Laplace Transforms 1

NTTF

CONTROL SYSTEMS

LAPLACE TRANSFORMS 1

Control Systems - Laplace transforms 1 2NTTF

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.


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


ExampleUndefined 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).


ExampleUndefined 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


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


ExampleDefined 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.


Solution• The Transform is determined using the

transform table. The corresponding entries from the table of transform pairs are

2222,

20

1,

1,

100

10

ssss


ExampleDefined 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.


Solution• The transform of each term is the product of

the constant term and the corresponding entry from the transform table.

or


Example• Find the Laplace transform of each of the

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


Solutiona. 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)


Examplea. Determine the Laplace transform of t.

b. Determine the Laplace transform of t3.

Solution

a. From the Laplace transform table ,

b. From the transform table ,


Examplea. Determine the Laplace transform of sin 10t.

b. Determine the Laplace transform of sin t.

c. Determine the Laplace transform of cos 0.lt.


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

Solution

2

1][ 2

seL t

Therefore


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.


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.


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.


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).


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.


Inverse Transform• Term Containing Division by s

– It relates to a constant term in the time domain.


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


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


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.


Inverse Transform• Other Terms Containing s

1. Find an entry in the transform table similar to the s-domain term under consideration. The entry should have identical s terms in both the numerator and denominator but can differ in the numerical constant term.

2. If no match can be found, use partial fractions to factor the s-domain term into simpler terms.

3. 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.

4. If necessary, take the constant terms outside the transformation (rule 1)

5. Using one-to-one correspondence with the table entry, convert the s-domain term into time domain.


Example• Find the inverse Laplace transform of the

following.– a. M(s)– b. P(s)– c. X(s)


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)



Solution

• On checking the entries in Table, entry 14 is found to be a close match

Therefore, the answer is cos 10t

NTTF

Laplace Transform Pairs




Example• Find the inverse Laplace transform of the

following.

A. 2

B. sI(s)

C. s2X(s)+Y(s)

D. 4X(s)/s


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

• C

D

Documents

Laplace Transforms 1