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EE2003 Circuit Theory

Chapter 13

Magnetically Coupled Circuits

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Magnetically Coupled Circuit Chapter 13

13.1 What is a transformer?

13.2 Mutual Inductance

13.3 Energy in a Coupled Circuit

13.4 Linear Transformers

13.5 Ideal Transformers

13.6 Ideal Autotransformers

13.6 Applications

Introduction

• The circuits we have considered so far may be regarded as

conductively coupled, because one loop affects the

neighboring loop through current conduction

• When two loops 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

• The transformer is an electrical device designed on the

basis of the concept of magnetic coupling

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Introduction

• The transformer uses magnetically coupled coils to transfer

energy from one circuit to another

• Transformers are used in power systems for stepping up or

stepping down ac voltages or currents

• They are used in electronic circuits such as radio and

television receivers for such purposes as impedance

matching, isolating one part of a circuit from another, and

again for stepping up or down ac voltages and currents

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Introduction • In electromagnetics, electric circuit analysis is applied at low

frequencies

• The principles of electromagnetics (EM) are applied in various

allied disciplines, such as electric machines, electromechanical

energy conversion, radar meteorology, remote sensing, satellite

communications, bioelectromagnetics, electromagnetic

interference and compatibility, plasmas, and fiber optics

• EM devices include electric motors and generators, transformers,

electromagnets, magnetic levitation, antennas, radars,

microwave ovens, microwave dishes, superconductors, and

electrocardiograms 5

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13.1 What is a transformer? (1)

• It is an electrical device designed on the basis of the concept of magnetic coupling

• It uses magnetically coupled coils to transfer energy from one circuit to another

• It is the key circuit elements for stepping up or stepping down ac voltages or currents, impedance matching, isolation, etc.

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13.2 Mutual Inductance (1) • Let us first consider a single inductor, 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,

• But the flux 𝛷 is produced by current 𝑖 so that any change in 𝛷 is caused by a change in the current

• The inductance 𝐿 is commonly called self-inductance, because it relates the voltage induced in a coil by a time-varying current in the same coil.

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13.2 Mutual Inductance (1)

• Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H)

12 21

div M

dt

dt

diMv 2

121

When two inductors (or coils) are in a close proximity to each other, the magnetic flux

caused by a time-varying current in one coil links with the other coil, thereby

inducing voltage in the latter. This phenomenon is known as mutual inductance.

is known as the mutual inductance of coil 2 with respect to coil 1. 21M

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13.2 Mutual Inductance (2) • To determine the polarity of mutual voltage, the dot

convention is applied in circuit analysis

• By this convention, a dot is placed in the circuit at one end of each of the two magnetically coupled coils to indicate the direction of the magnetic flux if current enters that dotted terminal of the coil

• If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil

Illustration of the dot convention.

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13.2 Mutual Inductance (3)

)connection aiding-(series

221 MLLL

Dot convention for coils in series; the sign indicates the polarity of the mutual voltage; (a) series-aiding connection, (b) series-opposing connection.

)connection opposing-(series

221 MLLL

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13.2 Mutual Inductance (4)

Time-domain analysis of a circuit containing coupled coils.

Frequency-domain analysis of a circuit containing coupled coils

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13.2 Mutual Inductance (5)

Example 1

Calculate the phasor currents I1 and I2 in the circuit shown below.

A04.1491.2I A;39.4901.13I 21 Ans:

*Refer to in-class illustration, textbook

13.3 Energy in a Coupled Circuit (1)

• The energy stored in an inductor is given by

• at 𝑖2 = 0, the power in coil 1 is

• and the energy stored in the circuit at 𝑖1 = 𝐼1 is

• If 𝑖2 > 0 and 𝑖1 = 𝐼1, the power in the coils is

(1)

13.3 Energy in a Coupled Circuit (1)

• the energy stored in the circuit is

• The total energy stored in the coils when both 𝑖1 and 𝑖2 have reached constant values is

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(2)

From (1) and (2)

13.3 Energy in a Coupled Circuit (1)

• The coupling coefficient, 𝑘, is a measure of the magnetic coupling between two coils; 0≤k≤1.

• The instantaneous energy stored in the circuit is given by

21LLkM

21

2

22

2

112

1

2

1iMiiLiLw

The positive sign is selected for the mutual term if both currents enter or leave the dotted terminals of the coils; the negative sign is selected otherwise.

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13.3 Energy in a Coupled Circuit (2)

Example 2

Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductors at time t = 1s if v=60cos(4t +30°) V.