1. The Laplace transform a quick review!
L.R.Linares.
This lecture is dedicated to the memory of Charles Schultz
(all cartoons are copyright issue of UPS)
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From x to +
Replacing multiplications by sums, and divisions by subtractions!
How?
Transformation!!!
Transforming the operands into their logarithms.
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Log tables
By Napier, 1628
Log … a transformation!
We have to multiply a and b…
We “transform” the numbers into logs:
We add the logs.
We “transform” back the resulting log into the final result.
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4. Transform first number into a log
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5. Transform second number into log
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Add the two logs
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7. Convert the sum into a number
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Log … a transform?
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I did all that,
sooo…???
Transforming…
…derivatives into products.
…integrals into divisions.
This would convert differential equations into algebraic equations.
Heaviside operator…
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I do it better!!! HEH!
Transforming…
…derivatives into products.
…integrals into divisions.
This would convert differential equations into algebraic equations.
…Laplace Transform!!!
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Numbers? Functions!
Logarithm transforms numbers into numbers.
Laplace transforms functions of time into functions of …s
What is s?
s is a complex number s+jw (complex frequency)
From the time-domain to the Laplace-domain (the frequency domain!)
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Laplace, from time to…
Exponential
decay
Oscillation
Angular frequency
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Differentiating…
In the frequency domain, s, it is easy…
A time
derivative
A multiplication
by s
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Integrating…
In the frequency domain, s, it is easy too…
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But… wasn’t it done by p?
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Laplace includes ICs!
Yeah! Laplace transform includes the initial conditions:
Just a number, a constant value!
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…yes it does!
Yeah! Laplace transform includes the initial conditions:
Just a number, a constant value!
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The plan…
Take a circuit’s differential equation…
…transform it into an algebraic equation, using Laplace transform.
Solve this algebraic equation for the function we’re looking for (we obtain the “transformed” solution, of course!)
Transform back this solution to the “time domain”…and they lived happily ever after!
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Hmmm…tranform back?
Transform back you said? How!!!??
There is an integral inverse Laplace transformation (which we will NOT use!).
We create look-up tables of transforms…
…and we look up for the transform that we obtained, and read from the table the function of time that corresponds to it!
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Some plan, eh?
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The plan…graphically!
differential
equations
solution
integration
algebraic
equations
Laplace
Inverse
Laplace
algebraic
solution
Solve alg. Eq.
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Notation
Laplace transform of f(t)
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…and more notation
inverse Laplace transform of F(s)
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Essential Functions
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The unit step
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The “shifted” step
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The “inverted” step
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The pulse
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The “shifted” pulse
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The ramp
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The “shifted” ramp
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The “inverted” ramp
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33. The impulse…aka “weirdo-func”
Take a pulse with an area of …
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34. The impulse…aka “weirdo-func”
Make it narrower…
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35. The impulse…aka “weirdo-func”
…and narrower…
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36. The impulse…aka “weirdo-func”
…and waaaaayyyy narrower…
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37. The impulse…aka “weirdo-func”
The area under it is still one, eh?
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So … good…but how?
Laplace transform is a general transformation… it transforms a function of one variable into a function of another variable.
Laplace transform is an integral transformation, since the transformation process involves an integral.
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So … good…but how?
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So … from zero, eh?
From “just before” zero, zero minus!
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Transform u(t)
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Transform u(t)
This is the Laplace transform of the unit step, u(t)
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Transform expo
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Transform expo
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Are you still awake?
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We were on a break!
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A break on Euler’s
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Transform the cosine
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49. Remember Integration by parts?
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50. Transform the ramp r(t)
Integrating by parts!
ZERO!!!
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51. Transform the weird…ahem! ...the impulse!
Peppermint Patty
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52. Transform the weird…ahem! ...the impulse!
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53. Laplace look-up list (partial)
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Reading assignment!
From the textbook (Alexander and Sadiku, Ed 4th), read section 1, 2, 3 and 4 from Chapter 15 (pp. 675 to 697). And work out the corresponding end of chapter exercises.
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Reading assignment!
From the textbook (Linear Circuit Analysis by DeCarlo/Lin), read section 1, 2, 3, 4 and 5 from Chapter 13 (pp. 493 to 508). And work out the corresponding end of chapter exercises.
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