Chap 2 - Circuit Analysis
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Transcript of Chap 2 - Circuit Analysis
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2.1 Branch-Current Analysis2.2 Mesh Analysis2.3 Nodal Analysis2.4 Mesh with Current Sources2.5 Nodal with Voltage Sources2.6 Bridge Network
Chapter 2: Circuit Analysis
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Introduction
Consider this network• Circuit simplification is difficult because there are
2 or more sources, and they are neither in series nor parallel.
• There will be an interaction between sources that will not permit the reduction techniques that is used to find quantities such as the effective resistance.
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Steps required for Branch-Current Analysis: -
1. Assign a distinct current of arbitrary direction to each branch of the circuit.
2. Add the polarities for each voltage drop across resistor.
3. Apply KVL for each mesh.
4. Apply KCL to a node that includes all the branch currents.
5. Solve the equations for branch currents.
2.1 Branch-Current Analysis
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Example :
Determine the current in each branch of the network using branch-current analysis
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Solution:• Assign distinct current
of arbitrary direction to each branch of the network (one branch one current)
• Add the polarities for each R
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Solution(continue…):Kirchhoff’s Voltage Law:
Loop 1:
- E1 + V1 + V3 = 0
I1R1 + I3R3 = E1
Loop 2:
- V3 – V2 + E2 = 0
I2R2 + I3R3 = E2
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Solution(continue…):Kirchhoff’s Current Laws
I1 + I2 = I3
Solve the equations
2I1 + 0I2 + 4I3 = 2 ---(1)
0I1 + I2 + 4I3 = 6 ---(2)
I1 + I2 - I3 = 0 ---(3)
Observation: 3 branches, 3 equations
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• Mesh Analysis – defines a unique array of currents (Mesh or Loop current) to the network using KCL
• Steps required for Mesh Analysis: -1. Assign current in clockwise direction to each closed loop
of network.2. Insert polarities for each resistor.3. Apply KVL to each closed loop.4. Solve the resulting equations.
Note: if a resistor has two or more current passing through it, the netcurrent = the mesh current of the closed loop + mesh
currents from other loops in same direction - mesh currents from other loops in opposite direction.
2.2 Mesh Analysis
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Example : find the branch currents I1, I2, I3.
1.Assign current in clockwise direction
123
51015
01010515
:1mesh For
21
21
211
ii
ii
iii
12or 12
102010
0461010
:2mesh For
2121
21
2212
iiii
ii
iiii
Solution:+ -
+
-
+ -
+
-
2. Insert polarities for each R according to the direction of current.
3. Apply KVL to each closed loop in clockwise direction.
-
+
Observation: 2 meshes, 2 equations.
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2.3 Nodal Analysis• Nodal analysis - provides nodal voltages by using KCL.
• Steps required for Nodal Analysis: -1. Determine the no. of nodes (junction of 2 or more
branches).2. Select a reference node (Ground), and label all other
nodes.3. Apply KCL at each node (except the reference node).4. Solve the resulting equations.
Note: A network of N nodes require (N-1) equations to find (N-1) nodal voltages where by the Reference node is eliminated.
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Example : Calculate the node voltages.
1. Determine the no. of nodes.2. Select a reference node (Ground),
and label all other nodes. 3. Apply KCL at each node except the
reference node.4. Solve the resulting equations.
203
522
0
45
:1
21
121
121
321
vv
vvv
vvv
iii
nodeFor
Solution:
6053
260120336
0510
4
:2
21
221
221
5142
vv
vvv
vvv
iiii
nodeFor
Observation: (3-1) nodes, 2 equations.
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Example:
For the circuit below, obtain v1 and v2.
Observation: 4 meshes, (3-1) nodes
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At node 1,
60 = - 8v1 + 5v2 (1)
At node 2,
36 = - 2v1 + 3v2 (2)
Solving (1) and (2),
v1 = 0 V, v2 = 12 V
2
vv6
5
v
10
v 2111
2
vv63
4
v 212
Solution:
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2.4 Mesh with Current SourcesCase 1: If the current source exists only in one of the mesh, find the
current directly.Case 2: If the current source exists between two or more meshes, it
forms a supermesh. Exclude the current source (replace with an open circuit) and the elements that connected in series with it. Apply KVL in the supermesh and KCL in the nodes of the excluded
elements.
Consider the following case 1
Applying Mesh analysis in usual wayLoop 1:
06410 211 iii
Loop 2:
A 52 i A 21 i
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Supermesh is formed when two meshes have a (dependent or independent) current source in common.
Consider the following case 2
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Apply KVL on the supermesh loop figure (b):
20146
0410620
21
221
ii
iii
Applying KCL:612 ii
Solving Equation:
A 8.2
A 2.3
2
1
i
i
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Consider the following circuitFind the nodal voltages forthe given circuit.
2.5 Nodal with Voltage SourcesCase 1: If the voltage source is connected to a reference node, set the
voltage equal to the voltage source.Case 2: If the voltage source is connected between two non-reference
nodes, it forms a supernode. Exclude the voltage source and replace with a short circuit. Apply KCL in the supernode and KVL to determine the nodes voltages.
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• Define the node voltage• Replace voltage source with short circuit • Apply KCL to the new network
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244-6
:SupernodetheFor
21
2121
232131
VV
IIII
IIIIII
SS
SS
2112
:circuit original theRefer to
VV
12
250250
:equations Resulting
21
21
V V
V.V.
*challenge: use source conversion or mesh analysis
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Summary: Mesh-Nodal Analysis
(i) Mesh• To find mesh currents • Suitable to be used with
circuit with many series-connected elements, voltage sources & Supermesh
Supermesh• Replace current sources
with an open circuits
(ii) Nodal• To find node voltages • Suitable to be used with
circuit with many parallel-connected elements, current sources & Supernode
Supernode• Replace voltage sources
with a short circuits
Mesh Analysis vs. Nodal Analysis
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2.6 Bridge Network
• Belong to the family of complex networks because the elements are neither in series nor in parallel.
• Bridge configuration may be analyzed by using either mesh or nodal analysis.
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052152
054254
2024243
213
312
321
III
III
III
0852
05114
20249
321
321
321
III
III
III
A 667.2
A 667.2
A 4
3
2
1
I
I
I
Note that:A 0325 III R
A bridge network is said tobe balanced, if the current or voltage through the bridgearm is 0A or 0V.