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AMJAD HASOON Process Control Lecture1 Laplace Transformations Introduction Laplace transform techniques provide powerful tools in numerous fields of technology such as Control Theory where knowledge of the system transfer function is essential and where the Laplace transform comes into its own. Definition : Let f(t) be defined for t ≥ 0. The Laplace transform of f(t), denoted by F(s) or L{f(t)}, is an integral transform given by the Laplace integral The parameter s is assumed to be positive and large enough to ensure that the integral converge. In more advanced applications s may be complex and in such cases the real part of s must be positive and large enough to ensure convergence. In determining the transform of an expression, you will appreciate that the limits of the integral are substituted for t, so that the result will be an expression in s. Note: The Laplace transform is an operation that transforms a function of t (i.e., a function of time domain), defined on [0, ∞], to a function of s (i.e., of frequency domain). F(s) is the Laplace transform, or simply transform, of f(t). Together the two functions f(t) and F(s) are called a Laplace transform pair. Example(1): Find the Laplace transform of f(t)=1 L{f(t)}=F(s)= ∫ =- ∫ =∫ [ ]=- [ ]=- Example: Find the Laplace transform of f(t) = a, where a is a constant. 1 AMJAD HASOON Process Control Example(3): let f(t)=t L{f(t)}=F(s)= ∫ =∫ By using integral by part ∫ ∫ Let u=t dv= du=dt v=- ∫ ∫ = ∫ ∫ { + ∫ =0 = So L{f(t)}= = Example(4) : let f(t)= L{f(t)}=F(s)= ∫ =∫ =∫ = = ∫ { 2 Lecture1 AMJAD HASOON Process Control Lecture1 Example(4): let f(t)= L[ ]= ∫ = ∫ ∫ ) ∫ = =0- ∫ [ ={ ] [ =- H.w( : ∫ ] = find laplace for Some properties of the Laplace Transform 1.L{0}=0 2.L{f(t) =L{f(t)}+L{g(t)} 3. L{cf(t)}=c L{f(t)} c is constant 3 AMJAD HASOON Process Control Theorem 1:Shifting Theory: L{f(t) =F(s+a) T Example(5): find the Laplace transform for f(t)= ; then { Since the cos bt= Example(6): find the Laplace transform for f(t)= t L{ H.W: find LT for: 1. 2. , 3. Theorem 2: Multiplying by t and tn 4 Lecture1 AMJAD HASOON Process Control Lecture1 H.W: Determine the Laplace transform of( t2sint ) Laplace transform of derivative ] L[ L[ ] L[ ] Example(7):Find th L.T of the function x(t) which satisfy the following differential equation and initial condition : x(0) –s Sol.:[ ̇ ] [ ̈ ̇ ]+5[sx(s)- x(0)]+2x(s)= [ x(s) +4 x(s)+5sx(s)+2x(s)]= X(s)= H.w : Find th L.T of the function y(t) which satisfy the following differential equation and initial condition ̈ ̇ ;y(0)=1. ̇ 5 AMJAD HASOON Process Control Laplace transform of integral L∫ Example(8): find x(s) for the following X(t)=∫ L{cosat}= L{∫ = H.W: Find the L.T of the following: [1] ∫ [2] ∫ 6 Lecture1 AMJAD HASOON Process Control 7 Lecture1