# BEE1113 ELECTRIC CIRCUIT I CHAPT

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```					            CHAPTER 5: TRANSFORMER
AND MUTUAL INDUCTANCE

•          Review of Magnetic Induction
•          Mutual Inductance
•          Linear & Ideal Transformers

AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual   1
Inductance
Introduction

• 1 coil (inductor)
– Single solenoid has only self-inductance (L)

• 2 coils (inductors)
– 2 solenoids have self-inductance (L) & Mutual-
inductance

AHBMH      DEE2113 : Chapter 5 - Transformer & Mutual   2
Inductance
1 Coil

• A coil with N turns produced  = magnetic flux
• only has self inductance, L

AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual   3
Inductance
Self-Inductance

• Voltage induced in a coil by a time-varying current in
the same coil

d di    di
vN       L
di dt    dt
d
LN
di
AHBMH      DEE2113 : Chapter 5 - Transformer & Mutual   4
Inductance
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
AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual           5
Inductance
Mutual voltage (induced voltage)

Voltage induced in coil 1:
di1
1  L1
dt
Voltage induced in coil 2 :

di1
2  M 21
dt
M21 : mutual inductance of coil 2 with respect to coil 1

AHBMH         DEE2113 : Chapter 5 - Transformer & Mutual            6
Inductance
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.

AHBMH         DEE2113 : Chapter 5 - Transformer & Mutual           7
Inductance
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
AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual           8
Inductance
Mutual voltage (induced voltage)

Voltage induced in coil 2:
di2
2  L2
dt
Voltage induced in coil 1 :

di 2
1  M12
dt
M12 : mutual inductance of coil 1 with respect to coil 2

AHBMH         DEE2113 : Chapter 5 - Transformer & Mutual            9
Inductance
Dot Convention

• Not easy to determine the polarity of mutual voltage –
4 terminals involved
• Apply dot convention

AHBMH        DEE2113 : Chapter 5 - Transformer & Mutual            10
Inductance
Dot Convention

AHBMH   DEE2113 : Chapter 5 - Transformer & Mutual   11
Inductance
Dot Convention

AHBMH   DEE2113 : Chapter 5 - Transformer & Mutual   12
Inductance
Frequency Domain Circuit

For coil 1 :       V  ( Z1  jL1 )I1  jMI 2
For coil 2 :         0   jMI 1  ( ZL  jL 2 )I 2

AHBMH           DEE2113 : Chapter 5 - Transformer & Mutual      13
Inductance
Example 1

Calculate the phasor current I1 and I2 in the circuit

AHBMH           DEE2113 : Chapter 5 - Transformer & Mutual      14
Inductance
Exercise 1

Determine the voltage Vo in the circuit

AHBMH     DEE2113 : Chapter 5 - Transformer & Mutual   15
Inductance
Energy In A Coupled Circuit
Energy stored in an inductor:
1 2
w  Li                 Unit : Joule

2
Energy stored in a coupled circuit:

1      1
w  L1i1  L 2i 2  Mi1i 2
2        2

2      2
Positive sign: both currents enter or leave the dotted terminals
Negative sign: one current enters and one current leaves the dotted terminals

AHBMH            DEE2113 : Chapter 5 - Transformer & Mutual                     16
Inductance
Energy In A Coupled Circuit
Coupled Circuit
M
i1                             i2

+                                           +

v1
. L1
.
L2        v2

-                                          -

AHBMH    DEE2113 : Chapter 5 - Transformer & Mutual                  17
Inductance
Energy In A Coupled Circuit
Energy stored must be greater or equal to zero.

1 2 1 2
L1i1  L 2i 2  Mi1i 2  0
2       2
L1L2  M  0                                     or   M  L1L2

Mutual inductance cannot be greater than the geometric mean of self inductances.

AHBMH          DEE2113 : Chapter 5 - Transformer & Mutual                   18
Inductance
Energy In A Coupled Circuit
The coupling coefficient k is a measure of the magnetic
coupling between two coils 0  k  1

M
k                                             or   M  k L1L2
L1L 2
Where:

0  k 1                                or         0  M  L1L2

AHBMH      DEE2113 : Chapter 5 - Transformer & Mutual                       19
Inductance
Energy In A Coupled Circuit

Perfectly coupled : k = 1

Loosely coupled : k < 0.5                                        Tightly coupled : k > 0.5
- Linear/air-core transformers                                   - 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
AHBMH             DEE2113 : Chapter 5 - Transformer & Mutual                                 20
Inductance
Example 2
Consider the circuit below. Determine the coupling
coefficient. Calculate the energy stored in the
coupled inductor at time t=1s if v  60 cos(4t  300 )V

AHBMH        DEE2113 : Chapter 5 - Transformer & Mutual           21
Inductance
Exercise 2

For the circuit below, determine the coupling coefficient
and the energy stored in the coupled inductors at t=1.5s.

AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual          22
Inductance
Linear Transformers
Transformer is linear/air-core if:
1. k < 0.5
2. The coils are wound on a magnetically linear material
(air, plastic, wood)

Input impedance:
V                    2 M 2
Zin   R 1  jL1 
I1              R 2  jL 2  ZL
Reflected impedance:
2 M 2
ZR 
R 2  jL 2  ZL

AHBMH           DEE2113 : Chapter 5 - Transformer & Mutual                23
Inductance
Linear Transformers

An equivalent circuit of linear                              An equivalent T circuit
transformer
L a  L1  M                        Lb  L2  M              Lc  M

AHBMH           DEE2113 : Chapter 5 - Transformer & Mutual                             24
Inductance
Linear Transformers

An equivalent circuit of linear                              An equivalent П/ circuit
transformer

L1L 2  M              2
L1L 2  M              2
L1L2  M            2

LA                                  LB                             LC 
L2  M                               L1  M                            M

AHBMH          DEE2113 : Chapter 5 - Transformer & Mutual                               25
Inductance
Example 3
Calculate the input impedance and current I1.

Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω

AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual       26
Inductance
Exercise 3
For the linear transformer below, find the
T-equivalent circuit and П equivalent circuit.

AHBMH     DEE2113 : Chapter 5 - Transformer & Mutual     27
Inductance
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 → ∞)
• Coupling coefficient is unity (k=1)
• Lossless primary and secondary coils (R1 = R2 = 0)

AHBMH           DEE2113 : Chapter 5 - Transformer & Mutual            28
Inductance
Ideal Transformer
A step-down transformer is one whose secondary voltage is
less than its primary voltage (n<1, V2<V1)

A step-up transformer is one whose secondary voltage is
greater than its primary voltage (n>1, V2>V1)

V2 N 2                                I 2 N1 1
    n                                 
V1   N1                               I1 N 2 n

AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual              29
Inductance
Ideal Transformer

The complex power in the primary winding :

V2
S1  V I     nI 2 *  V2 I 2  S 2
*
1 1
*

n

The input impedance :

ZL
Z in  2
n

AHBMH       DEE2113 : Chapter 5 - Transformer & Mutual   30
Inductance
Example 4

An ideal transformer is rated at 2400/120 V, 9.6 kVA
and has 50 turns on the secondary side. Calculate :
a) The turns ratio
b) The number of turns on the primary side
c) The currents ratings for the primary and
secondary windings

AHBMH         DEE2113 : Chapter 5 - Transformer & Mutual       31
Inductance
Exercise 4

The primary current to an ideal transformer rated at
3300/110 V is 3 A. Calculate :
a) The turns ratio
b) The kVA rating
c) The secondary current

AHBMH           DEE2113 : Chapter 5 - Transformer & Mutual     32
Inductance

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