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1. Department of Electronic Engineering
National Taipei University of Technology

2. •
•
•
•
•
• T π
•
2/28 Department of Electronic Engineering, NTUT

3. • N1
(Flux Linkage)
L1 ( )
Department of Electronic Engineering, NTUT3/28
( ) ( ) ( )1 1 1 1 1t N t L i tλ φ= =
+
−
( )1i t
( )1v t
( )1 tφ

4. ( )
• ( )
( ) ( ) ( )1 1 1
1 1 1
d t d t di t
v t N L
dt dt dt
λ φ
= = =
( ) ( ) ( ) ( )1 1 1 1 11 12t N t N t tλ φ φ φ= = +  
( ) ( ) ( )1 11 12t t tφ φ φ= +
( )
( ) ( ) ( )1 11 12
1 1 1
d t d t d t
v t N N
dt dt dt
λ φ φ
= = +
( )
( )
( ) ( )
( )
( )11 1 12 2
1 1
1 2
d t di t d t di t
N N
di t dt di t dt
φ φ   
= +   
   
( ) ( )1 2
1 12
di t di t
L M
dt dt
= +
( )1 tφ
+
−
+
−
1
( )11 tφ
( )12 tφ
( )1i t
( )1v t ( )2v t
( )2i t
2
1 :
Department of Electronic Engineering, NTUT4/28

5. • 1
• 1 2
• 2
• 2 1
• :
( )
( )
11
1 1
1
d t
L N
di t
φ
=
( )
( )
12
12 1
2
d t
M N
di t
φ
=
( )
( ) ( )1 2
1 1 12
di t di t
v t L M
dt dt
= +
( )
( )
21
21 2
1
d t
M N
di t
φ
=
( )
( )
22
2 2
2
d t
L N
di t
φ
=
( )
( ) ( )1 2
2 21 2
di t di t
v t M L
dt dt
= +
( )1 tφ
+
−
+
−
1
( )11 tφ
( )12 tφ
( )1i t
( )1v t ( )2v t
( )2i t
2
( )2 tφ
+
−
+
−
1
( )22 tφ
( )21 tφ
( )1i t
( )1v t ( )2v t
( )2i t
2
12 21M M M= =
Department of Electronic Engineering, NTUT5/28

6. •
( )
• ( )
( )1i t ( )1 tφ
1 2
_
+
( )1 tφ ( )1 tφ′
( )1i t ( )2i t
( )2v t
1 2
Department of Electronic Engineering, NTUT6/28

7. 1 ( )
• ( )
+
−
M
+
−
( )1i t
( )1v t ( )2v t1L 2L
( )
( )1
11 1
di t
v t L
dt
=
( )
( )2
12 0
di t
v t M
dt
= =
( )
( )2
22 2 0
di t
v t L
dt
= =
( )
( )1
21
di t
v t M
dt
=
+
−
M
+
−
( )1i t
( )1v t ( )2v t1L 2L
( )
( )1
11 1
di t
v t L
dt
=
( )
( )2
12 0
di t
v t M
dt
= =
( )
( )2
22 2 0
di t
v t L
dt
= =
( )
( )1
21
di t
v t M
dt
= −
( )
( )1
2
di t
v t M
dt
=
( )
( )1
2
di t
v t M
dt
= −
( ) ( ) ( )1 11 12v t v t v t= +
( )
( )1
11 1
di t
v t L
dt
= =
( ) ( ) ( )2 22 21v t v t v t= +
( )
( )1
21
di t
v t M
dt
= =
( ) ( ) ( )1 11 12v t v t v t= +
( )
( )1
11 1
di t
v t L
dt
= =
( ) ( ) ( )2 22 21v t v t v t= +
( )
( )1
21
di t
v t M
dt
= = −
Department of Electronic Engineering, NTUT7/28

8. 1 ( )
+
−
M
+
−
( )1i t
( )1v t ( )2v t1L 2L
( )
( )1
11 1
di t
v t L
dt
=
( )
( )2
12 0
di t
v t M
dt
= =
( )
( )2
22 2 0
di t
v t L
dt
= =
( )
( )1
21
di t
v t M
dt
= −
( ) ( ) ( )1 11 12v t v t v t= +
( ) ( ) ( )2 22 21v t v t v t= +
+
−
M
+
−
( )1i t
( )1v t ( )2v t1L 2L
( )
( )1
11 1
di t
v t L
dt
=
( )
( )2
12 0
di t
v t M
dt
= =
( )
( )2
22 2 0
di t
v t L
dt
= =
( )
( )1
21
di t
v t M
dt
=
( ) ( ) ( )1 11 12v t v t v t= +
( ) ( ) ( )2 22 21v t v t v t= +
( )
( )1
2
di t
v t M
dt
=
( )
( )1
2
di t
v t M
dt
= −
( )
( )1
11 1
di t
v t L
dt
= =
( )
( )1
21
di t
v t M
dt
= = −
( )
( )1
11 1
di t
v t L
dt
= =
( )
( )1
21
di t
v t M
dt
= =
Department of Electronic Engineering, NTUT8/28

9. 2 ( )
• v1(t) v2(t) i1(t) i2(t)
( )
( )1
11 1
di t
v t L
dt
=
( )
( )2
12
di t
v t M
dt
= −
( )
( )2
22 2
di t
v t L
dt
= −
( )
( )1
21
di t
v t M
dt
=
( ) ( ) ( )1 11 12v t v t v t− = +
( ) ( ) ( )2 22 21v t v t v t= +
+
−
M
+
−
( )1i t
( )1v t ( )2v t1L 2L
( )2i t
( )
( )1
11 1
di t
v t L
dt
=
( )
( )2
12
di t
v t M
dt
= −
( )
( )2
22 2
di t
v t L
dt
=
( )
( )1
21
di t
v t M
dt
= −
( ) ( ) ( )1 11 12v t v t v t= +
( ) ( ) ( )2 22 21v t v t v t= +
+
−
M
+
−
( )1i t
( )1v t ( )2v t1L 2L
( )2i t
( ) ( )1 2
1
di t di t
L M
dt dt
= −
( ) ( )1 2
2
di t di t
M L
dt dt
= −
( ) ( )1 2
1
di t di t
L M
dt dt
= −
( ) ( )1 2
2
di t di t
M L
dt dt
= − +
Department of Electronic Engineering, NTUT9/28

10. 3 −
( )
( ) ( )
( )
( )1 2
1 1 1
di t di t di t
v t L M L M
dt dt dt
= − = −
( )
( ) ( )
( )
( )1 2
2 2 2
di t di t di t
v t M L M L
dt dt dt
= − = −
( ) ( ) ( ) ( )
( )
1 2 1 2 2
di t
v t v t v t L L M
dt
= − = + −
1 2 2aL L L M= + −
+
−
+
−
+
−
M( )1i t ( )2i t( )i t
( )1v t ( )2v t1L 2L
( )v t
( )1i t ( )2i t
+−+ −
+ −
M( )1v t ( )2v t
1L 2L
( )v t
• v1(t) v2(t) i1(t) i2(t)
−
+
+
−
+
−
M( )1i t ( )2i t( )i t
( )1v t ( )2v t1L 2L
( )v t
( )1i t ( )2i t
−++ −
+ −
M( )1v t ( )2v t
1L 2L
( )v t
( )
( ) ( )
( )
( )1 2
1 1 1
di t di t di t
v t L M L M
dt dt dt
= − = −
( )
( ) ( )
( )
( )1 2
2 2 2
di t di t di t
v t M L L M
dt dt dt
= − + = −
( ) ( ) ( ) ( )
( )
1 2 1 2 2
di t
v t v t v t L L M
dt
= + = + −
1 2 2aL L L M= + −
Department of Electronic Engineering, NTUT10/28

11. 3 −
( )
( ) ( )
( )
( )1 2
1 1 1=
di t di t di t
v t L M L M
dt dt dt
+ = +
( )
( ) ( )
( )
( )1 2
2 2 2=
di t di t di t
v t M L L M
dt dt dt
+ = +
( ) ( ) ( ) ( )
( )
1 2 1 2 2
di t
v t v t v t L L M
dt
= + = + +
1 2 2aL L L M= + +
+
−
+
−
+
−
M( )1i t ( )2i t( )i t
( )1v t ( )2v t1L 2L
( )v t
( )1i t ( )2i t
+ −+ −
+ −
M( )1v t ( )2v t
1L 2L
( )v t
( )
( ) ( )
( )
( )1 2
1 1 1=
di t di t di t
v t L M L M
dt dt dt
+ = +
( )
( ) ( )
( )
( )1 2
2 2 2=
di t di t di t
v t M L L M
dt dt dt
− + = +
( ) ( ) ( ) ( )
( )
1 2 1 2 2
di t
v t v t v t L L M
dt
= − = + +
1 2 2aL L L M= + +
−
+
+
−
+
−
M( )1i t ( )2i t( )i t
( )1v t ( )2v t1L 2L
( )v t
( )1i t ( )2i t
− ++ −
+ −
M( )1v t ( )2v t
1L 2L
( )v t
Department of Electronic Engineering, NTUT11/28

12. 4 −
•
( )
( ) ( )1 2
1
di t di t
v t L M
dt dt
= + ( )
( ) ( )1 2
2
di t di t
v t M L
dt dt
= +
( )
( )1 2
2
1 2
di t L M
v t
dt L L M
−
=
−
( )
( )2 1
2
1 2
di t L M
v t
dt L L M
−
=
−
( ) ( ) ( )1 2i t i t i t= +
( ) ( ) ( )
( )1 2 1 2
2
1 2
2di t di t di t L L M
v t
dt dt dt L L M
+ −
= + =
−
( )
( )2
1 2
1 2 2
di tL L M
v t
L L M dt
−
=
+ −
2
1 2
1 2 2
a
L L M
L
L L M
−
=
+ −
M
+
−
( )i t
( )1i t
( )2i t
1L 2L( )v t
,
M
+
−
( )i t
( )1i t
( )2i t
1L 2L( )v t
2
1 2
1 2 2
a
L L M
L
L L M
−
=
+ +
2
1 2 0L L M− ≥
1 2M L L≤
Department of Electronic Engineering, NTUT12/28

13. 5
•
is(t) = cost v2(t)
( )
+
−
( )si t
( )2v t
2 F
1 Ω 3 H 1 H 2 Ω
( )2i t
( )1i t
( )1v t 1 HM =
1 1 23V j I j Iω ω= −
2 1 2V j I j Iω ω= − − 2 1 2
1 3
2 2
I V V
j jω ω
= +
1 1 2
1 1
2 2
I V V
j jω ω
= +
( ) 1 2 11 2 2 sj V j V I Iω ω+ − = −
1 2 2
1
2 2
2
j V j V Iω ω
 
− + + = − 
 
1 2
1 1
1 2 2
2 2
sj V j V I
j j
ω ω
ω ω
   
+ + + − =   
   
1 2
1 1 3
2 2 0
2 2 2
j V j V
j j
ω ω
ω ω
   
− + + + =   
   
KCL:
I1 I2
1 2
3 5
1 1
2 2
j V j V
 
+ − = 
 
1 2
5 1 1
+ 0
2 2 2
j V j V
 
− + = 
 
2
10
0.408 78.2
24 5
j
V
j
= = ∠
+
( ) ( )( )2 0.408cos 78.2 Vv t t= +
1 0sI = ∠ 1ω =( ) cossi t t= , ,
Department of Electronic Engineering, NTUT13/28

14. (Ι)
• 1
(Unity Coupled Coil)
•
+
−
+
−
( )1i t
1N
( )1v t
( )2v t
2N
( )2i t
( )1 tφ
Department of Electronic Engineering, NTUT14/28

15. (II)
•
• (Turns Ratio)
a 1 ; a 1
2
1
N
a
N
=
( ) ( ) ( ) ( )2
2 1 2 1
1
N
v t v t v t av t
N
= ⇒ =
( ) ( )2
2 1
1
N
v t v t
N
= ( ) ( )1 2i t ai t= −
+
−
+
−
( )1v t ( )2v t
( )1i t ( )2i t
1 2:N N
−
+
+
−
+
−
( )1i t ( )2i t
( )1v t ( )2v t( ) 2
2
1
N
i t
N
( ) 2
1
1
N
v t
N
Department of Electronic Engineering, NTUT15/28

16. (III)
•
( ) ( )2
2 1
1
N
v t v t
N
= − ( ) ( )2
1 2
1
N
i t i t
N
=
+
−
+
−
( )1v t ( )2v t
( )1i t ( )2i t
1 2:N N
−
+
+
−
+
−
( )1i t ( )2i t
( )1v t ( )2v t( ) 2
2
1
N
i t
N
( ) 2
1
1
N
v t
N
−
Department of Electronic Engineering, NTUT16/28

17. (I)
Department of Electronic Engineering, NTUT
(a)
+
−
+
−
+
−
( )1i t ( )2i t
1 2:N NsR
( )1v t ( )2v t LR( )sv t
+
−
sR 1 2:N N
( )sv t
2
1
2
L
N
R
N
 
 
 
LR
(b)
+
−
sR
( )sv t
2
1
2
L
N
R
N
 
 
 
(c)
• RL
2
1
2
N
N
 
 
 
17/28

18. (II)
• L
• C
RLC
+
−
+
−
( )1v t
sR
( )sv t
( )1i t
+
−
( )2v t LZ
( )2i t
1 2:N N
Department of Electronic Engineering, NTUT
2
1
2
N
L
N
 
 
 
2
2
1
N
C
N
 
 
 
18/28

19. (III)
• (a)
(b)
•
2
2
1
N
N
 
 
 
+
−
a
b
(b)
+
−
a
b
(a)
1 2:N NsR
( )sv t
2
2
1
s
N
R
N
 
 
 
( ) 2
1
s
N
v t
N
Department of Electronic Engineering, NTUT19/28

20. 6
• Vs I1
RL V0
I1 I2
I1
−
+
−
+ +
−
sV
1R
1I
:1a
2I
LR 0V
1V 2V
2R
( )1 2 1 2 2 1sR R I R I V V+ − = −
( )2 2 2 2LR I R R V− + + =
2 1
1
V V
a
= 2 1I aI=,
( ) 2 1 1
1
1 0La R aR I V
a
− + − =  
( )1 2 1 11 sR a R I V V+ − + =  
( )
( )1 2 2
1 2
A
1
s
L
V
I
R a R a R
=
+ − +
( )
( )0 2 1
1 2
V
1 2 2
L
L L s
L
aR
V I R aI R V
R a R a R
= = =
+ − +
Department of Electronic Engineering, NTUT20/28

21. 7
• a-b
1.
(a)
b
+
−
aa
b
(a)
b
+
−
a
b
2.
(b)
3. a-b (c)
a
( ) 3sinsv t t=
25 Ω
4 Ω
1: 5
12sint
4 Ω 1: 5 25 Ω 100 Ω 25 Ω
60sint
125 Ω
60sint
(b) (c)
Department of Electronic Engineering, NTUT21/28

22. (Primary Winding) (Secondary Winding)
+
−
+
−
M1R 2R
sV 1I 2I1L 2L LVLZ
( )1 1 1 2 sR sL I sMI V+ − = ( )
2 2
1
L
s
R sI Z
I V
s
+ +
=
∆ ( )2 s
sM
I V
s
=
∆
( ) ( )
2 2
1 1
1 2 2
s
i
L
V s M
Z s R sL
I R sL Z
= = + −
+ +
( ) 2
1 2 2
i
L
I sM
H s
I R sL Z
= =
+ +
( )
( )
L L
v
s
V sMZ
H s
V s
= =
∆
( ) ( ) ( ) ( )2 2
1 2 1 2 2 1 1 2L Ls L L M s R L R Z L s R R Z∆ = − + + + + +  
( )2 2 2 1 0LR sL I sMI V+ − + =
,
Department of Electronic Engineering, NTUT22/28

23. •
(–M) M
( ) ( )
2 2
1 1
1 2 2
s
i
L
V s M
Z s R sL
I R sL Z
= = + −
+ +
( ) ( ) ( )1i fZ s Z s Z s= +
( )
2 2
2 2
f
L
s M
Z s
R sL Z
= −
+ +
( )1 1 1Z s R sL= +
s jω= L L LZ R jX= +
( )
( ) ( )
2 2
2 2
f f f
L L
M
Z j R jX
R R j L X
ω
ω
ω
= = +
+ + +
( )
( ) ( )
2 2
2
2 2
2 2
L
f
L L
M R R
R
R R L X
ω
ω
+
=
+ + +
( )
( ) ( )
2 2
2
2 2
2 2
L
f
L L
M L X
X
R R L X
ω ω
ω
− +
=
+ + +,
Department of Electronic Engineering, NTUT23/28

24. 8
• (a) (a)
I1 (b)
(b)
(a)
(b)
+
−
(a)
100 0°
∠
1I
j
j2j 1 +
−
2j
1I
100 0°
∠
1
2
1
2
j−
fZ
( )
2 2
2 2
f
L
s M
Z s
R sL Z
= −
+ +
( )
2
2 2
1 1 1
1 2 2
f
M
Z j
j L R j
ω
ω
= = = −
+ +
( ) ( )2
1 1
1 1
100 0 20 60 A 63.2 71.6 A
2 2
I j j j I
 
= ∠ + − = − = = ∠ − 
 
( )
2
2
1
f
M
Z j
j L j
ω
ω
= = = −
( 1j )( )2iZ j j j= − = Ω
(b)
Department of Electronic Engineering, NTUT24/28

25. T
•
T π
(c) T
T
+
−
+
−
(a)
( )1i t ( )2i t
( )1v t ( )2v t
M
1L 2L
+
−
+
−
+
−
+
−
( )1i t M ( )2i t
( )1v t ( )2v t( )2di t
M
dt
( )1di t
M
dt
+
−
+
−
( )2i t( )1i t( )1v t ( )2v tM
1L M− 2L M−
( )
( ) ( )1 2
1 1
di t di t
v t L M
dt dt
= +
( )
( ) ( )1 2
2 2
di t di t
v t M L
dt dt
= +
( ) ( )
( )
( ) ( )1
1 1 1 2
di t d
v t L M M i t i t
dt dt
= − + +  
( ) ( ) ( ) ( )
( )1
2 2 1 2
di td
v t M i t i t L M
dt dt
= + + −  
(b)
(c)
Department of Electronic Engineering, NTUT25/28

26. 9
• (a) ω = 500 /
0.6 T I1
V2
+
−
+
−
M
(a)
( )100 0 VsV °
= ∠
1 40 mHL = 2 70 mHL =
1 15R = Ω
2 10R = Ω
1I
2V30LR = Ω
2 70mHL =1 40mHL = 0.6k =
( )( )0.6 40 70 31.7mHM = =
( ) ( )1 1 2 2 2 2 sR j L M j M R I j M R I Vω ω ω+ − + + − + =  
( ) ( )2 2 2 2 2 0j M R R j M j L M R Iω ω ω− + + + + − + =  
( ) ( )1 225 20 10 15.9 100j I j I+ − + =
( ) ( )1 210 15.9 40 35 0j I j I− + + + =
( )2 2 39.3 13.70 VLV I R= = ∠−
, ,
+
−
(b)
30LR = Ω
31.7 mHM =
2 10R = Ω
1I 2I
1 15R = Ω
( )100 0 VsV °
= ∠
( )1
8.3 mH
L M−
=
( )2
38.3 mH
L M−
=
+
−
2V
( )1 3.72 30.3 AI = ∠ − ( )2 1.31 13.70 AI = ∠ −,
Department of Electronic Engineering, NTUT26/28

27. π
• π
Y ∆ T π
sL R
Y ∆
T
+
−
+
−
( )2i t( )1i t( )1v t ( )2v t3L
1L 2L
+
−
+
−
( )1i t ( )2i tBL
( )1v t ( )2v tAL CL
1 2 2 3 3 1
2
A
R R R R R R
R
R
+ +
= 1 2 2 3 3 1
3
B
R R R R R R
R
R
+ +
= 1 2 2 3 3 1
1
C
R R R R R R
R
R
+ +
=
1 1L L M= − 2 2L L M= − 3L M=
2
1 2
2
A
L L M
L
L M
−
=
−
2
1 2
B
L L M
L
M
−
=
2
1 2
1
C
L L M
L
L M
−
=
−
, ,
, , , ,
Y ∆
T π
Department of Electronic Engineering, NTUT27/28

28. •
•
•
• AC
Department of Electronic Engineering, NTUT28/28