## What is an Inductor?

An actor is a passive electrical component that consists of a coil bent in the form of wire which uses electromagnetism to produce electric current through the coil. When the electric current flows through a conductor that is the coil, a magnetic flux is developed around the conductor. This flow of current through the coil creates a magnetic field based on Fleming's Right-hand thumb rule. If there is any decrease in the current passing through the inductor, then the magnetic field will also decrease and energy is released through the generation of electric current. There is also a secondary voltage inducing in the same coil due to the magnetic flux, since it opposes any changes in the electric current flowing through the circuit. An inductor is also called choke, coil, or solenoid. A choke is designed to block certain frequencies when the DC current passes. A winding coil is coated with insulation to give extra insulation and protect the circuit. They are usually tightly wrapped around a central core which is cylindrical in nature so that the magnetic flux can be induced. The solenoid is a three-dimension coil that is used to exhibit electromagnetism when the magnetic field is developed due to the current flowing through the coil. The sign convention of the inductor is the same as the power dissipating device. When the current flows into the positive side of the voltage across the inductor, the value is positive and the inductor dissipates power whereas when the inductor releases the power back to the circuit, the current has a negative sign convention.

The current flowing through the inductor exerts a magnetic flux that is directly proportional to it. This is contrary to the working of a capacitor that opposes the voltage variation across the conducting plates. In the case of an inductor, the inductor opposes the rate of change of current flowing through its coil due to the self-inductance o of the magnetic field.

The inductance is the capacity of the inductor to resist the change in the current flowing through it with its magnetic flux and the number of turns of the coil.

## Mathematical Formula

The voltage across the inductor can be found by the product of the rate of change of current and the value of the inductance. It is mathematically represented as${v}_{L}=-L\frac{di}{dt}$. The sign convention is negative because this is known as back emf. The L is the value of the self-inductance and di/dt is the rate of change of current.

The SI unit of Inductance is Henry. The inductance is proportional to the area of the cross-sectional of the core, the number of turns of the wire, and the material used in the core.

The steady-state DC current flowing through the inductor makes zero induced voltage across the coil, so in this case, the inductor acts like a short circuit.

## Self Inductance and Mutual Inductance

When a current is passed through the coil of the inductor, the magnetic flux indicates that an opposite emf, ‘back emf’ is produced. This is called self-induced current or self-inductance. But in the case of two coils placed next to each other, one of the coils is called the primary coil and the other one is called the secondary coil. When the current flows through one of the coils producing magnetic flux linked with both the coils, an opposing back emf is produced across the coil. This is called mutual inductance. The behavior of the circuit changes as per the source voltage, whether AC voltage or a DC source voltage is connected.

NOTE: The resistance of an ideal inductor is zero. The reactance of an ideal inductor is positive for all frequency and inductance values. The resistance is tested with an ohmmeter and the increase in ohmmeter denotes the resistance value.

**Lenz's Law**

It is a phenomenon in Electromagnetism that tells about the induced current flowing in the direction in such a way that it opposes the change that induced it. It is the same as Faraday's law of electromagnetic induction but it deals with the back emf induced in the inductive circuit.

## Experimenting the Voltage and the Inductor

If an inductor is connected with a switch instead of a resistor and a voltage source. Before the switch is turned on, the initial current through the inductor is zero since the time is zero. After the switch turns on, since the source is connected to the emf or voltage source, the current starts flowing. The current slowly increases while the inductor integrates its voltage. So the current across the inductor can be represented as a mathematical equation:

${i}_{L}=\frac{1}{L}{\displaystyle \underset{0}{\overset{t}{\int}}v(x)dx}+i(0)$

The inductive reactance is known as the opposition exerted by the inductor in an AC circuit

due to the flow of ac current. So the corresponding formula is ${X}_{L}=\text{}2\pi \times f\times L$

The capacitive reactance is the opposition exerted by a capacitor to the flow of an ac current in the ac circuit. It is inversely proportional to the capacitance and the frequency.

So the corresponding formula is ${X}_{c}=\frac{1}{2\pi \times f\times C}$

## AC Source Inductor

In AC circuits, when the voltage source is a DC source then the equation of the voltage across the inductor is directly proportional to the change in current so the current will be constant and the derivative of a constant is 0, so the current flowing will not be able to induce voltage. Hence AC source is necessary for magnetic flux to be produced and induce the current. When the voltage of an AC source is connected to the inductor, then the input voltage increase and drops with the frequency producing the back emf. The emf is proportional to the rate of the change in current through the coil, where it has its maximum value when the voltage range is between the 0 and 180 degrees of the sinusoidal wave graph and the minimum voltage range shift is when the AC sinusoidal wave is at its minimum peak voltage point. The voltage and current lag by 90 degrees for an inductive circuit. The phasor equation would be

${I}_{L}=\text{}{I}_{m}sin(\omega t-90\xb0)$

where I_{m }is the maximum value of current, I_{L} is the inductive current and w is the angular velocity with t as the time.

The AC circuits contain two or more components connected in series and parallel configurations. The series configuration is classified into three types. RL series circuit, RC series circuit, and RLC series circuit.

In a pure resistance circuit, the voltage is in phase with the current. In pure inductance circuit, the voltage leads current by 90º. In a pure capacitance circuit the voltage lags the current by 90º. The RL series circuit is a resistor-inductor circuit where only the resistor is connected in series with an inductor. The phase angle is given by $\varphi ={\mathrm{tan}}^{-1}\left(\frac{{X}_{L}}{R}\right)$

In an RC series circuit, the resistor is connected in series with a capacitor. The phase angle is given by $\varphi ={\mathrm{tan}}^{-1}\left(\frac{{X}_{C}}{R}\right)$

In RLC series circuit, all the three resistor, inductor, and capacitor are connected in series. The phase angle is given by $\varphi ={\mathrm{tan}}^{-1}\left(\frac{{X}_{L}-{X}_{C}}{R}\right)$

- Here when the inductive reactance is greater than the capacitive reactance, the circuit behaves like RL circuit. The applied voltage is$v={v}_{m}\mathrm{sin}\omega t$. The current through the inductor is $I={I}_{m}\mathrm{sin}(\omega t-\varphi )$

- If the inductive reactance is lesser than the capacitive reactance, then the circuit behaves like an RC circuit. The current through the inductor is $I={I}_{m}\mathrm{sin}(\omega t+\varphi )$

- If the inductive reactance is equal to the capacitive reactance, then the circuit behaves purely resistive circuit. The current equation becomes $I={I}_{m}\mathrm{sin}\omega t$

## Practice Problem

1. An inductor has a resistance of 100 Ohms to an AC source of 100 voltage and frequency of 50 Hz. what is the value of the inductance?

Ans. It is given that resistance of inductor is 100 Ohms, and the frequency of the source is 50 Hz. The formula used is

$\begin{array}{l}{X}_{L}=2\pi fL\\ \end{array}$

Substitute the values of frequency and resistance of the inductor to calculate the inductance.

$$\begin{array}{c}L={X}_{L}/2\pi f\\ L=100\text{Ohm/2}\times \text{3}\text{.14}\times \text{50Hz}\\ \text{L=0}\text{.318H}\end{array}$$2. A steady-state current of 4A passes through a solenoid coil of 0.5 H. What would be the average back emf, voltage induced in the coil if the switch in the circuit was opened for 10ms and the current through the coil is 0 A.

Ans. It is given that the switch is the circuit was opened and the current falls to 0 A. The initial current was 4 A and the final current is 0 A. The time taken is 10 ms. The formula used to calculate the back emf is

$\begin{array}{l}{v}_{L}=-L\frac{di}{dt}\\ \\ \end{array}$Substitute 4 A for change in the current, 10 ms for the change in the time and 0.5 H for inductance to calculate the value of average back emf introduced.

$$\begin{array}{l}{v}_{L}=0.5\text{H}\times \frac{4\text{A}}{10\text{ms}}\\ {v}_{L}=0.5\text{H}\times \frac{4\text{A}}{10\times {10}^{-3}\text{s}}\\ {v}_{L}=200\text{V}\end{array}$$## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelors in Science (Physics)
- Masters in Science (Physics)

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