**Inductor: **

An inductor is a passive component of an electrical circuit that opposes current. It is a coil of wire wrapped around a magnetic material. Applied voltage induces current across the inductor. When current flows through the inductor, it generates a magnetic field. Magnetic fields don’t change. Therefore, the inductor tries to prevent the current flowing through it from changing.

**Reactance:**

Reactance is defined as an opposition to current flow in an electrical circuit. It is denoted by **?**.

**Inductive Reactance X**_{L}:

_{L}:

Inductive reactance is the reactance offered by an inductor: the greater the reactance, the smaller the current.

## In a dc circuit inductive reactance would be zero (short-circuit), at high frequencies an inductor has infinite reactance (open-circuit).

**Inductive Reactance Units** **| SI unit of inductive reactance**

Inductive reactance acts as an opposition to the current flow in the circuit. So the SI unit of inductive reactance is the same as that of resistance, i.e. Ohms.

**Symbol for Inductive reactance**

Inductive reactance is denoted by **?**_{L }or **X _{L}**.

**Derivation of Inductive Reactance **

Suppose we have the following electric circuit with inductance L connected to an AC voltage source. This source creates an alternating current that flows inside the inductor if the switch is closed. So, the electric current in the circuit at any moment is given by,

[Latex]I=I_{0}cos\left ( \omega t \right )[/Latex]

Where I_{0}= peak value of the current

ω= angular frequency

Now, if we apply Kirchhoff’s second law or Kirchhoff’s loop law in this circuit, we get,

[Latex]V-L\frac{\mathrm{d} I}{\mathrm{d} t}=0[/Latex]

[Latex]V=L\frac{\mathrm{d} I}{\mathrm{d} t}[/Latex]

So, the voltage across the inductor V is equal to the inductance multiplied by the derivative of electric current I with respect to time.

[Latex]\frac{\mathrm{d} I}{\mathrm{d} t}=\frac{\mathrm{d} }{\mathrm{d} t}\left ( I_{0}cos(\omega t) \right )=- I_{0}\omega sin(\omega t)[/Latex]

[Latex]V=-L I_{0}\omega sin(\omega t)=L I_{0}\omega sin(\omega t+90^{\circ})[/Latex]

If cos(ωt+90°)= 1, then V=V_{0}=LI_{0}ω (peak voltage)

We know by Ohm’s law,

Inside a resistor,

[Latex]V_{0}=I_{0}R[/Latex]

where R= resistance

As the inductive reactance is similar to resistance, we can get an analogous equation-

[Latex]V_{0}=I_{0}\X_{L}[/Latex]

where ?_{L}=inductive reactance

By comparing V_{0} found in the previous equation, it can be concluded that,

**[Latex]\X _{L}=\omega L=2\pi fL[/Latex]**

where f=frequency

**Inductive Reactance formula**

The inductive reactance of a coil is,

**?**_{L}=ωL or **?**_{L}=2?fL

Where ω is the angular frequency, f is the frequency of the applied voltage, and L is the inductance of the coil.

**Derivation of Inductive Reactance**

**Inductive reactance in series**

In the above circuit, three inductances L_{1}, L_{2} and L_{3} are connected in series. Therefore, if we apply Kirchhoff’s law,

[Latex]V-\left ( L_{1}+L_{2}+L_{3} \right )\frac{\mathrm{d} I}{\mathrm{d} t}=0[/Latex]

[Latex]V=\left ( L_{1}+L_{2}+L_{3} \right )\frac{\mathrm{d} I}{\mathrm{d} t}=\left ( L_{1}+L_{2}+L_{3} \right )I_{0}\omega cos\left ( \omega t+90^{\circ} \right )[/Latex]

Taking the peak value, we can say that,

[Latex]V_{0}=I_{0}\omega \left ( L_{1}+L_{2}+L_{3} \right )[/Latex]

So, total inductance L=L_{1}+L_{2}+L_{3}

Therefore, inductive reactance in series connection, ** ?_{L}=ω**(L

_{1}+L

_{2}+L

_{3}+…..L

_{n})

**Inductive reactance in parallel**

In the above circuit, three inductances, L_{1}, L_{2} and L_{3}, are connected in parallel. If the total inductance is L, by Kirchhoff’s law, we can say,

[Latex]V=L(\frac{\mathrm{d} I_{1}}{\mathrm{d} x}+\frac{\mathrm{d} I_{2}}{\mathrm{d} x}+\frac{\mathrm{d} I_{3}}{\mathrm{d} x})=L\left ( \frac{V}{L_{1}}+\frac{V}{L_{2}}+\frac{V}{L_{3}} \right )[/Latex]

So, [Latex]\frac{1}{L}=\frac{1}{L_{1}}+\frac{1}{L_{2}}+\frac{1}{L_{3}}[/Latex]

Therefore, inductive reactance in parallel connection, [Latex]\mathbf{X_{L}=\omega \left ( \frac{1}{L_{1}}+\frac{1}{L_{2}}+\frac{1}{L_{3}}+…..\frac{1}{L_{n}}\right )^{-1}}[/Latex]

**Inductance and inductive reactance**

Magnetism and electricity co-exist in electrical circuits. If a conductor is placed in a continuously changing magnetic field, a force is generated in the conductor. It is called the electromotive force or EMF. The ability to create voltage for the change in current flow is called **inductance**.

EMF helps the current flow in the circuit. While current passes through the inductor coil, it tries to oppose the current. This reaction is known as **inductive reactance**.

**What is the difference between inductance and inductive reactance ?**

Inductance | Inductive reactance |

Inductance [Latex]L=\frac{V}{\frac{\mathrm{d} I}{\mathrm{d} t}}[/Latex] | Inductive reactance [Latex]X_{L}=\omega L[/Latex] |

Unit of inductance is Henry or H. | Unit of inductive reactance is ohm or Ω |

Dimension of inductance is [ML^{2}T^{-2}A^{-2}] | Dimension of inductive reactance is [ML^{2}T^{-3}I^{-2}] |

It does not depend upon frequency. | It is dependent upon frequency. |

The greater the inductance, the more the induced EMF and current will be. | The greater the inductive reactance, the lesser the current will be. |

**Inductive Reactance in DC circuit**

In a DC circuit, power frequency is equal to zero. Hence ?_{L} is also zero. The inductor would behave like a short circuit in the steady-state.

**Relation between inductance and reactance**

Reactance **? **consists of two components-

- Inductive reactance or
**?**_{L} - Capacitive reactance or
**?**_{C}

Therefore

## Total inductive reactance formula

[Latex]X=X_{L}+X_{C}=\omega L+\frac{1}{\omega C}[/Latex]

**Difference between inductance and reactance**

Inductance | Reactance |

Inductance [Latex]L=\frac{V}{\frac{\mathrm{d} I}{\mathrm{d} t}}[/Latex] | Reactance [Latex]X=\omega L+\frac{1}{\omega C}[/Latex] |

Unit of inductance is Henry or H. | Unit of reactance is ohm or Ω |

Dimension of inductance is [ML^{2}T^{-2}A^{-2}] | Dimension of inductive reactance is [ML^{2}T^{-3}I^{-2}] |

It is independent of frequency. | It is dependent on frequency. |

Inductance is directly proportional to current. | Reactance is inversely proportional to current. |

**The inverse of inductive reactance is susceptance**

The quantity reciprocal to inductive reactance is known as inductive susceptance. It is denoted by B_{L}.

[Latex]X=\omega L+\frac{1}{\omega C}[/Latex]

Inductive susceptance is similar to conductance G, which is the inverse of resistance.

So the unit of B_{L} is also siemen or S.

Physically inductive susceptance represents the capability of a purely inductive electrical circuit to allow the flow of current through it.

**Reactance and susceptance **

Reactance measures a circuit’s reaction against the change in current with time, while susceptance measures how susceptible the circuit is in conducting the varying current.

**Resistance, reactance, capacitance, inductance impedance-comparison **

Parameters | Resistance | Reactance | Capacitance | Inductance | Impedance |

Definition | The measure of obstruction caused by the conductortowards the current is known as resistance. | The characteristic of the inductor and the capacitor to oppose any change in current is called reactance. | The capacity of a conductor to store electric charge is known as capacitance. | The property of a conductor to generate an EMF due to the change in current is known as inductance. | Impedance is the entire opposition in an electrical circuit caused by the inductor, the capacitor and the resistor. |

Symbol | Resistance is represented by R | Reactance is represented by ? | Capacitance is represented by C | Inductance is represented by L | Impedance is represented by Z |

Unit | Ohm | Ohm | Farad | Henry | Ohm |

General Expression | Resistance in a circuit with voltage v and current i is, [Latex]R=\frac{V}{I}[/Latex] | Reactance in a circuit with voltage source’s angular frequency ω is, [Latex]X=\omega L+\frac{1}{\omega C}[/Latex] | The capacitance of a parallel plate capacitor with medium permittivity ϵ, A plate area and d separation between plates is, [Latex]C=\frac{\epsilon A}{d}[/Latex] | The inductance of a coil with induced voltage V is,[Latex]L=\frac{V}{\frac{\mathrm{d} I}{\mathrm{d} t}}[/Latex] | The total impedance of a circuit can be written as Z=Z_{R}+Z_{C}+Z_{L} |

**Capacitive reactance**

Just like the inductive reactance, capacitive reactance is the impedance caused by the capacitor. It is denoted by Xc. When DC voltage is applied in an RC circuit, the capacitor starts charging. Subsequently, current flows, and the capacitor’s internal impedance obstructs it.

Capacitive reactance [Latex]X_{C}=\frac{1}{\omega C}=\frac{1}{2\pi fC}[/Latex]

**What is the difference between inductive reactance and capacitive reactance ?**

**Capacitive Reactance vs Inductive Reactance**

Capacitive reactance | Inductive reactance |

The reactance of the capacitor | The reactance of the inductor |

It is denoted by X_{C} | It is denoted by X_{L} |

[Latex]X_{C}=\frac{1}{\omega C}[/Latex] | [Latex]X_{L}=\omega L[/Latex] |

When a sinusoidal AC voltage is applied to a capacitor, the current leads the voltage by a phase angle of 90° | When a sinusoidal AC voltage is applied to an inductor, the current lags the voltage by a phase angle of 90° |

It is inversely proportional to the frequency. | It is directly proportional to the frequency |

In DC supply, the capacitor behaves like an open circuit. | In DC supply, the inductor behaves like a short circuit. |

At high frequency, the capacitor acts as a short circuit. | At high frequency, the inductor acts as an open circuit. |

**AC circuit in LR series combination**

There are two components in the above circuit- resistor R and inductor L. Let the voltage across the resistor is Vr, and the voltage across the inductor is VL.

The phasor diagram shows that total voltage V, resistor voltage V_{r} and inductor voltage V_{L} forms a right-angled triangle.

By applying Pythagoras theorem, we get,

V^{2}=V_{r}^{2}+V_{L}^{2}

[Latex]tan\varphi =\frac{V_{L}}{V_{r}}[/Latex] where φ=phase angle

**How to find inductive reactance ? | Important formulas**

[Latex]X_{L}=2\pi fL[/Latex]

[Latex]Z=\sqrt{R^{2}+X_{L}^{2}}[/Latex]

[Latex]I_{rms}=\frac{V}{Z}[/Latex]

Power [Latex]P=V_{rms}I_{rms}cos\varphi[/Latex]

**Calculate the inductive reactance | Inductive reactance calculation example**

## Find the AC voltage required for 20 mA current to flow through a 100 mH inductor. Supply frequency is 500 Hz.

Given: i= 20 mA f=400 Hz L=100mH

As the series is purely inductive, the impedance in the circuit, Z=X_{L}

We know, X_{L}=ωL=2?fL=2 x 3.14 x 400 x 0.1=251.2 ohm

Therefore, supply voltage V=iX_{L}= .02 x 251.2= 5.024 volts

## Calculate X_{L} of a 5 mH inductor when 50 Hz Ac voltage is applied. Also find I_{rms} at each frequency when V_{rms} is 125 volts.

X_{L}=2?fL=2 x 3.14 x 50 x 5 x .001 = 1.57 ohm

[Latex]I_{rms}=\frac{V_{rms}}{X_{L}}[/Latex]= 79.6 A

**Calculate inductive reactance using voltage and current**

## A resistance of 20 ohm, inductance of 200 mH and capacitance of 100 µF are connected in series across 220 V, 50 Hz mains. Determine X_{L}, X_{C} and current flowing through the circuit.

We know, V=220 volt R=20 ohm L=0.2 H f=50 Hz

X_{L}=2?fL=2 x 3.14 x 50 x 0.2=62.8 ohm

[Latex]X_{C}=\frac{1}{2\pi fC}[/Latex]=1/(2 x 3.14 x 50 x 0.0001)=31.8 ohm

Therefore total impedance,

[Latex]Z=\sqrt{R^{2}+X_{L}^{2}}[/Latex]= (20)2+(62.8-31.8)2=36.8 ohm

So, current [Latex]I=\frac{V}{Z}[/Latex]= 5.95 A

**Resistance-Reactance-Impedance: Comparative study**

Resistance | Reactance | Impedance |

Opposes electron flow | Opposes change in current | Combination of reactance and resistance |

[Latex]R=\frac{V}{I}[/Latex] | [Latex]X=X_{L}+X_{C}[/Latex] | [Latex]Z=\sqrt{R^{2}+X_{L}^{2}}[/Latex] |

Measured in ohm | Measured in ohm | Measured in ohm |

Does not depend upon frequency | Depends upon frequency | Depends upon frequency |

**Leakage reactance in induction motor**

Leakage reactance is the impedance caused by the leakage inductor in an induction motor. A rotating magnetic field develops in the induction motor due to the applied 3-phase power. Most of the magnetic flux lines generated by the stator winding travel across the rotor. Though a very few flux lines close in the air gap and fail to contribute to magnetic field strength.This is the leakage flux.

Due to this leakage flux, a self-inductance is induced in the winding. This is known as **leakage reactance**.

**Sub-transient reactance of induction motor**

In a short circuit, the magnetic flux generated in the damper winding reduces steady-state reactance. It is known as **sub-transient reactance**. The term ‘sub-transient’ suggests that the quantity operates even faster than the ‘transient’.

**FAQs**

**To what is inductive reactance proportional?**

Inductive reactance is directly proportional to the frequency.

**What is inductive reactance and how does it affect an AC circuit ?**

Unlike DC, in the AC circuit, the current varies with respect to time.

**What happens when the capacitive reactance is greater than the inductive reactance?**

If X_{C} is more than X_{L}, then the overall reactance is capacitive.

**What is induction?**

The change in magnetic field causes voltage and current in the circuit. This phenomenon is known as induction.

**What does inductance do in a circuit?**

Inductance opposes the change in current flowing through the circuit.

**What is inductance of a coil?**

The inductance of a coil originates from the magnetic field due to varying current.

**Why is L used for inductance?**

According to the initials, I should have been used for representing inductance. But as I is already being used for current, L is used for inductance to honour scientist Heinrich Lenz for his extraordinary contribution in the field of electromagnetism.

**Can self inductance be negative?**

Self-inductance is purely a geometric quantity, and it depends upon the external circuitry. Therefore it cannot be negative. The minus sign in Lenz law indicates the opposing nature of EMF towards the magnetic field.

**Do Motors have inductance?**

Back EMF is a crucial factor in motors. Both AC and DC motors make use of a low AC voltage source to measure inductance.

**What is unit of inductance?**

SI unit of inductance is volt -second per ampere or Henry.

**Why does inductor block AC and allows DC?**

The inductor creates an EMF when current flows through it. In AC, the EMF is very high as the frequency is increased. Therefore the opposition is also significant. But in DC supply, there’s no EMF, and consequently no opposition takes place. So it is said that the inductor blocks AC and allows DC.

**Does inductor allow DC current?**

Inductor allows DC current as there’s no opposite force acting in the circuit.

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