Laplace Transforms 1

CONTROL SYSTEMS LAPLACE TRANSFORMS 1

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Forward Transformation Procedure • Transform of an Equation – The transform of an equation is obtained by taking the transform of the expressions on each side of the equation.

• Transform of an Expression – 1. Identify the terms, which are connected by sum (+) or difference (-) signs. – 2. Categorize the terms and take the transform of each term. – Constant Term: Term containing only constants and no variables of time. NTTF

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Example • Find the Laplace transform of 12, 256, 0.25, and 1/40 Solution • The transformed term is represented by the constant divided by s. The corresponding terms are

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Example Undefined Variable Term without Any coefficient • Find the Laplace transform of e, x, x(t), y, and i(t). Solution • The transform is represented by the capital letter representing the variable, followed by s enclosed in parentheses: E(s), X(s), x(s), y(s), I(s). NTTF

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Example Undefined Variable Term with Constant Coefficient • Term containing an undefined variable of time multiplied or divided by a constant. • Find the Laplace transform of 10e, 0.5x, x(t)/25, 2y(t)/3

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Solution • The transform is represented by the constant term (as is ) followed by the capital letter representing the variable with s enclosed in paranthesis. • The transformed terms are 10E(s), 0.5X(s), X(s)/25, 2Y(s)/3

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Example Defined Variable Term without Any Constant Coefficient • A term that is defined as a variable of time and is not multiplied or divided by a constant. • Find the Laplace transform of sin 1Ot, t, e-20t and sin πt.

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Solution • The Transform is determined using the transform table. The corresponding entries from the table of transform pairs are

10 , 2 s + 100

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1 , 2 s

1 , s + 20

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π 2 2 s +π

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Example Defined Variable Term with Constant Coefficient • A term defined as a variable of time and multiplied and/or divided by a constant. • Find the Laplace transform of 5 sin 1Ot, 1OOt 2.5e-20t and 6 sin πt.

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Solution • The transform of each term is the product of the constant term and the corresponding entry from the transform table.

or

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Example • Find the Laplace transform of each of the following. – – – –

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a. i(t) b. v(t) c. a d. x

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Solution •

L[i(t)] = I(s) –

The Laplace transform of a function of time is indicated simply by changing the letter to uppercase and replacing t in parentheses with s.

b. Similarly, L[v(t)] = V(s). c. L[a] = A(s) By convention, a is a function of time and can also be expressed as a(t).

d. L[z]=Z(s) NTTF

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Example • •

Determine the Laplace transform of t. Determine the Laplace transform of t3. Solution • From the Laplace transform table , •

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From the transform table ,

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Example • • •

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Determine the Laplace transform of sin 10t. Determine the Laplace transform of sin t. Determine the Laplace transform of cos 0.lt.

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Example • Determine the Laplace transform of e-2t. Solution

Therefore

L[e

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− 2t

1 ]= s+2

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Inverse transformation • Inverse transformation is performed to convert the s-domain terms back to real time (time domain). • Inverse Transform of an Equation – Inverse transformation of an equation is done by taking the inverse transform of the expressions on each side of the equation.

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Inverse Transformation • Inverse Transform of an Expression – The inverse transformation process can be as simple as the forward transformation process. – Ideally, an entry can be found in transform table that matches the problem at hand. – It then becomes a simple matter of substitution of the terms. – Unfortunately, it is rare to find an exact match, and some manipulation of terms is needed. NTTF

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Inverse Transform Process • In general, the following approach is recommended. • Identify the terms, which are connected by sum (+) or difference (- ) signs. • Categorize the terms as follows and take the inverse transform of each term – Constant Term: A term containing only constants and no s term (variables of time). A purely constant term represents an impulse function in time. NTTF

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Example • Determine the inverse Laplace transform of 1,2.5, and 10. Solution • The corresponding time-domain term (inverse transformation) is a product of a constant with the delta function. (See Chapter 6 for the delta/impulse function.) • The inverse transformed terms are δ(t), 2.5δ(t), and 10δ(t). NTTF

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Inverse Transform • In the following discussions, multiplication or division by a constant is not considered. • A constant coefficient does not change the forward or inverse transformation process and can be considered transparent to both. • The only time a constant term is treated differently is when it appears on its own and not as a coefficient to another term. NTTF

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Inverse Transform • Term Containing Division by s – It relates to a constant term in the time domain.

Example • Determine the inverse Laplace transform of Solution • The corresponding time-domain term in each case is expressed by simply removing the s term from the expression: 1,256,0.25, and 1/40 NTTF

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Inverse Transform • Undefined Variable Term – A term containing a capital letter representing an undefined variable followed by s enclosed in parentheses.

Example • Determine the inverse Laplace transform of

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Solution • The time-domain term is expressed as the lowercase letter representing the variable. • To explicitly indicate that the inverse quantity is in the time domain, it can be expressed as the lowercase variable followed by t enclosed in parentheses.

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Inverse Transform •

Other Terms Containing s –

– –

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Find an entry in the transform table similar to the sdomain term under consideration. The entry should have identical s terms in both the numerator and denominator but can differ in the numerical constant term. If no match can be found, use partial fractions to factor the s-domain term into simpler terms. Manipulate the s term so that the coefficient of the s term is the same as in' a suitable entry in transform pair table. This can be done by dividing and/or multiplying both numerator and denominator by a suitable constant. If necessary, take the constant terms outside the transformation (rule 1) Using one-to-one correspondence with the table entry, convert the s-domain term into time domain. Control Systems - Laplace transforms 1

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Example • Find the inverse Laplace transform of the following. – a. M(s) – b. P(s) – c. X(s)

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Solution • a. L-1[M(s)] = m(t) – Simply changing the uppercase letter to lowercase and replacing s in parentheses with t indicate the inverse Laplace transform of a function.

• b. Similarly, L-1[P(S)] = p(t). • c. L-1[X(s)] = x(t)

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Example • Determine the inverse Laplace transform of Solution • On checking the entries in Table, entry 14 is found to be a close match

Therefore, the answer is cos 10t NTTF

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Laplace Transform Pairs

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Example • • • • •

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Find the inverse Laplace transform of the following. 2 sI(s) s2X(s)+Y(s) 4X(s)/s

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Solution • A f (t) = L-1 [2] = 2L-1

[l ] = 2δ(t) [δ(t) is the impulse function ]

•B

f (t) = L-1 [sI(s)] = di(t)/dt

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