Heaviside's function

Heaviside's Unit Step function in Laplace transform.

1. Heaviside’s Unit Step Function

2. Introduction The Unit Step Function (Heaviside Function) In engineering applications, we frequently encounter functions whose values change abruptly at specified values of time  t . 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 which is also known as the  Heaviside Unit Step function .

3. 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 :-

4. Laplace transform of Unit Step function H(t) By definition of Laplace transform ̅  

5. 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

6. Laplace Transform of Shifted Unit Step Function H(t - a)  

7. Unit Impulse Function Rectangular Pulse A common situation in a circuit is for a voltage to be applied at a particular time t = a and removed later, at  t = b . We write such a situation using unit step functions as 1 for a < t < b We can represent it graphically as :- 0 otherwise u(t) = t=a t=b 1 u(t)

8. Laplace Transform of Impulse Function (0)  

9. 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. F(t).H(t) F(t).H(t – a) F(t – a).H(t) F(t – a).H(t – b) F(t) from t = a to t = b

10. Case 1 : F(t).H(t) We know that Therefore multiplying f(t) with H(t), we get Hence by taking the product f(t).H(t) the part of f(t) to the left of the origin is cut off. Example : Let  

11. Case 2 : F(t).H(t - a) We know that Therefore multiplying f(t) with H(t - a), we get Hence by taking the product f(t).H(t-a) the part of f(t) to the left of the t = a is cut off. Example: Let   Here a = 2

12. Case 3 : F(t - a).H(t) We know the curve is same as only difference is that the origin is shifted at a. Hence the shape of the curve remains unchanged. Therefore   will represent the curve on the right of origin.  

13. Case 4 : F(t - a).H(t - b) f(t - a).H(t - b) will give the part of the shifted curve f(t - a) to the right of t = b cutting off the part before t = b Since f(t-a) is the curve f(t) with origin shifted to a. Here H(t-b) is zero before t = b and unity after t = b. Therefore  

14. Case 5 : Representation of the part of the curve f(t) from t = a to t = b We see that H(t-a) is a unit function on the right of t=a and H(t-b) on the right of t=b. So the function [H(t-a)- H(t-b)] is zero before t=a and after t=b. Therefore here H(t)=   Hence the only remaining part of f(t). [H(t-a)- H(t-b)] is between t=a and t=b called as filter function.

15. Applications of Heaviside’s Unit Step Function

16. 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.

17. 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: 4u c (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 – u c (t) = {1 – 0 = 1} ; if = {1 – 1 = 0} ; if