# Series Connected Coils

Serial Connected Inductors can be connected serially, just like resistors.

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These interconnections of inductors produce more complex networks, the general inductor of which is a combination of singular inductors. However, there are certain rules for connecting inductors in series or parallel, and they are based on the fact that there is no mutual inducing or magnetic coupling between individual inductors.

In the Training of Serial Connected Resistors, we found that the different values of the resistances connected to each other in series were not only "added" to each other, this also applies to inducing. Inductors in the series are simply "collected" because the number of coil rotations effectively increases, the total circuit inductle L_{T} is equal to the sum of all individual inductees added together.

The current flowing from the first inductor passes through all the inductors in turn because there is no other way to pass. That's why there's a common current flowing through the serial inductors:

I_{L1} = I_{L2} = I_{L3} = I_{AB} … like.

In the example above, the L_{1}, L_{2,} and L_{3} inductors are all serially connected between points A and B. The sum of the separate voltage drops in each inductor can be found using Kirchoff's Voltage Law (KVL), where we know from previous tutorials on V_{T} = V_{1} + V_{2} + V_{3} and induced induction that the self-induced emk along an inductor is given as follows: V = L di/dt.

Therefore, in our example above, taking the values of separate voltage drops on each inductor, the total inducing for the series combination is given as follows:

By dividing the above equation by di/dt, we can reduce the equation to give a final expression to calculate the total inductee of a circuit when connecting the inductors in series:

L_{total} = L_{1} + L_{2} + L_{3} + ….. + L_{n}

Then the total inductee of the series chain can be found simply by simply collecting individual inducts of inductors in the series, just as putting together the resistances in the series. However, the above equation applies only when there is no mutual inducing or magnetic coupling between two or more inductors (they are magnetically isolated from each other).

An important point to remember about inductors in serial circuits is that the total inductee (L_{T)}of any two or more inductors connected to each other in series will always be GREATER than the value of the largest inductor in the series chain.

## Serial Connected Inductors Question Sample 1

Three inductors of 10mH, 40mH and 50mH are connected in a series combination without mutual inducing between them. What is the total inductee of the series combination?

## Mutually Connected Inductors

When inductors are serially connected to one's magnetic field to the other, the effect of mutual inducing increases or decreases the total inductee depending on the amount of magnetic coupling. The effect of this mutual inducing depends on the distance between the coils and their orientation to each other.

Mutually connected serial inductors can be classified as "Auxiliary" or "Opposite" rather than total inductees. If the magnetic flux produced by the current flows in the same direction from the coils, the coils are called CumulativeLy Connected. If the current flows from the coils in opposite directions, it is said that the coils are connected differentially, as shown below.

### Cumulative Connected Serial Inductors

Although the current flowing from the two coils, which are cumulatively connected between points A and D, is in the same direction, the equation above needs to be changed to take into account the interaction between the two coils for voltage drops throughout each of the coils. The self-induced inducation of each coil, L_{1} and L_{2,}respectively, will be the same as the previous one, but with the addition of M, which indicates mutual inducing.

The total emk, which is then induced into the cumulatively connected coils, is given as follows:

Where: 2M represents the effect of the L_{1} coil on L_{2} and similarly the effect of the L_{2} coil on L_{1.}

By dividing the above equation by di/dt, we can reduce it to give a final statement to calculate the total inducing of a circuit when the inductors are cumulatively connected:

L_{total} = L _{1} + L _{2} + 2M

If one of the coils is inverted so that the same current flows only in opposite directions from each coil, the mutual inductanx M located between the two coils will have a canceling effect on each coil, as shown below.

### Differential Connected Serial Inductors

The emk, which is induced to coil 1 due to the effect of the mutual inducing of coil 2, is against the self-induced emk in coil 1, since now the same current passes through each coil in opposite directions. To take into account this cancellation effect, a minus sign with M is used when the magnetic field of the two coils is connected differentially, which gives us the final equation to calculate the total inductee of a circuit when the inductors are connected differentially:

L_{total} = L _{1} + L _{2} – 2M

Then the final equation for inductively connected inductors in series is given as follows:

## Serial Connected Inductors Question Sample 2

Two inductors of 10 mH, respectively, are connected in a series combination, thereby helping each other by giving their magnetic fields cumulative coupling. Mutual inducings are given in 5mH. Calculate the total inductee of the series combination.

## Serial Connected Inductors Question Sample 3

The two series-connected coils have a self-induced inducation of 20mH and 60mH respectively. If the total induct of the combination is 100mH, determine the amount of mutual inducing that exists between the two coils, assuming that they help each other.

## Summarize

We now know that we can serially connect inductors to find the L_{T} total inductive value equal to the sum of individual values, similar to connecting resistors in series. However, when connecting inductors in series, they can be affected by mutual inductance.

Mutually connected serial inductors are classified as "auxiliary" or "opposite" rather than total inductive, depending on whether the coils are connected cumulatively (in the same direction) or differentially (in the opposite direction).

In the next tutorial about inductors, we will see that the position of the coils also affects the total inducing (L_{T)}of the circuit when connecting inductors in parallel.