LECTURE 12 TRANSFORMERS (2) · PDF fileThe power transformer is an essential component of the power system. ... •Eddy current losses occurring in the laminations, resulting in an

Copyright © 2017 Hassan Sowidan

ECE 330

POWER CIRCUITS AND ELECTROMECHANICS

LECTURE 12

TRANSFORMERS (2)

Acknowledgment-These handouts and lecture notes given in class are based on material from Prof. Peter

Sauer’s ECE 330 lecture notes. Some slides are taken from Ali Bazi’s presentations

Disclaimer- These handouts only provide highlights and should not be used to replace the course textbook.

9/26/2017


POWER TRANSFORMER

The power transformer is an essential component of

the power system. It steps up the voltage at the

generating point, transmits it over long distances, and

then steps it down at the

sub-transmission and distribution

levels for use by both commercial

units and individual homes.

Source: torbosquid.com

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POWER TRANSFORMER

A physical transformer consists of two windings

mounted on a core, generally of a ferromagnetic

material of high permeability.

Source: electronichub.org Source: electricaleasy.com

9/26/2017 3


POWER TRANSFORMER

• The construction of the core is such as to keep the

leakage flux to a minimum.

• The winding to which power is supplied is called the

“primary” winding, and the other one is called the

“secondary” winding.

• In the case of power transformers, the terms high-

voltage (HV) and low-voltage (LV) windings are also

used, depending on the voltage levels.

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POWER TRANSFORMER

We assume it to be an ideal transformer. This implies:

• No leakage flux.

• Winding resistances are neglected.

• Core has infinite permeability. Hence, reluctance of

the core is zero and negligible current is required to

set up the magnetic field.

• Magnetic core is lossless

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POWER TRANSFORMER

• Power transformers are usually excited with

sinusoidal voltages and currents.

• It is interesting to find the maximum flux, flux

density, and current that a transformer can handle.

9/26/2017

v1(t) v2(t)

i1(t) i2(t)

N1 N2

6

1 1 1

11

1 1

1 1max

1 1

( ) cos ( )

1( ) cos ( ) sin ( )

2

m

mm

m m

dv t V t N

dt

Vt V t dt t

N N

V V

N f N


POWER TRANSFORMER

• Also,

• Example: Coil 1 has 100 turns, operating at 60 Hz

and Vrms = 230 V. The core area is 50 cm2 and the

core length is 25 cm. Find the maximum flux and

flux density.

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1 1 max

11 1 1 max 1 max

2

2 4.442

m

mrms

V fN

VV V fN fN

3

max

2maxmax

8.64 10

1.728 /

Wb

B Wb mA

7


POWER TRANSFORMER

• Given that H = 600 A.turns/m at Bmax, find the

maximum current.

Source: electricaleasy.com

9/26/2017

max 1.5H

i AN

8


EQUIVALENT CIRCUIT MODEL

Consider the two-winding transformer. Let be the

voltage across . Let the resistances of the primary

and secondary windings be and , respectively.

Viewing the two windings as coupled coils, the

differential equations are written as

Source: machineequipmentonline.com

9/26/2017 9

2v

LR

1R 2R



• Non-ideal:

• Add and subtract M di1/dt to the first equation and

M di2/dt to the second equation:

9/26/2017

1 21 1 1 1

2 12 2 2 2

2 2

0 L

L

di div R i L M

dt dt

di diR i L M i R

dt dt

v i R

1 1 21 1 1 1

2 2 12 2 2 2

2 2

( )

0 ( ) L

L

di di div R i L M M M

dt dt dt

di di diR i L M M M i R

dt dt dt

v i R

10



What if or < 0? There is something

wrong.

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11 1 1 1 1 2

22 2 2 2 1 2

2 2

( ) ( )

0 ( ) ( ) L

L

di dv R i L M M i i

dt dt

di dR i L M M i i i R

dt dt

v i R

2( )L M1( )L M

11



• We need to refer all impedances from side 2 to side 1:

(since M either sees i1 or i2).

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1 21 1 1 1

2

2 2 22 1 22 2

222

0

( )

L

L

di idv R i L aM

dt dt a

id

i di iaa R a L aM a R

a dt dt a

iav a R

a

222 2 2, ,

ii Z a Z M a M

a

12



• The equivalent circuit becomes:

• If we wish to preserve RL , the circuit becomes:

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Losses:

Hysteresis losses due to the nonlinear multi-valued

nature of the relationship of the actual core.

• the maximum flux density, and (often assumed

to be 1.6) and are constants depending on the

material of the core

• add Rc in parallel with a M.

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= n

h h mHysteresis loss P K f B W

mB n

14

i



• Eddy current losses occurring in the laminations,

resulting in an -type loss.

• is a constant that depends on the resistivity of the

material of the laminations

• The sum of these two losses represents the constant

losses for the transformer and depends only on the

value of

9/26/2017

2 2=e e mEddy current loss P K f B W

mB

eK

15



• depends on the applied voltage

Losses are represented by means of a resistance in

parallel with the magnetizing inductance .

9/26/2017

mBmV

cR

aM

16



• What does mean? (and similarly the

secondary term)?

• Therefore, is a leakage term.

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1 1 1 11 1 2 21

1 11 2 21 1 11 11 21 1

1 1 1 1

,

l

L i N Mi N

N aN N NL aM

i i i i

1( )L a M

17

1( )L a M



• Leakage reactance of winding1

magnetizing reactance referred to

winding1

Leakage reactance of winding 2

Leakage reactance of winding 2

referred to side 1

9/26/2017

11( ) XL aM

1( ) XmaM

22( ) XMLa

2

2 2

2(a ) a XL a M

18


READING MATERIAL

9/26/2017

• Reading material: Sections 3.4.3 and 3.4.4.

• Next time: Sections 3.4.5-3.4.7.

19

Documents

LECTURE 12 TRANSFORMERS (2) · PDF fileThe power transformer is an essential component of the power system. ... •Eddy current losses occurring in the laminations, resulting in an