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IM 2 Feedback and Control System

To find the inverse Laplace transform of a complicated function, we ca...
Course

Electronics Engineering (CR 061)

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Academic year: 2022/2023
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Samar State University

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LESSON CONTENT

In practice, one usually does not need to perform a contour integration in the complex plane. Instead, a “dictionary” of Laplace transforms pairs is generated and some simple rules allow one convert between the time domain solution, f(t), and the frequency or s-plane solution, F(s).

Some Laplace Transform Pairs

  1. Unit Step Signal

1 𝑡 < 0 1 ( (( (( (( (( (( (( (( (( (( (( (( (( (( (( ) = () = { (((((((((((((( () = 2 𝑡 > 0 2

Where:

  1. Exponential Signal:

Where:

Superposition (Sum and Difference):

) =((((((((((((((( 1 ((((((((((((((() ± 𝑡 ((((((((((((((( ) (((((((((((((( ( ) = 1 ((((((((((((((() ± 2 ((((((((((((((() Differentiation: 𝑡 ) =((((((((((((((( 1 ((((((((((((((( ) (((((((((((((( ( ) = 1 ((((( (((((((((() − 1 (0) 𝑡𝑡 For the 2nd derivative:

In general, the nth derivative is:

Laplace Transform Solution of a Differential Equation

Example Consider a system represented by a differential equation 𝑡 2 𝑡 1222222222222222 2 + 22(22(22(22(22(22(22(22(22(22(22(22(22(22( 2 + 3 2 = 3( )

𝑡𝑡 with zero initial condition Solution: y(0)=0 and y’(0) =0 for zero initial conditions Note that ℒ(y’) = s ℒ(y) – y(0)

𝑡 2 𝑡 𝑡𝑡 ℒ 2 + 12ℒ + 32 ℒy=32u(t) ⅆⅆⅆⅆⅆⅆⅆⅆⅆⅆⅆⅆⅆⅆ 𝑡𝑡

ℒ(y”) + 12ℒ(y’) + 32ℒ(y) = 32 𝑡

sℒ(y’) – y’(0) + 12{s ℒ(y) – y(0)} + 32 ℒ(y) = 32 𝑡

s{s ℒ(y) – y(0)} – y’(0) + 12{s ℒ(y) – y(0)} + 32 ℒ(y) = 32 𝑡

𝑡 2 ℒ(y) – 0 – 0 + 12sℒ(y) – 0 + 32 ℒ(y) = 32 𝑡

ℒ(y)(𝑡 2 + 12s + 32) = 32 , dividing both sides by )(𝑡 2 + 12s + 32) 𝑡 ℒ(y) or Y(s) 𝐀𝐀

3

) =((((((((((((((( ((((((((((((((( 2 + 2222222222222 2 2 + 5)

Transfer Functions - Is an algebraic expression for dynamic relation between a selected input and output to the process model - Can only be derived for a linear model because Laplace Transform can be applied only to linear equations. Let us begin by writing a general nth- order, linear, time-invariant differential equation.

𝑡𝑡 𝑡𝑡𝑡

Where c(t) is the output, r(t) is the input, and the ai’s, bi’s and the form of the differential equation represent the system. Taking the Laplace transform of both sides, 𝑡𝑡𝑡𝑡 ((((((((((((((() + 𝑡 ) +((((( ((((((((( () = 𝑡 𝑡𝑡𝑡 ) + (((((((((((((( ()

Is a purely algebraic expression. If we assume that all initial conditions are zero, then it reduces to (((((((((((((((𝑡𝑡𝑡 ) =(((((((((((((( (𝑡𝑡𝑡+

Now form the ratio of the output transform, C(s), divided by the input transform, R(s):

Notice that the equation separates the output, C(s), the input R(s), and the system, the ratio of polynomials is s on the right. We call this ration, G(s), the transfer function and evaluate it with zero initial condition.

Also, we can find the output, C(s) by using ((((((((((((((((((((((((((((((((((((((((((( ) =)))))))))))))) ()( ) The basic block diagram representation for this system is shown below.

𝑡𝐀𝑡 𝑡𝑡−1𝑡(𝐀

𝐀

) 𝑡𝐀𝑡(𝐀

𝐀

) 𝑡𝑡 −1𝑡(𝐀

𝐀

)

𝑡

𝑡𝑡

) +𝑡𝑡

𝑡 (𝑡𝑡 𝑡𝑡

𝑡 ) =(𝑡𝑡𝑡𝑡

The following are the test inputs used in the analysis of system response.

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IM 2 Feedback and Control System

Course: Electronics Engineering (CR 061)

95 Documents
Students shared 95 documents in this course

University: Samar State University

Was this document helpful?
LESSON CONTENT
In practice, one usually does not need to perform a contour integration in the complex plane. Instead, a
“dictionary” of Laplace transforms pairs is generated and some simple rules allow one convert
between the time domain solution, f(t), and the frequency or s-plane solution, F(s).
Some Laplace Transform Pairs
1. Unit Step Signal
1𝑡 < 0 1
) = ) = { ( (( (( (( (( (( (( (( (( (( (( (( (( (( (( ( ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
2𝑡 > 0 2
Where:
2. Exponential Signal:
Where:
Superposition (Sum and Difference):
) =( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1) ±( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 𝑡) ) =( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1) ±( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 2)((((((((((((((( Differentiation: 𝑡
) = ((((((((((((((( 1) ) =( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1) −( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1(0)
𝑡𝑡
For the 2nd derivative:
In general, the nth derivative is:

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