# Series and Parallel Inductors with Effects of Mutual Induction

Published on 24/2/2012 & updated on 27/10/2018**inductors are connected in series**the equivalent inductance of the combination will be the simple sum of the inductance of all individual inductors. This is just like the equivalent resistance of series connected resistors. But in the case of inductors, we sometimes may have to consider the effect of mutual inductance between the inductors. Then the for calculating the inductance of each inductor we consider both the self inductance and mutual inductance of the inductor. The mutual inductance will be either added or subtracted from the self inductance depending on the polarity of magnetically coupled inductors. We will learn about effect of mutual inductance later in this article. Now, without considering mutual inductances we can write the equivalent inductance of series connected inductors as,

When **inductors are connected in parallel** the reciprocal of the equivalent inductance of the combination will be the sum of the reciprocal of individual inductances. This is just like the equivalent resistance of parallel connected resistors. Here also we may have to consider the effect of mutual inductance in the same way if required. We will learn the effect of mutual inductance on parallel inductors later in this article. Without considering the effect of mutual inductance we can write

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An inductor is passive circuit element. Let us find out the equivalent inductance of **series connected and parallel connected inductors**.

## Series Connected Inductors

Let us consider n number of**inductors connected in series**as shown below.

Let us also consider that,
the inductance of inductor 1 and voltage drop across it are L_{1} and v_{1} rspectively,
the inductance of inductor 1 and voltage drop across it are L_{2} and v_{2} rspectively,
the inductance of inductor 1 and voltage drop across it are L_{3} and v_{3} rspectively,
the inductance of inductor 1 and voltage drop across it are L_{4} and v_{4} rspectively,
the inductance of inductor 1 and voltage drop across it are L_{n} and v_{n} rspectively.
Now, applying, Kirchhoff's Voltage Law, we get, total voltage drop (v) across the **series combination of the inductors**,

The votage drop across an inductor of inductance L can be expressed as, Where, i is the instanteous current through the inductor. As all inductors of the combinations are connected in series, here, the current through each of the inductors is same, and say also it is i. So, from above KVL equation, we get,

This equation can be rewritten as,
Where, L_{eq} is equivalent inductance of the **series combined inductors**. Hence,

## Parallel Connected Inductors

Let us consider n number of**inductors connected in parallel**as shown below. Let us also consider that, the inductance of inductor 1 and current through it are L

_{1}and i

_{1}rspectively, the inductance of inductor 1 and current through it are L

_{2}and i

_{2}rspectively, the inductance of inductor 1 and current through it are L

_{3}and i

_{3}rspectively, the inductance of inductor 1 and current through it are L

_{4}and i

_{4}rspectively, the inductance of inductor 1 and current through it are L

_{n}and i

_{n}rspectively.

Now, applying, Kirchhoff's Current Law, we get, total current (i) entering in the **parallel combination of the inductors**,
The current throgh an inductor of inductance L can be expressed as,

Where, v is the instanteous voltage across the inductor.
As all inductors of the combinations are connected in parallel, here, the voltage drop across each of the inductors is same, and say also it is v. So, from above KCL equation, we get,
This equation can be rewritten as,
Where, L_{eq} is equivalent inductance of the **parallel combined inductors**. Hence,

## Effect of Mutual Induction in Series Connected Inductors

Whenever more than one inductors come in closer, there may be mutual induction between them. If more than one inductors are connected in series and flux of one inductor links the other then we have to consider mutual inductance during calculations of equivalent inductance. For this purpose, we use the dot convention. Here each of the inductors is marked with a dot at one end. The current entering through the dotted terminal of one inductor will induce a voltage in the other inductor with positive polarity at the dotted terminal of the later. Let us consider the following example. As inductors are in series same current will flow to these inductors. Hence when current enters through the dotted terminal of inductor 1, the current enters through the dotted terminal of inductor 2. Inductor 2 will induce the voltage across inductor 1 with positive polarity at dotted end of inductor 1. Current enters through the dotted terminal of inductor 1 will induce the voltage across inductor 2 with positive polarity at dotted end of the inductor 2. As the both mutually induced emfs are in the direction self induced emf the equivalent impedances the mutual inductance will simply be added to self inductance to calculate equivalent impedance.Here in this second example as per dot convention provided in the figure below current is entering through the dotted terminal of one inductor and the same current is leaving the dotted terminal of the other inductor. In that case polarity of mutually induced emf differs the self induced emf. The equivalent inductance of the combination will be

## Effect of Mutual Induction in Parallel Connected Inductors

Now we will come to the effect of mutual inductance in parallel connected inductors. Here in this example dots are given to the same side of two inductors. When current is entering through the dotted terminal of inductor 1 the emf induced has the positive polarity at dotted end of inductor 2.Similarly, when current enters through the dotted terminal of inductor 2, the emf induced in the inductor 1 has positive polarity at dotted end of inductor 1. So equivalent inductance is

Similarly, when two parallel inductors are dotted in the following manner the equivalent inductance will be

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