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Linear Circuits
An introduction to linear electric components and a study of circuits containing such devices.
Dr. Bonnie FerriProfessorSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Concept Map
2
Background Resistive Circuits
Reactive Circuits
Frequency Analysis
Power
1 2
3 4
5
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Resistive vs Reactive Circuit
3
Time
Volta
ge
Reactive Circuits
Concept Map
4
Background Resistive Circuits
Frequency Analysis
Power
Methods to obtain circuit equations (KCL, KVL, mesh, node, Thvenin)
Current, voltage, sources, resistance
Capacitors Inductors Differential
Equations
RC Circuits RL Circuits 2-order Diff Eqn RLC Circuits Applications
Reactive Circuits
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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Capacitance
Describe the behavior of capacitors by calculating:the charge stored on the capacitor platesthe current flowing through the capacitorthe voltage across the capacitorthe capacitance of the capacitor
Describe the construction of a capacitor Find charge stored on a capacitor Find the current through a capacitor Find the voltage across a capacitor Calculate the capacitance of a capacitor Explain how current flows through a capacitor
Lesson Objectives
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Capacitors
6
Capacitors and Charge
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Current and Voltage
8
Calculating Capacitance
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Material Approximate r (or k)Air 1Teflon 2.1Paper 3.9Glass 4.7Rubber 7.0Silicon 11.7Water 78.5 (varies by T)
Permittivity of Common Materials
10
Current Through A Capacitor
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Identified how capacitors work Calculated charge stored on a capacitor Identified the relationship between current and
voltage on a capacitor Calculated capacitance Explained how current flows through a
capacitor
Summary
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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Capacitors
Present how capacitors work in a systemIdentify behavior in DC circuitsGraphically represent the relationships between current, voltage, power, and energy
Analyzing capacitors in series/parallel Analyze DC circuits with capacitors Calculate energy in a capacitor Sketch current/voltage/power/energy curves
Lesson Objectives
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Capacitors in Parallel
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Capacitors in Series
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Behavior in DC Circuits
8
Stored Energy
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Graphs
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Calculated capacitance for capacitors in parallel/series configurations Identified how capacitors in DC circuits behave like
open circuits Derived an equation for the energy stored by a
capacitor as an electric field Showed graphically the relationships between
voltage/current/power/energy in capacitors
Summary
11
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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Inductance
Introduce inductors and describe how they workCalculate current and voltage for inductors
Describe the construction and behavior of an inductor Find current through an inductor Find voltage across an inductor Explain how a voltage is created across an
inductor
Lesson Objectives
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Inductors
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Current and Voltage
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Ampres Law
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How Inductors Work
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Voltages Across a Wire
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Presented the equations for current and voltage in inductors Introduced Ampres Law and showed how
inductors work in context of this law Showed how a voltage is created across an
inductor as currents change in a system
Summary
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1
Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Inductors
Present how inductors work in a systemIdentify behavior in DC circuitsGraphically represent the relationships between current, voltage, power, and energy
Analyze inductors in series/parallel Analyze DC circuits with inductors Calculate the energy in an inductor Sketch current/voltage/power/energy curves
Learning Objectives
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Inductors in Series
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Inductors in Parallel
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Behavior in DC Circuits
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Stored Energy
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Graphs
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Calculated inductance for inductors in parallel/series configurations Identified how inductors in DC circuits behave like
short circuits Derived an equation for the energy stored by an
inductor as a magnetic field Showed graphically the relationships between
voltage/current/power/energy in inductors
Summary
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Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
First-OrderDifferentialEquations
Solve and graph solutions to first-order differential equations
Examine first-order differential equations with a constant input Write the solution Sketch the solution
Lesson Objectives
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ODE: Include functions of variables and their derivatives.
Ordinary Differential Equations
6
dy
dt+ 2y = 4 dy
dt 2y = 4sin(t)
d 2y
dt2+ 2
dy
dt+ 4y = f (t)
dv
dt+ 2v = i(t)
Models of Physical Systems
7
System Model
Inputs, f(t) Outputs, y(t)dy
dt+ ay = f (t)
Thermal System
Heat source Temp Biological
SystemProduction Population Linear
Circuit
Voltage or current source
Voltage or current
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Has solution:
Solution to First-Order Differential Equation
8
dy
dt+ ay = K, y(0)
y(t) = Ka
(1 eat )+ y(0)eat, t 0
If a 0, eat 0 y(t) Ka
= steady-state
Graph of Response
9
y(t) = Ka
(1 eat )+ y(0)eat, t 0
K
a
y(0) Ka
e
at, t 0
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time, , for exponential transient to decay to of its initial value (or 63% to its final value)
Time Constant
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e1 0.37
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (sec)
y(t)
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (sec)y(
t)
Sample Problems
11
Pause
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Summary
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Discussed how various physical phenomena are modeled by differential equations Showed the solution to a generic first-order
differential equation with a constant input and initial condition Introduced the transient and steady-state
responses Showed how to sketch the response and plot
the time constant
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1
Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
RC Circuits
Generate a differential equation that describes the behavior of a circuit with resistors and capacitorsSolve the differential equation for step inputs (or switching constant inputs)Graph the behavior
Generate a differential equation from a circuit Identify initial and final conditions Solve the differential equation Graph the result
Lesson Objectives
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Behavior of RC circuits
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Example 1: Initial and Final Conditions
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Example 1: Differential Equation
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Example 1: Graph
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Example 2: Initial and Final Conditions
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Example 2: Differential Equation
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Example 2: Graph
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Got some intuition about how RC circuits behave Identified initial and final conditions Found differential equations for the circuit and
solved them Graphed the results
Summary
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Dr. Bonnie FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Module 3Lab Demo: RC Circuits
Summary of Reactive Circuits Module
RC Circuit
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vs
R +
-
vc~ CvR +
-vc
C-+-
+
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RC Circuit
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vs
R +
-
vc~ CvR +
-vc
C-+-
+
RC Circuit
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vs
R +
-
vc~ CvR +
-vc
C-+-
+
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RC Circuit
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vs
R +
-
vc~ CvR +
-vc
C-+-
+
RC Circuit
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vs
R +
-
vc~ CvR +
-vc
C-+-
+
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Lab Demo: RC Circuits
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An oscilloscope is used to measure and record voltage signals versus time.
A function generator allows you to input voltage signals into a circuit
Inputting a square wave into a circuit allows you to capture RC circuit transient behavior and to measure the time constant
Summary
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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
RL Circuits
Use differential equations to show the behavior of an RL circuit as the system changes.
Generate a differential equation from a circuit Identify initial and final conditions Solve the differential equation Graph the result
Lesson Objectives
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Behavior of RL circuits
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Example 1: Initial and Final Conditions
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Example 1: Differential Equation
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Lv
t
5V
Example 1: Graph
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Lv
t
5V
2%).(63
5%).(86
1.84V
0.68V
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Example 2: Initial and Final Conditions
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Example 2: Differential Equation
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Li
t
4mA
mA21
Example 2: Graph
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2%).(63
5%).(86
1.16mA
0.11mA
Li
t
4mA
mA21
Got some intuition about how RL circuits behave Identified initial and final conditions Found differential equations for the circuit and
solved them Graphed the results
Summary
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Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Second-OrderDifferentialEquations
Recognize types of second-order responses
Examine second-order differential equations with a constant input:
Determine the steady-state solution Determine the type of transient response Recognize the characteristics of the
plot of the solution
Lesson Objectives
5
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ODE: Include functions of variables and their derivatives
Ordinary Differential Equations
6
d 2y
dt2+ 2
dy
dt+ 4y = f (t)
Models of Physical Systems
7
System Model
Inputs, f(t) Outputs, y(t)
Vibratory System
Force PositionRLC
Circuit
Voltage or current source
Voltage or current
d 2y
dt2+ 2
dy
dt+ 4y = f (t)
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Solution: y(t) = steady-state + transient
Solutions to Second-Order Differential Equation
8
d 2y
dt 2+ a1
dy
dt+ a2y = K, y(0)and
dy
dt t=0
2aKty )(
Three possible forms depending on roots of (s2+a1s+a2)=0.
Transient Response
9
(s2+a1s+a2)=0.
trtr eKeK 2211 +
rtrt teKeK 21 +
)sin( +btKeat
((real and distinct roots, r1 and r2)
((real and equal roots, r and r)
((complex roots, ajb)
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0 5 10 150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time (sec)
y(t)
Sample ProblemsOverdamped
0,0)0(,1450
2
2
===++=tdt
dyyydtdy
dtyd
Sample Problems
11
0 5 10 150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time (sec)
y(t)
Underdamped0,0)0(,148.0
02
2
===++=tdt
dyyydtdy
dtyd
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Summary
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Examined generic 2nd order differential equation Vibratory systems, RLC circuits
Showed steady-state solution Showed generic transient solutions to
underdamped and overdamped responses Showed characteristic plots of under
damped and overdamped responses to a constant input applied at t=0
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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
RLC CircuitsPart 1
Use differential equations to show the behavior of an RLC circuit as the system changes.
Generate a second-order differential equation from a RLC circuit Identify initial and final conditions Solve the differential equation Recognize if a system is
underdamped/overdamped
Lesson Objectives
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Example 1: Initial and Final Conditions
6
Example 1: Differential Equation
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Example 1: Transient
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Characteristic Equation:
Example 1: Steady State
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Example 1: Solving for Constants
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Example 1: Final Solution
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Looked at an overdamped case Identified initial and final conditions Found and solved representative differential
equation Plotted the results
Summary
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1
Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
RLC CircuitsPart 2
Use differential equations to show the behavior of an RLC circuit as the system changes.
Generate a second-order differential equation from an underdamped RLC circuit Identify initial and final conditions Solve the differential equation Recognize if a system is
underdamped/overdamped Identify the effect of damping on a second-order
system
Lesson Objectives
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Example 2: Initial and Final Conditions
6
Example 2: Differential Equation
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Example 2: Final Solution
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Damping Ratio
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Got some intuition about how RLC circuits behave and contrasted overdamped and underdampedcases Identified initial and final conditions Found and solved representative differential
equations Plotted the results Animated response as the resistance changes to
show the effect of damping on the system
Summary
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1
Dr. Bonnie FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Lab Demo: RLC Circuit
Transient response of an RLC circuit
RLC Circuit Measurements
4
vs
20k +
-vc0.01f
3.3mH
+15v
-15v
+
-
Buffer(Op Amp)
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Lab Demo: RLC Circuits
5
An underdamped RLC circuit has a small R value and results in large peaks An overdamped RLC circuit has a
large R value
Summary
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Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Lab Demo:Applications of Capacitance
Show common applications of capacitance
Applications of Capacitance
dwlC =
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Lab Demo: Applications of Capacitance
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Summary
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Showed capacitive sensors such as touch pads and capacitive microphone, and antenna tuner
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Thanks to Allen Robinson, James Steinberg, Kevin Pham, and Al Ferri for help with demonstration ideas.
Thanks to Marion Crowder for videotaping the demonstration.
Capacitance drawings done by Nathan Parrish.
Credits
7
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Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Lab Demo:Applications of Inductance
Show common applications of inductance
Lab Demo: Applications of Inductance
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Summary
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Discussed energy exchange in inductors mechanical to electrical and vice versa Moving conductor in magnetic field induces
current Changing current in coiled wire causes a
magnetic field Showed inductance applications Passive Sensing (guitar pick-up) Active Sensing (metal detector) Actuation (solenoid, speaker)
Thanks to Allen Robinson, James Steinberg, Kevin Pham, and Al Ferri for help with demonstration ideas. Thanks to Ken Connor and Don Millard for the guitar string experiment.
Thanks to Marion Crowder for videotaping the demonstration.
Inductance drawings done by Nathan Parrish.
Credits
6
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1
Dr. Bonnie FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Module 3Reactive Circuit Wrap Up
Summary of Reactive Circuits Module
Reactive Circuits
Concept Map
3
Background Resistive Circuits
Frequency Analysis
Power
Methods to obtain circuit equations (KCL, KVL, mesh, node, Thvenin)
Current, voltage, sources, resistance
Capacitors Inductors Differential
Equations
RC Circuits RL Circuits 2-order Diff Eqn RLC Circuits Applications
Reactive Circuits
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Understand the basic structure of a capacitor and its fundamental physical behavior
Be able to use the i-v relationship to calculate current from voltage or vice versa
Be able to reduce capacitor connections using using parallel and series connections
Be able to calculate energy in a capacitor Be able to sketch current/voltage/power/energy
curves
Important Concepts and Skills
4
Be able to describe the construction and behavior of an inductor Be able to use the i-v relationship to find current through an inductor from
the voltage across it, and vice versa Be able to explain how a voltage is created across an inductor Be able to analyze inductors in series/parallel Be able to calculate the energy in an inductor Be able to sketch current/voltage/power/energy
curves
Important Concepts and Skills
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Be able to write a differential equation governing the behavior of the circuit
Be able to calculate the time constant, steady-state value, and sketch the response
Important Concepts and Skills
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Given a constant input, be able to determine the steady-state value, time constant, and sketch the response
Be able to write the differential equation that governs the behavior
Be able to predict the type of response (underdamped, overdamped, critically damped)
Be able to compute the damping factor and the resonant frequency
Know that the smaller the damping factor, the larger the oscillations
Important Concepts and Skills
7
Be able to identify the steady-state value Be able to predict the type of response from the roots
(underdamped, critically damped, overdamped)
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Know the purposes of an oscilloscope and a function generator Know several applications of inductors and capacitors when they are
used with non-electrical components
Important Concepts and Skills
8
Concept Map
9
Background Resistive Circuits
Reactive Circuits
Frequency Analysis
Power
1 2
3 4
5
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Do all homework for this module Study for the quiz Continue to visit the forum
Reminder
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