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Mayurbhanj School Of Engineering, Baripada, Baripada

Branch : Electrical Engineering�Semester : 3^rd

Subject : Circuit & Network Theory

Chapter : 02

Topic : Coupled Circuits

Faculty : ER ABHISEK ACHARYA

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CONTENTS

SELF AND MUTUAL INDUCTANCE.
COUPLING COEFFICIENT (K)
DOT CONVENTION
SERIES AND PARALLEL CONNECTION OF COUPLED INDUCTORS

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OBJECTIVES

To understand the basic concept of self inductance and mutual inductance.
To understand the concept of coupling coefficient and dot determination in circuit analysis.

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SELF AND MUTUAL INDUCTANCE

When two loops with or without contacts between them affect each other through the magnetic field generated by one of them, it called magnetically coupled.
Example: transformer

An electrical device designed on the basis of the concept of magnetic coupling.
Used magnetically coupled coils to transfer energy from one circuit to another.

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a) SELF INDUCTANCE

It called self inductance because it relates the voltage induced in a coil by a time varying current in the same coil.
Consider a single inductor with N number of turns when current, i flows through the coil, a magnetic flux, Φ is produces around it.

i(t)

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According to Faraday’s Law, the voltage, v induced in the coil is proportional to N number of turns and rate of change of the magnetic flux, Φ;

But a change in the flux Φ is caused by a change in current, i.

Hence;

SELF INDUCTANCE

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Thus, (2) into (1) yields;

From equation (3) and (4) the self inductance L is

define as;

The unit is in Henrys (H)

SELF INDUCTANCE

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b) MUTUAL INDUCTANCE

When two inductors or coils are in close proximity to each other, magnetic flux caused by current in one coil links with the other coil, therefore producing the induced voltage.

Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor.

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Consider the following two cases:

Case 1:

Two coil with self – inductance L₁ and L₂ which are in close proximity which each other . Coil 1 has N₁ turns, while coil 2 has N₂turns.

i₁(t)

Φ₁₂

V₁

V₂

Φ₁₁

L₂

L₁

N₁ turns

N₂ turns

MUTUAL INDUCTANCE

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Magnetic flux Φ₁ from coil 1 has two components;

* Φ₁₁ links only coil 1.

* Φ₁₂ links both coils.

Hence; Φ₁ = Φ₁₁+ Φ₁₂ ……. (6)

Thus;

Voltage induces in coil 1

MUTUAL INDUCTANCE

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Voltage induces in coil 2

Subscript 21 in M₂₁ means the mutual inductance on coil 2 due to coil 1

MUTUAL INDUCTANCE

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Case 2:

Same circuit but let current i₂ flow in coil 2.

The magnetic flux Φ₂ from coil 2 has two components:

* Φ₂₂ links only coil 2.

* Φ₂₁ links both coils.

Hence; Φ₂ = Φ₂₁ + Φ₂₂ ……. (9)

i₂(t)

Φ₂₁

V₁

V₂

Φ₂₂

L₂

L₁

N₁ turns

N₂ turns

MUTUAL INDUCTANCE

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Thus;

Voltage induced in coil 2

Voltage induced in coil 1

Subscript 12 in M₁₂means the Mutual Inductance on coil 1 due to coil 2

MUTUAL INDUCTANCE

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Since the two circuits and two current are the same:

Mutual inductance M is measured in Henrys (H)

MUTUAL INDUCTANCE

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COUPLING COEFFICIENT (k)

It is measure of the magnetic coupling between two coils.
Range of k : 0 ≤ k ≤ 1

k = 0 means the two coils are NOT COUPLED.
k = 1 means the two coils are PERFECTLY COUPLED.
k < 0.5 means the two coils are LOOSELY COUPLED.
k > 0.5 means the two coils are TIGHTLY COUPLED.

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k depends on the closeness of two coils, their core, their orientation and their winding.
The coefficient of coupling, k is given by;

COUPLING COEFFICIENT (k)

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DOT CONVENTION

Required to determine polarity of “mutual” induced voltage.

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.

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Φ₁₂

Φ₂₁

Φ₂₂

Φ₁₁

Coil 2

Coil 1

DOT CONVENTION

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Dot convention is stated as follows:

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.

Conversely, Dot convention may also be stated as follow:

if a current LEAVES the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is NEGATIVE at the dotted terminal of the second coil.

DOT CONVENTION

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The following dot rule may be used:

when the assumed currents both entered or both leaves a pair of couple coils by the dotted terminals, the signs on the L – terms.

if one current enters by a dotted terminals while the other leaves by a dotted terminal, the sign on the M – terms will be opposite to the signs on the L – terms.

DOT CONVENTION

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Once the polarity of the mutual voltage is already known, the circuit can be analyzed using mesh method.
Application of the dot convention
Example 1

The sign of the mutual voltage v₂ is determined by the reference polarity for v₂ and the direction of i₁. Since i₁enters the dotted terminal of coil 1 and v₂ is positive at the dotted terminal of coil 2, the mutual voltage is M di₁/dt

i₁(t)

V₁

V₂(t) = M di₁/dt

L₂

L₁

DOT CONVENTION

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Example 2

Current i₁ enters the dotted terminal of coil 1 and v₂ is negative at the dotted terminal of coil 2. the mutual voltage is –M di₁/dt

i₁(t)

V₁

V₂(t) = -M di₁/dt

L₂

L₁

DOT CONVENTION

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Same reasoning applies to the coil in example 3 and example 4.
Example 3

Example 4

i₂(t)

V₁= -M di₂/dt

V₂(t)

L₂

L₁

i₂(t)

V₁= M di₂/dt

V₂(t)

L₂

L₁

DOT CONVENTION

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Dot convention for coils in series

L₂

L₁

(+)

L₂

L₁

(-)

Series – aiding connection

Series – opposing connection

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Examples of the sets of equations derived from basic configurations involving mutual inductance

Circuit 1

Solution:

−

R₂

R₃

R₁

V_s

I₁

I₂

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Circuit 2

Solution:

−

R₂

-jc

R₁

V_s

I₁

I₂

EXAMPLES:

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Circuit 3

Solution:

R₂

−

R₁

V_s

I₁

I₂

EXAMPLES:

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Circuit 4

Solution:

−

R₂

-jc

R₁

V_s

I₁

I₂

EXAMPLES:

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Circuit 5

Solution:

−

M₂

R₂

R₁

-jd

V_s

I₁

I₂

M₁

M₃

EXAMPLES: