Read : Ch. 12 in Electric Circuits, 9 th Edition by Nilsson

Read: Ch. 12 in Electric Circuits, 10th Edition by NilssonHandout: Laplace Transform Properties and Common Laplace Transforms

Laplace Transforms – an extremely important topic in EE!

Key Uses of Laplace Transforms:• Solving differential equations• Analyzing circuits in the s-domain• Transfer functions• Frequency response• Applications in many courses

Testing:Some calculators can often be used to find Laplace transforms and inverse Laplace transforms. However, it is also easy to make mistakes with the calculators and if the student is not familiar with the material, the mistakes might easily go undetected. As a result:

Courses Using Laplace Transforms:• Circuit Theory II• Electronics• Control Theory• Discrete Time Systems (z-

transforms)• Communications• Others

No calculators allowed on Test #3

1Chapter 12 EGR 272 – Circuit Theory II

Notation:F(s) = L{f(t)} = the Laplace transform of f(t).f(t) = L-1{F(s)} = the inverse Laplace transform of F(s).

Uniqueness: Every f(t) has a unique F(s) and every F(s) has a unique f(t).

Note:Transferring to the s-domain when using Laplace transforms is similar to transferring to the phasor domain for AC circuit analysis.

2Chapter 12 EGR 272 – Circuit Theory II 2Chapter 12 EGR 272 – Circuit Theory II

f(t) F(s)L

L -1

Definition:

(one-sided Laplace transform)

wheres = + jw = complex frequency = Re[s] and w = Im[s]sometimes complex frequency values are displayed on the s-plane as follows:

-st

0

F(s) f(t)e dt

jw

s-planeNote: The s-plane is sometimes used to plot the roots of systems, determine system stability, and more. It is used routinely in later courses, such as Control Theory.


Convergence: A negative exponent (real part) is required within the integral definition of the Laplace Transform for it to converge, so Laplace Transforms are often defined over a specific range (such as for > 0). Convergence will discussed in the first couple of examples in this course to illustrate the point, but will not be stressed afterwards as convergence is not typically a problem in circuits problems.

Determining Laplace Transforms - Laplace transforms can be found by:1) Definition - use the integral definition of the Laplace transform2) Tables - tables of Laplace transforms are common in engineering and math

texts. The table on the following page will be provided on tests.3) Using properties of Laplace transforms - if the Laplace transforms of a few

basic functions are known, properties of Laplace transforms can be used to find the Laplace transforms of more complex functions.



Table of Laplace Transforms (to be provided on tests)

Example: If f(t) = u(t), find F(s) using the definition of the Laplace transform. List the range over which the transform is defined (converges).

Example: If f(t) = e-at u(t), find F(s) using the definition of the Laplace transform. List the range over which the transform is defined (converges).


Example: Find F(s) if f(t) = cos(wot)u(t)

(Hint: use Euler’s Identity)

Example: Find F(s) if f(t) = sin(wot)u(t)

jx

+jx -jx

+jx -jx

Euler's Identities

e = cos(x) jsin(x)

e + ecos(x) = 2

e - esin(x) = 2 j


Laplace Transform PropertiesLaplace transforms of complicated functions may be found by using known transforms of simple functions and then by applying properties in order to see the effect on the Laplace transform due to some modification to the time function. Ten properties will be discussed as shown below.Table of Laplace Transform Properties (will be provided on tests)1 Linearity L{af(t)} = aF(s)2 Superposition L {f1(t) + f2(t) } = F1(s) + F2(s)3 Modulation L {e-atf(t)} = F(s + a)4 Time-Shifting L {f(t - )u(t - )} = e-sF(s)5 Scaling 1 sf(at) F

a a

L6 Real Differentiation d f(t) sF(s) - f(0)

dt

L7 Real Integration

0

1f(t)dt F(s)s

t

L

8 Complex Differentiation dtf(t) F(s)ds

L9 Complex Integration

s

f(t) F(s)dst

L

10 Convolution L {f(t) * g(t)} = F(s)·G(s)


Laplace Transform Properties:1. Linearity:

Example: Use the results of the last two examples plus the two properties above to find F(s) if f(t) = 25(1 – e-3t )u(t)

L {f1(t) + f2(t) } = F1(s) + F2(s)2. Superposition:

L {af(t)} = aF(s)


Laplace Transform Properties: (continued)3. Modulation:

This means that if you know F(s) for any f(t), then the result of multiplying f(t) by e-at is that you replace each s in F(s) by s+a.

Example: Find V(s) if v(t) = 10e-2t cos(3t)u(t)

L {e-atf(t)} = F(s + a)


n)(modulatio 9 2s

2s10 )cos(3t)u(t10e

table)from nsform(known tra 3 s

10s (t)10cos(3t)u

:Solution

22t-

22

L

LThis solution shows a good way to use Laplace transform properties:1) Begin with a

known transform2) Apply property3) Apply property4) Etc

Example: Find I(s) if i(t) = 4e-20t sin(7t)u(t)


Laplace Transform Properties: (continued)4. Time-Shifting:

Example: Find L {4e-2(t - 3) u(t - 3)}

Note: Be sure that all t’s are in the (t - ) form when using this property.

Example: Find L {10e-2(t - 4)sin(4[t - 4])u(t - 4)}

L {f(t - )u(t - )} = e-sF(s)


Example: Find F(s) if f(t) = 4e-3t u(t - 5) using 2 approaches:A) By applying modulation and then time-shifting

B) By applying time-shifting and then modulation


Note: Properties can sometimes be applied in different orders. However, one of the methods may be easier. Practice helps in deciding which method to use.

Example: Find L {4e-3tcos(4[t - 6])u(t - 6)}


Laplace Transform Properties: (continued)5. Scaling:

Example: Find F(s) if f(t) = 12cos(3t)u(t)

In other words, the result of replacing each (t) in a function with (at) is that each s in the transform is replaced by s/a and the transform is also divided by a.

1 sf(at) Fa a

L

Note: This is not a commonly used property.


Laplace Transform Properties: (continued)6. Real (Time) Differentiation:

Example: Find L {f ’(t)}

d f(t) sF(s) - f(0)dt

L

Example: Find L {f ’’(t)}

Example: Find L {f ’’(t)}

Example: Find L {f n(t)}


Example: Find the Laplace transform of the familiar relationship:

dvi(t) Cdt

7. Real (Time) Integration:

0

1f(t)dt F(s)s

t

L

0 0

f(t)dtdt t t

LExample: Find


Example: Find the Laplace transform of the familiar relationship: t

0

1v(t) i(t)dt v(0)C


Laplace Transform Properties: (continued)8. Complex Differentiation:

Example: Find L {tu(t)}

Example: Find L {t2 u(t)}

Example: Find L {t3 u(t)}

Example: Find L {t n u(t)}

dtf(t) F(s)ds

L


Example: Find L {3te-2t cos(4t)u(t)}

Example: Find L {tu(t - 2)}


Laplace Transform Properties: (continued)9. Complex Integration:

Example: Find

s

f(t) F(s)dst

L

1u(t)t

L

Note: This is not a commonly used property. Multiplying by t is common (such as with repeatedroots), but dividing by t is rare.


Laplace Transform Properties: (continued)10. Convolution:

f(t) * g(t) reads as “f(t) convolved with g(t)”Convolution is defined by the difficult integral relationship shown below.Evaluating this integral is covered in a later course.

Laplace transforms are often used to bypass the convolution integral (illustrated on the following page). Since L {f(t) * g(t)} = F(s) G(s) we can determine f(t)*g(t) using

t

0

f(t) * g(t) g( )f(t - )d

f(t)*g(t) = L -1 {F(s)G(s)}

L {f(t) * g(t)} = F(s)·G(s)


The method of using Laplace transforms to bypass the convolution integral is illustrated by the diagram below.

f(t)*g(t) = L -1 {F(s)G(s)}

Given:f(t), g(t)

Find:f(t)*g(t) = ?

F(s), G(s) F(s)·G(s)

TakeLaplace

TransformTake

InverseLaplace

Transform

Multiply

Evaluatedifficult

convolutionintegral

t

0

f(t) * g(t) g( )f(t - )d


Example: If f(t) = 2e-2tu(t) and g(t) = 4e-3tu(t), find f(t)*g(t) using Laplace transforms.


Impulse Function(t) = impulse functionThe impulse function is defined as:

Delta functions are often illustrated as shown below.

-

(t) 0 for t 0

(t)dt 1 (i.e., the area equals 1)

t 0

(t)

Area = 1 (not height)

t 0

2(t-4)

Area = 2 (not height)

4

IllustrationTo illustrate the concept that the area under (t) = 1 (not the height =1), consider the function f(t) shown below:

f(t)

t -A/2 +A/2 0

1/A (t) lim f(t)

A 0


Example: When do impulse functions occur? Consider the example shown below. Sketch the capacitor current.

v(t)

t

i(t)

t

+ v(t) -

i(t)

C


Laplace Transform of an impulse functionThe Laplace transform of an impulse function can be found using the definition of the Laplace transform:

L {(t)} = 1


0

st- dt(t)e (t) L

1 (t)dt dt(t)e (t)00

0

L

Since (t) only exits at t=0, e-st only needs to be evaluated at t = 0 (this is sometimes called the sifting property), so:

so

Example: Find f(t) for F(s) shown below. (Hint: Order of N(s) = 2 and order of D(s) = 2, so begin with long division)

25 6s s6 2s F(s) 2

2


Impulse functions can occur in real circuits. Constant terms in F(s) will correspond to impulse functions in f(t). We will soon see that if the order of N(s) is not less than the order of D(s), we should begin by using long division before finding an inverse Laplace transform.

Laplace Transforms of waveformsPiecewise-continuous waveforms can be expressed using unit functions. Laplace transforms of these expressions can then be found.Example: Find F(s) for f(t) shown below.

Example: Find F(s) for f(t) shown below.

0 4

12

t

f(t)

0 4

6

t

f(t)

2


Read : Ch. 12 in Electric Circuits, 9 th Edition by Nilsson

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