Transcript ECE_3340_FT_LT_part2.ppsx

ECE3340
Review of Numerical Methods for Fourier
and Laplace Transform Applications
Part 2 Laplace
PROF. HAN Q. LE
Note: PPT file is the main outline of the chapter topic –
associated Mathematica file(s) contain details and assignments
Remember this question?
Which one is periodic and which one is not?
(a)
(b)
This is for real
Which one is strictly periodic and which one is not?
Remember these circuits?
Consider this circuit that has no resistors, no dissipative elements (things
that absorb power or energy permanently and not give back to the circuit).
Assume perfect capacitor and perfect inductor.
Then switch A is opened at time tA<=0;
simultaneously or after tA, switch B is
instantaneously closed at t=0.
Switch A was connected for a
long time t<0 and switch B
was open.
A
B
+
-
V=5 V
0.3 mF
+
47 mH
vout
-
Q.3 What is the voltage vout before and after the switches are activated?
Sketch your guess what it looks like (only “guessing” is asked, no solving
equation or anything complicated).
Then switch A is opened at time tA<=0;
simultaneously or after tA, switch B is
instantaneously closed at t=0.
Switch A was connected for a
long time t<0 and switch B
was open.
A
B
+
-
V=5 V
0.3 mF
+
47 mH
vout
-
What happens if we have a resistor in here?
Not just with circuits…
https://www.youtube.com/watch?v=99ZE2RG
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https://www.youtube.com/watch?v=38XPT9s
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https://www.youtube.com/watch?v=sPLawCXv
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https://www.youtube.com/watch?v=b_ujQiEyT
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What do those things discussed above
- with transient behavior from an
initial condition, have in common
regarding numerical computation
methods?
Numerical application of Laplace
transform
When design a circuit, we want to know how it works, what it does
for the intended application. We want numerical simulation results
such as this:
input
output
This is what “numerical methods” is about. This is your objective:
learn how to use computers to apply to ECE problems.
Numerical Discrete Fourier Transform (DFT) can also be applied…but
Overview

(or the “big” perspective)
We have learned Fourier transform, Laplace transform, and
their applications in scientific/engineering problem –
specifically for ECE, circuit and signal analysis. (Math 3321)

Fourier transform is most applicable for harmonic, steady
state phenomena (or when a starting time is not important).

Laplace transform is most applicable when the initial
condition is essential.

In computation and digital applications, we use numerical
methods such as DFT or Z-transform especially for digital
control of physical analog systems.
we
were
here
now we go
here
Learning objectives: review and practice numerical
computation of these methods with specific examples
of circuits and signal processing.
Checklist of reviewing/learning
topics

Harmonic signal analysis: Fourier transform



Power spectral density
Frequency response of a system (circuit)

Transfer function

Bode plot (Nyquist plot if time permits)
Filters

Analog

Digital
Similar concepts –
for both Fourier
and Laplace
whether you have learned all these things
and only need a refresher or you have not
learned much, it’s good to go through the
complete list of topics, do exercises and
be proficient at numerical methods
Concept Review: Signal Processing



All electronics around us involve signal processing.
Signal represents information. That information can be
something we generate. The APP is an example of
sound signal. Other common types: texts, images, all
sensors (as discussed previously)
Electronics deal with signals: preprocessing, postacquisition, analog, digital.
Concept Review: Signal Processing
(cont.)

Signal processing is a general concept, not a single specific
operation. It includes:




signal synthesis or signal acquisition
signal conditioning (transforming): shaping, filtering, amplifying
signal transmitting (telecomm) or applying for control (robotics)
signal receiving and analysis: transforming the signal, converting
into information, for implementing certain controls.
Signal processing is mathematical operation; electronics are
simply tools.
 Some computation are high-level signal processing: dealing
directly with encode or embedded information rather than raw
signal.

An APP to illustrate basic concepts of signals
ECE1100
APP
CARRIER AND SIGNAL MODULATION
RUN STATUS CONTROL
Concept review: numerical methods
Signal and AC circuit problems
Harmonic
function
Fourier
transform
• RLC or any time-varying linear
circuits. Applicable to linear
portion of circuits that include
nonlinear elements
• Signal processing
• signal analysis (spectral
decomposition)
Complex
number
&analysis
Phasors
• filtering, conditioning (inc
amplification)
• synthesizing
Concept: analog processing vs.
digital processing
simplicity, capable of high speed and high frequency
mid-range speed and frequency
Signal input
Microprocessor
(DSP)
Filtered
signal
output
User input
hard-wired/dedicated
(use “recipes”)
software-implemented
direct compatibility with
num. methods
Example: a transient circuit
Application of Laplace transform
Then switch A is opened at time tA<=0;
simultaneously or after tA, switch B is
instantaneously closed at t=0.
Switch A was connected for a
long time t<0 and switch B
was open.
A
B
+
-
V=5 V
0.3 mF
+
47 mH
vout
-
Link to review of LT
ECE generic
APP Laplace transforms of simple functions
RUN STATUS CONTROL
Note: checklist of review
and learning topics:
• LT of derivatives and integral
• LT integro-differential eqs. with
initial conditions)
• LT transfer function of a linear
system
Use APP to review LT
of common functions
Another example of Laplace transformed circuit
use the mesh
current
method
(alternative:
node voltage
method)
impedance
matrix
This is also known as a linear system – we’ll learn more later on in linear
algebra
APP demo: generic Laplace transform response
function for an analytic system
• Circuit analysis
• Open loop and closed loop
transfer function for control
Objectives: apply LT and Ztransform with numerical
algorithm and software code
to solve problems as in
examples above
Note: if need more time to study & do work on FFT, we can
reduce work load on LT/ZT
Use the APP to review learning topics:
- structure of a typical LT transfer function
- poles and zeroes
- polynomial decomposition
- responses (over damped, critically damped, under damped)
- not needed: root locust
An introduction
to the next
subject
Direct editing poles or zeros, by
selecting natural frequency and
damping coefficient.
(example: poles for Butterworthor Chebyshev-type filters)
Observing response function for
any new pair of roots.
Inspect poles and zeroes with 3D
plot. Root locust design option in
a different APP
pole, zero and polynomial numerical computation are obviously crucial
for applications (we’ll learn that next)
some other examples
6th order Butterworth poles
4th order Chebyshev 1
3rd order Bessel poles
total response function (white) and
component response functions (color)
use this for project on filter (see later slides)
Further example: Laplace-transformed and Ztransformed for control – for knowledge only
P
I
D
Open loop TF
Application in control,
especially robotics
Root Locus Analysis and Design
It’s good to be aware of control theory
application and ZT num. method (ZT APP),
but we won’t do computer work on this
Response and error
A quick summary of what we learn
on numerical Fourier and Laplace
selecting pulse shape and
its digitized sampling rep.
FFT output
Assignment: more general pulse shape; choose Butterworth, Chebyshev,
or Bessel filter (LP, HP, or BP); calculate output – use LT APP response
function. May not use Mathematica built-in filter functions.
example circuit for illust only – not actual assignment.
FFT signal * transfer function
inverse FFT for output
Mathematica built-in filter functions work only on analytic signals
- it cannot handle arbitrary shape functions
- cannot handle random noises
- OK to use to for sanity check of numerical algorithm with known test
functions.
- “may” contain bugs
example only
Advanced project
• AM or FM signal with HF carrier
• additive white gaussian noise (or some interference
signal)
• carrier bandpass filter (select one type)
• output signal
We see that Fourier and Laplace
transforms are highly useful for so
many applications.
Can they be always applied? to any
problem? IOW, are there limitations?
The system has to be linear (or can be
linearized with approximation)
Are there other types of transforms
useful for scientific/engineering
applications?
Yes, and numerical methods are also
developed for them:
• variations and generalizations of Fourier and
Laplace (numerous)
• wavelet transform
• Hilbert transform and other transforms for signal
processing, compression.
An introduction to next subjects
What is a common feature in the
formulation of those problems that we
apply FT and LT to?
Differential (or integro-differential)
equations
• All the examples discussed above involve
differentiation
• Solutions are obtained by solving the
differential equations (DE)
• Fourier and Laplace transforms are
approaches to solve the DEs indirectly: solve
in the frequency domain, then convert back
to the time domain
• Most applicable for linear time-invariant
systems with analytic models of response
• Not applicable for non-linear systems.
We will see that there are many cases
in which, even if the transform
approach is theoretically applicable, it is
not necessarily practical or
advantageous with numerical methods
We will follow up with this in next
topics of learning