Chapter 4
Basic Nodal and Mesh
Analysis
4.1 Nodal Analysis
4.2 The Supernode
4.3 Mesh Analysis
4.4 The Supermesh
4.5 Nodal vs. Mesh Analysis: A Comparison
4.6 Computer-aided Circuit Analysis
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4.1 Nodal Analysis
- Method
•
•
•
•
We select one node in the circuit to be a reference node.
Node voltage = The voltage of each of the nonreference nodes with respect to the
reference node.
Use Node voltages as unknown variables to set up Independent KCL equations
(node equations). Find node voltages.
We can find all branch voltages and currents if we know node voltages
3.1
2
→ 3.1
5
2
1.4
→ 1.4
1
1
5 ,
0
5
5
0
5
2
3
Nodal analysis is based on KCL
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4.1 Nodal Analysis
Example 4.1 Determine the current through 15 resistor
Node 1: 2
→5
Node 2: 4
2
→
20 ,
60
4
60
0
20
Example 4.2 Determine the nodal voltages as referenced to the bottom node
Node 1:
→
8 3
0.5833
0.3333
11
Node 2:
→
3
0.3333
1.4762
0.1429
Node 3:
→
25
0.25
5,412 ,
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0.1429
0.5929
7.736 ,
3
25
46.32
3
4.1 Nodal Analysis
Example 4.3 Determine the power supplied by the dependent source
Node 1: 15
→ 3
2
30
2
Node 2: 3
→ 3
40 75 20
→
3
3
4500
20
15
8
0
75
Example 4.4 Determine the power supplied by the dependent source
Node 1: 15
→ 3
Node x : 3
→ 5
50
7
30
7
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2
30
3
0
3
55.1
4
4.2 The Supernode
To solve the circuit (a) on the left side, a branch
current between v2 and v3 should be assigned.
 Four equations and four unknowns
- Supernode
•
•
•
A closed region which encloses each
voltage source
KCL is available to the supernode
All current flowing into the region sum to
zero
Example 4.5 Determine the node voltage v1
Node 1 : 8 3
→ 0.5833
0.3333
11
Supernode 2-3 : 3
→ 0.5833
25
1.3333
0.45
22
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1.071
5
4.2 The Supernode
Example 4.6 Determine the node-to-reference voltages
12
Node 2 : 14
→ 2
→ 2.5
.
2.5
0.5
0.5
Supernode 3-4 : 0.5
→ 0.1
→
0.5
14
10
0.5
1.4
1.4
1.2
1.2
2.4
.
0
0.2
0.2
→ 0.2
∴
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1.2
12 ,
0.2
0 →
4 ,
0 ,
2
6
4.3 Mesh Analysis
•
Loop and mesh
•
•
•
•
•
•
Loop : Any closed path in the circuit
Mesh : A loop that does not contain any other loops within it
Mesh Current = The current that flows only around the perimeter of the
mesh. (Independent loop current)
Use Mesh Current as unknown variables to set up Independent KVL equations
(mesh equations)
We can find all branch voltages and currents if we know mesh currents.
We restrict attention only to planar circuits (No branch passes over or under
any other branch)
Non-planar circuit
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4.3 Mesh Analysis
Loop and Mesh
No path, no loop
Loop, No mesh
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No path, no loop
Loop, No mesh
Loop and Mesh
Loop and Mesh
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4.3 Mesh Analysis
Loop 1:
42
6
3
Loop 2: 3
0 → 9
4
6 ,
Loop 1:
0 → 3
6
3
0 → 9
3
Loop 2: 10
4
3
0 → 3
7
3
Loop 2: 10
4
6 ,
42
10
4 ,
6
3
42
10
2
42
Loop 1:
42
7
4 ,
6 ,
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10
3
0 → 9
0 → 3
4 ,
2
3
42
7
10
2
9
4.3 Mesh Analysis
Example 4.7 Determine the power supplied by the 2 V source
Loop 1:
Loop 2: 2
5
4
2
2
2
5
1
43
,
38
2
,
19
0 → 6
2
0 → 2
7
2
47
38
2
7
3
2.474
Example 4.8 Determine the three mesh currents
Loop 1 : 7
→ 3
Loop 2 : 1
→
1
2
6
Loop 3 : 2
→ 2
2
3
6
3
3 ,
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6
1
2
0
3
0
0
3
6
2 ,
0
6
3
10
4.3 Mesh Analysis
Example 4.9 Determine the current i1
Loop 1 : 5 4
4
→ 8
8
5
Loop 2 : 2
→ 6
3
4
4
4
0
0
3
,
Example 4.10 Determine the current i1
Loop 1: 5 2
4
5 2 4
5
→4
Loop 2 : 2
→ 6
3
4
4
4
4
0
4
0
0
3
5
4
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4.4 Supermesh
•
•
Circuits with current sources
The current sources reduce the # of simultaneous mesh equations
Example 4.11
Using conventional method
Loop 1:
7
Loop 2: 2
→
1
0 → 3
1
6
Loop 3:
7
0
3
0
3
0 → 3
2 ,
7
9 ,
4
15
,
6
0
0.5
Using supermesh
Loop 1:
7
→
Loop 2: 1
→
7
1
4
4
2
6
3
7
1
3
0
3
0
0
2 ,
9 ,
15
6
3 equations and 3 unknowns
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4.4 Supermesh
Example 4.12 Evaluate the three unknown currents
3
15
15
9
1
3
1
3
→ →
6
11 ,
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1
3
2
1
3
3
9
2
3
3
0 → 6
15
0
3
15
17
13
4.5 Nodal vs. Mesh analysis: A Comparison
- In choosing nodal or mesh analysis,
•
•
•
•
•
•
Non-planar circuit  Nodal analysis
N nodes  N-1 KCL equations (Each supernode reduce the # of equations by 1)
M meshes  M KVL equations (Each supermesh reduce this number by 1)
Controlling quantity in dependent sources : nodal voltage  nodal analysis
mesh current  mesh analysis
Location of source : current source in mesh line  mesh analysis
voltage source connected to ref.  nodal analysis
Asking current  mesh analysis, asking voltage  nodal analysis
Find ix
Node 1:
→ 0.875
Node 2:
→
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20.31
12.5
8
0.5
Node 3: 8
→ 0.1
25.89 ,
0.5
0.933
0.1
0.3
8
2
8
2.79
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4.5 Nodal vs. Mesh analysis: A Comparison
Loop 1:
100
→ 12
Loop 2: 2
→
8
4
4
100
3
4
4
9
Loop 3: 3
→
0
3
3
10
18
0
0
8
80
5
0
2.79
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4.6 Computer-aided Circuit Analysis
-
To analyze a complex circuit, a computer software package is useful
SPICE (Simulation Program with Integrated Circuit Emphasis)
Pspice : SPICE with intuitive graphical interface by MicroSim Corp.
Loop 1:
42
6
Loop 2: 3
3
0 → 9
4
6 ,
10
3
0 → 3
4 ,
42
7
10
2
Hand analysis using KCL, KVL, and ohm’s law
Orcad schematic capture software
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Circuit after simulation run
16
4.6 Computer-aided Circuit Analysis
-
Circuit simulation example with dependent source
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4.6 Computer-aided Circuit Analysis
-
Node-based Pspice schematic creation (Text-based input format)
Input Deck
Output Deck
Homework : 4장 Exercises 4의 배수 문제 (57번 문제까지)
-
Due day : 4장 수업 끝나고 일주일 후 수업시작 전까지 제출.
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이 자료에 나오는 모든 그림은 McGraw·hill 출판사로부터 제공받은 그림을 사
용하였음.
All figures at this slide file are provided from The McGraw-hill company.
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