# Series and parallel combination of an Inductor

An inductor is nothing but a coil of wire wound around a central iron core. It is a passive component, which stores energy in the form of an electromagnetic field. When a current I flows through the coil of wire, the magnetic field is produced around the core, proportional to the flow of current.

Let ϕ be the flux produced due to the flow of current I in the coil. If so, the inductance of the coil is given by the ratio of flux produced to the current flowing through the coil.

Inductance is the property of an inductor, which opposes the change in current through the coil. Because the current through the coil will induce an emf that opposes the cause producing it, as per Lenz’s Law. It is represented with the letter ‘L’ and the unit of inductance is Henry.

The voltage across the inductor is proportional to the rate of change of current. It is expressed as,

where L is the value of inductance in Henry.

From this, the current equation can be written as,

## Inductor in Series

Consider three inductors L_{1}, L_{2} and L_{3} are connected in series as shown below. Let V be the supply voltage applied across the three series inductor and I be the current flowing through the circuit.

Each inductor carries the same amount of current I. Let V_{1}, V_{2}, V_{3} be the voltage across the inductors L_{1}, L_{2}, L_{3} respectively. So, we can write,

V = V_{1} + V_{2} + V_{3} ——–> (1)

The voltage across each inductor is given by,

Thus equation (1) becomes

If L_{eq} is the effective inductance of the entire inductor circuit, then the voltage equation is given by

From equations (2) and (3), we can write,

L_{eq} = L_{1} + L_{2} + L_{3}

If ‘n’ number of inductors are connected in series, then the equivalent inductance is given by the expression,

*The effective or equivalent inductance of the inductor in series is the sum of the inductances of individual inductors.*

## Inductor in Parallel

Let us consider three inductors L_{1}, L_{2} and L_{3} are connected in parallel with one another. The supply voltage V is given across the parallel combination of three inductors. Since all the elements are in parallel, the voltage is same between points 1 and 2.

Let I be the current flowing from the supply. This current has three paths along L1, L2, L3 from the node point 1, so the corresponding currents are I_{1}, I_{2}, I_{3}, as shown in the figure.

The current through the parallel inductor is written as,

The current through each inductor is given by,

Now, equation (4) becomes,

If L_{eq} is the equivalent inductance of the entire circuit, then the total current in the electric circuit is given by,

From equation (5) and (6), we can write,

If there are n number of inductors connected in parallel, then the equivalent inductance or effective inductance is given by,

## Solved Problem

Let us solve a problem to find the equivalent inductance of the given circuit.

In the above example, inductor L4 is in parallel with a series combination of L5 and L6. The equivalent inductance value of these three inductors L4, L5 and L6 are

Now, the circuit get reduced as below,

If you look at the circuit, inductor L2 is in parallel with the series combination of inductors L3 and L7. The corresponding equivalent inductance value is

Again the circuit is reduced as below,

In the reduced circuit, the inductors L1 and L8 are in series.

Thus the equivalent inductance is given by,

When two inductors are connected in either series or parallel, the magnetic flux produced from one inductor will link with the other inductor. This property is called as mutual inductance.

In the solved examples of inductors in series and parallel the mutual inductance is taken as zero.

Yes, In the given example, self-inductance alone is considered. A problem considering both self-inductance and mutual inductance will be explained in a separate article.

Inductance is the property of a coil which oppose the change in current through it. Inductor oppose momentary change in current through it, in the same way a capacitor oppose any momentary change in voltage across it.

It is not oppose the current flowing through it, it is for a resistor.

Dear Dr. V. Muralidhara, What you have said is right. It has been corrected.

Thank you for your kind comments.