1 mutual inductance

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Analysis and Simulation

of Electric Circuits (EME 208)

Spring 2011



2

Syllabus Mutual Inductance

RC & RL Circuits (Revision) RLC Circuits

Laplace Transform Application

The Frequency Response Non-sinusoidal Periodic Inputs

The Two Port Networks



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Assumed Knowledge AC Circuits; R,L, and C Circuits

(EME 207)

Differential Equations

The Laplace Transform

The Fourier Series



4

Resourses

These Notes

Nilsson and Riedel; ³Electric Circuits´ Alexander and Sadiku; ³Fundamentals

of Electric Circuits´

Exercise Sheets



5

Course Evaluation

Laboratory Attendance and Reports; (20 Points)

Two Mid-Term Exams; (20 Points) Assignments; (10 Points)

Final Exam; (50 Points)



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Chapter 1The Mutual Inductance

Magnetically Coupled Circuits



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Contents:

1. Capacitance and Inductance

2. What is a transformer?

3. Mutual Inductance

4. Energy in a Coupled Circuit



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Capacitors (1) A capacitor is a passive element designed to store energy

in its electric field.

A capacitor consists of two conducting plates separatedby an insulator (or dielectric).



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Capacitors (2) If i is flowing into the +ve

terminal of C

Charging =>i is +ve

Discharging => i is ±ve

The current-voltage relationship of capacitor according toabove convention is

t d

vd C i ! )(

10

0

t vt d iC

vt

t

! ´and



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Capacitors (3) The energy, w, stored in the

capacitor is

A capacitor is an open circuit to dc (dv/dt = 0). (i=Cdv/dt).

its voltage cannot change abruptly.

2

21 vw !



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Inductors (1) An inductor is a passive element designed to store

energy in its magnetic field.

An inductor consists of a coil of conducting wire.



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Inductors (2)

Inductance is the propertywhereby an inductorexhibits opposition to thechange of current flowingthrough it, measured inhenrys (H).

The unit of inductors is Henry (H), mH (10 ±3) and QH

(10 ±6).

t d

id Lv !



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Inductors (3) The current-voltage relationship of an

inductor:

The energy stored by an inductor:

)()(1 00

t it d t vit

t ! ´

2

21 i Lw !

An inductor acts like a short circuit to dc (di/dt = 0) and its

current cannot change abruptly.

t d id v !



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Current and Voltage Relations

+

+

+



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What is a transformer?

It is an electric device designed on the basisof the concept of magnetic coupling

It uses magnetically coupled coils to transferenergy 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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Ideal Transformer An ideal transformer consists of two coils wound on

the same core.

n N

N n

N

N 1

I

I

V

V

2

1

1

2

1

2

1

2 !!!!

(a)Ideal Transformer

(b)

Circuit symbol

V2>V1 step-up transformer

V2<V1

step-down transformer



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Mutual Inductance (1) It is the ability of one inductor to induce a voltage across a

neighboring inductor, measured in henrys (H).

d t

d i M v

1212 !

d t

d i M v

2121 !

The open-circuit mutual voltage across coil 2

The open-circuit mutual voltage across coil 1



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Mutual Inductance (2)

The mutual voltage is positive if both currents enter (or leave) thedotted terminal of the corresponding coils.

The mutual voltage is negative if one current enters while the otherleaves the dotted terminal of the corresponding coils.

Illustration of the dot convention.

THE ¶DOT· CONVENTION



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The Mutual voltage



20

=

=

=

=



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Coils in Series and Dot Markings

)connectionaiding-(series

221 M L L L !

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

1 2 2

(series-opposing connection)

L L L M !



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Time and Frequency Domains

Time-domain analysis of a circuit

containing coupled coils.

Frequency-domain analysis of a circuit containing coupled coils



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Example (1)Calculate the phasor currents I1 and I2 in the circuitshown below.



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Example (1)For coil 1, KVL gives

(j 4 + j 5) I1 j 3 I2 = 12

or

j I1 j 3I2 = 12

For coil 2, KVL givesí j 3 I1 + ( 12 + j 6 ) I2 = 0

orI1 = ( 12 + j 6 )I2 / j3

= ( 2 í j 4 ) I2

2

12I 2.91 14.04 A

4 j

! ! � r

1I 13.01 49.39! � r



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Coupling Coefficient & Energy The coupling coefficient, k, is a measure of the

magnetic coupling between two coils; 0k1.

The instantaneous energy stored in the circuit is given by

21 L Lk M !

2 2

1 1 2 2 1 2

1 1

2 2

w L i L i M i i! s



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Example (2)

2I 3.254 160.6! �



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Example (2)2 3.254cos(4 160.6 )i t !

o

1 mutual inductance

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