An inductor is a passive two-terminal electrical component that stores electrical energy in the form of a magnetic field. Capacitors and inductors both stores electrical energy but in a different form. Capacitors store electrical energy in the form of an electric field. Both can be used to smooth DC current.
An inductor is nothing but a coil of a wire wound around an iron or some kind of magnetic material. The function of the material in which the wire is wound around is to increase the magnitude of the magnetic field, thus increasing the storage capacity of the inductor. An inductor has a symbol of a coil of wire because it is really nothing but a coil.
Inductors are mostly used in Alternating current (AC) electronic equipment. Like a capacitor, the inductor dissipates negligible energy as heat.

Inductor symbols

Picture of a variety of inductors

Surface-mounted inductors(SMD) images

Theory of an inductor

An inductor resists the change in alternating current flow. An inductor is a frequency sensitive component, if a DC source is applied to an inductor, the inductor will allow current to pass through it but when an Alternating current is applied, it will resist the current flow, the resistance is related to the frequency of the applied AC signal, if it is of high value the AC current would get completely blocked.

ω = 2πf. f is the frequency of the applied AC signal L = is the Inductance of the inductor.
Inductance is the characteristics of an inductor to generate a magnetic field for a given current. L is the symbol of an inductance it has a unit H, which is henry, inductors have values that typically range from 1 µH to 1H. Inductance can be defined mathematically as:

The relationship between the voltage across an inductor and the current flowing through it is given as:

When the magnetic field around the coil of the inductor changes with time, a voltage is produced across the leads of the inductor, this is called a back EMF

This back EMF tends to oppose the increasing current in the coil of the inductor.

Effect  of a DC to an inductor
When a DC is applied to an inductor, the current will pass through the inductor and the inductor will have a magnetic field around it. This is called the electromagnet. When the DC source is removed the magnetic field dies.

Effect of an AC to an inductor
When an AC is applied to an inductor, a voltage known as a back EMF will be developed by the inductor to resist the changing current, depending on the value of the inductor, small current pass through, if the varying AC signal has a very high frequency the inductor will block the current from passing through. The resistance of an inductor to the flow of AC signal is termed reactance.

Energy stored by an inductor
the energy stored by an inductor is given as:

In the figure below the inductor blocked AC but allows DC to pass-through.

In the figure below an inductor is used to bypass DC to ground but allows an AC to pass. Since inductor blocks AC the only way for the AC to pass is straight ahead.

Types of an inductor
Inductors come in many shapes. Some inductors are designed for a specific purpose, like Antenna coil which is used in a high-frequency application.
There are Air core inductors that have air as their core, these inductors have low inductance. There are Toroidal shaped inductors that have a donut shape, these inductors have a high value of inductance.

Inductors that are purposely used to limit AC but allows DC to pass are called chokes, some choke inductors look like a color-coded resistor.

The Transformer

When a coil of a wire is connected to an AC power and another coil of wire is brought near it, a voltage is induced in the second coil, the magnitude of the induced voltage depends on the number of turns of the coil in the second coil. This is called mutual inductance. This phenomenon (mutual inductance) is very important.
An alternating current of whatever value can be raised or lowered. DC voltage cannot be transformed by a transformer because mutual inductance doesn’t occur at 0 frequency.
A transformer is an electrical device that raises or lowers the voltage supplied to it.

The number of turns of coil of a transformer is proportional to the voltage induced in the coil. There is a very useful formula which is handy when winding a custom transformer. Conversely, the number of turns is inversely proportional to the current in the coil, the higher the turns(Voltage) the lower the current.

Author:https://en.wikipedia.org/wiki/User:BillC

A transformer has an input terminal and an output terminal, the output terminal may contain many independent coils, that have different voltage. The input coil is called a primary coil and the output coil is called a secondary coil.
A transformer that transforms its input voltage to a higher voltage than the input is called a step-up transformer
A transformer that transforms its input voltage to a lower voltage than the input is called a step-down transformer.

A transformer that gives an output voltage of the same value as the input ( does not step-up or step-down ) is called an isolation transformer.

Transformer voltage, current, and the number of turns relationships.

There are many types of transformers, the most common ones are listed below:

Common transformers:

Center-tap transformer

This transformer has a tap in its primary or secondary coil, tapping on the secondary winding gives the transformer two or more output voltages.

Multiple winding transformer

This special transformer has multiple winding in its secondary coil which are electrically isolated from each other, each coil has its voltage based on its number of turns.

Auto-transformer

This transformer has a single coil with a tap at the center, that makes it primary and secondary.

Isolation transformer

This kind of transformer has an equal number of turns in both primary and secondary coils. It is used to isolate a circuit from another since the two circuits are only coupled magnetically not electrically and there is no way a DC can pass from one circuit to another.

A transformer can be used for impedance matching, to connected different devices that have entirely different impedance, eg high impedance amplifier and low impedance loudspeaker.

Inductors in series
When two or more inductors are connected end to end their final inductance could b calculated as.

Inductors in parallel

When two or more inductors are connected side by side there final inductance could be calculated as

RL circuits

Current growth in an RL circuit

When a resistor and an inductor are connected in series and connected to a voltage source. The current in the inductor will not just rise to the final value, the current begins rising steadily based on the inductance of the inductor or the value of the resistor in series. The time that takes the current in the inductor to reach its final value depends on the inductor and the resistor.

Current decay in an RL circuit

From the above experiment when the voltage source is removed and the resistor and the inductor are connected as in the figure below. The current inside the inductor will not decay instantaneously but it decays smoothly to zero. The time that takes the current in the inductor to decay to zero depends on the value of the inductor and the resistor.

Mathematically

LC circuit

If a capacitor and an inductor are connected as in the figure below. There will be an energy transfer between the capacitor and the inductor. This arrangement creates an electrical oscillation.

Assume initially the capacitor is fully charged and the inductor is uncharged. When connected the capacitor starts discharging through the inductor smoothly until the capacitor becomes fully uncharged, at this moment the inductor becomes fully charged and the capacitor is uncharged, the moment the inductor becomes fully charged it starts discharging through the uncharged capacitor after a short while the inductor becomes fully uncharged while the capacitor becomes fully charged this whole process keeps repeating until the energy becomes damped.

The oscillation dies down over time due to energy lost as heat in the coil of the inductor and in the connecting wires.

Capacitor

A capacitor is an electrical component that stores electrical energy, it is a two-terminal passive component. Capacitor stores electrical energy in the form of an electric field. A capacitor could be thought of like a bucket that stores electrical energy.
There are many different practical capacitors available. It depends on the dielectric present between the two positive and negative plate of the capacitor. A capacitor dissipates a negligible amount of electrical energy.
A capacitance of a capacitor is the ability to store electrical energy, some capacitors have a higher capacitance than others, Farad is the SI unit of a capacitance of a capacitor, the capacitance of a capacitor ranges from picofarad to millifarad. One farad = 1coulumb/voltage.
Capacitors are widely used in electronic circuits. Ceiling fans have a capacitor, capacitors in electronic circuits are used to block DC current, it is also used as a filter, like a tuning device. A resistor combined with a capacitor in series makes a timing circuit.

Author:Eric Schrader from San Francisco, CA, United States https://commons.wikimedia.org/wiki/File:Capacitors_(7189597135).jpg
Capacitor symbols Fixed and variable

Theory of a capacitor

When two conductors are separated by an insulator, be it air or some kind of an insulator, a capacitor is formed.

Practical capacitors are made up of two plates separated by a dielectric. A dielectric is an insulator.

When a voltage is applied across a capacitor the dielectric becomes polarized and develops the electric field, the conductive plates hold equal and opposite charges on their facing surfaces.

Author: Papa November , https://en.wikipedia.org/wiki/File:Capacitor_schematic_with_dielectric.svg

A capacitance of a capacitor is defined as

The charge stored by a capacitor is directly proportional to the applied voltage. the C is the proportionality constant, which is the capacitance of the capacitor

The charge stored by a parallel plate capacitor is defined as

This means as the area of the plates is increased the capacitance also increases. Furthermore, if the dielectric material placed has a higher permittivity, the capacitance increases also.

The energy stored by a capacitor
Capacitors store electrical energy in the form of an electric field. When a voltage is applied across a capacitor, the capacitor stores the electrical energy, the voltage across the plates of the capacitor starts to rise until their voltage reached the applied voltage. A given capacitor with 22uF 400V connected to 400V power supply, the voltage of the capacitor will keep rising until it reached 400V. the capacitor is charged to 400V!!. So care must be taken when handling a circuit with capacitors that are connected on a power source, even when the power is cut.

Charge and discharge of a capacitor

When a capacitor is connected across a DC source, one of the plates becomes positively charged and the other becomes negatively charged.

Capacitors block DC.

When a capacitor is connected across an alternating current, one of the plates become positively charge and the other becomes negatively charged, the capacitor continuously charges and discharge, Alternating current flows. However, this current is not a conventional current it is called displacement current due to the displacement charge. Currently doesn't actually flow through a capacitor.

The displacement current is given as.

and the displacement voltage is.

Charging a capacitor through a resistor

When a capacitor is connected in series with a resistor, and voltage is applied across them. The rate at which the capacitor charges depends on

i) The capacitance of the capacitor

ii) The value of the resistor in series with the capacitor

This makes a capacitor a very useful component in the timing circuit. The combination is called the RC circuit. The time at which the capacitor charge is proportional to the RC values.

When a capacitor is charging through a resistor to a final voltage. The instantaneous voltage is given as

Discharging a capacitor through a resistor

When a charged capacitor is connected to a resistor, the capacitor discharges through the resistor. The rate at which the capacitor discharges depends on the value of the capacitor and the resistor. The instantaneous voltage of the capacitor is

Capacitor circuits

Capacitors in parallel
When two or more capacitors are connected side by side as in the figure below they are said to be in parallel, the potential difference across both the capacitors is the same.

And the sum of the capacitance in parallel is.

Capacitors in series
When two or more capacitors are connected end to end they are said to be in series. The voltage across the capacitors is not the same. And the resultant capacitance is

And the sum of the capacitance in series is.

RC filter

The RC filter is a frequency-dependent voltage divider. They are used to block either a high-frequency signal or low-frequency signal depending on the RC arrangement.

 High pass filter

This is a capacitor-resistor voltage divider circuit. This circuit blocks low-frequency signal (DC) from passing while allowing high frequency signal AC.

Low pass filter

This is a resistor-capacitor voltage divider circuit. This circuit blocks High-frequency signal (AC) while allowing a low-frequency signal to pass.

Types of a capacitor

There are generally two types of capacitors Polarized and non-polarized.

A polarized capacitor has to be connected correctly in a circuit with a positive leg connected to the higher potential in the circuit and the negative leg to the lower potential in the circuit.

The practical form of a capacitor

Electrolytic capacitor
Generally, they are polarized capacitors, they contain an electrolyte as a dialectic. They fall into a category, 1 Aluminum capacitor, Tantalum Capacitor, Niobium Capacitor.

Ceramic capacitor
Ceramic capacitors have a ceramic as a dielectric. Usually, it is a mixture of ceramic and titanium dioxide. Some ceramic capacitor is mixed with barium titanate.

Mica capacitor
These type of capacitors are reliable and they have a high precision they are also called silver mica capacitors, they have mica as a dielectric material, they are used in a high-frequency application. Their apacitance ranges between 1pF-0.1μ

Air Capacitor
These have air as a dielectric, they are used as variable capacitors
Paper capacitor