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Analysis Methods Overview
Solving Linear Equations
Nodal Analysis
Supernodes (Nodal Analysis with Voltage Sources)
Mesh Analysis
Supermeshes (Mesh Analysis with Current Sources)
This is a very important chapter.
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Review of Basic Concepts: Current
i4 i5i3i2i1
What goes in, has to come out
Kirchhoffs current law
Similar to conservation of mass
Conservation of electrons
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Review of Basic Concepts: Voltage
10 V-
+
-
++ - + -
2 k2 k
5 k 7 kv1 v2
v3 v4
The voltage drop from one node to another is the same, nomatter
what path is chosen
Kirchhoffs voltage law
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Resistors in Parallel with Voltage Sources
CircuitRVs vo-
+
CircuitVs vo-
+
What is vo in each case?
What effect does the resistor have on the current pumped into
thecircuit?
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Resistors in Series with Current Sources
CircuitIs CircuitIs
Rio
io
What is io in each case?
What effect does the resistor have on the voltage seen by
thecircuit?
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Solving Linear Equations
Much of our circuit analysis will focus on finding a set of
linearequations and solving these equations
Need as many equations as there are unknowns
Three possible approaches
Algebra (elimination, substitution, etc.)
Cramers rule
Linear algebra
Last is easiest and least susceptible to errors
Requires use your scientific calculators
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Example 1: Solving Linear Equations
i1 = i2 + i3
i4 = i3 + 2m
10 = (1k)i1 + (5k)i2
(5k)i2 = (2k)i3 + (10k)i4
Rewrite so variables are in consistent order on left side and
constantsare on the right side
i1 i2 i3 = 0 i3 + i4 = 2m
(1k)i1 + (5k)i2 = 10+ (5k)i2 (2k)i3 (10k)i4 = 0
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Example 1: Continued (1)
i1 i2 i3 = 0 i3 + i4 = 2m
(1k)i1 + (5k)i2 = 10+ (5k)i2 (2k)i3 (10k)i4 = 0
In Matrix form this becomes
1 1 1 00 0 1 11k 5k 0 00 5k 2k 10k
i1
i2
i3
i4
=
02m100
or
Ai = b
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Example 1: Continued (2)
Ai = b where
A =
1 1 1 00 0 1 11k 5k 0 00 +5k 2k 10k
i =
i1
i2
i3
i4
b =
02m100
Your calculator should be able to solve this directly
You should only need to enter A and b
Your calculator will return a vector i
Simultaneously solves for all the unknown variables
Much faster than Cramers rule or brute-force algrebra
Read the manuals for your calculators
This will save you time (homework & exams) and reduce
errors
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Example 1: Continued (3)
Linear Equations:1 1 1 00 0 1 11k 5k 0 00 5k 2k 10k
i1
i2
i3
i4
=
02m100
Calculator should return:
i1
i2
i3
i4
=
+0.909+1.8180.909+1.091
mA
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Network Terminology
Planar Circuit A circuit that can be drawn on a plane with
nocrossing branches
Node Point or portion of a circuit where 2 or more elements
arejoined
Essential Node Point or portion of a circuit where 3 or
moreelements are joined
Branch Path that connects 2 nodes
Essential Branch Path that connects 2 essential nodes w/o
passingthrough an essential node
Loop Path with last node same as starting node that does not
crossitself
Mesh Loop that does not enclose any other loops
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Example 2: Terminology
20 V 2 A
R1 R2
R3 R4 R4
R6 R7 R8
35ip
ip
Identify the following informationNodes: Essential
Nodes:Branches: Essential Branches:EBs with Unknown Current:
Meshes:
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Example 3: Circuit Analysis The Hard Way
10 V
i1 i3
i2 i42 mA
1 k 2 k
5 k 10 k
Can solve with KCL & KVL. Four unknowns.
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Nodal Analysis: Introduction
There is an another way to solve for currents and voltages
Easier
More methodical
Still based on Ohms law, KVL, & KCL
Nodal analysis is one of two key methods
Mesh analysis is the other
We will discuss nodal analysis first
Based on KCL
Must understand terminology introduced earlier
Use to solve for voltages
All voltages have a common reference point
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Nodal Analysis Steps
1. Identifiy essential nodes
2. Pick a reference node
3. Label all other essential nodes
4. Apply KCL to all labelled nodes
5. Solve linear equations for all node voltages
6. Solve for variables of interest
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Nodal Analysis: Step 1 Identify Essential Nodes
10 V 2 mA
1 k 2 k
5 k 10 k
Some essential nodes may include portions of the circuit (pieces
ofwire)
Circle the entire node to prevent errors
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Nodal Analysis: Step 2 Pick a Reference
10 V 2 mA
1 k 2 k
5 k 10 k
Second step is to pick a reference node
Is often easiest to choose the node that interconnects the
mostbranches
Must be an essential node
Usually is at bottom of circuit
Label with the same symbol used for ground
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Nodal Analysis: Step 3 Label Other Essential Nodes
10 V 2 mA
1 k 2 k
5 k 10 k
Also a bit easier if voltages are labeled
All voltages are measured relative to the reference node
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Nodal Analysis: Step 4 Apply KCL All Labeled Nodes
10 V 2 mA
1 2
-
+
v2-
+
v1
1 k 2 k
5 k 10 k
50 k
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Nodal Analysis: Step 5 Solve Linear Equations
Linear Equations:
Solution (from calculator):
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Nodal Analysis: Step 6 Solve for Variables of Interest
10 V 2 mA
1 2
-
+
v2-
+
v1
i1 i3
i2 i4
1 k 2 k
5 k 10 k
50 k
i1 =
i2 =
i3 =
i4 =
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Nodal Analysis: Review of Steps
1. Identify essential nodes
2. Pick a reference
Must be an essential node
Always label with the ground symbol
Best to pick essential node with most branches
Often at the bottom of the circuit diagram
3. Label other essential nodes
4. Apply KCL to all labelled nodes except reference node
5. Solve linear equations
Generates voltage at each node (relative to reference node)
6. Solve for variables of interest
Usually easy after Step 5
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Nodal Analysis: Use of Laws
All three laws are used
KCL is applied at each labelled node except the reference
node
Ohms law is used to determine the current in branches
thatcontain resistors
KVL is used to determine the voltage drop across the
resistors
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Example 4: Nodal Analysis
144 V-
+
v2-
+
v1 3 A
4
5 10
80
Solve for v1 and v2.
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Example 4: Workspace
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Example 5: Nodal Analysis
20 mA-
+
v2-
+
v1-
+
v3 5 V2 k
2.7 k2.7 k
3.3 k
4.7 k
10 k
Solve for v1, v2, and v3.
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Example 5: Workspace
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Example 6: Dependent Voltage Source
50 V-
+
10
10
30 39 78
80 k
v/5
v
Solve for v.
What effect does the 10 resistor have on the circuit?
What is the current flowing through the dependent source?
How can we apply KCL at the essential nodes without
thisinformation?
Ans: One extra variable
Implies we need an extra equation
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Example 6: Continued
50 V-
+
10
10
30 39 78
100 k
v/5
v
Solve for v.
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Example 6: Workspace
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Nodal Analysis and Supernodes
Supernodes eliminate the need to introduce an extra
variable(unknown current)
Necessary when a voltage source is between two labeled
nodes(excluding reference node)
Still need to use voltage source to generate one of the
equations
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Example 7: Dependent Source Continued
50 V-
+
10
10
30 39 78
160 k
v/5
v
Solve for v. Use a supernode.
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Example 7: Workspace
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Example 8: Dependent Voltage Source
20 V
+ -
1 2 4
20 40 80 3.125v
v
35i
i
Find the power developed by the 20 V source.
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Example 8: Workspace
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Example 9: Nodal Analysis
11 mA
i1
20 Vi2
10 Vi3
250
500
1 k
25 k
Solve for i1, i2, and i3.
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Example 9: Workspace
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Example 10: Nodal Analysis
1 A
3i
i
-
+
v
1
1
2
2 4
Solve for v.
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Example 10: Workspace
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Mesh Analysis: Introduction
Recall: There is an easier way to solve for currents and
voltagesthan applying KVL and KCL directly
Nodal analysis is one of two key methods
Mesh analysis is the other
Applies KVL to solve for currents
More abstract
Work with imaginary currents
Only applies to planar circuits
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Mesh Analysis: Step 1 Label Meshes
40 V 64 V
ia
ic
ib
1.5 2
3 4
45
Find the branch currents ia, ib, and ic.
Recall: A mesh is a loop that does not enclose any other
loops
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Mesh Analysis: Step 2 Apply KVL to Each Mesh
40 V 64 V
ia
ic
ib
1.5 2
3 4
45
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Mesh Analysis: Step 3 Solve Linear Equations[50 4545 50.5
] [i1
i2
]=
[4064
]
i1 = 9.8 A
i2 = 10 A
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Mesh Analysis: Step 4 Solve for Variables of Interest
40 V 64 V
ia
ic
ib
1.5 2
3 4
45
ia =
ib =
ic =
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Mesh Analysis: Review of Steps
Step 1 Label Meshes
Step 2 Apply KVL to Each Mesh
Step 3 Solve Linear Equations
Step 4 Solve for Variables of Interest
Usually easy after Step 3
Limitation: Only works with planar circuits
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Example 11: Mesh Analysis
12 V
110 V 70V
2
3
4
6
10 12
Find the total power developed in the circuit.
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Example 11: Workspace
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Example 12: Mesh Analysis
18 V 15 V3 A
2
3
6
9
Find the total power dissipated.
Problem: What is the voltage across the 3 A source?
Solutions
1 Add it as a variable
2 Use a supermesh
Second option requires less work
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Example 12: Mesh Analysis
18 V 15 V3 A
2
3
6
9
Find the total power dissipated. Add a variable.
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Example 12: Workspace
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Example 13: Mesh Analysis
18 V 15 V3 A
2
3
6
9
Find the total power dissipated. Use a supermesh.
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Example 13: Workspace
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Example 14: Mesh Analysis
200 V
4.3 id
ie
ib
id
ia
ic
10
10
25
50
100
Find the branch currents ia ie.
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Example 14: Workspace
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Example 15: Mesh Analysis
1.5 mA
8 V
2 k
3 k
4 k
4 k 4 k
5 k
7 k
3i
i
Solve for i
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Example 15: Workspace
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Nodal versus Mesh Analysis
You should know how to do both
Which is more efficient depends on the problem
Will learn which to use with experience
Nodal analysis used more often
On exams, I will specify which method to use
Concise Summary:
Nodal Analysis Mesh AnalysisMethod KCL KVLSolve For Node
Voltages Mesh CurrentsSuper Conditions Voltage Sources Current
Sources
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