4-2sinusoids in circuitsece2040.ece.gatech.edu/videos/module4handouts.pdf · 1/5/2014 1 school of...
TRANSCRIPT
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9/22/2013
1
Linear Circuits
An introduction to linear electric components and a study of circuits containing such devices.
Dr. Bonnie H. FerriProfessor and Associate ChairSchool 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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2
Resistive vs Reactive Circuits
3
Time
Volta
ge
Concept Map
4
Background Resistive Circuits
Reactive Circuits
Frequency Analysis
Power
Methods to obtain circuit equations (KCL, KVL, mesh, node, Thvenin)
RC, RLC circuits
Frequency Domain
Impedance AC Circuit
Analysis
Transfer Function
Frequency Response
Filters
Frequency Analysis
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School of Electrical and Computer Engineering
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
Sinusoids in Circuits
Review sinusoidal properties and introduce their representation in circuits
Identify sinusoid properties Examine sinusoids in circuits
(Alternating Current)
Lesson Objectives
4
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Sinsoids
5
Amplitude: VmPeriod: T secFrequency (Hz): Frequency (rad/sec):Phase Angle:
v(t) = Vmcos(t + )v(t)
Vm
-Vm
Circuit Responses
6
T Tvin
vout
If the output is from the input, output phase input phase
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Cosines and Sines
7
cos(100t)sin(100t)
sin(t) = cos(t 90o)-sin(t) = cos(t +90o)
Sinusoids and Capacitors
8
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Reviewed sinusoid properties Frequency (Hz, rad/sec), amplitude, phase
Identified sinusoid behavior in linear circuits AC Phase lag/lead
Summary
9
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1
School of Electrical and Computer Engineering
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
Phasors
Use phasors to represent sinusoids
Introduced sinusoids in circuits Alternating Current (AC)
Previous Lesson
4
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Introduce phasors to represent sinusoids
Lesson Objectives
5
Why? Easier than solving differential equations!
Phasorsv(t)=Vmcos(t + )
Polar: V = Vm Rectangular: V = a+bj
6
Im
Re
i(t)=Imcos(t + )Polar: I = Im Rectangular: I = a+bj
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Signal Phasor in Polar Form Phasor in Rectangular Formv(t) = 10cos(100t 45o) V = 10-45o V = 102 j102
v(t) = 10cos(1000t + 90o) V = 1090o V = 0 + 10j = 10j
i(t) = 10cos(500t) I = 100o I = 10 + 0j = 10
i(t) = 10sin(1000t + 20o) = 10cos(1000t + 20o 90o)
I = 10-70o I = 3.42 9.40j
Examples
7
Adding Sinusoids with Phasors
8
Phasorv1 (t) = 7cos(1t+30o) 730o 6.1 + 3.5jv2 (t) = 3cos(1t-60o) 3-60o 1.5 - 2.6jv1(t) + v2(t)
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Multiplying:VI = V1 I2 = V I 1+2
Dividing:V/I = V1 I2 = V /I 1-2
Multiplying/Dividing Phasors
9
V = 530oI = 2-60o
Sinusoids must have same frequencies Adding/subtracting phasors rectangular Multiplying/dividing phasors polar
Summary
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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Impedance
Identify impedances a mathematical tool to analyze reactive circuits with sinusoidal inputs.
Be able to describe impedance Calculate impedances of resistors,
capacitors, and inductors Identify the relationship between voltage and
current based on and impedance value
Lesson Objectives
5
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2
Definition of Impedance
6
Impedance of an Inductor
7
Inductor impedance purely imaginaryScales based on frequencyPositive imaginary, so current lags voltage
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Impedances
8
In-phase Current leads voltage Current lags voltageFrequency invariant Voltage attenuates for
high frequencyCurrent attenuates for high frequency
Defined impedance and calculated impedance of linear devices Described the relationship between the
current and the voltage given impedance
Summary
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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
AC Circuit Analysis
Identify how past techniques apply to impedances in AC circuit analysis.
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
Lesson Objectives
5
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Impedance is Linear
6
Impedances in Series
7
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Impedances in Parallel
8
Kirchhoffs Laws
9
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4
Source Transformations
10
Superposition
11
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5
Kirchhoffs Laws Superposition Node-voltage Mesh-current Thvenin and Norton Equivalent Circuits Source Transformations
Valid Impedance Techniques
12
Example
13
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6
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
Summary
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School of Electrical and Computer Engineering
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
Transfer Functions
Transfer functions characterize the input to output relationship of a system.
Introduce transfer functions to characterize a circuit to find sinusoidal output
Lesson Objectives
5
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Behavior of Sinusoids in Linear Systems
6
y(t) = Aoutcos(t + out)x(t) = Aincos(t + in)
xin youtLinear Circuit
Transfer Function
7
x(t) = Ain(t + in) H() y(t) = Aoutcos(t + out)
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Series RC
8
+
Vo-
Vi
Series RC
9
+Vo-
Vi
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RLC Example
10
+
Vo-
Vi
Series RLC
11
Vi+Vo-
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Using the Transfer Function
12
Vi+Vo
-
R = 20k, L = 3.3mH, C = 0.12F, f = 50Hz
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
Summary
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Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
FrequencySpectrum
Understanding and displaying the frequency content of signals
Introduce the frequency spectrum as a way of showing the frequency content of signals Introduce both linear and log scales for displaying
frequency content
Lesson Objectives
5
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2
Summation of Sines
6
0 1 2 3 4 5 6 7 80
0.5
1
Frequency (Hz)
Am
plitu
de
0 1 2 3 4 5 6 7 80
0.5
1
Frequency (Hz)
Am
plitu
de
0 1 2 3 4 5 6 7 80
0.5
1
Frequency (Hz)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-1
0
1
Time (sec)
x 1(t
)
0 0.5 1 1.5 2 2.5 3-1
0
1
Time (sec)
x s(t
)
0 0.5 1 1.5 2 2.5 3-1
0
1
Time (sec)
x 2(t
)
x1 = sin(22t)
x2 = 0.2sin(26t)
xs = x1+x2
Summation of Sines
7
0 0.5 1 1.5 2 2.5 3-1
0
1
Time (sec)
x 1(t
)
0 0.5 1 1.5 2 2.5 3-1
0
1
Time (sec)
x 2(t
)
0 0.5 1 1.5 2 2.5 3-2
0
2
Time (sec)
x s(t
)
x1 = 0.2sin(22t)
x2 = sin(26t)
xs = x1+x2
0 1 2 3 4 5 6 7 80
0.5
1
Frequency (Hz)
Am
plit
ude
0 1 2 3 4 5 6 7 80
0.5
1
Frequency (Hz)
Am
plitu
de
0 1 2 3 4 5 6 7 80
0.5
1
Frequency (Hz)
Am
plitu
de
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Harmonics
8
)cos()( 01
0 k
N
kk tkAAtx ++=
=
Frequency (rad/sec)0 0 20 30
Frequency Spectrum (Log Scale)
9
Frequency (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
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Example Spectra
10
0 50 100 150 2000
0.5
1
1.5
Frequency (rad/sec)
Mag
nitu
de
0 50 100 150 200-100
-80
-60
-40
-20
0
20
Frequency (rad/sec)
Mag
nitu
de (
deci
bels
)
0 5 10 15 200
0.5
1
1.5
Time (sec)
x(t)
A is a plot of the frequency content of signals
include a fundamental frequency and multiples of it Log scale is often preferred Units are or dB
Summary
11
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1
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Lab Demo: Guitar String FrequencySpectrum
Understanding and displaying the frequency content of signals
Demonstrate the use of a , a common measurement instrument for computing and displaying the frequency spectrum
Lesson Objectives
5
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2
is an instrument to measure and compute the frequency spectrum
Guitar string produces a tone and
Summary
7
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1
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
FrequencyResponse: Linear Plots
Understanding and displaying the frequency response of systems
Introduce the frequency response as a way of showing how a system processes signals of different frequencies
Lesson Objectives
5
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2
Frequency Response
6
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Mag
nitu
de0 200 400 600 800 1000
-100
-80
-60
-40
-20
0
A
ngle
(de
g)
vs
R
-
+vcC+-
Transfer Function
)tan()(
)(1
1)(
1
1)(
2
RCaHRC
HRCjH
=+
=
+=
Circuit Response
7
Vin Vout0 0.05 0.1 0.15 0.2 0.25
-2
-1
0
1
2
Time (sec)
v(t)
0 0.05 0.1 0.15 0.2 0.25-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
v(t)
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Mag
nitu
de
Frequency Domain
50 800
1
50 800
1
Time Domain
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A circuit has the frequency response plot shown. What is steady-state response, vo(t), to an input of vin(t) = 2 + cos(200t)?
Example
8
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Mag
nitu
de0 200 400 600 800 1000
-100
-80
-60
-40
-20
0
A
ngle
(de
g)
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
Summary
9
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Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
FrequencyResponse: Bode Plots
Understanding and displaying the frequency response of systems
Introduce the Bode plot as a way of showing the frequency response
Lesson Objectives
5
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2
Bode Plots
6
Frequency (rad/sec) or f (Hz)1 10 100 1000
Frequency (rad/sec) or f (Hz)1 10 100 1000
Linear Plot and Bode Plot
7
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Bode Plot First-Order Characteristics
8
Bode Plot of RLC Circuit, Overdamped
9
vs
+
-vcC
L
vs+
--
R
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Bode Plot of RLC Circuit, Underdamped
10
Example
11
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)?
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A is a plot of the transfer function versus frequency A 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
Summary
12
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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 Frequency Response
Transient response of an RLC circuit
RLC Circuit Schematic
4
vs
20k +
-vc0.01f
3.3mH
+15v
-15v
+
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2
Lab Demo: RLC Circuit Frequency Response
5
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
Summary
6
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1
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Lowpass and Highpass Filters
Introduce lowpass and highpass filters
Introduce filtering concepts Show the properties of lowpass and highpass filters
Lesson Objectives
5
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2
An is a circuit that has a specific shaped frequency response to attenuate (or filter) signals with specific frequency content
Analog Filters
6
Lowpass Filter
Highpass Filter
Lowpass Filter Example
7
Vin Vout0 0.05 0.1 0.15 0.2 0.25
-2
-1
0
1
2
Time (sec)
v(t)
0 0.05 0.1 0.15 0.2 0.25-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
v(t)
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Mag
nitu
de
Frequency Domain
50 800
1
50 800
1
Time Domain
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3
Pass low frequency components and attenuate high frequency components
Lowpass Filters
8
Linear Plot
Magn
itude
KDC
B
0.707KDC
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
Mag
nitu
de
RC circuitR = 1000, C = 10F
Lowpass Filter Example
9
0.05 0.1 0.15 0.2 0.250
5
10
15
20
Time (sec)
Vou
t
0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
Time (sec)
Vin
CircuitVin Vout
2
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4
Bode Plots of Lowpass Filters
10
Linear Plot
Magn
itude
KDC
B
0.707KDC
Bode Plot
Magn
itude
(dB)
20log10(KDC)3dB
Example Lowpass Filter Bode Plot
11
Bode Plot
Magn
itude
(dB)
20log10(KDC)3dB
101
102
103
104
105
-80
-60
-40
-20
0
Mag
nitu
de (
dB
)
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5
Passes high frequency components and attenuates low frequency components
Highpass Filter
12
Linear Plot
Magn
itude
vin R+
-vo
C
1+
=
RCj
RCjH )(
Highpass Filter Example
13
0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
Time (sec)
Vin
0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
Time (sec)
Vou
tCircuitVin Vout
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RLC Filters
14
vinR +
-voC
L
+= RCjLCH )()( 21
1
Lowpass Filter
Highpass Filter
+
= RCjLC
LCH )()( 22
1vin
R
+
-vo
C
L
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
Summary
15
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1
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Bandpass and Notch Filters
Show schematics and characteristics of notch and bandpass filters
Introduce characteristics of notch and bandpass filters
Lesson Objective
5
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2
Summary of RC Filters
6
vinR +
-voC
vin R+
-vo
C
Summary of RLC Filters
7
vin
R +
-vo
CL
vin
R
+
-vo
C
L
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3
RLC Bandpass Filter
8
+
= RCjLC
RCjH)1(
)(2
Passband
LC1
LR
-3dB
Example Bandpass Filter
9
+
= RCjLC
RCjH)1(
)(2
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4
Notch RLC Filter
10
+
= RCjLC
LCH)1(
1)(
2
2
LC1
Example Notch Filter
11
+
= RCjLC
LCH)1(
1)(
2
2
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5
Different filter characteristics can be found from RC and RLC circuits
passes frequencies in a bandrejects frequencies in a band
Summary
24
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1
Dr. Bonnie FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Lab Demo: Guitar String Filtering
Lowpass filtering of the guitar string signal
Tone Control
4
vin
R1
+
-vo
C
R2+
-
R1 = 10kR2 = 47kC = 0.022f
100k pot for tone 100k pot for volume
Stupidly Wonderful Tone Control 2, www.muzique.com/lab/swtc.htm
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Input Output Relationship
5
CircuitH()Vin Vout
0 0.05 0.1 0.15 0.2 0.25-2
-1
0
1
2
Time (sec)
v(t)
0 0.05 0.1 0.15 0.2 0.25-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
v(t)
Linear Scale: Ai |H()| = Ao|Input| x |H| = |Output|
Bode Scale: |Input|dB+ |H|dB = |Output|dB
Lab Demo: Guitar String Filtering
6
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3
Frequency Response of Lowpass Filter
7
Input and Output Spectra
8
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4
Input/Output Relationship Linear Scale: |Input| x |H| = |Output| Bode Scale: |Input|dB + |H|dB = |Output|dB
First-Order filter: -20dB/dec rolloff Passive filters Made of R, L, and C components Require no power supply
Active filters Made of R, C, and operational amplifiers Require a power supply
Summary
9
Ken Conner from RPI Stupidly Wonderful Tone Control 2,
www.muzique.com/lab/swtc.htm
Credits
10
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1
Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering
School of Electrical and Computer Engineering
Module 4 Frequency Analysis Wrap Up
Summary of the Module
Concept Map
3
Background Resistive Circuits
Reactive Circuits
Frequency Analysis
Power
Methods to obtain circuit equations (KCL, KVL, mesh, node, Thvenin)
RC, RLC circuits
Phasors Impedance AC Circuit
Analysis
Transfer Function
Frequency Response
Filters
Frequency Analysis
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2
Resistive vs Reactive Circuits
4
Time
Volta
ge
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
Important Concepts and Skills
5
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3
Understand impedance Be able to
calculate impedances of resistors, capacitors, and inductors identify the relationship between
voltage and current based on impedance value
Important Concepts and Skills
6
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 (Ohms Law, KCL, KVL, Mesh, Node, Thvenin, Norton)
Important Concepts and Skills
7
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4
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 transferfunctions of a first-order system on a linear scale
Important Concepts and Skills
8
Know the definition of a frequency spectrum
Be able to 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
Important Concepts and Skills
9
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5
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
Important Concepts and Skills
10
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
Important Concepts and Skills
11
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Do all homework for this module Study for the quiz Continue to visit the forum
Reminder
12
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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 Frequency Response
Transient response of an RLC circuit
RLC Circuit Schematic
4
vs
20k +
-vc0.01f
3.3mH
+15v
-15v
+
-
-
9/22/2013
2
Lab Demo: RLC Circuit Frequency Response
5
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
Summary
6