Inductors are an important device used in electronics engineering for circuit designing and analysis. There are different configurations in which we can place an inductor two of the most important of which are series and parallel. In this article, we will study series and parallel inductor.

We will see the inductance, current, and flux linkage in each type of circuit. We will also see the total inductance in each case and discuss the concept of mutual inductance. Later, we will study the applications of each type of circuit. We will also see some examples to enhance the understanding of the concepts. The article concludes with some frequently asked questions that readers can refer to.

Inductors in Series

As the name suggests, inductors in series refer to the configuration in which inductors are placed next to each other forming a series connection. Due to this configuration, the inductors share the same current and develop voltages across them which are proportional to the inductance of each inductor.

Later we will discuss the connection, total inductance, Voltage, Current, and flux linkage in each conductor in a series circuit along with the applications of Inductors in Series. Let us look at the configuration of inductors in a series.

Inductor in Series

Mathematical Equation of Inductors in Series

Let us see how the total inductance of a series circuit vary

L_eq = L₁ + L₂ + L₃+………………………….L_n for n inductors in series

And current developed across each inductor will be the same which can be given by

I₁ = I₂ = I₃= I₄…………………..I_n

Inductors in Parallel

As the name suggests, inductors in parallel refer to the configuration in which inductors are placed parallel to each other, it results in inductors in parallel configuration. Due to this configuration, the voltage developed across the inductors will be the same and the current developed across them will be inversely proportional to the inductance of each inductor.

Later we will discuss the connection, total inductance, Voltage, Current, and flux linkage in each conductor in a parallel circuit along with the applications of Inductors in Series. Let us look at the configuration of inductors in a series.

Inductors in Parallel

Mathematical Equation of Inductors in Parallel

Let us see how the total inductance of parallel circuits vary

1 / L_eq = 1 / L₁ + 1 / L₂ + 1 / L₃+………………….1 / L_n for n inductors in series

And voltage developed across each inductor will be the same which can be given by

V₁ = V₂ = V₃ = V₄…………………………………V_n

Series Inductor

Let us see some of the properties of inductors in series configuration.

Connection

Series Inductor

A series inductor connection has been shown above. Notice how the inductors are connected end-to-end resulting in a path flow of charges. Due to this form of connection, the current flowing is the same throughout the connection.

Total Inductance in Series

Let us see the derivation of the mathematical equation seen above for inductors in series.

Total Inductance in Series

From the above circuit diagram, we can see that the voltage developed across each inductor is v_i. We can calculate the total voltage as the sum of voltages across each inductor

From KVL v = v₁ + v₂ eq 1

The voltage Across Each Conductor

The voltage across each inductor can be given by the product of inductance and the rate of change of current in the circuit.

v_i = L_i .di/dt

Therefore, we can write the equation 1 as

v = L₁ .di /dt + L₂.di /dt ∴ v = (L₁ + L₂) .di/dt eq 2

v is the voltage across the whole circuit therefore, we can re-write the eq 2 as

v = L_eq. di /dt eq 3

On comparing eq 2 and eq 3

(L₁ + L₂) .di /dt = L_eq. di /dt ∴ L_eq = L₁ + L₂

Clearly, for N inductors we can say that

L_eq= L₁+ L₂+ L₃+…………………………………………….L_n

Current and Flux Linkage

In a series circuit, the current across each inductor will be the same and we denote it by i in the derivation

Applications of Series Inductor

Let us some of the applications of Series Inductor:

1. Inductors in series are used in filtering circuits to block some frequencies. Inductors along with capacitors are used to form different types of filters with different resonant frequencies that help in blocking certain frequencies.
2. Radio Frequency applications use series inductors to optimize the power transfer. This is usually done by impedance matching which is performed in between the circuits at different points.
3. Inductors have a magnetic field associated with them. This property of inductors is used for storing energy in inductors in the form of a magnetic field. This energy will be released by changing the current.
4. Inductors are used in circuit designing mainly in telecommunication where we are required to eliminate noise. They help in generating smooth waveforms with DC power supplies
5. Tuning circuits make use of series inductors due to their ability to block and pass certain frequencies. Depending on our needs, we can make a combination of different inductors and define the frequencies that are allowed therefore helping in making tuning circuits.

Parallel Inductor

Let us see some of the properties of inductors in parallel configuration.

Connection

Total Inductance in Parallel

A parallel inductor connection has been shown above. Notice how one end of the inductors is connected to one common point and the other end is connected to another common point resulting in the same potential development across each. Due to this form of connection, the voltage developed is the same across all inductors.

Total inductance in Parallel

Let us see the derivation of the mathematical equation seen above for inductors in parallel.

Total Inductance in Parallel

From the above circuit diagram, we can see that the voltage developed across each inductor is the same. We can calculate the total current as the sum of current across each inductor

From KCL i = i₁ + i₂ eq 1

The Voltage Across Each Inductor

In parallel, circuit the voltage across each inductor will be the same and we, denote it by V in the derivation

Current and Flux Linkage

The current across each inductor can be given by the following equation

i_L = 1 / L ∫V.dt

If we assume the initial conditions to be zero, we can write the equation 1 as

i = 1 / L₁ ∫V.dt + 1 / L₂ ∫V.dt eq 2

where total current can be represented in terms of L_eq as

i = 1 / L_eq ∫V.dt eq 3

On comparing eq 2 and eq 3

1 / L_eq ∫V .dt = 1 / L₁ ∫V.dt + 1 / L₂ ∫V.dt ∴ 1 / L_eq = 1 / L₁ + 1 / L₂

Clearly, for N inductors we can say that

1 / L_eq = 1 / L₁ +1 / L₂ + 1 / L₃ ……………………………… 1 / L_n

Applications of Parallel Inductor

Let us some of the applications of parallel Inductor.

1. Parallel Inductors can be used to make oscillator circuits with variable resonant frequency. Inductors and capacitors can be used to make resonant circuits with desired frequency.
2. Similar to series inductors, parallel circuits can also be used for making filters that block certain frequencies from passing. This is seen mostly in radio-frequency applications.
3. Since inductors in parallel configuration have the same voltage across them therefore they are used as voltage regulator circuits to stabilize the output voltage.
4. When parallel inductor configurations are connected to other components like motors, they can reduce the current flowing in the circuit thereby preventing the motor from getting destroyed due to any major current spike.
5. One major application of parallel circuits is their use in fault protection. They can control the inflowing current by controlling the inductance and preventing sensitive components from getting damaged.

Mutual Inductance

Mutual inductance is seen in the case of two coils that are in proximity to each other. The proximity between these coils tends to affect the magnetic field in other coils resulting in a flux linkage. Let us formally define mutual inductance

Mutual Inductance can be defined as a property of a coil due to its magnetic field which affects the properties like current and voltage in other coils known as secondary coils.

Let us see the effect of mutual inductance in series connection and parallel connection of inductors.

Mutual Inductance in Series Conductor

In a series configuration of inductors, the flux of one inductor links the nearby inductors in proximity therefore, mutual inductance is introduced to take into consideration that magnetic field. Here is a diagram showing the mutual inductance in the case of inductors in series.

Mutual Inductance in Series Conductor

In this series configuration, due to the same current flowing through both the inductors, we can say that current enters both the inductors at the same time denoted by the dot. The current flowing in inductor 1 (L₁) will produce a voltage across inductor 2 and the current flowing in inductor 2 (L₂) will induce a voltage in inductor 1.

Note that this will induce a voltage in the direction of the self-induced voltage. This is why the inductance due to the mutual interaction will be added to the self-inductance to calculate the equivalent expression.

Therefore, the voltage across L₁ and L₂ are

For inductor 1

V₁ = L₁ .di / dt + M.di /dt

For inductor 2

V₂ = L₂ .di / dt + M.di /dt

Also if we consider the total inductance of the system as L_eq then, we can write

V₁ + V₂ = L_eq . di /dt

L₁ .di /dt + M. di /dt + L₂. di /dt + M.di /dt = L_eq. di /dt

L₁ + L₂ + 2 M = L_eq

Note that if the polarity of mutually induced emf is opposite from the self-induced emf the 2M will be subtracted instead of adding.

Mutual Inductance in Parallel Conductor

In a parallel configuration of inductors, the flux of one inductor links the nearby inductors in proximity therefore, mutual inductance is introduced to take into consideration that magnetic field. Here is a diagram showing the mutual inductance in the case of inductors in parallel.

Mutual Inductance in Parallel Inductor

In this parallel configuration, due to the same voltage across both the inductors, we can say that current enters both the inductors at the same time denoted by the dot. The current flowing in inductor 1 (L₁) will produce a voltage across inductor 2 and the current flowing in inductor 2 (L₂) will induce a voltage in inductor 1.

Note that this will induce a voltage in the direction of the self-induced voltage. This is why the inductance due to the mutual interaction will be added to the self-inductance to calculate the equivalent expression.

The total inductance in this case will be

L_eq = ( L₁ L₂ – M² ) / (L₁ + L₂ + 2 M )

Note that if the polarity of mutually induced emf is opposite from the self-induced emf the 2M will be subtracted instead of adding.

Difference Between Series and Parallel Inductor

Let us compare the two types of inductors discussed in this article

Parameter	Series Inductor	Parallel Inductor
Definition	When inductors are connected in series next to each other it results in series configuration.	inductors are placed parallel to each other, it results in inductors in parallel configuration.
Total inductance	Ln=L1+ L2+……….Ln	1/Ln= 1/L1+ 1/L2+……….1/Ln
Current Developed	The same current flows through each inductor.	Different current flows through each inductor depending on the inductance value.
Voltage Developed	Different voltage is developed across each inductor depending on the inductance value.	The same voltage is developed across each inductor.
Mutual Inductance	The mutual inductance in series is calculated using Lnet=L1 + L2 + 2 M	The mutual inductance in parallel is calculated using Leq=( L1L2 – M2 ) / (L1 + L2 + 2 M )
Uses	This circuit is use in AC generators and transformers to conserve energy.	This circuit is used to limit the net current.

Solved Examples of Series and Parallel Inductor

We will discuss series and parallel inductor through solved examples.

Example 1

Two inductors 4 H and 12 H are connected in parallel. If this combination is connected in series with an inductor of inductance 7 H. The equivalent inductance of the circuit will be.

The net inductance will be series combination of 7H with parallel combination of 4 H and 8 H

∴ L_eq = L₁. L₂ /( L₁ + L₂ ) + L₃

∴ L_eq = 4 .12 / (4 + 12) + 7

∴ L_eq = 3 + 7

∴ L_eq = 10 H

Example 2

Find the equivalent inductance of the circuit shown in looking from terminals ab.

Equivalent Inductance in Series and Parallel

In the given circuit, inductors 20 H, 12 H, and 10 H are connected in series, thus their equivalent inductance is,

L₁ = 20 + 12 + 10 = 42 H

Also The series combination of 20 H, 12 H, and 10 H is connected in parallel with the inductor of 7 H, then L₂ = 42 × 7 / (42 + 7 ) = 6 H

Finally, this combined 6 H, 4 H, and 8 H are connected in series, thus, the equivalent inductance of the circuit is

∴ L_eq = 6 + 4 + 8 = 18 H

Conclusion

We have seen how we can arrange inductors in series and parallel configurations depending on our needs. We even saw how the circuits differ from each other and offer different flux linkage. We have even derived some important expressions for each of the circuits. It is necessary to understand how the voltage and current distribution are different in series and parallel combinations of Inductor circuits therefore, some examples were solved to ensure the clarity of concepts among readers. In case of any doubts, readers are advised to go through the article again and refer to the frequently asked questions.

Series and Parallel Inductor – FAQs

What is an inductor and how does it differ from a capacitor?

Inductors and capacitors, both are devices used to store energy. The difference is that an inductor stores energy in its magnetic field and a capacitor stores its energy in the electric field.

Is the inductor an active or passive component and why?

Since inductors can store energy and then release this energy when required therefore it is a passive component.

What is the voltage in a series inductor circuit?

In a series inductor circuit, the current across each inductor is the same and voltage is proportional to the inductance. Therefore, the net voltage is greater than the individual voltages of inductors.