INDUCTOR

An inductor is a passive element designed to store energy in its magnetic field. They find numerous applications in electronics and power systems. For example, they are used in transformers, radios, TVs, radars and electric motors.

Any conductor of electric current has inductive properties and can be thought of as an inductor. But in order to enhance the inductive effect, a practical inductor is usually formed into a cylindrical coil with many turns of conducting wire as shown in the figure.

It was in the early 1800s, that English experimentalist Michael Faraday and American inventor Joseph Henry discovered almost simultaneously that a changing magnetic field could produce a voltage in a neighboring circuit. They showed that this voltage was proportional to the time rate of change of current which produced the magnetic field. That constant of proportionality is called inductance and is denoted by L.

\[v=L\frac{di}{dt}\]

The unit of inductance is Henry (H).

Now, what is Inductance?

Inductance is the property of the coil by virtue of which it opposes any change in the current flowing through it.

The inductance of an inductor depends on its physical dimensions and construction. Inductance of a solenoid (inductor) is given by

\[L=\frac{\mu {{N}^{2}}A}{l}\]

Where,    N = number of turns

               l = length

               A = cross-sectional area

               $\mu $= permeability of the core

Inductance can be increased by increasing N,$\mu $, and A or by reducing l.

TYPES OF INDUCTORS-

Inductors are available in different types ranging from large high current iron- cored chokes to tiny low current coils.

Air-cored coils are wound on a tubular insulating material such as cardboard, fiber, hard rubber, Bakelite etc. Such coils find use in electronic circuits working at high frequencies. Their inductance values are in millihenry (mH) and microhenry ($\mu $H) range.

Solenoidal wound inductor

Toroidal inductor

Chip inductor

Iron-cored chokes may have laminated iron core or powdered iron cores. They are mainly used at AC power frequency (50Hz) and at audio frequencies (20Hz to about 10 kHz). Such chokes have inductance of a few henry's.

HOW TO MAKE AN INDUCTOR VARIABLE?

An inductor can be made variable, by having tapped coils (taps are brought out at several points in the winding), by having a slider (similar to a rheostat), or by having a movable core. Standard symbols of different types of inductor are shown in the figure given below.

ENERGY STORED IN AN INDUCTOR-

We know induced voltage v, in an inductor is given by

\[v=L\frac{di}{dt}\]

\[\Rightarrow di=\frac{1}{L}vdt\]

Integrating gives,\[i=\frac{1}{L}\int\limits_{-\infty }^{t}{v(t)dt}\]

or\[i=\frac{1}{L}\int\limits_{{{t}_{o}}}^{t}{v(t)dt+i({{t}_{0}})}\]

Where i(t₀) is the total current for -$\infty $<t<t₀ and i(-$\infty $)=0. This argument is reasonable because there must be time in the past when there was no current in the inductor.

Power delivered to inductor is p = vi = $\left( L\frac{di}{dt} \right)i$

Energy stored is $w=\int\limits_{-\infty }^{t}{pdt}=\int\limits_{-\infty }^{t}{\left( L\frac{di}{dt} \right)idt}$

\[=L\int\limits_{-\infty }^{t}{idt}=\frac{1}{2}L{{i}^{2}}(t)-\frac{1}{2}L{{i}^{2}}(-\infty )\]

Since, i($-\infty $)=0,

                                                     \[w=\frac{1}{2}L{{i}^{2}}\]

PROPERTIES OF AN INDUCTOR-

Following are the important properties of an inductor –

1. Voltage across an inductor is zero when the current is constant which can be inferred from the equation, $v=L\frac{di}{dt}$ . Thus, inductor acts like a short circuit to DC.
2. The current through an inductor cannot change instantaneously. Thus, an inductor opposes abrupt change in current through it.

   

   Current through an inductor (i) allowed, (ii) not allowed; an abrupt change is not possible. 

3. An ideal inductor does not dissipate energy. The energy stored in it can be retrieved at a later time. It takes power from the circuit when storing energy and delivers power to the circuit when returning previously stored energy.
4. A practical, non-ideal inductor has a significant resistive component as shown in the figure given below.

   

   Circuit model of a practical inductor

This is due to the fact that inductor is made of a conducting material such as copper, which has some resistance. This resistance is called the winding resistance R_w, and it appears in series with the inductance of inductor.

The presence of R_w, makes it both an energy storage device and an energy dissipation device.

Since R_w is small, it is ignored in most cases. The non-ideal inductor also has a winding capacitance C_w due to the capacitive coupling between the conducting coils. C_w is small and can be ignored in most cases, except at high frequencies.