10/11/2013
Linear Circuits
Dr. Bonnie Ferri
Professor
School of Electrical and
Computer Engineering
An introduction to linear electric components and a study of circuits
containing such devices.
School of Electrical and Computer Engineering
Concept Map
1
Background
2
Resistive
Circuits
5
Power
3
Reactive
Circuits
4
Frequency
Analysis
2
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Concept Map: Power
Background
Resistive
Circuits
Max power
transfer
Power
Reactive
Circuits
Frequency
Analysis
Impedance,
phasors
3
Concept Map: Power
Background
Resistive
Circuits
Max power
transfer
Power
• Apparent power
Power
• Reactive
power
• Power factor
Reactive
Circuits
Frequency
Analysis
Impedance,
phasors
4
2
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Root Mean Square
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and Computer
Engineering
Introduce the root mean square statistic and how to calculate it.
School of Electrical and Computer Engineering
Lesson Objectives
Identify the equation for calculating root mean
square (RMS) value
Calculate the RMS values of simple periodic
functions
Find peak value from RMS
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Average of a Sinusoid
7
Root Mean Square
8
4
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Root Mean Square Example
9
Example
The voltage that goes into your home is
described by the root-mean-square
voltage. In the US, the voltage is
sinusoidal with 120V rms at 60 Hz.
What is the peak amplitude?
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Summary
Defined the root mean square calculation
Calculated the RMS values of
Sinusoidal functions
Triangular functions
Applied to home power voltages
11
Power Factor and
Power Triangles
Part 1
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and Computer
Engineering
Gain an understanding of the way that sinusoidal power is analyzed.
School of Electrical and Computer Engineering
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Lesson Objectives
Identify average power in resistive and
reactive devices
Calculate complex power
13
Instantaneous Power
14
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Average Power
15
Average Power
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Complex Power
17
What Complex Power Represents
18
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Summary
Calculated complex power
Identified the meaning behind complex power
19
Power Factor and
Power Triangles
Part 2
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and Computer
Engineering
Gain an understanding of the way that sinusoidal power is analyzed.
School of Electrical and Computer Engineering
10
10/11/2013
Lesson Objectives
Use power triangles
Calculate
Power angle and power factor
Real and reactive power
Apparent power
21
Review of Complex Power
22
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Power Factor
23
Complex Power for Impedances
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Complex Power for Impedances
25
Complex Power for Impedances
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Complex Power for Impedances
27
Complex Power for Impedances
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Implications
Only real power is being transformed to
heat/light/etc.
Reactive power causes increased current, so
more power is consumed by resistive
transmission lines
Private customers generally only charged for real
power, industrial customers charged for both
29
Summary
Defined
Power angle and power factor
Real and reactive power
Apparent power
Illustrated using power triangles
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Transformers
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and Computer
Engineering
Present transformers, a circuit device commonly used in power
applications.
School of Electrical and Computer Engineering
Lesson Objectives
Identify physical transformers and their
circuit representations
Describe the physical function of
transformers
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Transformer
Primary Winding
Secondary Winding
33
Relationship of Magnetic Field and Current
Ampere’s Law
Faraday’s Law of Induction
Transformers are AC
devices
34
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Two Transformer Models
Linear Transformer Model
Used primarily for communications applications
Uses impedances for analysis
Ideal Transformer Model
Used primarily for power transfer
Uses voltages and number of coil turns
35
Summary
Introduced transformers as a circuit device
Described the physical behavior of these
devices
Introduced two analysis models
36
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Linear Model of
Transformers
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and Computer
Engineering
Present linear model for analyzing transformers.
School of Electrical and Computer Engineering
Lesson Objectives
Identify the linear model of transformers
Use circuit analysis to analyze the behavior of
a transformer system
Apply this analysis to solving a transformer
circuit problem
38
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Linear Transformer
Reflected
impedance
39
Transformer Example
40
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Ideal Transformers
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and Computer
Engineering
Use the ideal transformer model for transformer analysis.
School of Electrical and Computer Engineering
Lesson Objectives
Identify the assumptions used for the ideal
transformer model
Use the ideal transformer model for doing
simple circuit analysis
Describe the importance of transformers in
power transmission
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k : Coefficient of Coupling
43
The Ideal Transformer
Coupling coefficient k=1
L1 = L2 = ∞
Losses from coil
resistances are
negligible
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Properties of the Ideal Transformer
45
Example
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Implications
Transformers allow a change from one
voltage to another voltage
High-voltage low-current power transmission
allows long-distance power distribution
systems
47
Summary
Showed the ideal transformer model
Used the model to solve an example
system
Identified the usefulness of transformers
for power transmission
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10/11/2013
Linear Variable
Differential
Transformer
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and Computer
Engineering
Explore LVDT sensors – devices which use mutual inductance for
measurement.
School of Electrical and Computer Engineering
Lesson Objectives
Explain how LVDT sensors work
Identify relative position measured by a
LVDT based on magnitude and phase
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Linear Variable Differential Transformer
Amplitude shows
displacement
Phase shows
direction
51
Benefits of LVDT
Capable of very high precision
Completely electrically shielded
Can operate in extreme conditions
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Summary
Described the behavior of LVDT sensors
Described how to identify the position by
measuring the voltage and phase
Described the benefits of such a sensor
53
Summary
Presented the linear model
Derived the phenomenon of reflected
impedance
Used circuit analysis to analyze an example
transformer circuit
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Concept Map: Power
Background
Max power
transfer
Resistive
Circuits
Power
•
•
•
•
Reactive
Circuits
Frequency
Analysis
Apparent power
Power
Reactive
power
Power factor
Transformers
Impedance,
phasors
55
Important Concepts and Skills
Be able to calculate the root-mean square of a periodic
function
Recognize that RMS is invariant to frequency
Use known RMS value equations to find RMS values
given peak values
Use known RMS value equations to find peak value
given RMS values
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Important Concepts and Skills
Calculate the complex power from either equations or phasors
Generate power triangles
Using power triangles, be able to find
○
○
○
○
○
Apparent power, |S|
Real (or average) power, P
Reactive power, Q
Power factor
Power angle
57
Important Concepts and Skills
Using the phase angle, identify if a load is resistive,
capacitive, or inductive
From equations, identify if a load is resistive,
capacitive, or inductive
From a plot of current and voltage, identify if a load is
reactive, capacitive, or inductive
Recognize if a system is “leading” or “lagging”
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Important Concepts and Skills
Calculate the impedance which gives maximal power
transfer
Calculate the average power consumed when the load
gives maximal power transfer
Find the optimal purely resistive load for constrained
maximal power transfer
Calculate average power for purely resistive load
59
Important Concepts and Skills
Describe the physical effects which make transformers work
Use the linear model to analyze a circuit with a transformer
Use the ideal model to analyze a circuit with a transformer
Identify circumstances when a transformer is an appropriate device to be
used in a system
Explain how the use of transformers allow long-distance power
distribution
Describe why transformers do not typically function for direct current
systems
Identify, from amplitude and phase, the relative displacement for an LVDT
transformer
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