Automatic Control(E.E- 412) Chapter 1 Laplace Transform Dr. Monji Mohamed Zaidi

1)Laplace Transform  Complex Variable  Definition  Laplace transforms of usual functions  Laplace transform Properties  Laplace Transform Theorems 2)Inverse Laplace transform  Partial-Fraction Expansion Method Course Outline

3 basic knowledge A complex number has a real part and an imaginary part, both of which are constant. If the real part and/or imaginary part are variables, a complex quantity is called a complex variable s = σ + jω (1) where σ is the real part and ω is the imaginary part. A complex function G(s), a function of s, has a real part and an imaginary part or The magnitude of G(s) is Where G x et G y are real quantities. and the angle θ de G(s) is The angle is measured counter clockwise from the positive real axis. The complex conjugate of G(s) is The power series expansions of cos θ et de sin θ are, respectively :

basic knowledge Thus : Since the power series expansions of e θ is : We see that This is known as Euler’s theorem. By using Euler’s theorem, we can express sine and cosine in terms of an exponential function. Noting that e −jθ is the complex conjugate of e jθ

Let f(t) be a function of time t such that s a complex variable, L an operational symbol indicating that the quantity that it prefixes is to be transformed by the Laplace integral and F(s) the Laplace transform of f(t). Then the Laplace transform of f(t) is given by : L With respect to the Laplace transform definition, If a function f(t) has a Laplace transform, then the Laplace transform of Af(t), where A is a constant, is given by Since Laplace transformation is a linear operation, if functions f1 (t) and f2(t) have Laplace transforms, F1 (s) and F2(s) respectively, then the Laplace transform of the function αf1 (t) + βf2(t) is given by Laplace Transform Definition

Exponential Function Consider the exponential function : Where A and α are constants. The Laplace transform of this exponential function can be obtained as follows :

Step Function Physically, a step function is a signal of constant magnitude A applied to the system at time t = 0 The Laplace transform of this function is given by : The Laplace transform of the unit-step function (A = 1 ) is defined by

Ramp Function Consider the ramp function defined as follow : The Laplace transform of this function is given by : The Laplace transform of the unit-Ramp function (A = 1 ) is defined by

Sinusoidal Function Where A and ω are constants Referring to the expression the Laplace transform of the sinusoidal function can be written Similarly, the Laplace transform of A cos ωt can be derived as follows :

Pulse Function Consider the pulse function defined as follows : Where A and t 0 are constants. The pulse function here may be considered a step function of height that begins at t = 0 and that is superimposed by a negative step function of height beginning at t = t 0 ; that is Then the Laplace transform of f(t) is obtained as :

Impulse Function The impulse function is a special limiting case of the pulse function Since the height of the impulse function is and the duration is t 0, the area under the impulse is equal to A. As the duration t 0 approaches zero, the height approaches infinity, but the area under the impulse remains equal to A Since, Then

Impulse Function Thus the Laplace transform of the impulse function is equal to the area A under the impulse. The impulse function whose area is equal to unity is called the unit-impulse function or the Dirac delta function. The unit- impulse function occurring at t = t 0 is usually denoted by δ(t − t 0 ) where δ(t − t 0 ) satisfies the following :