E.E. Essential Knowledge Sereies My Online Courses: https://www.byparams.com/courses 電路分析方法

1. Department of Electronic Engineering
National Taipei University of Technology

2. •
•
• (Node Voltage Analysis)
• (Mesh Current Analysis)
Department of Electronic Engineering, NTUT2/26

3. •
N KVL
N =
Department of Electronic Engineering, NTUT
1 2 Nv v v v= + + +⋯
1 1 2 2, , , N Nv R i v R i v R i= = =⋯
1 2 Nv R i R i R i= + + +⋯
1 2 N
v
i
R R R
=
+ + +⋯
1 2
1
N
s N n
n
R R R R R
=
= + + + = ∑⋯
+
+
−
+
−
+
−
v
i
v1
−
vN
v2
R1
R2
RN
3/26

4. •
N
•
Department of Electronic Engineering, NTUT
1 2
1 2, , , N
N
s s s
RR R
v v v v v v
R R R
= = =⋯
1 2 1 2: : : : : :N Nv v v R R R=⋯ ⋯
+
+
−
+
−
+
−
v
i
v1
−
vN
v2
R1
R2
RN
4/26

5. 1
• 1: i v1 v2 v3 ?
Department of Electronic Engineering, NTUT
(a)(a)(a)(a)
+
−
16 Ω
+
−
6 Ω
+
−
8 Ω
+
−
2 Ω 12 Ω 4 Ω
v1
i
′a
b c
′b ′c
a
v2 v3
( )2
30 Vt
e−
(b)(b)(b)(b)
+
−
+
−
+
−
i
a b
′a ′b
2 Ω 12 Ω
16 Ω 4 Ωv1 v2( )2
30 Vt
e− (a) (b)
5/26

6. 1
Department of Electronic Engineering, NTUT
(b)(b)(b)(b)
+
−
+
−
+
−
i
a b
′a ′b
2 Ω 12 Ω
16 Ω 4 Ωv1 v2( )2
30 Vt
e−
(c)(c)(c)(c)
+
−
+
−
i
a
′a
( )2
30 Vt
e−
2 Ω
8 Ω v1
(b) (c)
• (c) KLC
2
30 2 8 0t
e i i−
− + + =
2
3 t
i e−
=
( )2 2
1
8
30 24 V
2 8
t t
v e e− − 
= ⋅ = 
+ 
( )2
2 1
4
6 V
12 4
t
v v e− 
= ⋅ = 
+ 
( )2
3 2
8
4 V
8 4
t
v v e− 
= ⋅ = 
+ 
6/26

7. • :
N i = i1 + i2 + ... + iN
• R1 R2
Department of Electronic Engineering, NTUT
1 1 2 2, , , N Ni G v i G v i G v= = =⋯
1 2 Ni G v G v G v= + + +⋯
1 2 N
i
v
G G G
=
+ + +⋯
11 2
1 1 1 1 1N
np N nR R R R R=
= + + + = ∑⋯
1 2
1 2
1 2
||eq
R R
R R R
R R
= =
+
+
−
vi G1 G2 GN
i1 i2
iN
7/26

8. •
Department of Electronic Engineering, NTUT
1 2
1 2
1 2
, , ,p p pN
N
p p p N
R R RGG G
i i i i i i i i i
G R G R G R
= = = = = =⋯
1 2 1 2
1 2
1 1 1
: : : : : : : : :N N
N
i i i G G G
R R R
= =⋯ ⋯ ⋯
+
−
vi G1 G2 GN
i1 i2
iN
8/26

9. 2
• 2: i1 i2 ?
Department of Electronic Engineering, NTUT
6 Ω3 Ω
6 Ω 3 Ω 4 Ω 6 Ω
a b
a′ b′
(a)(a)(a)(a)
(A)12
i1 i2
3 Ω
3 Ω3 Ω6 ΩA12
a b
a′ b′
(b)(b)(b)(b)
i1
(a) (b)
9/26

10. 2
Department of Electronic Engineering, NTUT
3 Ω
3 Ω3 Ω6 ΩA12
a b
a′ b′
(b)(b)(b)(b)
i1
a
A12
a′
(c)(c)(c)(c)
6 Ω2 Ω
i1
( )1
2
12 3 A
2 6
i
 
= ⋅ = 
+ 
• (c)
( )2 1
4 3
A
4 6 6 4
i i
 
= ⋅ = 
+ + 
• (a)
10/26

11. (Node Voltage Analysis)
• KCL
•
•
va , vb , vc va , vb , vc
Department of Electronic Engineering, NTUT
a : 1 2 1 0si i i+ − = ( )1 2 1 0a b a sG v v G v i− + − =
1 3 4 0i i i− + + = ( ) ( )1 3 4 0a b b b cG v v G v G v v− − + + − =b :
4 5 2 0si i i− + + = ( )4 5 2 0b c c sG v v G v i− − + + =c :
( )1 2 1 10a b sG G v G v i+ − + =
( )1 1 3 4 4 0a b cG v G G G v G v− + + − =
( )4 4 5 20 b c sG v G G v i− + + = −
is1
a b
d
G2
i2 G3
i3
i5
i4i1
G1 G4 c
is2
G5
11/26

12. •
• G
Department of Electronic Engineering, NTUT
v1 . . . vN N ( )
Gjj j j
Gjk = Gkj , j ≠ k ， j k
j k
ij j
G G G
G G G
G G G
v
v
v
i
i
i
N
N
N N NN N N
11 12 1
21 22 2
1 2
1
2
1
2
−−−− −−−−
−−−− −−−−
−−−− −−−−
























====












......
......
......
......
...
...
...
...
...
12/26

13. 3
• 3: v1 , v2 , v3 , i1 , i2 , i3 , i4
Department of Electronic Engineering, NTUT
Cramer’s rule
A2
3 A
v1
i22
Ω
1
Ω
4
Ω
v2 v3
i3i4
i1
11 3G = ℧ 22 5G = ℧ 33 5G = ℧
12 21 1G G= = ℧
13 31 2G G= = ℧
23 32 2G G= = ℧
1
2
3
3 1 2 2
1 5 0 3
2 0 5 3
v
v
v
− − −    
    − =    
    − −    
3 1 2
1 5 0 50
2 0 5
− −
∆ = − =
−
2
2 1 2
3 5 0 65
3 0 5
− − −
∆ = = −
−
3
3 2 2
1 3 0 17
2 3 5
− −
∆ = − =
− −
4
3 1 2
1 5 3 56
2 0 3
− −
∆ = − = −
− −
1
1 1.3 Vv
∆
= = −
∆
2
2 0.34 Vv
∆
= =
∆
3
3 1.12 Vv
∆
= = −
∆
( )1 1 21 1.64 Ai v v= − = −
( )2 1 21 0.36 Ai v v= − =
3 3.36 Ai = −
4 1.36 Ai =
13/26
Ω
3

14. 3
•
Department of Electronic Engineering, NTUT
= 3 × 5 × 5 + (-1) × 0 × (-2) + (-1) × 0 × (-2)
−(-2) × 5 × (-2) − 0 × 0 × 3 − (-1) × (-1) × 5
= 75 + 0 + 0 − 20 − 0 − 5 = 50
502
051
213
−
−
−−
=∆
(4) 3 1 2
1 5 0
2 0 5
2
3
3
1
2
3
−−−− −−−−
−−−−
−−−−




















====
−−−−
−−−−










v
v
v
(1) For node 1：
2+1× (v1 v2)+2×(v1 v3) = 0
⇒ (1+2) × v1-1×v2 2 × v3 = 2
⇒ G11v1+G12v2+G13 = vs1
(2) For node 2：
1× (v1 v2) + 4 × (v2 0) 3= 0
⇒ −1× v1+(1+4)v2+ 0 × v3 = 3
⇒ G21v1+G22v2+G23v3 = vs2
(3) For node 3：
2× (v3 v1) + 3 × (v3 0) + 3= 0
⇒ −2× v1+ 0 × v2 +(2+3)v3 = 3
⇒ G31v1+G32v2+G33v3 = vs3
A2
3 A
v1
i22
Ω
1
Ω
4
Ω
v2 v3
i3i4
i1
14/26
Ω
3

15. •
M (M-1)
(M-1) (M-1)
(Supernode)
KCL
Department of Electronic Engineering, NTUT15/26

16. • v1 v2 v3 v4 v5 5
v1 = vs1 , v5 – v4 = vs2
v2 v3 v4
KCL
Department of Electronic Engineering, NTUT
2
3
( )1 2 4 2 1 1 2 3 4 5 0G G G v G v G v G v+ + − − − =
( )2 3 5 3 2 2 5 4 0G G G v G v G v+ + − − =
( ) ( )4 5 2 5 4 3 6 5 0G v v G v v G v− + − + =
+
−
1 2
3
4
5vs1
G1
G4
G6
G5
G3
G2 vs2
16/26

17. •
KVL
KCL
KCL
1 2 3 KCL ( )
Department of Electronic Engineering, NTUT
+
−
1 2
3
44 sv v=
( )1 2 3 1 2 2 3 3 1 0sG G G v G v G v G v+ + − − − =
( ) ( )2 1 2 5 2 1 3 0G v G G v v vβ− + + + − =
( ) ( )3 1 3 4 3 1 3 0G v G G v v vβ− + + − − =
vs
G1 G2
G3
G4
( )1 3v vβ −
G5
17/26

18. 4
• 4: v2 , v3 , v4 v1 = 100 V
Department of Electronic Engineering, NTUT
3
(a) 2
(b) 3
(c) 4
( ) ( )1 2 3 2 2 2 41 100 2 4i i i v v v v= + ⇒ − = + −
3 22v v= −
( ) ( ) ( )3 4 5 2 4 3 4 40 4 4 4 0 0i i i v v v v v+ + = ⇒ − + − + − =
2 2
3 3
4 4
7 0 4 100 12 V
2 1 0 0 24 V
1 1 3 0 4 V
v v
v v
v v
− =      
     = ⇒ = −     
     − = −      
+
−
+
−
1
Ω
4
32
V100
4v1
Ω
1
Ω
4
Ω
4
Ω
2
2v2
v3
v4
i5i1 i2
v2
i3
18/26

19. (Mesh Current Analysis)
•
KVL
IR
KVL
• 2 2
Ia Ib KVL
Department of Electronic Engineering, NTUT
1 (a-b-e-f-a) ⇒
2 (b-c-d-e-b) ⇒
+
−
+
−
vg1
a b c
def
I1
vg2
I2
I3
R1
R3
R2
Ia Ib
1 1 1 3 3 0gv I R I R− + + =
3 3 2 2 2 0gI R I R v− + + =
1 1 3 3 1gI R I R v+ =
2 2 3 3 2gI R I R v+ =
19/26

20. •
Department of Electronic Engineering, NTUT
b
+
−
+
−
vg1
a b c
def
I1
vg2
I2
I3
R1
R3
R2
Ia Ib
3 1 2I I I= − 1 2,a bI I I I= =
3 a bI I I= −
( )
( )
1 3 3 1
3 2 3 2
a b g
a b g
R R I R I v
R I R R I v
 + − =

− + + = −
11 3 3
23 2 3
ga
gb
vR R R I
vR R R I
+ −     
=      −− +     
20/26

21. •
Department of Electronic Engineering, NTUT
M
v1 j
i1 j
Rjj j
Rjk=Rk j j k
j k ij , ik
Rjk( Rk j) ij , ik Rjk( Rk j) 。
+
−
+
−
vg1
a b c
def
I1
vg2
I2
I3
R1
R3
R2
Ia Ib
11 12 1 1 1
21 22 2 2 2
1 2 3
M
M
M M M MM M M
R R R i v
R R R i v
R R R R i v
± ±     
     ± ±
     =
     
     
± ± ±     
⋯
⋯
⋮ ⋮ ⋱ ⋮ ⋮ ⋮
21/26

22. •
Ia , Ib R3
R3 R3 Ia Ib I3 = Ia + Ib
Department of Electronic Engineering, NTUT
+
−
+
−
vg1
a b c
def
I1
vg2
I2
I3
R1
R3
R2
Ia Ib
11 3 3
23 2 3
ga
gb
vR R R I
vR R R I
+     
=     +     
22/26

23. •
KVL
Department of Electronic Engineering, NTUT23/26

24. 5
• 5: i1 , i2 , i3
Department of Electronic Engineering, NTUT
KVL
KVL
+
−
V2
2 Ω 1 Ω 3 Ω
2 Ω
+
−
+ −
A2
+
−
+
−
3 Ω
+ −+ −
v1 v3 v5
v3v2 v6i1 i2 i3
( )1 1 22 3 2i i i+ − =
( ) ( )1 2 2 3
3 2
3 3 2
2 A
i i i i
i i
 − = + +

− =
( ) ( ) ( )
1
2 1 2 3
3
5 3 0 2
1 11 7
3 4 5 0 A , A , A
3 9 9
0 1 1 2
i
i i i i
i
−     
− −    − = ⇒ = = =    
    −    
24/26

25. 6
• 6: i1 , i2 , i3
Department of Electronic Engineering, NTUT
(a) 1: KVL
i1 – i3 = 1
(b) 2: KVL
i1 – i2 = 2v3 = 2(3i2)
(c) 3: KVL 2 3 4
KVL 2i1 + 3i2 + 4i3 = 0
4 Ω
+ −
A1
+
−
+
−
+ −
+ −
v4
2 Ω 3 Ω
v5
v2
v1
v3
1 Ω
i3
i1 i2
2v3 A
( ) ( ) ( )
1
2 1 2 3
3
1 0 1 1
28 4 17
1 7 0 0 A , A , A
45 45 45
2 3 4 0
i
i i i i
i
−     
−    − = ⇒ = = =    
        
25/26

26. •
(KCL)
(KVL)
•
•
( ) ( )
Department of Electronic Engineering, NTUT26/26