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1. Department of Electronic Engineering
National Taipei University of Technology

2. •
•
•
•
•
•
• Y ∆
•
Department of Electronic Engineering, NTUT2/47

3. (Linear Circuits)
• :
• :
y = Kx K
x y
R y = Kx
, where 0
dy
ax a
dt
= ≠
, where 0
dx
y b b
dt
= ≠
v Ri=
( ) ( )Kv K Ri R Ki= =
Department of Electronic Engineering, NTUT3/47

4. 1
• 1: i1
i2 vg1 ig2
KVL vg1 = i1 i2
KCL i1 + i2 = ig2
2
+
−
3 Ω
i1
6 Ω
i2
vg1 ig2
2 1 13 6 gi i v− + =
1 1 2
2 1 2
1 1
9 3
1 2
9 3
g g
g g
i v i
i v i

= +

 = − +

Department of Electronic Engineering, NTUT4/47

5. (Practical Sources) I
•
(a)
(a)
(b) :
• Rs Rs
Rs Rs
= 0
s
L
s L
v
i
R R
=
+
L s
L
s L
R v
v
R R
⋅
=
+
+
−
(a)
Rs
vs
(b)
+
−
+
−
Rs
iL
vs vL RL
Department of Electronic Engineering, NTUT5/47

6. II
•
Rg
+
−
is VRg
Department of Electronic Engineering, NTUT6/47

7. (Equivalent Circuit) I
•
• (a)
• (b)
+ − +
−
(a)
+
−vs
v1
1
s
s s
v vv
i
R R
−
= =
22
2
s g
s s
g g
i R vv
i i i i
R R
−
= − = − =
Rs i1
v1
+
−
(b)
i
i2
v2is Rg
i
–
Department of Electronic Engineering, NTUT7/47

8. II
• i1=i2 , v1=v2
i1= i2
v1= v2
•
21 s gs
s g
i R vv v
R R
−−
=
11 s gs
s g
i R vv v
R R
−−
=
+ − +
−
(a)
+
−vs
v
Rs i1
v1
+
−
(b)
i
i2
v2is Rg
i
ands s g s gv i R R R= =
Department of Electronic Engineering, NTUT8/47

9. 2
• 2: (a) I
(c)
+
− 8 V
I
(a)
+
−
3 Ω
4 A
I
12 Ω 1 Ω
6 Ω 6 Ω 2 Ω32 V
(b)
+
− 8 VA
3
32 I
12 Ω 1 Ω 2 Ω
6 Ω6 Ω3 Ω
12 Ω 3 Ω
6 Ω2 ΩA
3
32
Department of Electronic Engineering, NTUT9/47

10. 2
+
−
16 Ω
16 V
I
(g)
(a) (f) (g)
(e)
+
−
I
V
3
64
A
3
8
12 Ω2 Ω
2 Ω
(d)
+
−
I
V
3
64
12 Ω2 Ω
6 Ω 3 Ω A
3
8
(f)
+
−
+
−
I
2 Ω 12 Ω 2 Ω
V
3
16
V
3
64
( ) ( )
64 16
2 12 2 1 A
3 3
I I+ + = − ⇒ =
Department of Electronic Engineering, NTUT10/47

11. I
• (Superposition Theorem)
Department of Electronic Engineering, NTUT11/47

12. II
•
1)
2)
3) (1) (2)
4)
5) ( ) K
K
Department of Electronic Engineering, NTUT12/47

13. 3
• 3: (a)
I
(b) (c)
(b) (c) 4Ia 4Ib
5
(b):
+ −
+
−
4 Ω
(a)
I
4I
18 Ω4 A20 V
5 Ω
+ −
+
−
(b)
4 Ω 4Ia
18 Ω20 V
Ia
5 Ω
+ −
(c)
4 Ω 4Ib
18 Ω4 Α
Ib
5 Ω
4 − Ib
( ) ( )20 4 4 18 5 0.87 Aa a aI I I− = − + + ⇒ = −
( ) ( ) ( )4 4 5 4 18 3.13 Ab a b bI I I I= + − − ⋅ ⇒ =
( )2.26 Aa bI I I= + =
Department of Electronic Engineering, NTUT13/47

14. 4
• 4: (a) 5
+ −
(c) (d)
v+ −
(a)
+
−4 Ω
3 Ω
5 Ω
5 A
2 A
10 V
+ −
(b)
v1
5 Ω
4 Ω
3 Ω
5 A
v2
5 Ω
4 Ω
3 Ω2 A
+ −
−
+
v3
5 Ω
4 Ω
3 Ω
10 V
( )5
5 4
1.67 A
4 3 5
I Ω
⋅
= =
+ +
( )1 1.67 5 8.33 Vv = ⋅ =
( )5
2 3
0.5 A
4 3 5
I Ω
⋅
= =
+ +
( )2 0.5 5 2.5 Vv = ⋅ = ( )3
10 5
4.17 V
4 3 5
v
⋅
= =
+ +
( )1 2 3 15 Vv v v v= + + = ( )
2 2
15
45 W
5
v
p
R
= = =
Department of Electronic Engineering, NTUT14/47

15. III
• :
•
• p = i2R p = v2/R
( )
2 2
1
1
8.33
13.95 W
5
v
p
R
= = =
( )
2 2
2
2
2.5
1.25 W
5
v
p
R
= = =
( )
2 2
3
3
4.17
3.48 W
5
v
p
R
= = =
( )1 2 3 18.63 Wp p p p= + + =
Department of Electronic Engineering, NTUT15/47

16. IV
•
1)
2)
Department of Electronic Engineering, NTUT16/47

17. I
• (Thevenin’s Theorem)
• (voc eth) (Rth)
1) voc eth
(a)
2) Rth (
) (b)
( )
(a)
+
-
voc
a
b
(b)
(
)
Rth
a
b
Department of Electronic Engineering, NTUT17/47

18. II
•
+
-
(a)
+
-
+
-
(b)
voc
I
v
Rth I
v
a
b
a
b
Department of Electronic Engineering, NTUT18/47

19. 5
• 5: (a) v1
Rth
(b)
voc
:
1)
(c)
+
−
+
−
(a)
v1
a
b
6 Ω 4 Ω
3 Ω 2 Ω
2 A
12 V
(b)
Rth
a
b
6 Ω 4 Ω
3 Ω
(c)
+
− ′Voc
a
b
6 Ω 4 Ω
3 Ω12 V
( )6//3 4 6thR = + = Ω
( )
3
12 4 V
3 6
ocv′ = =
+
Department of Electronic Engineering, NTUT19/47

20. 5
a-b (e)
(f)
2 A 4 Ω 3 Ω 6 Ω
b c
1)
2) (d)
( )2 4 8 Vocv′′ = ⋅ =
( )2 4 8 Voc ab acv v v′′ = = = ⋅ =
( )12 Voc oc ocv v v′ ′′= + =
( )1
2
12 =3 V
6 2
v =
+
(d)
c
′′voc
a
b
4 Ω6 Ω
3 Ω
2 Α
(e)
+
−
6 Ω
12 V
+
−
(f)
+
−
v1
a
b
6 Ω
2 Ω12 V
Department of Electronic Engineering, NTUT20/47

21. I
• (Norton’s Theorem)
• (Isc) (Rth)
1) Isc:
2) Rth:
I
sc
a
b
Department of Electronic Engineering, NTUT21/47

22. II
•
• (a) a-b
(b)
(a) (b)
a
b
Isc Rth
a
b
(a)
Isc
a
b
oc
sc
th
v
I
R
=
oc ab th scv v R I= =
+
-
(b)
voc
Rth a
b
Isc
+
-
(c)
a
b
vocRthIsc
Department of Electronic Engineering, NTUT22/47

23. 6
• 6: (a) I
A. (a) Rth
B. a-b (b)
C. a-b (c)
D. (d) I
+
− Ω3V24
(a)
I
Rth
a
b
Ω2
Ω6 Ω3
+
− V24
(b)
I1
Isc
a
b
Ω6 Ω3
Ω3
A2.8
(c)
a
b
Ω5
(d)
I
a
b
A2.8 Ω5 Ω2
( )6//3 3 5thR = + = Ω
( )1
42
5.6 A
6 3//3
I = =
+
3
5.6 2.8 A
3 3
scI = =
+
5
2.8 2 A
5 2
I = =
+
Department of Electronic Engineering, NTUT23/47

24. III
• (
) Rth
•
Rth Rth
voc Isc
(Driving Point)
v( I ) I(
v)
oc
th
sc
v
R
I
=
th
v
R
I
=
Department of Electronic Engineering, NTUT24/47

25. IV
• (a)
(b)
v I+
−
+ −
Rth
(a)
+ −
+
−
S.C.
O.C.
v
(b) Rth
I
a
b
a
b
th
v
R
I
=
Department of Electronic Engineering, NTUT25/47

26. 7
• 7 (a) a-b
A. isc a-b (b)
+−
6 Ω4 ΩA10
2i1V
i1
a
b
(a)
3 Ω
2 110 sci i i= − −
( )1 1 14 10 2 6 0sci i i i− − − − + =
16 3 0sci i− + =
5 Asci =
A10
+−
V
(b)
4 Ω 6 Ω
3 Ω2i1
i2 i1 isc
Department of Electronic Engineering, NTUT26/47

27. 7
B. Rth
Rth
(c)
(d)
C. a-b
(e)
+−
A10
+
−
a
b(c)
(e)
16ocv i=
( )1 1 16 2 4 10i i i= + −
30 Vocv⇒ =
( )
30
6
5
oc
th
sc
v
R
i
= = = Ω
13 6v i i= +
( )1 1 16 2 4i i i i= + −
( )1 0.5 , 6 , 6th
v
i i v i R
i
= = = = Ω
3 Ω
V2i1i1
4 Ω 6 Ω
i110−i1
voc
(d)
+−
+
−
a
b
v
Rth
i
4 Ω 6 Ω
V2i1
i−i1 i1
3 Ω
a
b
5 A
6 Ω
Department of Electronic Engineering, NTUT27/47

28. 8
• 8 (a)
A. (b)
+−
a
b(a)
6 Ω
V2i1
i1
4 Ω
0, 0oc scv i= =
( )
1
th
v v
R v
i
= = = Ω
12
1
4 6
v i v−
+ =
16v i= ( ) ( )1 0.5 A , 3 Vi v= = ( )3thR = Ω
(b) Rth
+−
b
a
1A
+
−
v
Rth
v
V2i1
v−2i1
6 Ω4 Ω
i1
Department of Electronic Engineering, NTUT28/47

29. 8
(c) Rth
(d)
C. (d)
B. (c)
16v i=
( )1 1 16 2 4i i i i= + −
1 0.5 , 3i i v i= =
( )3th
v
R
i
= = Ω
v
+−
b
a
V
+
−
v
Rth
v
V2i1
i−i1
6 Ω4 Ω
i1
+
−
i
a
b
3 Ω
Department of Electronic Engineering, NTUT29/47

30. I
•
•
• ,
•
s
L
s L
v
i
R R
=
+
( )
2 2
2
2
s s L
L L L
s L s L
v v R
p i R R
R R R R
 
= = ⋅ = 
+ + 
0
L
dp
dR
=
( ) ( )
( )
2 2 2
4
2s L s s L s L
L s L
R R v v R R Rdp
dR R R
+ − +  =
+
( ) ( )
( )
2 2
4
2s L s L s s L
s L
R R R R v v R
R R
 + + − =
+
( )
( )
( )
22 2
3 3
s L ss s L s
s L s L
R R vR v R v
R R R R
−−
= =
+ +
s LR R= 0
L
dp
dR
= s LR R=
( )
2 2
2
4
s L s
Ls L
v R v
p
RR R
= =
+
+
−
Rs
+
−
vs
iL
RLvL
Department of Electronic Engineering, NTUT30/47

31. II
• Rs RL = Rs
RL
•
•
RL=Rth RL
+
−
Rth
+
−
eth RL = Rth
2
max
4
s
L
v
p
R
=
2
max
4
g gR i
p =
2
max
4
th
th
e
p
R
=
Department of Electronic Engineering, NTUT31/47

32. 9
• 9 (a) RL
+ −
30V
RL
6 Ω
3A
8 Ω
20V
1A
4 Ω
2 Ω
+
−20V
(a)
30V
6 Ω
3A8 Ω
20V
1A
4 Ω
2 Ω
20V + −
1A
3A
2A
+ −eth
+
−
(b)
(b)
8 4 2 6 20thR = + + + = Ω
30 20 8 1 4 2 2 2 6 3 72 Vth the e+ + = ⋅ − ⋅ − ⋅ − ⋅ ⇒ = −
( )
22
72
64.8 W
4 4 20
th
m
th
e
p
R
−
= = =
⋅
Department of Electronic Engineering, NTUT32/47

33. • (Millman’s Theorem)
• N a
• N
+
−
v1
+
−
+
−
b
a
+
−
v
v2 vN
R1 R2 RN
1 2
1 2
0N
N
v vv v v v
R R R
−− −
+ + + =⋯
( ) ( ) ( )
( ) ( ) ( )
1 1 2 2
1 21 1 1
N N
oc
N
v R v R v R
v v
R R R
+ + +
= =
+ + +
⋯
⋯ ( ) ( ) ( )1 2
1
1 1 1
th
N
R
R R R
=
+ + +⋯
Department of Electronic Engineering, NTUT33/47

34. 10
• 10
+
−
+
− +
−
4 Ω 4 Ω
3 Ω
2 Ω
24V32V40V
+
−
v
( )
40 32 0 24
184 4 3 2 4.5 V
1 1 1 1 4
4 4 3 2
v
+ + −
= = =
+ + +
Department of Electronic Engineering, NTUT34/47

35. Y ∆ I
• Y ∆
∆
R
Y
R Ry
∆
∆
=
′a ′b
′c
ba
c
R1 R2
R3
RA RC
RB
( ) 1 2//ab a b B A CR R R R R R R′ ′= ⇒ + = +
( ) 2 3//bc b c C B AR R R R R R R′ ′= ⇒ + = +
( ) 3 1//ca c a A B CR R R R R R R′ ′= ⇒ + = +
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
+ +
=
Department of Electronic Engineering, NTUT35/47

36. Y ∆ II
• Y ∆
• Y
Ry
R
R
=
∆
∆
′a ′b
′c
ba
c
R1 R2
R3
RA RC
RB
( ) 1 2//ab a b B A CR R R R R R R′ ′= ⇒ + = +
( ) 2 3//bc b c C B AR R R R R R R′ ′= ⇒ + = +
( ) 3 1//ca c a A B CR R R R R R R′ ′= ⇒ + = +
1
A B
A B C
R R
R
R R R
=
+ +
2
B C
A B C
R R
R
R R R
=
+ +
3
C A
A B C
R R
R
R R R
=
+ +
Department of Electronic Engineering, NTUT36/47

37. 11
• 11 (a) I vab
2 Ω
+
−
vab
4 Ω
4 Ω
2 Ω
3 Ω
5 Ω
I
2.5 Ωd b
+ −
200V
c a3.4 Ω
(a)
∆ (b) Y
+
−
d b
+ −
200V
c a
I1
IT
2.5 Ω
3.4 Ω
1 Ω
0.6 Ω
1.5 Ω
I2
vab
0.8 Ω
16 Ω
0.8 Ω
(b)
( ) ( ) ( )1 0.6 3.4 0.8 // 1.5 2.5 0.8 1.6 5TR = +  + + + +  + = Ω  ( )
200
40 AT
T
I
R
= =
( )1 2 2 20 ATI I I= = = ( )1 20.6 1.5 18 Vcdv I I= − + =
( )6 A
3
cdv
I = = ( )1 20.8 0.8 0 Vabv I I= − =
Department of Electronic Engineering, NTUT37/47

38. I
• Y - ∆ Y ( ) → ∆( )
•
1 2 2 3 3 1 1 2 2 3 3 1 1 2
12 1 2
3 3 1 2 1 2 3
1 1 1R R R R R R R R R R R R R R
R R R
R R R R R R R
  + + + +
= = = + +  
   
1
1,2, ,
1
,where 1,2, ,
N
ij i j
p p
i N
R R R j N
R
i j
=
=
= =
≠
∑
⋯
⋯
∆( )
23
1
23
1
Y( )
R1
R3 R2
R31 R12
R23
Department of Electronic Engineering, NTUT38/47

39. II
• N = 4
(1) (Star to Mesh Conversion)
= ( ) ( )
4 3
1 2
R1 R2
R4 R3
4 3
21
R23
R34
R41
R13
R24
R12
(a) N = 4
( )
(b) (1/2)N(N-1) = 6
( )
23 2 3
1 2 3 4
1 1 1 1
R R R
R R R R
 
= + + + 
 
34 3 4
1 2 3 4
1 1 1 1
R R R
R R R R
 
= + + + 
 
41 4 1
1 2 3 4
1 1 1 1
R R R
R R R R
 
= + + + 
 
13 1 3
1 2 3 4
1 1 1 1
R R R
R R R R
 
= + + + 
 
24 2 4
1 2 3 4
1 1 1 1
R R R
R R R R
 
= + + + 
 
12 1 2
1 2 3 4
1 1 1 1
R R R
R R R R
 
= + + + 
 
Department of Electronic Engineering, NTUT39/47

40. III
(2) (Mesh to Star Conversion)
N
R ( ) = R ( ) / N R ( ) = N R ( )
12 1 2
1 2 3 4
23 2 3
1 2 3 4
24 2 4
1 2 3 4
1 1 1 1
1 1 1 1
1 1 1 1
R R R R
R R R R
R R R R
R R R R
R R R R
R R R R
 
= + + + = 
 
 
= + + + = 
 
 
= + + + = 
 
⋮
12 23 24R R R R= = = =⋯
2312 24
1 2 3 4
4 4 4 4
RR R R
R R R R= = = = = = = =⋯
Department of Electronic Engineering, NTUT40/47

41. •
(
)
D
A B
C
C
D
10 Ω
A B
D
A B
C
(C, D )
(C, D )
Department of Electronic Engineering, NTUT41/47

42. 12
• 12 (a) A-B Req
(a)
C
A
4 Ω 4 Ω
6 Ω
2 Ω2 Ω
B
D10Ω
C
A
4Ω 4Ω
6Ω
2 Ω2 Ω
B
D
C
A
4Ω 4Ω
6Ω
2 Ω2 Ω
B
D
∆
C
A
RA RB
B
D
10 ΩRC
(b)
C
A
4 Ω 4 Ω
6 Ω
2 Ω2 Ω
B
D
10Ω
(C, D ) (C, D )
( )
24
13
eqR = Ω
Department of Electronic Engineering, NTUT42/47

43. 13
• 13 AC Req
E
CD
A B10 Ω
10 Ω12 Ω
12 Ω
E
CD
A B10 Ω
10 Ω12 Ω
12 Ω
E
CD
A
B
10 Ω
10 Ω12 Ω
12 Ω
(D-E, E-B )
(D-E, E-B )
( ) ( ) ( ) ( )
40
12 12 // 6 6 // 10 10
7
eqR = + + + = Ω
( ) ( ) ( )
40
12//6//10 12//6//10
7
eqR = + = Ω
Department of Electronic Engineering, NTUT43/47

44. •
(
)
C
5 Ω
A B
D
(a)
C D,
5 ΩA B
12 Ω
(b)
(C, D )
Department of Electronic Engineering, NTUT44/47

45. 14
• 14 AD
Req
1.
2. BC 10 Ω
AD
Req ( )
B
A
5 Ω 2 Ω
2 Ω
12 Ω
10 Ω12 Ω
C
D
B C,
A D5 Ω
12 Ω
2 Ω
B
A
5 Ω
2 Ω12 Ω
10 Ω
12 Ω
C
D
(a)
( ) ( ) ( )
35
12 12 2//2 //5
12
eqR = + + = Ω  
(B, C )
Department of Electronic Engineering, NTUT45/47

46. 15
• 15 R = 10
AB Req
( )
10 Ω
G
H
BA
D
C
F
E
5 Ω
5 Ω
5 Ω
F
G E× D
B H× C
A
5 Ω
5 Ω
10 Ω
10 Ω
(B-H, G-E )
10 Ω
GF
E D
H C
BA
( ){ } ( )
35
5 10 5 //5 5 5 //10
6
eq eqR R′= = + + + + = Ω  
Department of Electronic Engineering, NTUT46/47

47. •
•
•
• Y ∆
Department of Electronic Engineering, NTUT47/47