HardwareBee
https://www.nuvation.com/
  • Find ASIC Vendors
  • Design Services Directory
    • FPGA Design Services
    • Electronic Design Services
    • Embedded Software Companies
    • Add your company
  • Get Price Quotes From Vendors
    • Electronic Design Companies
    • FPGA Design Companies
    • Embedded Software Companies
    • Design & Manufacturing Companies
    • Get IC Device Prices
  • Emerging ICs Directory
    • UWB
      • Spark Microsystems
    • FPGA
      • Colonge Chip
      • Rapid Silicon
    • Radar
      • Acconeer
    • Add your IC
  • Get IC Prices
  • WikiBee
  • Resources
    • FPGA Academy
    • Embedded Academy
    • FPGA vs ASIC Calculator
    • Watt to dBm Converter
    • dBm to Watt Converter
  • Pricing
    • Get Your Company Listed
    • Book a Demo
    • Get a Monthly Lead List
HardwareBee
  • Design Services Directory
    • FPGA Design Services
    • Electronic Design Services
    • Embedded Software Companies
    • Add your company
  • Get Price Quotes From Vendors
    • Electronic Design Companies
    • FPGA Design Companies
    • Embedded Software Companies
    • Design & Manufacturing Companies
    • Get IC Device Prices
  • Emerging ICs Directory
    • UWB
      • Spark Microsystems
    • FPGA
      • Colonge Chip
      • Rapid Silicon
    • Radar
      • Acconeer
    • Add your IC
  • Get IC Prices
  • WikiBee
  • Resources
    • FPGA Academy
    • Embedded Academy
    • FPGA vs ASIC Calculator
    • Watt to dBm Converter
    • dBm to Watt Converter
  • Pricing
    • Get Your Company Listed
    • Book a Demo
    • Get a Monthly Lead List
11416 Views

Inductors In Parallel

04/06/2022, hardwarebee

Get a Price Quote

Inductors are said to be parallel if one or more inductors share the same two nodes. In this article we will discuss the topic around inductors in parallel and present the two common scenarios: with and without mutual inductance.

 

Inductors in Parallel – Background

 

When a current flows through the coil, an electromagnetic field is created in its surroundings. When there is a change in current, the electromagnetic field also changes. The electromagnetic field surrounding a coil expands and collapses with an increase and decrease in current. It changes its direction with the current. This ability is called self-inductance. “When current flows through an inductor, it produces a voltage in the other nearby inductors.” This ability is called mutual inductance.

 

Consider two coils (coil 1 and coil 2 as shown in the figure below), if the current in coil 1 is increasing, it will create an increasing magnetic field around the coil. Similarly, the decrease in current will result in a decreasing magnetic field. This induces a voltage in coil 2 in an opposite direction to that of the current. In the same way, increasing and decreasing current will produce a similar magnetic field. The magnetic field generated in coil 2 will result in induced voltage in coil 1 (in opposite direction to that of the current). Similarly, if there are more than two coils connected in parallel, they will be linked together with the same magnetic field. If two or more coils are parallel to each other, such that changing magnetic flux lines cut through each other, they are said to have common mutual inductance, and it is denoted by M. It is measured in Henries (H).

 

 

How does it react to AC and DC?

 

The voltage across the inductor is zero when there is a constant current (that is DC) flowing through it. Or in other words, they act as a short circuit to DC.

 

The current flowing through an inductor cannot change abruptly or instantaneously. For a time-varying current, it opposes the change in current flowing through it. The voltage across the inductor can change abruptly.

 

Inductors in Parallel without Mutual Inductance

 

Inductors in parallel share the same two nodes. The voltage across each coil remains the same, whereas there are multiple paths for current to flow. See the figure below. There are ‘N’ parallel-connected inductors. The total current flowing through the inductor is given by Kirchhoff’s Current Law.

 

 

 

The current-voltage relationship of the inductors is given below.

 

 

Substitute ‘i’ from equation 2 in equation 1; we get,

 

 

From the equivalent circuit, the initial current through Leq at t = t0 is the sum of inductor currents at t = t0.

 

 

From the equivalent circuit, the inductor current equation is,

 

 

From equation 3 and equation 4

 

 

Where Leq is the equivalent inductance of the parallel-connected inductors.

 

In parallel-connected resistors, the reciprocal of the equivalent resistance is the sum of reciprocals of individual resistances. The same is true for parallel-connected inductors. The equivalent inductance is lesser than the smallest inductance in that connection. Or in other words, inductors in parallel decrease the effective inductance of the circuit.

 

When considering a resistive parallel circuit, most of the current flows through a less resistive path. It is true for inductive circuits as well.

 

The resistors are electrically connected. These coils are not only electrically connected as well as magnetically coupled. The above formula is only valid when no mutual inductance is present between inductors.

 

While working with inductive circuits, the electromagnetic forces should be taken into account.

 

Inductors in Parallel with Mutual Inductance

 

In parallel inductors, we cannot ignore the effect of mutual inductance.

 

If two coils are connected in parallel (with self-inductance L1 and L2) and are nearby such that the magnetic flux in one coil links with the other coil. Consequently, an EMF is induced in the other coil. Based on the EMF induced two types of connections are possible.

 

  1. Parallel aiding connection
  2. Parallel opposing connection

 

Have a look at the figure, magnetic field lines from both coils touch each other.

 

According to Faraday’s Law,

 

 

Where,

N = Number of turns

Φ = Flux

 

Any change in flux Φ, by the time-varying current, can be expressed as

 

 

Where L is the self-inductance associated with each coil.

 

Each coil has its magnetic flux, which has two components. That is self-inductance and the other is mutual inductance.

 

Let’s suppose Φ1 is the flux associated with coil 1. Assume that coil 2 carries no current.

 

 

Φ11 is the flux link coil 1 and the voltage induced is given by,

 

 

Φ12 links coil 2 and voltage induced is given by,

 

 

is the open circuit induced voltage in coil 2. This voltage is induced because of the time-varying current in coil 1, which results in changing magnetic flux.

 

Repeat the same procedure for coil 2. Assume coil 1 carries no current. Φ2 is the flux associated with coil 2.

 

 

The induced voltage in coil 2 is given as,

 

 

The induced voltage in coil 1 because of Φ21

 

 

Mutual inductance is responsible for increasing and decreasing the inductance depending on the magnetic coupling between the two coils. The magnetic coupling, in turn, depends on the distance between two coils and their orientation to each other.

 

Parallel Aiding Inductors

 

In this type of connection, the EMF induced in both the coils are of the same polarity.

The total inductance is higher than the coils that do not have mutual inductance between them.

From KCL,

 

 

After differentiation,

 

 

The total voltage across L1, that self-induced and mutually induced.

 

 

Since inductors and voltage sources are in parallel,

 

 

Compare equation 6 and equation 7

 

 

Substitute the di1/dt value of  in equation 5 and eliminate it.

 

 

Equations 6, equation 7 and equation 8 are equal. Because of the parallel connection.

 

Compare equation 8 which is also equal to equations 6 and 7

 

 

Substitute di1/dt from equation 9

 

 

Parallel Opposing Inductors

 

In this type of connection, the EMF induced in both the coils are of the opposite polarity. The dot convention is used to show the connection is parallel opposing.

The whole derivation is similar to parallel aiding inductors. The only difference is induced voltage because mutual inductance is of opposite polarity. From equation 6 and equation 7, replace +M with -M.

 

 

Solved Example

 

Consider two coils, connected in parallel with each other.

 

 

Calculate the effective inductance between two coils if two are connected in,

 

  1. Parallel aiding connection
  2. Parallel opposing connection

 

For parallel aiding:

 

 

For parallel opposing:

 

 

Summary

 

From the above discussion, it is concluded that inductance diminishes on the parallel connection in the absence of mutual inductance.

 

 

Due to the presence of mutual inductance, the inductance calculated from the above-derived formula is invalid. There must be mutual inductance whenever two coils are placed in each other’s magnetic field. The mutual inductance may or may not decrease the total inductance.

linked in icon
Sign up for HardwareBee
* = required field

Recent Stories

CEO Talk: Maarten Kuper, QBayLogic
CEO Talk: Maarten Kuper, QBayLogic
Low ESR Capacitor: Ultimate Guide
Low ESR Capacitor: Ultimate Guide
The Ultimate Guide to PWM Controller
The Ultimate Guide to PWM Controller
What is a Piezo Driver IC and how to Choose one
What is a Piezo Driver IC and how to Choose one
Introduction to Hall Effect Sensor ICs
Introduction to Hall Effect Sensor ICs
The Ultimate Guide to: Oscillator ICs
The Ultimate Guide to: Oscillator ICs
Low Noise Amplifier: Ultimate Guide
Low Noise Amplifier: Ultimate Guide
ASIC Prototyping
Get 3 Quotes from Electronic Design Companies
Get 3 Quotes from FPGA Design Companies
Get 3 Quotes from Embedded SW Services
Get 3 Quotes from EMS Companies

Find Design Services

Get IC Prices

Get Price Offers From
  • Electronic Design Services
  • FPGA Design Services
  • Embedded Software Companies
  • PCB Layout Services
  • Printed Circuit Board Manufacturers
  • Design & Manufacturing Services
Welcome New Vendors
  • VVDN Technologies
  • Spark Product Innovation
  • QBayLogic
  • Fidus Systems
  • nao.design
Browse Vendor Directories
  • Electronic Design Companies
  • FPGA Design Companies
  • Embedded Software Services
  • Manufacturing Companies
Featured Vendor

Prevas

Recent Posts
  • CEO Talk: Maarten Kuper, QBayLogic
  • Low ESR Capacitor: Ultimate Guide
  • The Ultimate Guide to PWM Controller
  • What is a Piezo Driver IC and how to Choose one
  • Introduction to Hall Effect Sensor ICs
Most Popular Blog Posts
  • FPGA for AI (Artificial Intelligence): Ultimate Guide
  • PCB Stackup: Ultimate Guide and Examples
  • Cost of Electronic Manufacturing: A Guide
  • FPGA Video Processing: Ultimate Guide
  • The Ultimate Guide to Logic Chips

Never miss an update!

Follow us on LinkedIn

Do you need any price
information?

(Electronic design, FPGA design, Embedded SW services, PCB design, Turnkey)

Yes
No
This page is sponsored by
HardwareBee

Copyright 2017-2024, HardwareBee. All rights reserved.

  • About Us
  • Contact
  • Subscribe
  • News
  • Get Free Support
  • Get listed
  • Send a wiki/article
  • Advertise

Follow Us

Be sure to follow our LinkedIn company page where we share our latest updates LinkedIn
Partner with us Partner with us

Design and Manufacturing Services

  • Engineering Design Services
  • Electronic Design and Manufacturing
  • Electronic Product Development
  • Electronic Product Design
  • Electronic Consulting Services
  • Electronic Engineering Companies
  • Electronic Engineering Services
  • Electronic Product Design and Development
  • Electronics Design Services
  • Electronics Design Company
  • Electronic Design Consultants
  • Electronic Design Company
  • FPGA Design Company
  • FPGA Consultant
  • FPGA Design Services UK
  • Electronics Manufacturing services
  • Electronics Manufacturing Companies
  • Electronic Contract Manufacturing Companies
  • Electronic Manufacturing Services Companies
  • EMS Companies Directory
  • Electronic Design Services
  • FPGA Design Services
  • Embedded Software Companies
  • PCB Layout Services
  • Printed Circuit Board Manufacturers
  • Design and Manufacturing Services
X

Don’t miss anything, follow us on LinkedIn

https://www.linkedin.com/company/hardwarebee/

We are using cookies to give you the best experience on our website.

You can find out more about which cookies we are using or switch them off in .

Privacy Overview

This website uses cookies so that we can provide you with the best user experience possible. Cookie information is stored in your browser and performs functions such as recognising you when you return to our website and helping our team to understand which sections of the website you find most interesting and useful.

Strictly Necessary Cookies

Strictly Necessary Cookie should be enabled at all times so that we can save your preferences for cookie settings.

If you disable this cookie, we will not be able to save your preferences. This means that every time you visit this website you will need to enable or disable cookies again.

3rd Party Cookies

This website uses Google Analytics to collect anonymous information such as the number of visitors to the site, and the most popular pages.

Keeping this cookie enabled helps us to improve our website.

Please enable Strictly Necessary Cookies first so that we can save your preferences!

Additional Cookies

This website uses the following additional cookies:

(List the cookies that you are using on the website here.)

Please enable Strictly Necessary Cookies first so that we can save your preferences!