unitstepfunction

8/10/2019 unitstepfunction

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In engineering applications, we frequently encounter functions whose values change abruptly atspecified 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.

Shifted unit step function

Rectangular pulse

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(otherwise known as the Heaviside functionafterOliver Heaviside).

The Unit Step Function

Definition: The unit step function, u(t), is defined as

That is, uis a function of time t, and uhas value zerowhen time is negative (before we flip the switch);and value one when time is positive (from when we flip the switch).
http://www.intmath.com/laplace-transformation/1a-unit-step-functions-definition.php#shiftedhttp://www.intmath.com/laplace-transformation/1a-unit-step-functions-definition.php#shiftedhttp://www.intmath.com/laplace-transformation/1a-unit-step-functions-definition.php#rectangularhttp://www.intmath.com/laplace-transformation/1a-unit-step-functions-definition.php#rectangularhttp://www.intmath.com/laplace-transformation/heaviside.phphttp://www.intmath.com/laplace-transformation/heaviside.phphttp://www.intmath.com/laplace-transformation/heaviside.phphttp://www.intmath.com/laplace-transformation/heaviside.phphttp://www.intmath.com/laplace-transformation/1a-unit-step-functions-definition.php#rectangularhttp://www.intmath.com/laplace-transformation/1a-unit-step-functions-definition.php#shifted


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Example of Shifted Unit Step Function:

f(t)= u(t 3)

The equation meansf(t) has value of 0 when t< 3 and 1 when t> 3.

The sketch of the waveform is as follows:


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Rectangular Pulse

A common situation in a circuit is for a voltage to be applied at a particular time (say t = a)and

removed later, at t = b(say). Such a situation is written using unit step functions as:

V(t) = u(t a) u(t b)


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Write the fo l lowing funct ions in terms of u ni t step funct ion (s) . Sketch each

waveform.

(a) A 12-V source is switched on at t= 4 s.

Answer

Since the voltage is turned on at t= 4, we need to use u(t 4). We multiply by 12 since that is the

voltage.
http://www.intmath.com/laplace-transformation/ans-1a.php?a=0http://www.intmath.com/laplace-transformation/ans-1a.php?a=0


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(b)

(Assume a> 0.)

Answer

In words, the voltage has value 1 up until time t= a. Then it is turned off.

We have a "rectangular pulse" situation and need to use this formula:

V(t) = u(t a

) u(t

b)

In our example, the pulse starts at t = 0 [so we use u(t)] and finishes at t= a[so we use u(t a)].

So the required function is:


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(c) One cycle of a square wave,f(0) = 4,amplitude = 4,period = 2 seconds.

Answer

f(0) = 4means we start at value 4.

If the whole wave has period 2, and it is a square wave, then it means for half of the time, the value is(positive) 4 and the other half it is negative 4.

So for the first second, it has value 4, for the second second, the function value is -4.

We write this, using the "rectangular pulse" formula from before:


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The graph of this first cycle is:

(d) The unit Ramp function (i.e.f(t) = tfor t> 0)

Answer


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The unit ramp function has slope 1 [so the function is simply V(t) = t], starting from t = 0 [so we need tomultiply by u(t)], and passes through (0, 0).

So the voltage function is given by:

V(t) = t u(t)

The graph of the function is:


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(e) One cycle of a sawtooth waveform (i.e. for 0 < t< b.Assume a> 0.)

Answer

Our graph starts at t= 0 and has slope a / b. It finishes at t = b.

So our function will be:

The graph of our function:


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(f)

Answer

In this example, our function is V(t) = 2t+ 8which has slope 2 and V-intercept 8.

The signal is only turned on between t= 3 and t= 5. The rest of the time it is off.

So our voltage function will be:

The graph is as follows:


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