In this chapter we will study some types of voltage and current sources used in the solution of electrical circuits. There are several types, but let's focus basically on three types. Are they: the function impulse, the function step e the function ramp. These functions form a triad of related signal that are referred to as singularity functions. We are fundamentally interested in studying the behavior of a given electrical circuit when it is subjected to different types of stimulation. This is because we know that electrical circuits can replace by models mechanical, hydraulic, etc... to great advantage in terms of cost, time and detail in your analysis. Thus, for example, in the design of a car's suspension, we can idealize it through an electrical circuit and study its behavior in relation to different component values, as well as its response to various types of electrical stimuli. For this we use the sources which we will study next. After the detailed studies, we can transform the values of the components used in the electrical circuit to their mechanical, hydraulic equivalents, etc ...

2. Function Impulse

The function impulse obeys some peculiar characteristics. Thus, for a function to be considered an impulse function it must have the following characteristics when its parameter tends to zero:

1 - The amplitude of the function tends to infinity.

2 - The duration of the function tends to zero.

3 - The area under the curve representing the function does not depend on the parameter value.

There are many functions that satisfy these requirements. However, we are currently interested in a function called Dirac Delta function represented byd(t). Mathematically the impulse function is defined as follows:

∫^{ +∞}_{
-∞} K d(t) dt = K

This is valid if t = 0 and will be zero for t nonzero . Note that the integral of the function, that is, the area under the impulse function is constant. This area represents the intensity of the impulse. The function illustrated in the figure below on the left generates an impulse function when ε → 0.
Note that if we calculate the area under the curve we will find the value 1. In the
Figure 21-01, on the right, we have the symbol to represent the zero-centered impulse function..

Obviously, it is possible to represent the impulse function at a different time
of zero. In the figure below we see two examples.

In the Figure 21-02 we see the representation of two impulse functions where the first is offset to a = 2, being its intensity equal to K. The second is shifted to a = 7, and being its intensity equal to
K_{1}.

An important property of the impulse function is the filtering property , which can be expressed by the equation:

∫^{ +∞}_{
-∞}f(t) d(t - a) dt= f(a)

where we assume that the function f(t) be continuous in t= a, that is, the moment the impulse occurs. This equation shows that the impulse function filters out anything but the value of f(t) in t = a that is preserved. To demonstrate, note that d(t - a) is
zero at all times except in t= a, and so the integral can be written as:

I = ∫^{ +∞}_{
-∞}f(t) d(t - a) dt = ∫^{a + ε}_{a - ε} f(t)
d(t - a) dt

But, as f(t) is continuous at t = a, the function tends to the value f(a) when t → a and therefore:

I = ∫^{a + ε}_{a - ε} f(a) d(t - a) dt= f(a)∫^{a + ε}_{a - ε}d(t - a) dt= f(a)

Many books, and even teachers, use another representation for the impulse function. In this site we will use both notations. See below for equivalence.

função impulso ⟶K u_{0}(t - a) = Kd(t - a)

A Practical Look at the Impulse Function

It is common in the literature to assume that the impulse function exists between 0^{-} and 0^{+} when it is centered at zero . These times are extremely short. In practice, we could simulate using a known voltage source and a switch. Thus, when the key was turned on quickly, a voltage would appear at the output. But immediately after turning the key on, in an extremely short time, we would turn it off. This process gives rise to an output voltage equal to the source voltage, but the time this voltage exists is very short.

One question: Does this have any use?

Suppose we are interested in studying the behavior of an electrical system or a mechanical system. Imagine a pendulum . Yes, that piece of wire of known length where at one end is fixed a mass of known value and the other end is fixed to the ceiling, for example. Now imagine the pendulum at rest. If we do not act, the pendulum will continue to rest indefinitely. And in that case, there is nothing to study. So let's take action. Let's take a little "pat" sideways until the pendulum shifts
horizontal in relation to rest. Note that with this horizontal displacement the pendulum also gains a displacement vertical "h" and the pendulum gained a gravitational energy equal to m g h . With this initial energy the pendulum will swing and we can study its behavior.

Going back to the electrical case, it is evident that when we use an impulse source, the idea is to provide an initial energy to the system and with that we will have the so-called initial condition, a necessary condition to start studying the system behavior.

3. Step Function

In transient analysis, switching operations can cause sudden changes in circuit voltages and currents. These discontinuities can be represented by the step and impulse functions. The step function is represented in various ways in textbooks. In this site we will use the notation:

step function ⟶ u_{ -1}(t)

This notation indicates that the function is null for t < 0. For t> 0 presents a value constant and different from zero. Mathematically we can define it as

K u_{ -1}(t) = 0 if t < 0

K u_{ -1}(t) = Kif t > 0

In the Figure 21-03 we see the graphic illustration of the step function. Note the agreement with the above definition.

When K = 1 the function defined above is called UNIT step function. The
step function is not defined in t = 0 . When necessary to define the transition from
0^{-} to 0^{+}, we assume that it occurs linearly, that is, that in this interval we have:

K u_{ -1}(0) = 0.5 K

Like the impulse function, the step function can be time shifted. In the Figure 21-04 we see the graphical representation of a time-shifted function.

Note that when a > 0 , the step occurs to the right of the origin. So when
a < 0 , the step occurs to the left of the origin. Thus, a step function equal to
K for t < a can be written as:

K u_{ -1}(a - t) = Kwhen t < a

K u_{ -1}(a - t) = 0 when t > a

In the Figure 21-05 we see the graphic illustration of the step function. Note the agreement with the above definition.

4. Ramp Function

The ramp function is defined by increasing or decreasing lines. As the step function it can be represented in many ways. On this site we will use the two notations shown below for convenience.

ramp function ⟶ u_{ -2}(t) = t u_{ -1}(t)

Note that some books use the r (t) notation for the ramp function. The result is the same.

This notation indicates that the function is null for t < 0 and for t > 0 , has a
linearly increasing value or linearly decreasing, depending on the angular coefficient K of the line. Mathematically we can define it as:

K u_{ -2}(t) = 0 when t < 0

K u_{ -2}(t) = K when t > 0

In the Figure 21-06 we see the graphic illustration of the ramp function. Note the agreement with the above definition. Note that in this case the value of K is equal to
1. Since it represents the angular coefficient of the line, this means that the line of the graph makes an angle of 45° to the horizontal axis.

Like the previous two functions, the ramp function can also be shifted in time. In the
Figure 21-07 we see the illustration of this condition.

Note that when a > 0, the ramp function occurs to the right of the origin. So when a <0 , the ramp occurs to the left of the origin. Thus, a ramp function equal to K for t < a can be written as:

K u_{ -2} (a - t) = K (a - t) when t < a

K u_{ -2} (a - t) = 0 when t > a

In the Figure 21-08 we see the graphic illustration of the ramp function in this condition. Note the agreement with the above definition.

Therefore, by summing a sequence of ramps we can create triangular waves, sawtooth and many others. Use the imagination.

5. Relation Between the Three Functions

The three functions studied are related through integration and derivation. The two equations below define that the impulse function can be obtained by deriving the step function. And the step function can be obtained by deriving the ramp function.

On the other hand, the step function can be obtained by integrating the impulse function, while the ramp function can be obtained by integrating the step function. This is what illustrates the two equations below.

6. Product Between Functions

Using the filtering property we can realize product between functions by obtaining values that suit our interests. Let's suppose the function f(t) = sin 314 t. From the function we know that
f = 50 Hz, and therefore your period is 0.02 s. See the Figure 21-09 for a graph of this function.

Suppose for an experiment we will need a waveform as we can see in the Figure 21-10. Note that we have a positive part and a negative part. Thus, we must provide a step function that allows the positive part to pass and another step function that allows the negative part to pass.

For the positive side let's use the function u_{-1}(t - 5 ms) - u_{-1}(t - 10 ms)
and for the downside - u_{-1}(t - 10 ms) + u_{-1}(t - 20 ms). Thus, we generate two pulses that allow us to obtain the desired shape for the experiment.

In the Figure 21-11 we see the illustration of the pulse formation required to form the positive part of the waveform, that is, between the time of 5 ms and 10 ms. Note that the positive pulse, u_{-1}(t - 5 ms), starts after 5 ms from time zero. Since we want it not to exist after time 10 ms, then we must subtract the same amount in time equal to 10 ms. This is why we use the function represented by - u_{-1}(t - 10 ms).

Thus, from time 10 ms both functions cancel each other, generating the pulse we want. This process allows us to have the positive part of the waveform output. Now, to get the negative, we must have a pulse below the x axis , that is, negative. Then we use the function
- u_{-1}(t - 10 ms), which will generate a negative pulse from time 10 ms . And to cancel the pulse at time 20 ms, we use the same technique as before, adding the function
+ u_{-1}(t - 20 ms). Then, from time 20 ms , the two functions cancel each other, generating the necessary pulse to get the negative part of the waveform.
No less important is to describe it mathematically. Using the multiplication property of functions, we mathematically represent this function as: