Loading in 2 Seconds...

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

Loading in 2 Seconds...

- 149 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Read : Ch. 12 in Electric Circuits, 9 th Edition by Nilsson' - dorcas

Download Now**An Image/Link below is provided (as is) to download presentation**

Download Now

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Chapter 12 EGR 272 – Circuit Theory II

1

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

Handout: 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

Chapter 12 EGR 272 – Circuit Theory II

Chapter 12 EGR 272 – Circuit Theory II

2

2

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).

L

f(t)

F(s)

L-1

Note:

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

Chapter 12 EGR 272 – Circuit Theory II

3

jw

s-plane

Definition:

(one-sided Laplace transform)

where

s = + jw = complex frequency

= Re[s] and w = Im[s]

sometimes complex frequency values are displayed on the s-plane as follows:

Note: 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.

Chapter 12 EGR 272 – Circuit Theory II

4

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 transform

2) 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.

Chapter 12 EGR 272 – Circuit Theory II

6

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).

Chapter 12 EGR 272 – Circuit Theory II

7

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)

Chapter 12 EGR 272 – Circuit Theory II

8

Laplace Transform Properties

Laplace 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)

Chapter 12 EGR 272 – Circuit Theory II

9

Laplace Transform Properties:

1. Linearity:

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

L {f1(t) + f2(t) } = F1(s) + F2(s)

2. Superposition:

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)

Chapter 12 EGR 272 – Circuit Theory II

10

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.

Laplace Transform Properties: (continued)

3. Modulation:

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

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

This solution shows a good way to use Laplace transform properties:

Begin with a known transform

Apply property

Apply property

Etc

Chapter 12 EGR 272 – Circuit Theory II

12

Laplace Transform Properties: (continued)

4. Time-Shifting:

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

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

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

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

Chapter 12 EGR 272 – Circuit Theory II

13

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 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

Chapter 12 EGR 272 – Circuit Theory II

15

Laplace Transform Properties: (continued)

5. Scaling:

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.

Note: This is not a commonly used property.

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

Chapter 12 EGR 272 – Circuit Theory II

16

Laplace Transform Properties: (continued)

6. Real (Time) Differentiation:

Example: Find L{f ’(t)}

Example: Find L{f ’’(t)}

Example: Find L{f ’’(t)}

Example: Find L{f n(t)}

Chapter 12 EGR 272 – Circuit Theory II

17

Example: Find the Laplace transform of the familiar relationship:

7. Real (Time) Integration:

Example: Find

Chapter 12 EGR 272 – Circuit Theory II

18

Example: Find the Laplace transform of the familiar relationship:

Chapter 12 EGR 272 – Circuit Theory II

19

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)}

Chapter 12 EGR 272 – Circuit Theory II

20

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

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

Chapter 12 EGR 272 – Circuit Theory II

21

Note: This is not a commonly used property. Multiplying by t is common (such a with repeated

roots), but dividing by t is rare.

Laplace Transform Properties: (continued)

9. Complex Integration:

Example: Find

Chapter 12 EGR 272 – Circuit Theory II

22

Laplace Transform Properties: (continued)

10. Convolution:

L{f(t) * g(t)} = F(s)·G(s)

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

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

Chapter 12 EGR 272 – Circuit Theory II

23

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

Evaluate

difficult

convolution

integral

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

Given:

f(t), g(t)

Find:

f(t)*g(t) = ?

Take

Laplace

Transform

Take

Inverse

Laplace

Transform

F(s), G(s)

F(s)·G(s)

Multiply

Chapter 12 EGR 272 – Circuit Theory II

24

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

Chapter 12 EGR 272 – Circuit Theory II

25

Impulse Function

(t) = impulse function

The impulse function is defined as:

Delta functions are often illustrated as shown below.

Illustration

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

Chapter 12 EGR 272 – Circuit Theory II

26

+ v(t) -

i(t)

C

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

Chapter 12 EGR 272 – Circuit Theory II

27

Laplace Transform of an impulse function

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

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:

L {(t)} = 1

so

unctionscan easily be produced by real circuits. Constant terms in F(s) will correspond to impulse functions in f(t). One rule that will be proven next class related to inverse Laplace transforms is that if F(s) = D(s)/N(s), it is required that the order of D(s) be less than the order of N(s) in order to use Partial Factions Expansion (also to be discussed next class).

Chapter 12 EGR 272 – Circuit Theory II

28

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.

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)

Chapter 12 EGR 272 – Circuit Theory II

29

f(t)

f(t)

6

12

t

t

0

2

4

4

0

Laplace Transforms of waveforms

Piecewise-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.

Download Presentation

Connecting to Server..