2

E.m.f. equation of transformer

2

m

m

mm

mmm

mmm

m

fNE

fNE

NfNfEsmr

NfNE

tEtNtdt

dNe

t

dt

dNde

Φ=

Φ=

Φ=Φ=

Φ=Φ=

=Φ=Φ=

Φ=Φ

Φ=

22

11

44.4

44.4

44.42

2...

2

coscos)sin(

sin

π

πω

ωωωω

ω

Ideal Transformer

Conditions for the ideal transformer.

a) no leakage flux

b) all core losses (both hysteresis and eddy)

are zero

c) the windings have zero resistance

d)the permeability of the core is infinite

( the exciting current is negligible)

1

4

5

Ideal Transformer

• Relation between current that flows in the primary winding, ip(t) and current that flows in the secondary winding, is(t):

ati

ti

tiNtiN

S

P

SSPP

1

)(

)(

)()(

=

=

aI

I

aV

V

S

P

S

P

1=

=(2)

(3)

(4)

(5)

6

Transformer Impedance

• Primary Impedance:

• Primary Voltage:

• Primary Current:

• Primary impedance in terms of secondary impedance

P

P

LI

VZ ='

SPaVV =

a

II S

P=

LL

S

S

S

S

P

P

L

ZaZ

I

Va

aI

aV

I

VZ

2

2

'

/'

=

===

(15)

(16)

(17)

(18)

7

Example 1

A transformer coil possesses 4000 turns and

links an ac flux having a peak value of 2

mWb. If the frequency is 60 Hz, calculate

the effective value of the induced voltage E.

Ans: 2131V

8

Example 2

A coil having 90 turns is connected to a 120V, 60 Hz source. If the effective value of the magnetizing current is 4 A, calculate the following:

a. The peak value of flux

b. The peak value of the mmf

c. The inductive reactance of the coil

d. The inductance of the coil.

9

10

11

12

13

Equivalent Circuit a) referred to primary side b)

referred to secondary side

14

Approximate equivalent circuit

Approximate equivalent circuit c)

Referred to primary side (no

exicitation) d) Referred to secondary

side (no excitation).

15

16

17

18

19

Example 3

A not-quite-ideal transformer having 90 turns on the primary and 2250 turns on the secondary is connected to a 120 V, 60 hz source. The coupling between the primary and the secondary is perfect but the magnetizing current is 4 A. calculate:

a. The effective voltage across the secondary terminals

b. The peak voltage across the secondary terminals.

c. The instantaneous voltage across the secondary when the instantaneous voltage across the primary is 37 V.

Ans: 3000V, 4242 V, 925 V.

20

Example 4

An ideal transformer having 90 turns on the primary and 2250 turns on the secondary is connected to a 200 V, 50 Hz source. The load across the secondary draws a current of 2 A at a power factor of 80 per cent lagging. Calculate :

a. The effective value of the primary current

b. The instantaneous current in the primary when the instantaneous current in the secondary is 100 mA.

c. The peak flux linked by the secondary winding.

Ans: 50 A, 2.5 A, 10 mWb.

21

Example 5

Calculate voltage E and current I in the circuit

of Fig. shown below, knowing that the ideal

transformer T has a primary to secondary

turns ratio of 1;100.

Ans: 800 V, 2 A.

22

23

24

25

26

Determination of transformer parameters: Short

Circuit Test

• Magnitude of series

impedance:

• Power factor:

SC

SC

SEI

VZ =

SCSC

SC

IV

PPF == θcos

0

00

00θ

θ∠=

−∠

∠=

SC

SC

SC

SCSE

I

V

I

VZ

)()( 22

SPSPSE

eqeqSE

XaXjRaRZ

jXRZ

+++=

+=

27

Three Phase Transformer

• Three phase transformer consists of 3 transformers.

• It is connected independently or in combination of 3 transformers.

• Primary and secondary windings can be connected as wye (Y) or delta (∆)

• Thus, there are 4 types of connections:

– Wye – wye (Y-Y)

– Wye – delta (Y-∆)

– Delta – wye (∆-Y)

– Delta – delta (∆-∆)

28

3 phase Transformer

3 phase transformer

connected independently

29

Three Phase Transformer

Three phase transformer

connected to a common

core with three legs

30

Connection Y-Y

Three Phase

transformer Y-Y

connection

31

Transformer Connection Y-Y

aV

V

V

V

S

P

LS

LP ==φ

φ

3

3

32

Transformer Y-∆ Connection

aV

V

V

V

V

V

LS

LP

S

P

LS

LP

3

3

=

=φ

φ

33

Transformer ∆-Y Connection

aV

V

V

V

V

V

LS

LP

S

P

LS

LP

3

3

=

=φ

φ

34

Transformer ∆-∆ Connection

aV

V

V

V

S

P

LS

LP ==φ

φ

35

36