Laplace Transform Of Heaviside’s Unit Step Function.pptx

Laplace Transform Of Heaviside’s Unit Step Function ( Laplace Transformation)

1. Laplace Transform Of Heaviside’s Unit Step Function

2. Laplace Transform Of Heaviside’s Unit Step Function Definition : The unit step function is denoted as u(t) or H(t) and is defined as: That is, u is a function of time t, and u has value zero when time is negative and value one when time is positive. Graphically it can be represented as :-

3. Shifted Unit Step Function: In many circuits, waveforms are applied at specified intervals other than t = 0. Such a function may be described using the shifted /delayed unit step function. A function which has value 0 up to the time t = a and thereafter has value 1 is known as shifted unit step function and is written as: Graphically it can be represented as :-

4. Representation Of A Function Using Heaviside’s Functions: It is more convenient to represent a function with the help of unit step function A function f(t) can be represented in different ways using Heaviside’s function. i . F(t).H(t) ii. F(t).H(t – a) iii. F(t – a).H(t) iv. F(t – a).H(t – b) v. F(t) from t = a to t = b

5. Applications of Heaviside’s Unit Step Function: Where do we use it? The function is commonly used in the mathematics of control theory and signal processing. Heaviside’s unit step function represents unit output of a system with possible time lead or lag. It is used to calculate currents when electric circuit is switches on. It represents a signal that switches on at a specified time stays switched on indefinitely.

6. How do we use it? Heaviside functions can only take values 0 or 1, but we can also use them to get other kinds of switches. Example: 4uc(t) is a switch that is off until t = c and then turns on and takes a value 4. Now, suppose we want a switch that is on (with a value 1) and then turns off at t = c. We can represent this by 1 – uc (t) = {1 – 0 = 1} ; if 𝑡 < 𝑐 = {1 – 1 = 0} ; if 𝑡 ≥ 𝑐 Heaviside’s Unit Step Function:

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