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EENG223: CIRCUIT THEORY I
DC Circuits:
Methods of Analysis Hasan Demirel
EENG223: CIRCUIT THEORY I
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 non-reference 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.
EENG223: CIRCUIT THEORY I
Nodal Analysis • Steps to Determine Node Voltages:
Common symbols for indicating a reference node:
(a) common ground, (b) ground, (c) chassis.
EENG223: CIRCUIT THEORY I
Nodal Analysis • Typical circuit for nodal analysis
(a) Given circuit, (c) voltages v1 and v2 are assigned with respect to the reference node (i.e ground).
Reference Node
EENG223: CIRCUIT THEORY I
Nodal Analysis • Typical circuit for nodal analysis:
Apply KCL to each non-reference node.
233
3
23
2122
2
212
111
1
11
or 0
)(or
or 0
vGiR
vi
vvGiR
vvi
vGiR
vi
1 2 1 2
2 2 3
I I i i
I i i
R
vvi
lowerhigher
EENG223: CIRCUIT THEORY I
Nodal Analysis • Typical circuit for nodal analysis:
Apply KCL to each nonreference node.
3
2
2
212
2
21
1
121
R
v
R
vvI
R
vv
R
vII
232122
2121121
)(
)(
vGvvGI
vvGvGII
2
21
2
1
322
221
I
II
v
v
GGG
GGG
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.1: Calculate the node voltages in the
circuit shown below.
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.1: Calculate the node voltages in the
circuit shown below.
• At node 1:
203
220
2
0
45
21
121
121
321
vv
vvv
vvv
iii
(Multiply each term by 4)
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.1: Calculate the node voltages in the
circuit shown below.
• At node 2:
2 4 1 5
2
5 2
2 1 2
2 2 1
10 5
5
05
6 4
56 4
i i i i
i i
i i
v v v
v v v
(Multiply each term by 12)
6053
33260
21
122
vv
vvv
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.1: Calculate the node voltages in the
circuit shown below.
)2(6053:2
)1(203:1
21
21
vvnode
vvnode
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.1: Calculate the node voltages in the
circuit shown below. )2(6053:2
)1(203:1
21
21
vvnode
vvnode
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example: Calculate the node voltages in the circuit
shown below.
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.2: Determine the voltages at nodes 1, 2
and 3.
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.2: Determine the voltages at nodes 1, 2 and 3.
• At node 1:
243
3
2131
1
vvvv
ii x
(Multiplying by 4 and rearranging terms)
1223 321 vvv
EENG223: CIRCUIT THEORY I
Nodal Analysis
4
0
82
23221
32
vvvvv
iiix
• Example 3.2: Determine the voltages at nodes 1, 2 and 3.
• At node 2:
(Multiplying by 8 and rearranging terms)
074 321 vvv
EENG223: CIRCUIT THEORY I
Nodal Analysis
2
)(2
84
2
213231
21
vvvvvv
iii x
• Example 3.2: Determine the voltages at nodes 1, 2 and 3.
• At node 3:
(Multiplying by 8 and rearranging terms)
032 321 vvv
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.2: Determine the voltages at nodes 1, 2 and 3.
EENG223: CIRCUIT THEORY I
Nodal Analysis • Example 3.2: Determine the voltages at nodes 1, 2 and 3.
EENG223: CIRCUIT THEORY I
Nodal Analysis: Cramer's Rule
EENG223: CIRCUIT THEORY I
Nodal 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 nonreference nodes: a generalized node (supernode) is formed.
EENG223: CIRCUIT THEORY I
Nodal Analysis with Voltage Sources • A circuit with a supernode.
EENG223: CIRCUIT THEORY I
Nodal Analysis: 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.
EENG223: CIRCUIT THEORY I
Nodal Analysis: Supernode • Find the node voltages of the circuit below.
i1 i2
1 2
1 2 1 2
1 2
2 7 0
2 7 0 5 2 202 4 2 4
i i
v v v vv v
1 2 2v v
1 2
1 2
2 20
2
v v
v v
1 1
2
223 22 = 7.33V
3
22 16 +2= = 5.33V
3 3
v v
v
EENG223: CIRCUIT THEORY I
Nodal Analysis: Supernode • Find the node voltages of the circuit below.
EENG223: CIRCUIT THEORY I
Nodal Analysis: Supernode • Find the node voltages of the circuit below.
• Apply KCL to the two supernodes.
At supernode 1-2:
EENG223: CIRCUIT THEORY I
Nodal Analysis: Supernode • Find the node voltages of the circuit below.
• Apply KCL to the two supernodes.
At supernode 3-4:
EENG223: CIRCUIT THEORY I
Nodal Analysis: Supernode • Find the node voltages of the circuit below.
• Apply KVL around the loops:
Loop1:
Loop2:
Loop3:
EENG223: CIRCUIT THEORY I
Nodal Analysis: Supernode • Find the node voltages of the circuit below.
• In matrix form:
0
20
0
60
2103
0011
16524
2115
4
3
2
1
v
v
v
v
(KCL from supernode 1-2)
(KCL from supernode 3-4)
(KVL from loop1)
(KVL from loop2)
(KVL from loop3) Only 4 equations are enough (ignore the 5th equation)
>> A=[5 1 -1 -2 4 2 -5 -16 1 -1 0 0 3 0 -1 -2]; >> B= [60 0 20 0]'; >> V=inv(A)*B V = 26.6667 6.6667 173.3333 -46.6667
EENG223: CIRCUIT THEORY I
Nodal Analysis: Supernode • Find the node voltages of the circuit below.
• You can use Matlab to solve large matrix equations:
0
20
0
60
2103
0011
16524
2115
4
3
2
1
v
v
v
v
>> A=[5 1 -1 -2 4 2 -5 -16 1 -1 0 0 3 0 -1 -2]; >> B= [60 0 20 0]'; >> V=inv(A)*B V = 26.6667 6.6667 173.3333 -46.6667
V67.46
V33.173
V67.6
V67.26
4
3
2
1
v
v
v
v
We now use MATLAB to solve the matrix Equation. The Equation on the left can be written as AV =B → V= B/A= A-1B
Matlab Code
EENG223: CIRCUIT THEORY I
Mesh Analysis 1. Mesh analysis: another procedure for analyzing circuits,
applicable to planar circuits. 2. A Mesh is a loop which does not contain any other loops
within it. 3. Nodal analysis applies KCL to find voltages in a given circuit,
while Mesh Analysis applies KVL to calculate unknown currents.
EENG223: CIRCUIT THEORY I
Mesh Analysis • A circuit is planar if it can be drawn on a plane with no branches crossing
one another. Otherwise it is nonplanar. • The circuit in (a) is planar, because the same circuit that is redrawn(b) has
no crossing branches.
EENG223: CIRCUIT THEORY I
Mesh Analysis • A nonplanar circuit.
EENG223: CIRCUIT THEORY I
Mesh Analysis
• 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.
EENG223: CIRCUIT THEORY I
Mesh Analysis
• A circuit with two meshes.
EENG223: CIRCUIT THEORY I
Mesh Analysis
• A circuit with two meshes.
• Apply KVL to each mesh. For mesh 1,
• For mesh 2,
123131
213111
)(
0)(
ViRiRR
iiRiRV
223213
123222
)(
0)(
ViRRiR
iiRViR
EENG223: CIRCUIT THEORY I
Mesh Analysis
• A circuit with two meshes.
• Solve for the mesh currents.
• Use i for a mesh current and I for a branch current. It’s evident from the circuit that:
2
1
2
1
323
331
V
V
i
i
RRR
RRR
2132211 , , iiIiIiI
EENG223: CIRCUIT THEORY I
Mesh Analysis: Example 3.5
• A circuit Find the branch currents I1, I2, and I3 using mesh analysis.
EENG223: CIRCUIT THEORY I
Mesh Analysis: Example 3.5
• For mesh 1,
• For mesh 2,
• We can find i1 and i2 by substitution method or Cramer’s rule. Then,
123
010)(10515
21
211
ii
iii
12
010)(1046
21
1222
ii
iiii
2132211 , , iiIiIiI
EENG223: CIRCUIT THEORY I
Mesh Analysis: Example 3.6
• Use mesh analysis to find the current Io in the circuit below.
EENG223: CIRCUIT THEORY I
Mesh Analysis: Example 3.6
• Apply KVL to each mesh. For mesh 1,
• For mesh 2,
126511
0)(12)(1024
321
3121
iii
iiii
02195
0)(10)(424
321
12322
iii
iiiii
EENG223: CIRCUIT THEORY I
Mesh Analysis: Example 3.6
• For mesh 3,
• In matrix from:
• we can calculate i1, i2 and i3 by Cramer’s rule, and find I0.
02
0)(4)(12)(4
, A, nodeAt
0)(4)(124
321
231321
210
23130
iii
iiiiii
iII
iiiiI
0
0
12
211
2195
6511
3
2
1
i
i
i
EENG223: CIRCUIT THEORY I
Mesh Analysis: with Current Source
• Consider the following circuit with a current source.
EENG223: CIRCUIT THEORY I
Mesh Analysis: with Current Source
• Possibe cases of having a current source.
Case 1
• Current source exist only in one mesh
mesh 2:
mesh 1:
• One mesh variable is reduced
Case 2
• Current source exists between two meshes, a super-mesh is obtained.
EENG223: CIRCUIT THEORY I
Mesh Analysis: with Current Source
• Possibe cases of having a current source.
• A supermesh is considerred when two meshes have a (dependent/independent) current source in common.
EENG223: CIRCUIT THEORY I
Mesh Analysis: with Current Source
• Applying KVL in the supermesh below:
• KVL in the supermesh above: • Apply KCL at node 0 above:
EENG223: CIRCUIT THEORY I
Mesh Analysis: with Current Source
• Properties 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.
4. A supermesh requires the application of both KVL and KCL.
EENG223: CIRCUIT THEORY I
Mesh Analysis: with Current Source • Apply KVL in the supermesh (mesh 1 + mesh 2 + mesh 3):
• Apply KVL in mesh 4:
• Apply KCL at node P:
• Apply KCL at node Q:
0
5
5
0
3110
0011
5400
4631
4
3
2
1
i
i
i
i
• 4 equations for 4 variables. Using Cramer’s Rule
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • The analysis equations can be obtained by direct
inspection
a) For circuits with only resistors and independent current sources. (nodal analysis)
b) For planar circuits with only resistors and independent voltage sources. (mesh analysis)
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection a) For circuits with only resistors and independent
current sources. (nodal analysis).
• The circuit has two nonreference nodes and the node equations:
• In Matrix form:
• Consider the following example:
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • In general, if a circuit with independent current sources has N
nonreference nodes, the node-voltage equations can be written in terms of conductances as:
• G is called the conductance matrix, v is the output vector, and i is the input vector.
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection b) For planar circuits with only resistors and independent
voltage sources. (mesh analysis)
• The circuit has two meshes and the mesh equations :
• In Matrix form:
• Consider the following example:
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • In general, if a circuit has N meshes, mesh-current equations
can be expressed in terms of resistances as:
• R is called the resistance matrix, i is the output vector, and v is the input vector.
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • Example 3.8: Write the node-voltage matrix equations in
the following circuit.
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • Example 3.8: Write the node-voltage matrix equations in the
following circuit.
625.11
1
2
1
8
1 ,5.0
4
1
8
1
8
1
325.11
1
8
1
5
1 ,3.0
10
1
5
1
4433
2211
GG
GG
125.0 ,1 ,0
125.0 ,125.0 ,0
11
1 ,125.0
8
1 ,2.0
0 ,2.05
1
434241
343231
242321
141312
GGG
GGG
GGG
GGG
• The circuit has 4 nonreference nodes, so the diagonal terms of G are:
• The off-diagonal terms of G are:
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • Example 3.8: Write the node-voltage matrix equations in the
following circuit. • The input current vector i in amperes:
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
• The node-voltage equations can be calculated by:
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • Example 3.9: Write the mesh-current equations in in the
following circuit.
EENG223: CIRCUIT THEORY I
Nodal and Mesh Analysis by Inspection • Example 3.9: Write the mesh-current equations in in the
following circuit.
6 ,0 ,6612
,6410 ,4
543
21
vvv
vv
6
0
6
6
4
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
• The input voltage vector v in volts :
• The mesh-current equations are:
EENG223: CIRCUIT THEORY I
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.
EENG223: CIRCUIT THEORY I
Nodal versus Mesh Analysis: Summary • Both nodal and mesh analyses provide a systematic way of
analyzing a complex network. • The choice of the better method is dictated by two factors.
1. Nodal analysis: Apply KCL at the nonreference nodes.
(The circuit with fewer node equations)
2. A supernode: Voltage source between two nonreference nodes.
3. Mesh analysis: Apply KVL for each mesh.
(The circuit with fewer mesh equations)
4. A supermesh: Current source between two meshes.
EENG223: CIRCUIT THEORY I
Application: DC Transistor Circuits: BJT Circuit Models
(a)An npn transistor, (b) dc equivalent model.
EENG223: CIRCUIT THEORY I
Application: DC Transistor Circuits: BJT Circuit Models
Example 3.13: For the BJT circuit in Fig.3.43, =150 and VBE = 0.7 V. Find v0.
EENG223: CIRCUIT THEORY I
Application: DC Transistor Circuits: BJT Circuit Models
Example 3.13: For the BJT circuit in Fig.3.43, =150 and VBE = 0.7 V. Find v0.
• Use mesh analysis or nodal analysis
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