
א
٥٢
אא
٠٠٥
א
٠٤
אאא
٠٠٠٣
א،
J
אאא
–
אא
אא
J
אאאאא
J
אאא
K
א
KVA=
٥٢N
١
=
٠٠٥N
٢
=
٠٤V
١
=
٠٠٠٣V
5
.
12
40
500
2
1
2
1
=
=
=
N
N
V
V
V
V
240
5
.
12
3000
2
=
=
A
V
KVA
I
33
.
8
3000
10
*
25
3
1
1
=
=
=
A
I
N
N
V
KVA
I
13
.
104
33
.
8
*
40
500
1
1
2
2
2
=
=
=
=
1
1
44
.
4
N
f
E
φ
=
wb
027
.
0
500
*
50
*
44
.
4
3000
=
=
φ
٤
J
٣
א
٥
،
٠١١
L
٠٤٤
٠٦
א
٠٤
٨,
K
אא
K
א
א،
W
J
אא
J
א
–
א
אא
א
KVA= ٥V
١
/V
٢
=
٠١١/٠٤٤f=٦٠ Hz I
٢
=
٠٤A P.F=٠{٨ lead
אאא
W
volt
V
0
110
2
∠
=
A
I
8
.
0
cos
40
1
2
−
∠
=
A
I
o
87
.
36
40
2
∠
=
-The transformer ratio is
4
110
440
2
1
2
1
=
=
=
N
N
V
V
Thus
V
١
=
٠٤٤V
A
I
N
N
I
o
o
87
.
36
10
)
87
.
36
40
(
4
1
2
1
2
1
∠
=
∠
=
=
- The load impedance is
Ω
−
∠
=
∠
∠
=
=
o
o
I
V
Z
87
.
36
75
.
2
87
.
36
40
0
110
2
2
2
- The impedance of the load referred to the primary side is
Ω
−
∠
=
−
∠
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
o
o
Z
N
N
Z
87
.
36
44
87
.
36
75
.
2
(
)
4
(
2
2
2
2
1
2
or
Ω
−
∠
=
∠
=
=
′
o
o
I
V
Z
87
.
36
44
87
.
36
10
440
1
1
2
٤
J
٥
אאא
א
א א
א ،
אא
אאאא
،
אא
אאאאא
K
אאא
אאא
אאאאא
K
א
אא
K
אאא
R
١
א،
א
R
٢
K
א
א،אא
א
،אאאאאאא
אאא،אאאאאאא
אאאא
אאא
X
١
(Leakage reactance)
אא
X
١
=
٢πfL
١
K
א
אאא،אאאא
א א
א א
X
٢
(X
٢
=
٢πfL
٢
)
א
K
٤
J
٩
،אאא
אא
K
אאא
אא
K
אאאאאא
אאא
K
٤
J
٩
אא
٤
J
٠١
אאא
K
X
o
אאא
،
R
o
אאאאאא
K
אא
אאא
I
a
I
o
א،א
א
I
m
א
K
א אא
F
٤
J
٠١
E
א
אא
٤
J
١١
אאא
K
٤
J
١١
אאאאאאאא
)
(
אאא
K
א
אאא
W
٤
J
٠١
אאא
٤-١٦
2
2
1
2
V
N
N
V
=
′
٤-١٧
2
1
2
2
I
N
N
I
=
′
٤-١٨
2
2
1
2
2
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
N
N
R
R
٤-١٩
2
2
1
2
2
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
N
N
X
X
٤
J
١١
אאא
א
א
אאאא
٤
J
١١
א
אאא
אאאאא
F
א
E
אא
F
א
E
،
٤
J
٢١
אא،
K
אא
F
א
E
אאא
א
٤
J
٣١
K
٤
J
٢١
אאאאאא
٤
J
٣١
אאאאאא
אא
R
eq
١
אא
X
eq
١
٤
J
٢١
،
٤
J
٣١
אאאאאא
W
٠٢ ٤
2
1
1
R
R
R
eq
′
+
=
١٢ ٤
2
1
1
X
X
X
eq
′
+
=
א
אאא
W
٢٢ ٤
2
2
1
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
N
N
Z
Z
L
L
אאא
אאא
א
W
٣٢ ٤
1
1
2
1
V
N
N
V
=
′
٤-٢٤
1
2
1
1
I
N
N
I
=
′
٤-٢٥
2
1
2
1
1
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
N
N
R
R
٤-٢٦
2
1
2
1
1
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
N
N
X
X
אאא
אא
٤
J
٤١
אא،
Req
٢
אא
Xeq
٢
א
W
٧٢ ٤
2
1
2
R
R
R
eq
+
′
=
٨٢ ٤
2
1
2
X
X
X
eq
+
′
=
٤
J
٤١
אאא
א
٤
J
٤
א
٠٠١
٠٠٠٢
L
٠٠٤
אאא
W
R
١
=
٠{٠١ Ω X
١
=
٠{٠٣ Ω R
٢
=
٠{٢٥ Ω X
٢
=
٠{٧٥ Ω R
٠
=
٠٠٥Ω
X
o
=
٠٥١Ω
٠٩
א
٠٠٠٢
٨,
K
K
א
אאאאא
٤
J
٢١
F
KE
א
אא
،אאא
W
R
١
=
٠{٠١٥ Ω X
١
=
٠{٠٣٥ Ω
Ω
=
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
01
.
0
2000
400
25
.
0
2
2
2
1
2
2
N
N
R
R
Ω
=
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
03
.
0
2000
400
75
.
0
2
2
2
1
2
2
N
N
X
X
R
eq
=R
١
+R’
٢
=
٠{٠١٥H٠{٠١=٠{٠٢٥Ω X
eq
=X
١
+X’
٢
=
٠{٠٣٥H٠{٠٣=٠{٠٦٥ Ω
Z
o
=R
o
//JX
o
=
//٠٠٥J١٥٠
The voltage V
٢
=
٠٠٠٢V; thus
V
V
N
N
V
400
2000
400
2000
2
2
1
2
=
⎟
⎠
⎞
⎜
⎝
⎛
=
=
′
The current I
٢
=KVA*
٠١
٣
/V
٢
=
٠٩*١٠
٣
/
٠٠٠٢=٤٥ A
A
I
N
N
I
225
45
400
2000
2
1
2
2
=
⎟
⎠
⎞
⎜
⎝
⎛
=
=
′
The power factor=
٠{٨ lagging
o
I
87
.
36
225
2
−
∠
=
′
V
١
=V
’
٢
+I
’
٢
(R
eq
+JX
eq
)
)
065
.
0
025
.
0
(
87
.
36
225
0
400
J
o
+
−
∠
+
∠
=
o
o
96
.
68
07
.
0
*
87
.
36
225
0
400
∠
−
∠
+
∠
=
o
1
.
32
75
.
15
0
400
∠
+
∠
=
V
J
J
V
16
.
1
42
.
413
37
.
8
34
.
413
37
.
8
34
.
13
400
1
∠
=
+
=
+
+
=
Ω
∠
=
∠
∠
=
+
=
o
o
o
o
J
J
Z
3
.
73
7
.
143
7
.
16
522
90
75000
150
500
150
*
500
A
Z
V
I
o
o
14
.
72
88
.
2
3
.
73
7
.
143
16
.
1
42
.
413
1
−
∠
=
∠
∠
=
=
A
J
J
J
I
I
I
o
o
o
o
31
.
37
37
.
227
83
.
137
83
.
180
76
.
2
83
.
0
135
180
3
.
73
88
.
2
87
.
36
225
2
1
−
∠
=
−
=
−
+
−
=
−
∠
+
−
∠
=
+
′
=
א
אאאא
٤
J
٢١
F
E
אא
٤
J
٣١
אא
K
٤
J
٦
א
Transformer operation
٤
J
٦
J
١
אא
No load operation
א
א،אא
אאא
א
I
o
אא
אא
K
אאאאאאא
אאא
K
א
I
o
I
m
, I
a
א،
I
a
א
،א
I
m
אאאא
K
א
א
V
١
٩٢ ٤
t
E
V
m
ω
sin
1
=
אאא
I
m
א
٠٩
o
א
אאאאא
K
אא
W
٠٣ ٤
t
m
ω
cos
Φ
=
Φ
I
a
אאא
אאאא
٠٩
o
א
I
m
K
אאא
I
o
K
١٣ ٤
a
m
o
I
I
I
r
r
r
+
=
٤
J
٥١
א،אא
א
I
a
،
I
m
א
W
٢٣ ٤
o
o
m
o
o
a
I
I
I
I
φ
φ
sin
cos
=
=
φ
o
אאא
I
o
אאא
K
אאאא
R
١
X
١
א
אא،אאאא
W
٢٣ ٤
1
1
1
Z
I
E
V
o
r
r
r
r
+
=
2
1
2
1
1
X
R
Z
+
=
K
אאאאאא
F
א
E
אאאא،אא
א،אאא
א
אא
W
٤٣ ٤
o
o
o
I
V
P
φ
cos
1
=
אאא
אאא
R
o
אא
W
٥٣ ٤
o
o
a
o
I
V
I
V
R
φ
cos
1
1
=
=
א
X
o
אא
W
٦٣ ٤
o
o
m
o
I
V
I
V
X
φ
sin
1
1
=
=
א
I
o
אא
אא
אאא
K
٤
J
٥١
א
٤
J
٦
J
٢
אא
Load operation
א
Z
L
F
E
אאאא
I
٢
אאאא
I
٢
אאא
א،אאא
אאאאא
I’
٢
א
٤
J
٧١
K
א
אאאא
I
o
אאא
אא
אאאא
W
٧٣ ٤
a
I
I
I
r
r
r
+
′
=
2
1
א
I
٢
א
W
٤-٣٨
L
Z
V
I
2
2
=
I’
٢
א
٤
J
٧١
אא،
٤
J
٦١
K
٤
J
٦١
אא
٤
J
٦
J
٣
א
Transformer regulation
אאאא
א،
K
א
אאא
אא
K
א
W
٩٣ ٤
100
)
(
Re
)
(
2
)
(
2
)
(
2
x
V
V
V
PVR
gulation
Voltage
Percentage
rated
rated
noload
−
=
אאאא
V
٢
א
אאא
V’
١
א
Fא
٤
J
٤١
E
א א א ،
אא
V’
١
א
W
٠٤ ٤
100
)
(
Re
2
2
1
x
V
V
V
PVR
gulation
Voltage
Percentage
−
′
=
V’
١
א
W
١٤ ٤
)
(
2
2
2
2
1
eq
eq
JX
R
I
V
V
+
+
=
′
אאאא
אאא
W
٢٤ ٤
100
)
sin
cos
(
2
1
)
sin
cos
(
)
(
2
2
2
2
2
2
2
2
2
2
2
2
2
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
+
+
=
V
R
X
I
V
X
R
I
PVR
eq
eq
eq
eq
φ
φ
φ
φ
א
אאאאא
F
٤
J
٣١
E
אא
W
٣٤ ٤
100
)
(
Re
2
2
1
x
V
V
V
PVR
gulation
Voltage
Percentage
′
′
−
=
V
١
א
W
٤٤ ٤
)
(
1
1
2
2
1
eq
eq
JX
R
I
V
V
+
′
+
′
=
אאאא
אאא
W
٥٤ ٤
100
sin
cos
)
(
1
1
1
1
1
1
1
1
⎭
⎬
⎫
⎩
⎨
⎧
+
=
V
X
I
V
R
I
PVR
eq
eq
φ
φ
א
א
٤
J
٢٤
א
٤
J
٥٤
F
א
אא
אאא
KE
٤
J
٧
אאא
Equivalent circuit parameters
אאאא
٤
J
٥
،
אאא
R
١
אאא،
X
١
אא،
R
٢
،
אא
X
٢
אאאא،
R
o
א
X
o
K
א
N
١
/N
٢
א
K
א
R
١
=R’
٢
X
١
=X’
٢
אא
W
1
1
1
A
l
R
ρ
=
2
2
2
A
l
R
ρ
=
ρ
،אא
l
١
,A
١
،אאא
l
٢
,A
٢
אא
K
אא
W
1
2
2
1
2
1
A
l
A
l
R
R =
אא
א،
W
2
1
2
1
N
N
l
l =
אאאא
W
2
1
1
2
1
2
N
N
I
I
A
A
=
=
אאא
W
2
2
1
2
1
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
N
N
R
R
א
W
2
1
R
R
′
=
א
אאאא،אאא
א
٤
J
٠٢
،
א
٤
J
١٢
W
٦٤ ٤
2
1
2
2
R
R
R
eq
′
=
=
٧٤ ٤
2
1
2
2
X
X
X
eq
′
=
=
א א
،א אא
R
eq
,X
eq
, R
o
,X
o
K
אא
K
٤
J
٧
J
١
אא
No load test
אא
אאא
(open circuit test)
אא
א
K
אאא
א
،אאא
٤
J
٧١
K
א
Io
אאאא
Po
אא،
אאאאאאאא
K
א
Ro,Xo
:
o
o
o
o
V
I
P
φ
cos
=
o
o
o
o
V
I
P
=
φ
cos
o
o
a
I
I
φ
cos
=
o
o
m
I
I
φ
sin
=
a
o
I
V
R
1
=
m
m
I
V
R
1
=
٤
J
٧١
א
٤
J
٧
J
٢
אא
Short circuit test
אאאא
٤
J
٨١
،
אאא
אאאאא
V
sc
אא،אאאאאא
א
אאאא
I
٢
K
אאא
א
א
א
P
sc
אא،
א
I
sc
،
אא
V
sc
אאא،
R
eq
, X
eq
אאא
:
٤-٤٨
eq
sc
sc
R
I
P
2
=
٤-٤٩
2
2
eq
eq
sc
sc
sc
X
R
I
V
Z
+
=
=
א
٤
J
٨٤
Req
א،
٤
J
٩٤
Xeq
K
א
א
٤
J
٦٤
،
٤
J
٧٤
R
١,X١,R’٢,X’٢
א
א
R
٢,X٢
א
K
٤
J
٨١
א
٤
J
٥
א
٠٠٥
٠٣٢
L
٠٠٣٢
K
א
אא
K
א،אאאא
٠٥٢٢
אא
٤{
٩
،
א
٠٠٣٢
K
אאא
٠٢٢٨
אאא
אא
٥{
٤٩
K
אאאא
K
א
Open circuit test: V
o
=
٠٠٣٢V I
o
=
٩{٤A P
o
=
٠٥٢٢W
Short circuit test: V
sc
=
٤٩{٥V I
sc
=I
rated
P
sc
=
٠٢٢٨W
o
o
o
o
V
I
P
φ
cos
=
1156
.
0
2300
*
4
.
9
2250
cos
=
=
=
o
o
o
o
V
I
P
φ
o
o
36
.
83
)
1156
.
0
(
cos
1
=
=
−
φ
Ω
=
=
=
=
2116
1156
.
0
*
4
.
9
2300
cos
1
o
o
o
a
o
I
V
I
V
R
φ
Ω
=
=
=
=
33
.
246
9933
.
0
*
4
.
9
2300
sin
1
o
o
o
m
m
I
V
I
V
R
φ
A
I
I
l
f
sc
39
.
217
2300
10
*
500
3
.
=
=
=
Ω
=
=
=
1739
.
0
)
39
.
217
(
8220
2
2
sc
sc
eq
I
P
R
2
2
2
2
2
)
1739
.
0
(
39
.
217
5
.
94
)
(
eq
eq
eq
sc
sc
sc
X
X
R
I
V
Z
+
=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
=
As a result, we get X
eq
=
٠{٣٩٨٤Ω
Ω
=
=
=
′
=
08695
.
0
2
1739
.
0
2
2
1
eq
R
R
R
Ω
=
=
=
′
=
1992
.
0
2
3984
.
0
2
2
1
eq
X
X
X
Ω
=
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
′
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
0008695
.
2300
230
*
08695
.
0
2
2
1
2
2
2
2
2
1
2
2
N
N
R
R
N
N
R
R
Ω
=
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
′
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
′
001992
.
2300
230
*
1992
.
0
2
2
1
2
2
2
2
2
1
2
2
N
N
X
X
N
N
X
X
٤
J
٨
אאא
Losses and Efficiency
٤
J
٨
J
١
אא
א
אאא
Iron loss
א
Cupper loss
،
אא
אא،א
א
K
אאא
אא،
א א א
P
o
א א
א
،אא
،א
אאא
אאאא
K
P
iron
=P
o
٠٥ ٤
אאא
K
אאא
אאאא،א
אא
K
אאאא
אאאא
F
٣
٥
E٪
א
K
אאא
K
אאאא
אאאאאא
K
P
cu
=P
sc
١٥ ٤
אאאאאאאאא
אא
אאאאאאא
W
٢٥ ٤
eq
sc
sc
cu
R
I
P
2
)
(
=
٣٥ ٤
eq
l
f
l
f
cu
R
I
P
2
.
)
.
(
=
٤٥ ٤
2
.
)
(
)
.
(
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
sc
l
f
sc
cu
l
f
cu
I
I
P
P
אאאאאאא
K
א
א
K
٥٥ ٤
)
.
(
2
)
(
l
f
cu
x
cu
P
x
P
=
٤
J
٨
J
٢
א
אא
output power
א
אאא
א
אא،א
א
W
٦٥ ٤
100
2
2
x
P
P
P
P
cu
iron
+
+
=
η
٧٥ ٤
100
1
2
x
P
P
=
η
P
٢
،א
P
١
אאא
K
א
אא
W
٨٥ ٤
2
2
2
2
cos
φ
I
V
P
=
cosφ
٢
א
Power factor
K
P
٢
א
٤
J
٨٥
א
٤
J
٦٥
אאא
א
א
F
אאא
WE
٩٥ ٤
100
cos
cos
2
2
2
2
2
2
2
2
eq
iron
R
I
P
I
V
I
V
+
+
=
φ
φ
η
٤
J
٦
א
٠٠٥
אא،
٠٠٥٢
אא
אאא
٠٠٥٧
א
K
אאאא
א
٨,
K
א
KVA=
٠٠٥Piron =٢٥٠٠W Pcu(f.l.)=٧٥٠٠ W cosφ=٠{٨
P
٢=٥٠٠*١٠٣*٠{٨=٤٠٠٠٠٠ W
%
56
.
97
100
2500
7500
400000
400000
100
2
2
=
+
+
=
+
+
=
x
x
P
P
P
P
cu
iron
η
At half load
P
١(٢L٢)=١L٢*٥٠٠*١٠٣*٠{٨=٢٠٠٠٠٠W
Pcu(x)=x
٢Pcu(f.l)
Pcu(
١L٢)=١L٤*٧٥٠٠=١٨٧٥W
%
86
.
97
100
2500
1875
200000
200000
100
)
2
/
1
(
)
2
/
1
(
2
)
2
/
1
(
2
=
+
+
=
+
+
=
x
x
P
P
P
P
cu
iron
η
٤
J
٧
אאאא
٤
J
٥
٨,
K
א
W
V
٢
= ٠٣٢V P
i
= ٠٥٢٢W R
eq٢
=٣٧١٠٠٫٠Ω X
eq٢
=٨٩٣٠٠٫٠Ω
I
٢
=٠٠٥*١٠
٣
/٠٣٢=٢١٧٤A
Thus, substituting in Equation ٢-٤٢, we get
%
64
.
3
100
230
))
6
.
0
(
00173
.
0
)
8
.
0
(
00398
.
0
(
2174
2
1
230
))
6
.
0
(
00398
.
0
)
8
.
0
(
00173
.
0
(
2174
)
(
2
=
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
−
+
+
=
PVR
Also, substituting in Equation ٤-٥٩, we get
%
46
.
97
100
*
00173
.
0
*
2174
2250
8
.
0
*
2174
*
230
8
.
0
*
2174
*
230
2
=
+
+
=
η
٤
J
٩
אאא
Auto-transformer
אאאאאאאא
אא
K
א
א
אאאאאא
אאאאא
א
٤
J
٩١
K
אא
٤
J
٠٢
K
א
K
א
AB
٤
J
٩١
אאא
(N
١
)
א
BC
א
א
(N
٢
)
אא
א
K
אאאא
אא
K
W
٠٦ ٥
1
2
1
2
1
I
I
N
N
V
V
L
=
=
אאא
W
١٦ ٤
1
2
I
I
I
L
−
=
א
אאא
K
א
C
א
אאא
א
אא אא
٤
J
٠٢
K
א א א אא א
“Variac”
K
אאאאאאאא
١W
٥
{٢
אאאא
K
אאא
א
אאאא
אא
א
אאאאאאאא
K
אאאא
אא
K
א
א
א
א
K
٤
J
٩١
א
٤
J
٠٢
אא