EGR 277

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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Reading Assignment Sections 4.1-4.8 in Electric
Circuits, 9th Edition by Nilsson
Note Test 2 material starts here.

• Chapter 4 Methods of Analysis for Resistive
  Circuits
• In Chapters 2 and 3 KVL and KCL were applied in a
  somewhat arbitrary manner. No systematic
  procedure was introduced as to where to write the
  equations and how many equations to write. This
  arbitrary approach would be difficult to use for
  larger, more complex circuits.
• In this chapter we will develop methods for
  writing and solving simultaneous independent
  circuit equations.

Two methods are introduced in this chapter and
are commonly used throughout electrical
engineering 1. Node Equations (or Nodal
Analysis) result in a set of simultaneous,
independent KCL equations 2. Mesh Equations
(or Mesh Analysis) result in a set of
simultaneous, independent KVL
equations. Systematic procedures will be
introduced for writing these equations that will
give a clear approaches that can be used even for
very large circuits.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Node Equations First, a couple of
definitions Ground ( or reference) a node in
the circuit used as a reference point for
measuring node voltages. As far as node voltages
are concerned, the ground node is the 0V point in
the circuit. Ground symbols Node voltage
the voltage at a node with respect to ground.
For example, VA is the voltage at node A, meaning
that the positive terminal is at A and the
negative terminal is at the ground node. Note
Node voltages are relative measurements. The
value of a node voltage changes if a different
ground node is used.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Illustration The use of a ground essentially
means that a common node will be used as the
negative terminal for all voltages (node
voltages) in the circuit. This would be like
using a voltmeter to measure various voltages
where the negative lead of the meter always
stayed on the ground node and the positive lead
then moved to various other nodes to measure
node voltages.
Case 2 Using a voltmeter to measure node
voltage VB
Case 1 Using a voltmeter to measure node
voltage VA
Note In all cases the negative side of the
meter is connected to the ground node in order to
measure node voltages.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Component Voltages Recall that node voltages are
relative quantities. If a different reference is
used, the node voltages change. Component
voltages are not relative quantities. Component
voltages can be determine from node voltages as
illustrated below.
Vx VA - VB
(prove that this is true for any ground C)
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis

• Example The circuit below includes the value of
  the component voltages.
• Determine the corresponding node voltages (fill
  out the table).
• Show that Vx VA - VB for each possible set
  of node voltages.

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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Node equations Procedure 1) Label each
node. 2) Select a node as the reference (or
ground) node. Note It is generally easier to
pick the ground adjacent to a voltage
source. 3) If the circuit has no voltage sources,
skip to step 4. Otherwise A) For any voltage
source or group of voltage sources adjacent to
the ground, all node voltages adjacent to the
sources can be determined so no KCL equation will
be required. B) For any voltage source or group
of voltage sources not adjacent to the ground, a
supernode is required (to be discussed
later). 4) Write a KCL equation at each node not
adjacent to a voltage source and not at the
ground node. (Also write a KCL equation for each
supernode.) Express resistor currents in terms
of node voltages. 5) Solve the simultaneous KCL
equations. In general, the number of equations
required is
Node Equations nodes - voltage sources - 1
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Analyze the following circuit using
node equations. Use the result to find I1, V2,
and the power dissipated by the 8 ohm resistor.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Use node equations determine the
current I. (Answer I 1.92 A.)
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Use node equations determine the
current I.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Circuits containing dependent sources Key
Replace the control variable in terms of node
voltages.
Example Use node equations determine the
current I1 .
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Supernodes A supernode is a node formed by
drawing a surface around a voltage source or
group of voltage sources that are not adjacent to
the reference (this would be a node if the
voltage sources in the supernode were set to zero
or shorted).

• When supernodes are used, node equations are
  written as follows
• Write a KCL equation at each node not adjacent to
  the reference or adjacent to a voltage source
• Write one KCL equation for each supernode.

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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis

• Illustration - Supernode
• Note that it is not possible to place the ground
  adjacent to both voltage sources in the circuit
  shown, so a supernode is necessary.

• If the ground is placed at node D (adjacent to
  the 50V source) then a supernode is needed around
  the 20V source.
• Note that the voltage difference between nodes A
  and B is 20V, so

Supernode relationship VA - VB 20

• Think of the supernode as the node that would
  exist if the voltage source were replaced by a
  short (a wire).

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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Analyze the circuit shown using node
equations. Show that ?Pdel ?Pabs .
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Use node equations determine V and I.
Is a supernode required?
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Mesh Equations Mesh equations (or mesh analysis)
are a set of simultaneous KVL equations. Mesh
equations have one restriction Mesh analysis
can only be used if a circuit is planar. A
circuit is planar if it could be drawn on a 2D
surface with no crossovers.
Example Is the following circuit planar?
Example Is the following circuit planar?
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Mesh a minimal region in a circuit that is
bounded by circuit elements. A more informal
definition is that a mesh is like a window pane
in a window.
4 window panes
4 meshes
Circuit
Window
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Mesh current a current associated with a mesh.
Mesh currents are generally drawn all clockwise
(CW) or all counter clockwise (CCW).
Example Mesh currents IA, IB, IC, and ID are
shown in the circuit to the right.
Component currents note that a component
current is made up of either one or two mesh
currents.
Example Define the component currents I1, I2,
I3, and I4 in terms of mesh currents. I1 I2
I3 I4
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Expressing resistor voltages in terms of mesh
currents
Example Define the resistor voltages V1, V2,
and V3 in terms of mesh currents. V1 V2
V3
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Mesh Equations Procedure 1) Be sure that the
circuit is planar (redraw it if
necessary). 2) Label the mesh currents (generally
all CW or all CCW). 3) If the circuit contains
any current sources on the outer edge, the
corresponding mesh currents are defined. If the
circuit contains any internal current sources, a
supermesh is required (more information
later). 4) Write a KVL equation in each mesh with
no current sources and one KVL equation around
each supermesh. Express resistor voltages in
terms of mesh currents (see below). 5) Solve the
equations simultaneously. In general, the number
of mesh equations is
Mesh Equations meshes - current sources
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Use mesh equations to determine the
current I. Note that this problem was analyzed
in previous class using node equations and I was
determined to be 1.92 A.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Use mesh equations to analyze the
circuit shown below. Use the results to
determine V1, V2, I3, and the power absorbed by
the 80 ohm resistor.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Dependent Sources The key to using mesh analysis
on circuits with dependent sources is to redefine
the control variables in terms of mesh currents.
Example Use mesh equations to determine V1 and
i1 in the circuit shown below.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Supermesh If a circuit contains an internal
current source, a supermesh is required in order
to perform mesh analysis. A supermesh is the
new, larger mesh that is created by removing the
internal current source. A new mesh current is
not added. The supermesh simply shows the path
for a KVL equation around the supermesh.
Example 1) Note that the following circuit has
an internal current source, so a supermesh is
required.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
2) The supermesh is the new, larger mesh created
by removing the current source (as shown on the
following page).
3) Note that the supermesh defines a path for a
KVL equation. No new mesh current is
defined. 4) Also note that the internal current
source can be used to form a relationship between
currents IB and IC. In general, this is referred
to as the supermesh relationship.
Supermesh relationship IB - IC 4
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Analyze the circuit shown below using
mesh equations. Use the results to find the
voltage V1.
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Chapter 4, Sections 1-8 EGR 260 Circuit
Analysis
Example Analyze the circuit shown below using
mesh equations. Use the results to find the
voltage V1 and the current i1. Is a supermesh
required? Also discuss systems of units.