The inductor, a fundamental component in electrical and electronic circuits, is notable for its simplicity of manufacture
and the importance of its function. It consists of a conductive wire, typically copper, wound into a helical shape
on a nucleus that can have different geometries, such as circular, square or elliptical. The inductance of the inductor, which is
the property of generating an electromotive force opposite to the variation of the electric current that passes through it, is directly
influenced by this geometry. The standard unit of measurement for inductance is the henry (H), and for smaller values,
its submultiples are used: millihenry (mH), microhenry (µH) and nanohenry (nH). These components are essential
for magnetic energy storage, signal filtering and in many other applications that depend on its
ability to resist changes in electrical current.
As with capacitors, we can also vary the inductance of an inductor by changing the material
that fills its core. Thus, we can have inductors with air core, ferrite or other ferromagnetic material.
It should be emphasized that when an electric current, of any nature, circulates
by winding the inductor, it generates a magnetic flux inside it.
This magnetic flux can be constant or variable.
By definition, the inductance
of an inductor is given by:
eq. 04-01
In this equation, N is the number of inductor coils, while Φ is the
magnetic flux in webers and i is the electric current, in ampère,
that circulates through the inductor.
In circuit analysis, the voltage in the inductor always has a polarity opposite to the
source that generated it, and so the average voltage on the inductor is:
eq. 04-02
This equation means that if there is no variation in the electrical current circulating by the inductor,
so the voltage in your terminals will be equal to ZERO. This is a very important feature of the
inductor and we will see in more detail in item 5.
What characterizes a series association is that we connect to a node
only two components. So we can say that if the circuit was fed by a current source,
the current that would go through the circuit would be the same in any inductor of the circuit.
Figure 04-01
In the Figure 04-01, we see a series association.
We can replace all the inductors that are part of the circuit by a single equivalent inductor.
The value of the equivalent inductor is given by the equation below.
Leq = L1 + L2 + L3
Of course we can generalize to "n" serial inductors, and we use the eq. 04-03 below:
What characterizes a parallel association is that all inductors are
subjected to the same potential difference.
Figure 04-02
In the Figure 04-02, we see a parallel association of inductors.
We can replace all the inductors that are part of the circuit by a single equivalent inductor.
If we have "n" inductors we can calculate the value of the inductor equivalent
by the equation below:
In the mixed association, show in Figure 04-3, as the name itself is saying, we will have a circuit that contains
both, parallel and serial association. To solve it, we first found the result of the parallel between
L2 and L3 and subsequently added to the value of L1.
As well as capacitors can store energy in the electric field that arises
when you apply an electrical voltage between your plates, as we saw in Chapter 3, the inductors
can also accumulate energy in the magnetic field generated by the passing of the current
electrical through its winding. This energy only depends on the inductance and the
electrical current that circulates through the inductor and we can calculate it using the equation below.
In this item, we will see what is the behavior of an inductor in relation to the DC
or direct current. We'll consider that the inductor initially does not
circulate electrical current in its winding, so it has initial energy equal to zero.
When this is not the case, we will state the initial condition.
Below, one of the fundamental properties of an inductor is described.
Based on the property above, the inductor assumes special characteristics when subjected to
variations of electrical current in its terminals. Normally, a serial resistor is used with
the inductor to limit the electrical current circulating through it. So, when the inductor
is subjected, abruptly, to a variation of electrical voltage, it behaves like an open circuit.
In the Figure 04-04 we can see a classic circuit to study the behavior of the inductor.
Figure 04-04
In this circuit, we have a S key, which allows you to turn the power supply on and off
that feeds the circuit. When closed, apply an electrical voltage from the V voltage source
in the circuit formed by the resistor in series with the inductor. In the technical literature,
it represents the closing instant of the key S such as the time equal to
t = 0+.
The speed with which the electrical current circulates in the inductor, depends on the
values of the inductor inductance and the
resistor electrical resistance lying in series with inductor.
The values of these two components determine the time constant call of the
circuit and is represented by the greek letter τ (tau). So we can write that:
τ = L / R
Figure 04-05
By applying, abruptly an electrical voltage on the inductor, its inductance
does not allow an instantaneous variation of the current in the circuit. So if there's no current
circulating through the circuit, all the supply voltage will be over the inductor. So, VL = V.
When the circuit starts to drive electric current, it rises rapidly
at the beginning of the driving and reaches the final value of
i = V/R in an exponential way. The Figure 04-5 graphically shows this circuit
behavior. And this is closely related to the equation below.
eq. 04-06
Watch out for the fact that when the time grows, the current in the circuit tends to the final
value iL = V/R. And approximately after five
time constants, we can say that the circuit has reached the permanent regime.
From that moment, all the voltage of the source will be over the resistor and, of course,
the voltage on the inductor will be zero. This due to the
fact that the current becomes constant, that is, does not vary with respect to time, and as stated
previously, the voltage on the inductor is null. So to
calculations in electrical circuits, we should consider the inductor as a
short circuit when on permanent regime. We can calculate the voltage on the inductor at any
time using the equation below.
eq. 04-07
This was a brief approach to the behavior of an inductor when it is in a circuit that uses
only continuous current. We'll soon be dealing with more depth this problem by using
the solution of differential equations, as well as demonstrating where the above equations arose.
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