9/22/2013
Linear Circuits
Dr. Bonnie H. Ferri
Professor and Associate Chair
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
9/22/2013
Voltage
Resistive vs Reactive Circuits
Time
3
Concept Map
Resistive
Circuits
Background
Methods to obtain circuit
equations (KCL, KVL,
mesh, node, Thévenin)
Power
Frequency Analysis
Reactive
Circuits
RC, RLC
circuits
• Frequency • Transfer
Function
Domain
Frequency
•
• Impedance
AnalysisFrequency
Response
• AC Circuit
• Filters
Analysis
4
2
1/5/2014
Sinusoids in
Circuits
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Review sinusoidal properties and introduce their representation in
circuits
School of Electrical and Computer Engineering
Lesson Objectives
Identify sinusoid properties
Examine sinusoids in circuits
(Alternating Current)
4
1
1/5/2014
Sinsoids
v(t) = Vmcos(ωt + θ)
v(t)
Vm
Amplitude: Vm
Period: T sec
Frequency (Hz):
Frequency (rad/sec):
Phase Angle:
-Vm
5
Circuit Responses
T
∆T
vin
vout
If the output is
input,
input phase
from the
⇒ output phase
6
2
1/5/2014
Cosines and Sines
cos(100t)
sin(100t)
sin(ωt) = cos(ωt – 90o)
-sin(ωt) = cos(ωt +90o)
7
Sinusoids and Capacitors
8
3
1/5/2014
Summary
Reviewed sinusoid properties
Frequency (Hz, rad/sec), amplitude, phase
Identified sinusoid behavior in linear circuits
AC
Phase lag/lead
9
4
1/5/2014
Phasors
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Use phasors to represent sinusoids
School of Electrical and Computer Engineering
Previous Lesson
Introduced sinusoids in circuits
Alternating Current (AC)
4
1
1/5/2014
Lesson Objectives
Introduce phasors to represent sinusoids
Why?
Easier than solving differential equations!
5
Phasors
Im
v(t)=Vmcos(ωt + θ)
Polar: V = Vm∠θ
Rectangular: V = a+bj
θ
Re
i(t)=Imcos(ωt + θ)
Polar: I = Im∠θ
Rectangular: I = a+bj
6
2
1/5/2014
Examples
Signal
Phasor in Polar Form
Phasor in Rectangular Form
v(t) = 10cos(100t – 45o)
V = 10∠-45o
V = 10√2 – j10√2
v(t) = 10cos(1000t + 90o)
V = 10∠90o
V = 0 + 10j = 10j
i(t) = 10cos(500t)
I = 10∠0o
I = 10 + 0j = 10
i(t) = 10sin(1000t + 20o) =
10cos(1000t + 20o – 90o)
I = 10∠-70o
I = 3.42 – 9.40j
7
Adding Sinusoids with Phasors
Phasor
v1 (t) = 7cos(ω1t+30o)
7∠30o
6.1 + 3.5j
v2 (t) = 3cos(ω1t-60o)
3∠-60o
1.5 - 2.6j
v1(t) + v2(t)
8
3
1/5/2014
Multiplying/Dividing Phasors
Multiplying:
VI = V∠θ1 × I∠θ2 = V I ∠θ1+θ2
Dividing:
V/I = V∠θ1 ÷ I∠θ2 = V /I ∠θ1-θ2
V = 5∠30o
I = 2∠-60o
9
Summary
Sinusoids must have same frequencies
Adding/subtracting phasors
rectangular
Multiplying/dividing phasors
polar
10
4
9/22/2013
Impedance
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
Identify impedances – a mathematical tool to analyze reactive
circuits with sinusoidal inputs.
School of Electrical and Computer Engineering
Lesson Objectives
Be able to describe impedance
Calculate impedances of resistors,
capacitors, and inductors
Identify the relationship between voltage and
current based on and impedance value
5
1
9/22/2013
Definition of Impedance
6
Impedance of an Inductor
•Inductor impedance purely imaginary
•Scales based on frequency
•Positive imaginary, so current lags voltage
7
2
9/22/2013
Impedances
In-phase
Current leads voltage
Current lags voltage
Frequency invariant
Voltage attenuates for
high frequency
Current attenuates for
high frequency
8
Summary
Defined impedance and calculated
impedance of linear devices
Described the relationship between the
current and the voltage given impedance
9
3
9/22/2013
AC Circuit
Analysis
Nathan V. Parrish
PhD Candidate & Graduate
Research Assistant
School of Electrical and
Computer Engineering
Identify how past techniques apply to impedances in AC circuit
analysis.
School of Electrical and Computer Engineering
Lesson Objectives
Apply techniques from DC analysis to sinusoidal
systems
Find equivalent impedances for devices in
series/parallel
Use superposition for analysis: particularly for
systems with multiple frequencies
Be able to analyze a system using these techniques
5
1
9/22/2013
Impedance is Linear
6
Impedances in Series
7
2
9/22/2013
Impedances in Parallel
8
Kirchhoff’s Laws
9
3
9/22/2013
Source Transformations
10
Superposition
11
4
9/22/2013
Valid Impedance Techniques
Kirchhoff’s Laws
Superposition
Node-voltage
Mesh-current
Thévenin and Norton Equivalent Circuits
Source Transformations
12
Example
13
5
9/22/2013
Summary
Showed how DC analysis techniques are
applied in sinusoidal systems
Used superposition to analyze a system with
multiple frequencies
Solved an example system using these
techniques
14
6
1/5/2014
Transfer Functions
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and Computer
Engineering
Transfer functions characterize the input to output relationship of a
system.
School of Electrical and Computer Engineering
Lesson Objectives
Introduce transfer functions
to characterize a circuit
to find sinusoidal output
5
1
1/5/2014
Behavior of Sinusoids in Linear Systems
xin
Linear
Circuit
x(t) = Aincos(ωt + θin)
yout
y(t) = Aoutcos(ωt + θout)
6
Transfer Function
x(t) = Ain(ωt + θin)
H(ω)
y(t) = Aoutcos(ωt + θout)
7
2
1/5/2014
Series RC
+
Vo
Vi
-
8
Series RC
Vi
+
Vo
-
9
3
1/5/2014
RLC Example
+
Vo
Vi
-
10
Series RLC
Vi
+
Vo
-
11
4
1/5/2014
Using the Transfer Function
Vi
+
Vo
-
R = 20kΩ, L = 3.3mH,
C = 0.12µF, f = 50Hz
12
Summary
Introduced the concept of a transfer function
(output phasor)/(input phasor)
Showed how to calculate a transfer function for a
particular system
Impedance method (voltage divider law)
Showed how to use a transfer
function to compute the output
phasor
13
5
9/22/2013
Frequency
Spectrum
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Understanding and displaying the frequency content of signals
School of Electrical and Computer Engineering
Lesson Objectives
Introduce the frequency spectrum as a way of showing
the frequency content of signals
Introduce both linear and log scales for displaying
frequency content
5
1
9/22/2013
Summation of Sines
x1 = sin(2π2t)
1
0
-1
0
1
0.5
1
1.5
Time (sec)
2
2.5
A m plitud e
x (t)
1
0.5
0
0
3
x2 = 0.2sin(2π6t)
0
-1
0
0.5
1
1.5
Time (sec)
2
2.5
0
0
3
6
7
8
1
2
3
4
5
Frequency (Hz)
6
7
8
1
2
3
4
5
Frequency (Hz)
6
7
8
1
xs = x1+x2
0
0.5
1
1.5
Time (sec)
2
2.5
Am plitude
s
x (t)
3
4
5
Frequency (Hz)
0.5
1
-1
0
2
1
Am plitude
2
x (t)
1
1
0.5
0
0
3
6
Summation of Sines
x1 = 0.2sin(2π2t)
0
-1
0
0.5
1
1.5
Time (sec)
2
2.5
1
A m p litud e
1
x (t)
1
0
0
3
x2 = sin(2π6t)
0
0.5
1
1.5
Time (sec)
2
2.5
3
4
5
Frequency (Hz)
6
7
8
1
2
3
4
5
Frequency (Hz)
6
7
8
1
2
3
4
5
Frequency (Hz)
6
7
8
1
xs = x1+x2
0
0.5
1
1.5
Time (sec)
2
2.5
3
A m plitude
s
x (t)
2
0.5
0
0
3
2
-2
0
1
1
A m plitude
2
x (t)
1
-1
0
0.5
0.5
0
0
7
2
9/22/2013
Harmonics
x (t ) = A0 + ∑ Ak cos( kω0t + θ k )
N
k =1
0
ω0
2ω0 3ω0
Frequency ω (rad/sec)
8
Frequency Spectrum (Log Scale)
Frequency ω (rad/sec) or f (Hz)
1 10 100 1000
• Some frequency components are better viewed in
log scale
• Larger dynamic range while maintaining resolution
at the low amplitude range
• Historical usage, going back to time when graphs
drawn by hand
9
3
9/22/2013
Example Spectra
1.5
1
1
x(t)
Magnitude
1.5
0.5
0
0
5
10
Time (sec)
15
0.5
0
0
20
50
100
Frequency (rad/sec)
150
200
Magnitude (decibels)
20
0
-20
-40
-60
-80
-100
0
50
100
Frequency (rad/sec)
150
200
10
Summary
A
of signals
is a plot of the frequency content
include a fundamental frequency and
multiples of it
Log scale is often preferred
Units are
or dB
11
4
9/22/2013
Dr. Bonnie H. Ferri
Lab Demo: Guitar
String Frequency
Spectrum
Professor and Associate Chair
School of Electrical and
Computer Engineering
Understanding and displaying the frequency content of signals
School of Electrical and Computer Engineering
Lesson Objectives
Demonstrate the use of a
,a
common measurement instrument for computing
and displaying the frequency spectrum
5
1
9/22/2013
Summary
is an instrument to measure and
compute the frequency spectrum
Guitar string produces a
tone and
7
2
9/22/2013
Frequency
Response:
Linear Plots
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Understanding and displaying the frequency response of systems
School of Electrical and Computer Engineering
Lesson Objectives
Introduce the frequency response as a way of showing
how a system processes signals of different frequencies
5
1
9/22/2013
Frequency Response
C
+vs
vc
-
Transfer Function
H(ω) =
1
1+ jωRC
1
H(ω) =
1 + (ωRC)2
0.6
0.4
0.2
0
0
200
400
200
400
0
ω
600
800
1000
600
800
1000
-20
Angle (deg)
+
0.8
Magnitude
R
1
∠H(ω) = −a tan(ωRC)
-40
-60
-80
-100
0
ω
6
Circuit Response
Time Domain
1.5
2
1
v(t)
Vout
Vin
0
0.5
0
-0.5
-1
-1
-2
0
0.05
0.1
0.15
0.2
0.25
-1.5
0
Time (sec)
0.05
0.1
0.15
0.2
0.25
Time (sec)
Frequency Domain
1
0.8
1
M ag nitud e
v(t)
1
1
0.6
0.4
0.2
50
800
ω
0
0
200
400
ω
600
800
1000
50
800
ω
7
2
9/22/2013
Example
0.8
Magnitude
A circuit has the frequency response plot
shown. What is steady-state response,
vo(t), to an input of vin(t) = 2 + cos(200t)?
1
0.6
0.4
0.2
0
0
200
400
200
400
0
ω
600
800
1000
600
800
1000
Angle (deg)
-20
-40
-60
-80
-100
0
ω
8
Summary
A
is a plot of the transfer function
versus frequency
The frequency response can be used to determine the
steady-state sinusoidal response of a circuit at different
frequencies
9
3
1/5/2014
Frequency
Response:
Bode Plots
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Understanding and displaying the frequency response of systems
School of Electrical and Computer Engineering
Lesson Objectives
Introduce the Bode plot as a way of showing the
frequency response
5
1
1/5/2014
Bode Plots
1
10
100 1000
Frequency ω (rad/sec) or f (Hz)
1
10
100 1000
Frequency ω (rad/sec) or f (Hz)
6
Linear Plot and Bode Plot
7
2
1/5/2014
Bode Plot First-Order Characteristics
8
Bode Plot of RLC Circuit, Overdamped
L
+
-
- v
s
R
C
+
vc
-
9
3
1/5/2014
Bode Plot of RLC Circuit, Underdamped
10
Example
A circuit has the Bode plot shown. What is the
steady-state response of an input of
vs(t)=1+cos(100t-45o)+cos(3000t)?
11
4
1/5/2014
Summary
A
frequency
A
is a plot of the transfer function versus
is the frequency response on a log scale
Units are
or dB
RC Circuit
○ magnitude goes down by 20dB/decade
○ phase goes from 0o to -90o
RLC Circuit
○ magnitude goes down by 40dB/decade
○ phase goes from 0o to -180o
RLC with low damping has resonant peak
12
5
9/22/2013
Lab Demo: RLC
Circuit Frequency
Response
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 Schematic
+
-
+15v
-15v
vs
3.3mH 20kΩ
0.01µf
+
vc
-
4
1
9/22/2013
Lab Demo: RLC Circuit Frequency
Response
5
Summary
Low R means low damping and high
resonant peak
The Bode plot is generated by a sine
sweep
Input sinusoids of different frequencies and
calculate the gain (Ao/Ai) and phase for each
response
Compute and plot 20*log10(Ao/Ai) vs f
Plot phase vs f
6
2
9/22/2013
Lowpass and
Highpass Filters
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Introduce lowpass and highpass filters
School of Electrical and Computer Engineering
Lesson Objectives
Introduce filtering concepts
Show the properties of lowpass and highpass filters
5
1
9/22/2013
Analog Filters
An
is a circuit that has a specific shaped
frequency response to attenuate (or filter) signals with
specific frequency content
Lowpass Filter
Highpass Filter
ω
ω
6
Lowpass Filter Example
Time Domain
1.5
2
1
v(t)
Vout
Vin
0
0.5
0
-0.5
-1
-1
-2
0
0.05
0.1
0.15
0.2
0.25
-1.5
0
Time (sec)
0.05
0.1
0.15
0.2
0.25
Time (sec)
Frequency Domain
1
0.8
1
M ag nitud e
v(t)
1
1
0.6
0.4
0.2
50
800
ω
0
0
200
400
ω
600
800
1000
50
800
ω
7
2
9/22/2013
Lowpass Filters
Pass low frequency components and attenuate high
frequency components
Linear Plot
KDC
RC circuit
R = 1000Ω, C = 10µF
1
0.8
Magnitude
Magnitude
0.707KDC
0.6
0.4
0.2
0
0
ω
ωB
200
400
ω
600
800
1000
8
Lowpass Filter Example
12
10
20
Vin
6
Circuit
Vout
4
2
0
0.05
15
Vout
V in
8
10
5
0.1
0.15
Time (sec)
0.2
0.25
0
0.05
2
0.1
0.15
Time (sec)
0.2
0.25
ω
9
3
9/22/2013
Bode Plots of Lowpass Filters
Linear Plot
Magnitude
0.707KDC
ωB
ω
3dB
ω
Magnitude (dB)
KDC
Bode Plot
20log10(KDC)
10
Example Lowpass Filter Bode Plot
Bode Plot
0
3dB
ω
M agnitude (dB)
Magnitude (dB)
20log10(KDC)
-20
-40
-60
-80 1
10
10
2
3
10
ω
10
4
5
10
11
4
9/22/2013
Highpass Filter
Passes high frequency components and attenuates low
frequency components
Linear Plot
C
Magnitude
vin
H ( ω) =
R
+
vo
-
RCj ω
RCjω + 1
ω
12
Highpass Filter Example
12
12
10
Vin
V in
8
Circuit
Vout
6
6
4
4
2
2
0
0.05
8
V out
10
0.1
0.15
Time (sec)
0.2
0
0.05
0.25
0.1
0.15
Time (sec)
0.2
0.25
ω
13
5
9/22/2013
RLC Filters
Lowpass Filter
L
vin
R
+
vo
-
C
H(ω) =
1
(1− LCω2 ) + RCjω
Highpass Filter
R
C
+
vin
L
vo
H(ω) =
− LCω2
(1− LCω2 ) + RCjω
-
14
Summary
An
is a circuit that has a specific shaped frequency
response
A
passes low frequency component in signals and
attenuates high frequency components
A
passes high frequency components in signals and
attenuates low frequency components
15
6
9/22/2013
Bandpass and
Notch Filters
Dr. Bonnie H. Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Show schematics and characteristics of notch and bandpass filters
School of Electrical and Computer Engineering
Lesson Objective
Introduce characteristics of notch and
bandpass filters
5
1
9/22/2013
Summary of RC Filters
R
vin
C
vin
C
+
vo
-
R
+
vo
-
6
Summary of RLC Filters
L
R
vin
R
+
vo
-
C
C
+
vin
L
vo
-
7
2
9/22/2013
RLC Bandpass Filter
1
LC
R
L
H(ω) =
RCjω
(1− LCω2 ) + RCjω
-3dB
Passband
8
Example Bandpass Filter
H(ω) =
RCjω
(1 − LCω2 ) + RCjω
9
3
9/22/2013
Notch RLC Filter
1
LC
1− LCω2
H(ω) =
(1− LCω2 ) + RCjω
10
Example Notch Filter
H(ω) =
1 − LCω2
(1 − LCω2 ) + RCjω
11
4
9/22/2013
Summary
Different filter characteristics can be found from RC and
RLC circuits
passes frequencies in a band
rejects frequencies in a band
24
5
9/22/2013
Lab Demo:
Guitar String
Filtering
Dr. Bonnie Ferri
Professor and Associate Chair
School of Electrical and
Computer Engineering
Lowpass filtering of the guitar string signal
School of Electrical and Computer Engineering
Tone Control
R1
+
vin
-
R2
C
100kΩ pot
for tone
100kΩ pot
for volume
+
vo
-
R1 = 10kΩ
R2 = 47kΩ
C = 0.022µf
“Stupidly Wonderful Tone Control 2”,
www.muzique.com/lab/swtc.htm
4
1
9/22/2013
Input – Output Relationship
1.5
2
Vin
0
-1
Circuit
H(ω)
1
Vout
0.5
v(t)
v(t)
1
0
-0.5
-1
-2
0
0.05
0.1
0.15
0.2
0.25
Time (sec)
-1.5
0
0.05
0.1
0.15
0.2
0.25
Time (sec)
Linear Scale: Ai |H(ω)| = Ao
|Input| x |H| = |Output|
Bode Scale:
|Input|dB + |H|dB = |Output|dB
5
Lab Demo: Guitar String Filtering
6
2
9/22/2013
Frequency Response of Lowpass Filter
7
Input and Output Spectra
8
3
9/22/2013
Summary
Input/Output Relationship
First-Order filter: -20dB/dec rolloff
Passive filters
Active filters
Linear Scale: |Input| x |H| = |Output|
Bode Scale: |Input|dB + |H|dB = |Output|dB
Made of R, L, and C components
Require no power supply
Made of R, C, and operational amplifiers
Require a power supply
9
Credits
Ken Conner from RPI
“Stupidly Wonderful Tone Control 2”,
www.muzique.com/lab/swtc.htm
10
4
9/22/2013
Dr. Bonnie H. Ferri
Module 4
Frequency
Analysis Wrap Up
Professor and Associate Chair
School of Electrical and
Computer Engineering
Summary of the Module
School of Electrical and Computer Engineering
Concept Map
Resistive
Circuits
Background
Methods to obtain circuit
equations (KCL, KVL,
mesh, node, Thévenin)
Power
Frequency Analysis
Reactive
Circuits
RC, RLC
circuits
• Transfer
• Phasors
Function
• Impedance
Frequency
•
• AC Circuit
AnalysisFrequency
Response
Analysis
• Filters
3
1
9/22/2013
Voltage
Resistive vs Reactive Circuits
Time
4
Important Concepts and Skills
Be able to
identify sinusoid properties (amplitude, frequency, angular frequency,
period, phase)
find phasors of sinusoidal functions
add sinusoids using phasors
Understand and describe the
properties of sinusoids in
capacitors and inductors
5
2
9/22/2013
Important Concepts and Skills
Understand impedance
Be able to
calculate impedances of resistors, capacitors, and inductors
identify the relationship between
voltage and current based on
impedance value
6
Important Concepts and Skills
Given a source frequency, be able to
convert RLC circuits into equivalent circuits with impedances
find equivalent impedances for devices in series/parallel
solve for voltages and currents using
resistor analysis methods
(Ohm’s Law, KCL, KVL, Mesh, Node,
Thévenin, Norton)
7
3
9/22/2013
Important Concepts and Skills
Know
the definition of a transfer function
how a linear system responds to a sinusoid in steady state (how the amplitude and
phase change but the frequency stays the same)
the meaning of the plot of the transfer function in terms of finding an output
amplitude
Be able to
find the transfer functions of simple RL, RC and
RLC circuits
sketch the magnitude and angle of the transfer
functions of a first-order system on a linear scale
8
Important Concepts and Skills
Know
Be able to
the definition of a frequency spectrum
plot a frequency spectrum of a sum of sinusoids
Recognize high and low frequency content in a
signal in both the time domain and in the frequency
domain
9
4
9/22/2013
Important Concepts and Skills
Know
the what a frequency response is
and understand the graphical features of RC and
RLC circuits when plotted on linear scales and
on Bode scales
Be able to
sketch a frequency response from a transfer
function on linear scales
match time domain and frequency domain inputs
and corresponding outputs for a circuit with a
known frequency response
10
Important Concepts and Skills
Know
the motivation for using filters
the definition of a filter
the frequency response features of lowpass,
highpass, bandpass, and notch filters
Be able to
identify RC and RLC circuits as being lowpass,
bandpass, highpass, or notch
determine acceptable circuit parameters to
achieve desired bandwidth, corner
frequencies, and/or passband or rejection
frequencies
11
5
9/22/2013
Reminder
Do all homework for this module
Study for the quiz
Continue to visit the forum
12
6
9/22/2013
Lab Demo: RLC
Circuit Frequency
Response
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 Schematic
+
-
+15v
-15v
vs
3.3mH 20kΩ
0.01µf
+
vc
-
4
1
9/22/2013
Lab Demo: RLC Circuit Frequency
Response
5
Summary
Low R means low damping and high
resonant peak
The Bode plot is generated by a sine
sweep
Input sinusoids of different frequencies and
calculate the gain (Ao/Ai) and phase for each
response
Compute and plot 20*log10(Ao/Ai) vs f
Plot phase vs f
6
2