1. 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)

2. LAPLACE TRANSFORMS DEFINITION TIME (t) IS REPLACED BY A NEW INDEPENDENT VARIABLE (s) WE CALL s THE LAPLACE TRANSFORM VARIABLE

3. 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.

4. Basic Concepts

6. 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.

7. 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.

8.  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)

12. 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.

13. 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 constants 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{e at f(t)}=F(s-a)

14. 4. Laplace transform of derivatives L{f n (t)} = s n L{(t)} - s n-1 f(0) - s n-0 f’(0) - … - f n-1 (0). 5. Laplace transform of integrals L{f(t)}=F(s), then L{∫ t 0 f(u) du} = F(s)/s 6. Multiplication by t : L{t n f(t)} = (-1) n d n /ds n {F(s)} 7.Division by t: L{f(t)/t} = ∫ ∞ F(u)du s

15. Laplace Transform Formulas

16. 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.

17. 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

18. Application of laplace transform to solve the differential equations with constant coefficients Consider the general second order linear differential equation with constant coefficient as

19. Step3: Taking invers laplace transform to get completesolution y(t). This complete solution automatically takes care of initial conditions.

20. 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).

21. DEFINITION:  The unit step function, u(t) is defined as: 0 t<0 u(t)= 1 t>0 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).

22. 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

25. REFERENCES  Advanced Engineering Mathematics- R C Shah  Mathematics-III- R M Baphana  www.wikipedia/laplacetransform.com www.wikipedia/laplacetransform.com  www.wikipedia/inverselaplacetransform.com www.wikipedia/inverselaplacetransform.com