9/10/2013
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
1
9/10/2013
Voltage
Resistive vs Reactive Circuit
Time
3
Concept Map
Resistive
Circuits
Background
Current, voltage,
sources, resistance
Methods to obtain circuit
equations (KCL, KVL,
mesh, node, Thévenin)
Reactive Circuits
• Capacitors • RC Circuits
• Inductors • RL Circuits
Reactive
• Differential
• 2-order Diff Eqn
Circuits
Equations • RLC Circuits
• Applications
Power
Frequency
Analysis
4
2
9/10/2013
Capacitance
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Describe the behavior of capacitors by calculating:
•the charge stored on the capacitor plates
•the current flowing through the capacitor
•the voltage across the capacitor
•the capacitance of the capacitor
School of Electrical and Computer Engineering
Lesson Objectives
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
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Capacitors
6
Capacitors and Charge
7
2
9/10/2013
Current and Voltage
8
Calculating Capacitance
9
3
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Permittivity of Common Materials
Material
Air
Approximate εr (or k)
1
Teflon
2.1
Glass
4.7
Paper
Rubber
Silicon
Water
3.9
7.0
11.7
78.5 (varies by T)
10
Current “Through” A Capacitor
11
4
9/10/2013
Summary
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
12
5
9/10/2013
Capacitors
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Present how capacitors work in a system
•Identify behavior in DC circuits
•Graphically represent the relationships between current, voltage,
power, and energy
School of Electrical and Computer Engineering
Lesson Objectives
Analyzing capacitors in series/parallel
Analyze DC circuits with capacitors
Calculate energy in a capacitor
Sketch current/voltage/power/energy curves
5
1
9/10/2013
Capacitors in Parallel
6
Capacitors in Series
7
2
9/10/2013
Behavior in DC Circuits
8
Stored Energy
9
3
9/10/2013
Graphs
10
Summary
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
11
4
9/10/2013
Inductance
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Introduce inductors and describe how they work
•Calculate current and voltage for inductors
School of Electrical and Computer Engineering
Lesson Objectives
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
5
1
9/10/2013
Inductors
6
Current and Voltage
7
2
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Ampère’s Law
8
How Inductors Work
9
3
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Voltages Across a Wire
10
Summary
Presented the equations for current and
voltage in inductors
Introduced Ampère’s 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
11
4
9/10/2013
Inductors
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Present how inductors work in a system
•Identify behavior in DC circuits
•Graphically represent the relationships between current, voltage,
power, and energy
School of Electrical and Computer Engineering
Learning Objectives
Analyze inductors in series/parallel
Analyze DC circuits with inductors
Calculate the energy in an inductor
Sketch current/voltage/power/energy curves
5
1
9/10/2013
Inductors in Series
6
Inductors in Parallel
7
2
9/10/2013
Behavior in DC Circuits
8
Stored Energy
9
3
9/10/2013
Graphs
10
Summary
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
11
4
9/10/2013
Dr. Bonnie H. Ferri
First-Order
Differential
Equations
Professor and Associate Chair
School of Electrical and
Computer Engineering
Solve and graph solutions to first-order differential equations
School of Electrical and Computer Engineering
Lesson Objectives
Examine first-order differential equations with a
constant input
Write the solution
Sketch the solution
5
1
9/10/2013
Ordinary Differential Equations
ODE: Include functions of variables and
their derivatives.
dy
+ 2y = 4
dt
dy
− 2y = 4sin(ωt)
dt
dv
dy
d2y
+ 2v = i(t)
+
2
+
4y
=
f
(t)
dt
dt 2
dt
6
Models of Physical Systems
Inputs, f(t)
Heat
source
Thermal
System
Temp
System
Model
Production
Outputs, y(t)
Biological
System
Population
dy
+ ay = f (t)
dt
Voltage or
current
source
Linear
Circuit
Voltage
or
current
7
2
9/10/2013
Solution to First-Order Differential Equation
dy
+ ay = K,
dt
Has solution:
y(t) =
y(0)
K
(1− e−at ) + y(0)e− at , t ≥ 0
a
If a ≥ 0, e −at → 0
y(t) →
K = steady-state
a
8
Graph of Response
y(t) =
K
(1 − e−at ) + y(0)e− at , t ≥ 0
a
K
a

K  −at
 y(0) −  e , t ≥ 0

a
9
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9/10/2013
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
y(t)
y(t)
Time Constant
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
– time, τ, for
exponential transient to decay to e−1 ≈ 0.37
of its initial value (or 63% to its final value)
10
Sample Problems
Pause
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9/10/2013
Summary
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
12
5
9/10/2013
RC Circuits
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Generate a differential equation that describes the behavior of a
circuit with resistors and capacitors
•Solve the differential equation for step inputs (or switching
constant inputs)
•Graph the behavior
School of Electrical and Computer Engineering
Lesson Objectives
Generate a differential equation from a circuit
Identify initial and final conditions
Solve the differential equation
Graph the result
5
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9/10/2013
Behavior of RC circuits
6
Example 1: Initial and Final Conditions
7
2
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Example 1: Differential Equation
8
Example 1: Graph
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Example 2: Initial and Final Conditions
10
Example 2: Differential Equation
11
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9/10/2013
Example 2: Graph
12
Summary
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
13
5
9/10/2013
Module 3
Lab Demo:
RC Circuits
Dr. Bonnie Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Summary of Reactive Circuits Module
School of Electrical and Computer Engineering
RC Circuit
+ v
-
R
+
C
R
+
vc
-
vs
~
+
C
vc
-
3
1
9/10/2013
RC Circuit
+ v
-
R
+
C
R
+
vc
-
vs
~
+
C
vc
-
4
RC Circuit
+ v
-
R
+
C
R
+
vc
-
vs
~
+
C
vc
-
5
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9/10/2013
RC Circuit
+ v
-
R
+
C
R
+
vc
-
vs
~
+
C
vc
-
6
RC Circuit
+ v
-
R
+
C
R
+
vc
-
vs
~
+
C
vc
-
7
3
9/10/2013
Lab Demo: RC Circuits
8
Summary
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
9
4
9/10/2013
RL Circuits
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Use differential equations to show the behavior of an RL circuit as
the system changes.
School of Electrical and Computer Engineering
Lesson Objectives
Generate a differential equation from a circuit
Identify initial and final conditions
Solve the differential equation
Graph the result
5
1
9/10/2013
Behavior of RL circuits
6
Example 1: Initial and Final Conditions
7
2
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Example 1: Differential Equation
8
Example 1: Graph
vL
5V
1.84V
(63.2%)
0.68V
(86.5%)
t
9
3
9/10/2013
Example 2: Initial and Final Conditions
10
Example 2: Differential Equation
11
4
9/10/2013
Example 2: Graph
iL
4mA
1.16mA
(63.2%)
0.11mA
(86.5%)
1
2 mA
t
12
Summary
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
13
5
9/10/2013
Second-Order
Differential
Equations
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Recognize types of second-order responses
School of Electrical and Computer Engineering
Lesson Objectives
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
5
1
9/10/2013
Ordinary Differential Equations
ODE: Include functions of variables
and their derivatives
d 2 y dy
+ 2 + 4y = f (t)
dt 2
dt
6
Models of Physical Systems
Inputs, f(t)
Force
Vibratory
System
System
Model
Position
Outputs, y(t)
Voltage or
current
source
d 2 y dy
+ 2 + 4y = f (t)
dt 2
dt
RLC
Circuit
Voltage
or
current
7
2
9/10/2013
Solutions to Second-Order Differential
Equation
d2y
dy
dy
+
a
+
a
y
=
K,
y(0)and
1
2
dt 2
dt
dt t=0
Solution: y(t) = steady-state + transient
y(t ) →
K
a2
Three possible forms depending on
roots of (s2+a1s+a2)=0.
8
Transient Response
(s2+a1s+a2)=0.
((real and distinct roots, r1 and r2)
K1e r1t + K 2e r2t
((real and equal roots, r and r)
K1ert + K 2tert
((complex roots, a±jb)
Keat sin(bt + ϕ)
9
3
9/10/2013
Sample Problems
d2 y
Overdamped
dt
2
0.4
+5
dy
dy
+ 4 y = 1, y(0) = 0,
=0
dt
dt t =0
0.35
0.3
y(t)
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
Time (sec)
Sample Problems
d2 y
Underdamped
dt
0.4
2
0.35
+ 0.8
dy
dy
+ 4 y = 1, y(0) = 0,
=0
dt
dt t =0
0.3
y(t)
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
Time (sec)
11
4
9/10/2013
Summary
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
12
5
9/10/2013
RLC Circuits
Part 1
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Use differential equations to show the behavior of an RLC circuit
as the system changes.
School of Electrical and Computer Engineering
Lesson Objectives
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
5
1
9/10/2013
Example 1: Initial and Final Conditions
6
Example 1: Differential Equation
7
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Example 1: Transient
Characteristic Equation:
8
Example 1: Steady State
9
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Example 1: Solving for Constants
10
Example 1: Final Solution
11
4
9/10/2013
Summary
Looked at an overdamped case
Identified initial and final conditions
Found and solved representative differential
equation
Plotted the results
12
5
9/10/2013
RLC Circuits
Part 2
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
•Use differential equations to show the behavior of an RLC circuit
as the system changes.
School of Electrical and Computer Engineering
Lesson Objectives
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
5
1
9/10/2013
Example 2: Initial and Final Conditions
6
Example 2: Differential Equation
7
2
9/10/2013
Example 2: Final Solution
8
Damping Ratio
9
3
9/10/2013
Summary
Got some intuition about how RLC circuits behave
and contrasted overdamped and underdamped
cases
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
10
4
9/10/2013
Lab Demo:
RLC Circuit
Dr. Bonnie Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Transient response of an RLC circuit
School of Electrical and Computer Engineering
RLC Circuit Measurements
+ +15v
Buffer
(Op- Amp)
-15v
vs
3.3mH 20kΩ
0.01µf
+
vc
-
4
1
9/10/2013
Lab Demo: RLC Circuits
5
Summary
An underdamped RLC circuit has a
small R value and results in large
peaks
An overdamped RLC circuit has a
large R value
6
2
9/10/2013
Lab Demo:
Applications of
Capacitance
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Show common applications of capacitance
School of Electrical and Computer Engineering
Applications of Capacitance
C =
ε wl
d
1
9/10/2013
Lab Demo: Applications of
Capacitance
5
Summary
Showed capacitive sensors
such as touch pads and
capacitive microphone, and
antenna tuner
6
2
9/10/2013
Credits
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.
7
3
9/10/2013
Lab Demo:
Applications of
Inductance
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Show common applications of inductance
School of Electrical and Computer Engineering
Lab Demo: Applications of
Inductance
4
1
9/10/2013
Summary
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)
5
Credits
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.
6
2
9/10/2013
Module 3
Reactive Circuit
Wrap Up
Dr. Bonnie Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Summary of Reactive Circuits Module
School of Electrical and Computer Engineering
Concept Map
Resistive
Circuits
Background
Current, voltage,
sources, resistance
Methods to obtain circuit
equations (KCL, KVL,
mesh, node, Thévenin)
Reactive Circuits
• Capacitors • RC Circuits
• Inductors • RL Circuits
• Differential
Reactive
• 2-order Diff Eqn
Circuits
Equations
• RLC Circuits
• Applications
Power
Frequency
Analysis
3
1
9/10/2013
Important Concepts and Skills
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
4
Important Concepts and Skills
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
5
2
9/10/2013
Important Concepts and Skills
Given a constant input, be able to determine the steady-state value, time
constant, and sketch the response
Be able to write a differential equation governing
the behavior of the circuit
Be able to calculate the time constant, steadystate value, and sketch the response
6
Important Concepts and Skills
Be able to identify the steady-state value
Be able to predict the type of response from the roots
(underdamped, critically damped, overdamped)
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
7
3
9/10/2013
Important Concepts and Skills
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
8
Concept Map
1
Background
2
Resistive
Circuits
5
Power
3
Reactive
Circuits
4
Frequency
Analysis
9
4
9/10/2013
Reminder
Do all homework for this module
Study for the quiz
Continue to visit the forum
10
5