1

PART TWO: AC Circuits

Chapter 6Magnetically coupled circuits

phasors

THIS COURSE CAN BE FOUND AT:

http://users.utcluj.ro/~denisad

2/30

When two coils with or without contacts between them affect each other

through the magnetic field generated by one of them, they are said to be

magnetically coupled.

Inductive coupling is widely used throughout

electrical technology; examples include:

✓ Electric motors and generators

✓ Inductive charging products

✓ Induction cookers and induction heating

systems

✓ Induction loop communication systems

✓ Metal detectors

✓ Radio-frequency identification

✓ Transformers

✓ Wireless power transfer

NFC (Near Field Communication) works based on the principle

of inductive coupling, where loosely coupled inductive circuits share

power and data over a distance of a few centimeters. All NFC devices

operate at 13.56MHz. NFC devices share the basic technology with

proximity (13.56MHz) RFID (Radio-Frequency Identification) tags and

contactless smartcards, but have a number of key additional features.

Working principle of inductive coupling in NFC devices

Source: [www.wireless.intgckts.com]

6.1. INTRODUCTION

3/30

When two coils are in a close proximity to each other, the magnetic flux caused by current in

one coil links with the other coil, and inducing voltage in the latter. This phenomenon is known

as mutual inductance.

- Consider a coil with N turns, when current i flows through the coil, a magnetic flux φ is produced around it.

According to Faraday’s law, the voltage v induced in the coil is proportional to the number of turns N and the time

rate of change of the magnetic flux φ; that is,

6.2. MUTUAL INDUCTANCE

4/30

6.2. MUTUAL INDUCTANCE

5/30

6.2. MUTUAL INDUCTANCE

6/30

6.2. MUTUAL INDUCTANCE

7/30

6.2. MUTUAL INDUCTANCE

Examples illustrating how to apply the dot convention:

8/30

6.2. MUTUAL INDUCTANCE

6.2. MUTUAL INDUCTANCE

6.2.1. Series-Aiding Connection

9/30

6.2. MUTUAL INDUCTANCE

6.2.2. Series-Opposing Connection

10/30

11/30

6.2.3. Circuit Model for Coupled Inductors

6.2. MUTUAL INDUCTANCE

12/30

Example 1:

6.2. MUTUAL INDUCTANCE

13/30

6.2. MUTUAL INDUCTANCE

Example 2:

14/30

6.2. MUTUAL INDUCTANCE

6.2.4. Energy in a Coupled Circuit

15/30

6.2. MUTUAL INDUCTANCE

6.2.4. Energy in a Coupled Circuit

6.2. MUTUAL INDUCTANCE

6.2.4. Energy in a Coupled Circuit

16/30

17/30

6.2. MUTUAL INDUCTANCE

6.2.4. Energy in a Coupled Circuit

18/30

6.2. MUTUAL INDUCTANCE

6.2.5. Coupling Coefficient

The coupling coefficient k is a measure of magnetic coupling between two coils;

0 ≤ k ≤ 1.

Figure 1- Loose coupling Figure 2 – Tight coupling

Figure 1

Figure 2

19/30

20/30

6.2. MUTUAL INDUCTANCE

Example 3:

21/30

6.3. LINEAR TRANSFORMER

A transformer is a magnetic device that takes advantage of the phenomenon of mutual inductance.

✓ The coil that is directly connected to the voltage source is called the primary winding.

✓ The coil connected to the load is called the secondary winding.

✓ The resistances R1 and R2 are included to account for the losses (power dissipation) in the coils.

✓ The transformer is said to be linear if the coils are wound on a magnetically linear material.

6.3. LINEAR TRANSFORMER

22/30

23/30

6.3. LINEAR TRANSFORMER

6.3.1. T (or Y) Equivalent Circuits

24/30

6.3. LINEAR TRANSFORMER

6.3.2. Π (or Δ) Equivalent Circuits

25/30

6.4. IDEAL TRANSFORMER

➢ An ideal transformer is one with perfect coupling (k = 1). It consists of two (or more)

coils with a large number of turns wound on a common core of high permeability.

Because of this high permeability of the core, the flux links all the turns of both coils,

thereby resulting in a perfect coupling.

When a sinusoidal voltage is applied to the primary winding,

the same magnetic flux φ goes through both windings.

26/30

6.4. IDEAL TRANSFORMER

27/30

6.4. IDEAL TRANSFORMER

28/30

6.4. IDEAL TRANSFORMER

29/30

6.4. IDEAL TRANSFORMER

6.4.1. Impedance Transformation

30/30

References

[1] Charlews K. Alexander, Matthew N.O.Sadiku, Fundamentals of Electric

Circuits (Fifth Edition), published by McGraw-Hill, 2013

[2] Radu V. Ciupa, Vasile Topa, The Theory of Electric Circuits, published

by Casa Cartii de Stiinta, 1998

[3] Dan. D Micu, Laura Darabant, Denisa Stet et al., Teoria circuitelor

electrice. Probleme, published by UTPress, 2016