LECTURE 6 MUTUAL INDUCTANCE · 2018-01-31 · POWER CIRCUITS AND ELECTROMECHANICS LECTURE 6 MUTUAL INDUCTANCE ... •When two coils are magnetically coupled there is “mutual inductance.”

Copyright © 2017 Hassan Sowidan

ECE 330

POWER CIRCUITS AND ELECTROMECHANICS

LECTURE 6

MUTUAL INDUCTANCE

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.

1/31/2018


SOURCE ON PRIMARY SIDE

• When two coils are magnetically coupled there is

“mutual inductance.”

• Assuming an ac voltage source on one side and an

open circuit on the other (i2 = 0):

1/31/2018 2

11 1 21.

i1(t)

N1N2

v1(t) v2(t)

21

1l

2 2 21

2 21

2 2

N

d dv N

dt dt


SOURCE ON PRIMARY SIDE

• Let the flux and current be linearly related (no

saturation):

• Then,

• M21 is the mutual inductance.

• L1 is the self-inductance of coil 1.

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2 21 1 1 1 1, .M i L i

1 12 21 1 1, and .

di div M v L

dt dt


SOURCE ON SECONDARY SIDE

• Assuming an ac voltage source on one side and an

open circuit on the other (i1= 0):

• Let the flux and current be linearly related (no

saturation):

• M12 is the mutual inductance and

• L2 is the self-inductance of coil 2.

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22 2 12.l

i2(t)

N1N2

v1(t) v2(t)

12

2l

1 121 1 12 1 1,

d dN v N

dt dt

1 12 2 2 2 2, .M i L i

2 22 2 1 12, and .

di div L v M

dt dt


SOURCES ON BOTH SIDES

• Using superposition:

• Let M12= M21= M, then:

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i2(t)

N1N2

v1(t) v2(t)

i1(t)

12

2l211l

1 1 1 2

2 1 2 2

L i M i

M i L i

1 1 21 1

2 2 12 2

d di div L M

dt dt dt

d di div L M

dt dt dt

1 1 21 12 11 12

2 2 21 12 22 21

1 1 11 1 12 1 1 12 2

2 2 21 2 22 21 1 2 2

l

l

N N L i M i

N N M i L i


COUPLING COEFFICIENT

• Define the coupling coefficient as:

• If k = 0, no coupling and M = 0.

• If k =1, ideal coupling with zero leakage:

• Therefore, and

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1 2

Mk

L L

21 12 21 12

11 22 1 21 2 121 2 l l

Mk

L L

0 1k 1 20 .M L L

0.M is always


COUPLING COEFFICIENT

• What if i2 is reversed?

• What if v2 is reversed?

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1 1 21 1

2 2 12 2

'

'

d di div L M

dt dt dt

d di div L M

dt dt dt

1 1 21 1

2 2 12 2

'

''

d di div L M

dt dt dt

d di div L M

dt dt dt

2 2i i

2 2v v


EQUIVALENT CIRCUIT

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1

1 1N i

1

1l2 2N i

2

2l

21 2

21

12

12

2l


DOT MARKINGS

• Dots relate the flux direction between coils.

• If two fluxes are in the same direction, they add,

otherwise, they subtract.

• Depending which ends you connect the load to the

secondary coil you either get an

Output voltage in sync. With the

input voltage or in reverse phase.

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i2(t)

N1N2

v1(t) v2(t)

i1(t)

2l

21

1l


DOT MARKINGS

• The polarity markings are assigned such that a

positively increasing current in the dotted terminal in

one winding induces a positive voltage at the dotted

terminal of the other winding

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i2(t)

N1N2

v1(t) v2(t)

i1(t)

2l

21

1l


DOT MARKINGS

• Assume the following configuration.

1) Select one coil and one terminal and place a dot

on that terminal.

2) Assume a current is flowing and determine the

flux direction.

3) Place a test current in the second coil and

determine its flux direction.

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1

1'

2

2'

i1(t)

12


DOT MARKINGS

4) If and add, then place dot on 2 (where test current

enters).

5) If they subtract, then place dot on 2’ (where test current

leaves).

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1

1'

2

2'

i1(t)i2(t)

12

21

1

1'

2

2'

1 2


DOT MARKINGS

• Practical determination of dot locations:

1) Build the following circuit.

-Turn the switch on => pulse is generated where

di/dt is not zero on the secondary side.

-If the pulse causes the meter (V) to read

positive, then the dot on the secondary is on the

top terminal.

-If (V) reads a negative pulse,

then the dot is on the lower side.

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RVdc

+

-V


EXAMPLES

The dots in the circuit below are as shown.

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1 1'

2' 2

i1(t)

i2(t)


WRITING EQUATIONS WITH MUTUALLY

COUPLED COILS

Suppose we have two mutually coupled coils and the

dot markings are as shown

The self induced voltage due to the self inductance is

in the direction of the current and is a voltage drop.

The polarity of the mutually induced voltage depends

on the dot marking

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COUPLED COILS

Source: jacobs-university.de

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1 21 1 1 1=

di div i R L M

dt dt

2 12 2 2 2=

di div i R L M

dt dt


WRITING EQUATIONS WITH MUTUALLY COUPLED

COILS


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1 21 1 1 1=

di div i R L M

dt dt

2 12 2 2 2=

di div i R L M

dt dt


WRITING EQUATIONS WITH MUTUALLY COUPLED

COILS


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COUPLED COILS

• If the reference current in a coil leaves the dotted

(undotted) terminal, then the voltage induced at the

dotted (undotted) terminal of the other coil has a

negative sign.

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Documents

LECTURE 6 MUTUAL INDUCTANCE · 2018-01-31 · POWER CIRCUITS AND ELECTROMECHANICS LECTURE 6 MUTUAL INDUCTANCE ... •When two coils are magnetically coupled there is “mutual inductance.”