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CHAPTER 5 TRANSFORMER AND MUTUAL INDUCTANCE

- Review of Magnetic Induction
- Mutual Inductance
- Linear Ideal Transformers

Introduction

- 1 coil (inductor)
- Single solenoid has only self-inductance (L)
- 2 coils (inductors)
- 2 solenoids have self-inductance (L)

Mutual-inductance

1 Coil

- A coil with N turns produced ? magnetic flux
- only has self inductance, L

Self-Inductance

- Voltage induced in a coil by a time-varying

current in the same coil

2 coils

Mutual inductance of M21 of coil 2 with respect

to coil 1

- Coil 1 has N1 turns and Coil 2 has N2 turns

produced - ?1 ?11 ?12
- Magnetically coupled

Mutual voltage (induced voltage)

Voltage induced in coil 1

Voltage induced in coil 2

M21 mutual inductance of coil 2 with respect to

coil 1

Mutual Inductance

Mutual inductance is the ability of one inductor

to induce a voltage across a neighboring

inductor, measured in henrys (H)

- When we change a current in one coil, this

changes the magnetic field in the coil. - The magnetic field in the 1st coil produces a

magnetic field in the 2nd coil - EMF produced in 2nd coil, cause a current flow in

the 2nd coil. - Current in 1st coil induces current in the 2nd

coil.

2 coils

Mutual inductance of M12 of coil 1 with respect

to coil 2

- Coil 1 has N1 turns and Coil 2 has N2 turns

produced - ?2 ?21 ?22
- Magnetically coupled

Mutual voltage (induced voltage)

Voltage induced in coil 2

Voltage induced in coil 1

M12 mutual inductance of coil 1 with respect to

coil 2

Dot Convention

- Not easy to determine the polarity of mutual

voltage - 4 terminals involved
- Apply dot convention

Dot Convention

Dot Convention

Frequency Domain Circuit

For coil 1

For coil 2

Example 1

Calculate the phasor current I1 and I2 in the

circuit

Exercise 1

Determine the voltage Vo in the circuit

Energy In A Coupled Circuit

Energy stored in an inductor

Unit Joule

Energy stored in a coupled circuit

Positive sign both currents enter or leave the

dotted terminals Negative sign one current

enters and one current leaves the dotted terminals

Energy In A Coupled Circuit

Coupled Circuit

Energy In A Coupled Circuit

Energy stored must be greater or equal to zero.

or

Mutual inductance cannot be greater than the

geometric mean of self inductances.

Energy In A Coupled Circuit

The coupling coefficient k is a measure of the

magnetic coupling between two coils

or

Where

or

Energy In A Coupled Circuit

Perfectly coupled k 1

Loosely coupled k lt 0.5 - Linear/air-core

transformers

Tightly coupled k gt 0.5 - Ideal/iron-core

transformers

Coupling coefficient is depend on 1. The

closeness of the two coils 2. Their core

3. Their orientation 4. Their winding

Example 2

Consider the circuit below. Determine the

coupling coefficient. Calculate the energy stored

in the coupled inductor at time t1s if

Exercise 2

For the circuit below, determine the coupling

coefficient and the energy stored in the coupled

inductors at t1.5s.

Linear Transformers

Transformer is linear/air-core if

- k lt 0.5
- The coils are wound on a magnetically linear

material (air, plastic, wood)

Input impedance

Reflected impedance

Linear Transformers

An equivalent T circuit

An equivalent circuit of linear transformer

Linear Transformers

An equivalent ?/? circuit

An equivalent circuit of linear transformer

Example 3

Calculate the input impedance and current

I1. Take Z1 60 - j100 O , Z2 30 j40 O, and

ZL 80 j60 O

Exercise 3

For the linear transformer below, find the

T-equivalent circuit and ? equivalent circuit.

Ideal Transformer

- 1.An ideal transformer has
- 2/more coils with large numbers of turns wound

on an common core of high permeability. - Flux links all the turn of both coil perfect

coupling - 2. Transformer is ideal if it has
- Coils with large reactances (L1,L2, M ? 8)
- Coupling coefficient is unity (k1)
- Lossless primary and secondary coils (R1 R2 0)

Ideal Transformer

A step-down transformer is one whose secondary

voltage is less than its primary voltage (nlt1,

V2ltV1)

A step-up transformer is one whose secondary

voltage is greater than its primary voltage

(ngt1, V2gtV1)

Ideal Transformer

The complex power in the primary winding

The input impedance

Example 4

- An ideal transformer is rated at 2400/120 V, 9.6

kVA - and has 50 turns on the secondary side. Calculate

- The turns ratio
- The number of turns on the primary side
- The currents ratings for the primary and

secondary windings

Exercise 4

- The primary current to an ideal transformer rated

at - 3300/110 V is 3 A. Calculate
- The turns ratio
- The kVA rating
- The secondary current

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