EE2003 Circuit Theoryeng.staff.alexu.edu.eg/~bmokhtar/courses/circuit_II/... · 2017. 4. 11. · 2...
Transcript of EE2003 Circuit Theoryeng.staff.alexu.edu.eg/~bmokhtar/courses/circuit_II/... · 2017. 4. 11. · 2...
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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.