Methods of AnalysisMethods of Analysis Eastern Mediterranean UniversityEastern Mediterranean University 11
Methods of Analysis
Mustafa Kemal Uyguroğlu
Methods of AnalysisMethods of Analysis
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• Introduction
• Nodal analysis
• Nodal analysis with voltage source
• Mesh analysis
• Mesh analysis with current source
• Nodal and mesh analyses by inspection
• Nodal versus mesh analysis
3.2 Nodal Analysis3.2 Nodal Analysis
Steps to Determine Node Voltages:
1. Select a node as the reference node. Assign voltage v1, v2, …vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.
2. Apply KCL to each of the n-1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.
3. Solve the resulting simultaneous equations to obtain the unknown node voltages.
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Figure 3.1Figure 3.1
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Common symbols for indicating a reference node, (a) common ground, (b) ground, (c) chassis.
Figure 3.2Figure 3.2
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Typical circuit for nodal analysis
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322
2121
iiI
iiII
R
vvi lowerhigher
2333
23
21222
212
1111
11
or 0
)(or
or 0
vGiRv
i
vvGiRvv
i
vGiRv
i
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3
2
2
212
2
21
1
121
Rv
Rvv
I
Rvv
Rv
II
232122
2121121
)(
)(
vGvvGI
vvGvGII
2
21
2
1
322
221
III
vv
GGGGGG
Example 3.1Example 3.1
Calculus the node voltage in the circuit shown in Fig. 3.3(a)
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Example 3.1Example 3.1
At node 1
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2
0
45 121
321
vvv
iii
Example 3.1Example 3.1
At node 2
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6
0
45 212
5142
vvv
iiii
Example 3.1Example 3.1
In matrix form:
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5
5
4
1
6
1
4
14
1
4
1
2
1
2
1
v
v
Practice Problem 3.1Practice Problem 3.1
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Fig 3.4
Example 3.2Example 3.2
Determine the voltage at the nodes in Fig. 3.5(a)
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Example 3.2Example 3.2
At node 1,
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243
3
2131
1
vvvv
ii x
Example 3.2Example 3.2
At node 2
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4
0
8223221
32
vvvvv
iiix
Example 3.2Example 3.2
At node 3
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2
)(2
84
2
213231
21
vvvvvv
iii x
Example 3.2Example 3.2
In matrix form:
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0
0
3
8
3
8
9
4
38
1
8
7
2
14
1
2
1
4
3
3
2
1
v
v
v
3.3 3.3 Nodal Analysis with Voltage SourcesNodal Analysis with Voltage Sources
Case 1: The voltage source is connected between a nonreference node and the reference node: The nonreference node voltage is equal to the magnitude of voltage source and the number of unknown nonreference nodes is reduced by one.
Case 2: The voltage source is connected between two nonreferenced nodes: a generalized node (supernode) is formed.
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3.3 Nodal Analysis with Voltage Sources3.3 Nodal Analysis with Voltage Sources
56
0
8
0
42
32
323121
3241
vv
vvvvvv
iiii
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Fig. 3.7 A circuit with a supernode.
A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.
The required two equations for regulating the two nonreference node voltages are obtained by the KCL of the supernode and the relationship of node voltages due to the voltage source.
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Example 3.3Example 3.3
For the circuit shown in Fig. 3.9, find the node voltages.
2
042
72
02172
21
21
vv
vv
ii
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i1 i2
Example 3.4Example 3.4
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Find the node voltages in the circuit of Fig. 3.12.
Example 3.4Example 3.4
At suopernode 1-2,
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2023
106
21
14123
vv
vvvvv
Example 3.4Example 3.4
At supernode 3-4,
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)(34163
4143
342341
vvvv
vvvvvv
3.4 Mesh Analysis3.4 Mesh Analysis
Mesh analysis: another procedure for analyzing circuits, applicable to planar circuit.
A Mesh is a loop which does not contain any other loops within it
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Fig. 3.15Fig. 3.15
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(a) A Planar circuit with crossing branches,(b) The same circuit redrawn with no crossing branches.
Fig. 3.16Fig. 3.16
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A nonplanar circuit.
Steps to Determine Mesh Currents:
1. Assign mesh currents i1, i2, .., in to the n meshes.
2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.
3. Solve the resulting n simultaneous equations to get the mesh currents.
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Fig. 3.17Fig. 3.17
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A circuit with two meshes.
Apply KVL to each mesh. For mesh 1,
For mesh 2,
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123131
213111
)(
0)(
ViRiRR
iiRiRV
223213
123222
)(
0)(
ViRRiR
iiRViR
Solve for the mesh currents.
Use i for a mesh current and I for a branch current. It’s evident from Fig. 3.17 that
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2
1
2
1
323
331
VV
ii
RRRRRR
2132211 , , iiIiIiI
Example 3.5Example 3.5
Find the branch current I1, I2, and I3 using mesh analysis.
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Example 3.5Example 3.5
For mesh 1,
For mesh 2,
We can find i1 and i2 by substitution method or Cramer’s rule. Then,
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123
010)(10515
21
211
ii
iii
12
010)(1046
21
1222
ii
iiii
2132211 , , iiIiIiI
Example 3.6Example 3.6
Use mesh analysis to find the current I0 in the circuit of Fig. 3.20.
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Example 3.6Example 3.6
Apply KVL to each mesh. For mesh 1,
For mesh 2,
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126511
0)(12)(1024
321
3121
iii
iiii
02195
0)(10)(424
321
12322
iii
iiiii
Example 3.6Example 3.6
For mesh 3,
In matrix from Eqs. (3.6.1) to (3.6.3) become
we can calculus i1, i2 and i3 by Cramer’s rule, and find I0.
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02
0)(4)(12)(4
, A, nodeAt
0)(4)(124
321
231321
210
23130
iii
iiiiii
iII
iiiiI
00
12
21121956511
3
2
1
iii
3.5 Mesh Analysis with Current Sources 3.5 Mesh Analysis with Current Sources
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Fig. 3.22 A circuit with a current source.
Case 1
● Current source exist only in one mesh
● One mesh variable is reduced
Case 2
● Current source exists between two meshes, a super-mesh is obtained.
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A21 i
Fig. 3.23Fig. 3.23
a supermesh results when two meshes have a (dependent , independent) current source in common.
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Properties of a SupermeshProperties of a Supermesh
1. The current is not completely ignored
● provides the constraint equation necessary to solve for the mesh current.
2. A supermesh has no current of its own.
3. Several current sources in adjacency form a bigger supermesh.
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Example 3.7Example 3.7
For the circuit in Fig. 3.24, find i1 to i4 using mesh analysis.
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If a supermesh consists of two meshes, two equations are needed; one is obtained using KVL and Ohm’s law to the supermesh and the other is obtained by relation regulated due to the current source. 6
20146
21
21
ii
ii
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Similarly, a supermesh formed from three meshes needs three equations: one is from the supermesh and the other two equations are obtained from the two current sources.
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0102)(8
5
06)(842
443
432
21
24331
iii
iii
ii
iiiii
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3.6 Nodal and Mesh Analysis by 3.6 Nodal and Mesh Analysis by Inspection Inspection
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(a)For circuits with only resistors and independent current sources
(b)For planar circuits with only resistors and independent voltage sources
The analysis equations can be obtained by direct inspection
In the Fig. 3.26 (a), the circuit has two nonreference nodes and the node equations
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2
21
2
1
322
221
232122
2121121
)8.3()(
)7.3()(
III
vv
GGGGGG
MATRIX
vGvvGI
vvGvGII
In general, the node voltage equations in terms of the conductances is
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NNNNNN
N
N
i
i
i
v
v
v
GGG
GGG
GGG
2
1
2
1
21
22221
11211or simply
Gv = i
where G : the conductance matrix, v : the output vector, i : the input vector
The circuit has two nonreference nodes and the node equations were derived as
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2
1
2
1
323
331 vv
ii
RRRRRR
In general, if the circuit has N meshes, the mesh-current equations as the resistances term is
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NNNNNN
N
N
v
v
v
i
i
i
RRR
RRR
RRR
2
1
2
1
21
22221
11211
or simply
Rv = i
where R : the resistance matrix, i : the output vector, v : the input vector
Example 3.8Example 3.8
Write the node voltage matrix equations in Fig.3.27.
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Example 3.8Example 3.8
The circuit has 4 nonreference nodes, so
The off-diagonal terms are
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625.111
21
81
,5.041
81
81
325.111
81
51
,3.0101
51
4433
2211
GG
GG
125.0 ,1 ,0
125.0 ,125.0 ,0
111
,125.081
,2.0
0 ,2.051
434241
343231
242321
141312
GGG
GGG
GGG
GGG
Example 3.8Example 3.8
The input current vector i in amperes
The node-voltage equations are
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642 ,0 ,321 ,3 4321 iiii
6
0
3
3
.6251 0.125 1 0
0.125 .50 0.125 0
1 0.125 .3251 0.2
0 0 0.2 .30
4
3
2
1
v
v
v
v
Example 3.9Example 3.9
Write the mesh current equations in Fig.3.27.
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Example 3.9Example 3.9
The input voltage vector v in volts
The mesh-current equations are
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6 ,0 ,6612
,6410 ,4
543
21
vvv
vv
606
64
4 3 0 1 0
3 8 0 1 0
0 0 9 4 2
1 1 4 01 2
0 0 2 2 9
5
4
3
2
1
i
i
i
i
i
3.7 Nodal Versus Mesh Analysis3.7 Nodal Versus Mesh Analysis
Both nodal and mesh analyses provide a systematic way of analyzing a complex network.
The choice of the better method dictated by two factors.
● First factor : nature of the particular network. The key is to select the method that results in the smaller number of equations.
● Second factor : information required.
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BJT Circuit ModelsBJT Circuit Models
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(a)An npn transistor,(b) dc equivalent model.
Example 3.13Example 3.13
For the BJT circuit in Fig.3.43, =150 and VBE = 0.7 V. Find v0.
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Example 3.13Example 3.13
Use mesh analysis or nodal analysis
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Example 3.13Example 3.13
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3.10 Summery3.10 Summery
1. Nodal analysis: the application of KCL at the nonreference nodes
● A circuit has fewer node equations
2. A supernode: two nonreference nodes
3. Mesh analysis: the application of KVL
● A circuit has fewer mesh equations
4. A supermesh: two meshes
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HomeworkHomework
Problems 7, 12, 20, 31(write down required equations only), 39, 49, 53(write down required equations only)
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