EGR 277 Digital Logic

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Lecture 6 EGR 260 Circuit Analysis
Reading Assignment Chapter 3 in Electric
Circuits, 8th Edition by Nilsson

• Current Division
• Applies to parallel circuits only.
• Current divides between parallel Rs with the
  smallest R getting the most current.
• Show that

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Lecture 6 EGR 260 Circuit Analysis
Current Division - Special Case Two Resistors
Only Show that for two resistors only the
general form of current division can be expressed
as follows.
Example Find the current I1 using current
division (special form for two Rs only).
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Lecture 6 EGR 260 Circuit Analysis
Example Find the current I1 using current
division.
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Lecture 6 EGR 260 Circuit Analysis
Example Find the current I1 using repeated
current division.
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Lecture 6 EGR 260 Circuit Analysis
Example Find I1, V2, I3, I4, V5, and V6 in the
circuit shown below.
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Lecture 6 EGR 260 Circuit Analysis
Delta-to-Wye (?-Y) and Wye-to-Delta (Y-?)
Transformations One type of resistive circuit
that cannot be simplified through series and/or
parallel combinations is the bridge circuit.
This type of circuit contains resistors connected
into delta (?) and wye (Y) configurations. One
way to analyze this circuit is to use a ?-Y or a
Y-? transformation. Y and ? connections of
resistors are shown below
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Lecture 6 EGR 260 Circuit Analysis
If the wye and delta circuits to be equivalent,
then they should provide the same resistance
between each pair of terminals (a-b, b-c, and
c-a).
Development Determine the resistance seen at
each set of terminals and equate them as
follows Ra-b (Delta) Ra-b (Wye) Rb-c (Delta)
Rb-c (Wye) Rc-a (Delta) Rc-a (Wye)
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Lecture 6 EGR 260 Circuit Analysis
Solving the equations on the previous page yields
the following relationships
Note The equations above must be used along
with the the circuit diagrams provided. The
labeling of the resistors in the diagrams is
critical.
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Lecture 6 EGR 260 Circuit Analysis
Example Determine I in the circuit shown below
using a) ?-Y conversion
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Lecture 6 EGR 260 Circuit Analysis
Example Determine I in the circuit shown below
using b) Y-? conversion
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Lecture 6 EGR 260 Circuit Analysis
Applications of bridge circuits - the balanced
bridge A bridge circuit is balanced when the
following relationship exists
R1R4 R2R3
When the bridge is balanced, it can be shown that
I5 0
Now suppose that an ammeter is placed in series
with R5 to detect when I5 0 and an unknown
component is attached in place of R4. R2 can
then be adjusted until the meter reads zero (the
bridge is balanced) and Runknown can be
determined as follows
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Lecture 6 EGR 260 Circuit Analysis
Determining unknown component values A bridge
circuit with resistive values, sometimes called a
Wheatstone bridge, can be used to determine
unknown resistor values. Other bridge circuits
(such as the Scherring bridge) can be used to
determine unknown values of capacitors and
inductors in a similar manner. Strain
gauges Strain gauges are often attached to
surfaces to measure forces as the surface moves
(such as in the deflection of an airplane wing).
The force can be determined from the amount of
stretch in the wire by measuring the resistance
of the wire with a bridge circuit. As a wire
stretches note that resistance increases since
Note This marks the end of Test 1 material.
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Lecture 6 EGR 260 Circuit Analysis
Reading Assignment Sections 4.1-4.8 in Electric
Circuits, 7th Edition by Nilsson

• 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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Lecture 6 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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Lecture 6 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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Lecture 6 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)
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Lecture 6 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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Lecture 6 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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Lecture 6 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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Lecture 6 EGR 260 Circuit Analysis
Example Use node equations determine the
current I. (Answer I 1.92 A.)