CHAPTER
11
ADVANCED METHODS OF DC AND AC CIRCUIT
ANALYSIS
Learning Objectives
As a result of successfully completing this chapter, you should be able to:
1.
Explain why more sophisticated methods of circuit analysis are required.
2.
Solve for voltages and currents in circuits using the mesh analysis circuit technique.
3.
Solve for voltages and currents in circuits using the nodal (node voltage) analysis circuit technique.
4.
Perform delta-wye conversions.
5.
Explain why bridge circuits are used and determine if a bridge circuit is balanced.
From previous math courses, you should be able to;
a.
Set up simultaneous algebraic equations from application of the appropriate circuit laws.
b.
Use a calculator to obtain the solution to a set of simultaneous equations.
c.
Knowledge of the mechanics of solving simultaneous equations is assumed from previous courses.
d.
Insert the solutions back into the original or other appropriate equations to check the solution.
e.
This step will reinforce the concept of a simultaneous solution.
Chapter Outline
11.1 The Need for More Sophisticated Analysis Techniques
11.3 Nodal Analysis
11.4
11.3.1
DC Signals
11.3.2
AC Signals
Bridge Circuits and Delta-Wye Circuits
11.4.1
DC Signals
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Class Notes Ch. 11 Page 1
11.4.2
AC Signals
11.1 The Need for More Sophisticated Analysis Techniques

Superposition can be used to analyze circuits with multiple sources.

Superposition is straightforward.

Superposition can be cumbersome.
Example 11.1.1
Determine the current through each resistance in Figure 11.1.
Figure 11.1
Multisource Circuit for Example 11.1.1
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Class Notes Ch. 11 Page 2
Figure 11.2
Current Direction Assignment in Example 11.1.1
Given: DC multi-source circuit in Figure 11.1
Desired:
I1, I2, I3, I4, and I5, as shown in Figure 11.2, including current direction in each R
Strategy:
superposition, series-parallel circuit analysis
Solution:
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Class Notes Ch. 11 Page 3
Figure 11.3
Circuit in Figure 11.2 Redrawn with Only VS1 Activated
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Figure 11.4
Circuit in Figure 11.2 Redrawn with Only VS2 Activated
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Class Notes Ch. 11 Page 5
Figure 11.5
Circuit Figure 11.2 Redrawn with Only VS3 Activated
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Class Notes Ch. 11 Page 6
Figure 11.6
Actual Current Directions in Example 11.1.1 (all currents positive)
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Class Notes Ch. 11 Page 7
11.3 Nodal Analysis
Nodal circuit analysis is an organized method of applying KCL to solve for voltages at nodes.

Essential nodes must first be identified.
o

A reference node is identified
o

3 or more components connected
Usually the ground connection
Node Voltage
o
Defined between a node and the reference node

Label branch currents

Apply KCL to each essential node

Except the reference node
Review KCL conventions

KCL:  currents entering node =  currents leaving node.

A current entering a node is considered positive.

A current leaving a node is considered negative.
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Class Notes Ch. 11 Page 8
General Nodal Analysis
1.
Identify all essential nodes, including the reference node.
2.
Indicate all branch currents
o
Current flows toward the reference node.
o
A current source in a branch dictates the current direction in that branch.
3.
Write KCL for each essential node
4.
Express each branch current in terms of essential node voltages
5.
Substitute branch current equations into KCL equations.
6.
Solve the simultaneous equations for the node voltages.
7.
Determine other voltages and branch currents from those node voltages.
Example 11.1.1
Determine the node voltages in the circuit shown in Figure 11.1 using a) nodal analysis, b)computer simulation,
and c) determine the branch currents through all of the resistances from the node voltages.
Given:
DC multi-source circuit in Figure 11.1
Desired:
node voltages
I1, I2, I3, I4, and I5
Strategy:
nodal analysis  node voltages
node voltages  I1, I2, I3, I4, and I5
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Figure 11.13
1.
Figure 11.1 with Nodes and Branch Currents Labeled
Identify and label all non-trivial nodes, including a reference node.
a.
The reference node is arbitrarily selected.
b.
There are three non-trivial nodes (including the reference)
c.
V1 and V2 are to be found using nodal analysis.
d.
They are labeled on Figure 11.13
2.
Indicate all branch currents (including direction)—see Figure 11.13. The current direction in each branch
is arbitrary except through R4: I4 must flow toward the reference node because the branch consists only of
passive components (R) and it is connected directly between a non-trivial node and the reference node.
3.
Write KCL at each non-trivial node (except the reference node) in terms of the branch currents.
Node 1: -I1 + I2 – I3 = 0
Node 2: +I3 – I4 + I5 = 0
4.
Express each branch current in terms of adjacent non-trivial node voltages.
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Class Notes Ch. 11 Page 10
Figure 11.14 Branch Current Determinations in Terms of Node Voltage Using KVL
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Figure 11.14 (continued)
Branch Current Determination in Terms of Node Voltage Using KVL
Contemporary Electric Circuits, 2nd ed., ©Prentice‐Hall, 2009 Strangeway, Petersen, Gassert, Lokken
Class Notes Ch. 11 Page 12
Figure 11.14 (continued)
Branch Current Determination in Terms of Node Voltage Using KVL
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Figure 11.14 (continued)
Branch Current Determination in Terms of Node Voltage Using KVL
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Figure 11.14 (continued)
Branch Current Determination in Terms of Node Voltage Using KVL
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Figure 11.16
MultiSIM Computer Simulation Results for Example 11.3.1
11.3.2 Nodal Analysis AC Signals
Example 11.3.2
Determine the voltages across the 8 Ω and 4 Ω resistances and the current through the capacitor in the circuit shown
in Figure 11.17.
Given: Circuit in Figure 11.17
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Figure 11.17
AC Circuit for Example 11.3.2
Desired:
V x ,V y , I C
Strategy: Nodal Analysis
Solution:
Figure 11.18
The Circuit in Figure 11.17 with the Node Voltages and Branch Currents Identified
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Class Notes Ch. 11 Page 17
Results:
~
V X  15.655  169.175o
~
V Y  18.892  114.770o
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Class Notes Ch. 11 Page 18
11.4 Bridge Circuits and Delta-Wye Circuits
11.4.1 DC Signals
Figure 11.21 Bridge Circuit Configurations
Figure 11.22
Delta Configurations Within the Bridge Circuit
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Figure 11.23
One of Two Wye Configurations Within the Bridge Circuit
Figure 11.24
Delta-to-Wye Conversion in a Bridge Circuit
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Figure 11.25
Delta–Wye Conversions
How to convert from ∆ to Y?
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How to convert from Y to ∆?
Example 11.4.1
Determine the delta configuration that is equivalent to the wye configuration in Figure 11.26.
Given: wye configuration with:
R1 = 4 Ω
R2 = 3 Ω
R3 = 7 Ω
Desired:
RA , RB , RC in the delta configuration
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Figure 11.26 Delta–Wye Conversions for Example 11.4.1
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Figure 11.28
Bridge Circuit Use for Measurements

A bridge circuit in Figure 11.28, is useful for measurements.

Suppose one of the branches, say branch 4, is a sensor (temperature, strain, etc.).

The circle around R4 is used to indicate that it is not a standard resistor, but that it is the resistance of a
sensor.
The Bridge is Balanced when:
When the bridge is unbalanced:
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11.4.2 AC Signals
Delta-wye conversions are approached in the same manner as for DC circuits except impedances are used instead of
resistances
Figure 11.29
Delta–Wye Conversions with Impedances
How to convert from ∆ to Y?
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How to convert from Y to ∆?
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