LAPLACE TRANSFORM
Acharya Yash(130200111001)
130200111002
Aishwarya Jayadas(130200111003)
Asheesh Tiwari(130200111004)
Athira Pradeepan(130200111005)
130200111006
130200111007
130200111008
130200111009
130200111010
Safil Khira(140203111014)
Dave Hemang(140203111005)
Ronak Sarikhada(140203111029)
LAPLACE TRANSFORMS
• DEFINITION

F (s)  L[ f (t )]   f (t )e dt
0
• TIME (t) IS REPLACED BY A NEW INDEPENDENT
VARIABLE (s)
• WE CALL s THE LAPLACE TRANSFORM VARIABLE
 st
It is often more convenient to work in Laplace
domain than time domain
Time domain  ordinary differential equations in t
Laplace domain  algebraic equations in s
The solution of most electrical circuit problems can be
reduced ultimately to the solution of differential
equations.
Laplace transform is useful in solving linear differential
equation , ordinary as well as partial.
Basic Concepts
The Laplace transform of f(t) exists
when the following sufficient conditions
are satisfied :
f(t) is sectionally continuous in every
finite interval for t >=0.
f(t) is exponential order of α.
Let f(t) and g(t) be two functions which are
piecewise continuous with an exponential
order at infinity.
Assume that L{f(t)} = L{g(t)},then f(t)=g(t) for t
belongs to [0,B], for every B>0 , except may
be for a finite set of points.
 L(1)=1/s
Proof:
By definition :
 L(t^n)=n!/s^(n+1) where n=0,1,2,3…..
Put st=p,we obtain
L(t^n)=n!/s^(n+1)
Existence and Uniqueness of inverse
Laplace transform
The conditions requires for the existence of the inverse Laplace transform of
F(s) are:
1. lim F(s) =0
s—>∞
2. lim F(s) is finite
s—>∞
For most of the practical purposes the uniqueness of the inverse Laplace
transform is assumed. But there are some of the functions for which it may
not be unique.
Properties of Laplace transforms and
Inverse Laplace transform
1.
Linear Property of Laplace Transform
L{f(t)}=F(s) and L{g(t)}=G(s)
L{af(t)+bg(t)}=a L{f(t)}+bL{g(t)} ,a&b are
2. Change of scale property
L{f(t)}=F(s) then
L{f(bt)}=(1/b)F(s/b)
3. First Shifting Theorem
L{f(t)}=F(s), then L{eat f(t)}=F(s-a)
constants
4. Laplace transform of derivatives
L{fn (t)} = sn L{(t)} - sn-1 f(0) - sn-0 f’(0) - … - fn-1 (0).
5. Laplace transform of integrals
t
L{f(t)}=F(s), then L{∫ 0 f(u) du} = F(s)/s
6. Multiplication by t :
L{tn f(t)} = (-1)n dn /dsn {F(s)}
7.Division by t:
L{f(t)/t} = ∫∞ F(u)du
s
Laplace Transform Formulas
Convolution Theorem
 Convolution is very useful concept for many applications in various
engineering branch.
Let f(t) and g(t) be two piecewise continuous functions and of exponential
order α(α is constant) , then the convolution of f and g is denoted y f*g
and is defined as
f*g= ̥ʃˈ f(u)g(t-u)du
This intergral is known as convolution integral.
Properties Of Convolution
 The following are valid properties of convolution :
1.
( Commutative ): f*g=g*f
2.
(Associative) :f*(g*h)=(f*g)*h
3.
(Distributive) f*(g + h)=f*g + f*h
4.
f*0=0*f=0
5.
1*1=t
Application of laplace transform to solve the
differential equations with constant coefficients
Consider the general second order linear differential
equation with constant coefficient as
Step3:
Taking invers laplace transform to get completesolution y(t).
This complete solution automatically takes care of initial conditions.
THE UNIT STEP FUNCTION
(HEAVISIDE FUNCTION)
 In Engineering application, we frequently encounter at specified values of
time . One common example is when a voltage is switched on or off in an
electrical circuit at a specified value of time t.
 The value of t=0 is usually taken as a convenient time to switch on or off the
given voltage.
 The switching process can be described mathematically by the function
called the UNIT STEP FUNCTION(HEAVISIDE FUNCTION).
DEFINITION:
 The unit step function , u(t) is defined as:
0
t<0
1
t>0
u(t)=
That is ,u is a function of time, and u has value zero when time is
negative(before we slip the switch);and value one when time is
positive(from when we flip the switch).
Shifted Unit Step Function
 In many circuits , waveform are applied at specified intervals other
then t=0.Such a function may be described using the shifted (aka
delayed) unit step function.
Definition:
A function which has value 0 up to the time t=a and thereafter has value
1,is written
0 t<a
u(t –a)=
1 t>a
REFERENCES
 Advanced Engineering Mathematics- R C Shah
 Mathematics-III- R M Baphana
 www.wikipedia/laplacetransform.com
 www.wikipedia/inverselaplacetransform.com