Coupled Circuits

8/11/2019 Coupled Circuits

1/41

SELF INDUCTANCE:

V = L

where v = induced voltage

= rate of change of current

L = self inductance

dt

di

dt di


2/41

UNIT OF SELF INDUCTANCE IS Henry

INDUCTANCE CAN ALSO BE EXPRESSED AS:

Here L represents the magnitude of selfinductance

dt

d N

dt

di L

di

d N L


3/41

MUTUAL INDUCTANCE


4/41


5/41

The induced voltage across coil 2 is given by

It can also be expressed as

Where M = proportionality constant termedas mutual inductance between two coils

dt

d N v L 1222

dt di

M v L1

2


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dt d

N dt di

M 1221

1

122 di

d N M

2

211 di

d NMsimilarly


7/41

COEFFICIENT OF COUPLING:It is defined as the fraction of total flux thatlinks the coil

i.e.K = 12

1

21 2

M 2

N 1 N 2 21

i2

12i1

N 1 N

2

K 2i1

K 1i

2


8/41

K 2

N 1 1

i1

N 2 2

i2

M K 2 L1 L2

Q L N i

K

M

L1 L

2


9/41

ANALYSIS OF COUPLED CIRCUITS:


10/41

How to determine the sign of

mutual emf?

121

111 vdt di M

dt di L Ri

012

222

dt di

M dt di

L Ri


11/41

In the above equation the sign

of the mutual emf is positive


12/41

121

111 ? v

dt di M

dt di L Ri

0? 1222 dt di

M dt

di L Ri


13/41

The sign of mutual emf in aboveequation is


14/41

Equations in time domain form:(instantaneous form):

012222

1

21

111

dt

di M

dt

di L Ri

vdt di

M dt di

L Ri


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Equations in frequency domain form:

I 1 R1 j L1 I 1 Mj I 2 v1 I 1( R1 j L1) j MI 2 v1

I 2 R2 j L2 I 2 Mj I 1 0 I 2 ( R2 j L2 ) Mj I 1 0


16/41

Write the mesh equations in terms

of time and frequency domain


17/41

The equations are

vdt

di

M dt

di

L Ri

vdt

di M

dt

di L Ri

12222

21111


18/41

0)()()(

)()(

112212

1122

222

2211121

Rii

dt di

M dt

iid L

dt iid

M dt di

L Ri

vdt di

M dt iid

L Rii


19/41

Dot rule for coupled circuits :


20/41

Dotted equivalent is:


21/41

Dot Rule: When both assumed currents enter or leavea pair of coupled coils at the dotted

terminals, the signs of the M terms will bethe same as the signs of the L terms.If one current enters at a dotted terminal

and one leaves by a dotted terminal, thesigns of the M terms are opposite to thesigns of the L terms


22/41

M and L terms have the ----------------- sign


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26/41

Draw the dotted equivalent and

obtain I1

and I2

0010


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28/41

Series aiding of inductor:Leq =L1+L2+2M


29/41

Series opposing of inductor:Leq =L1+L2-2M


30/41

Parallel aiding of inductor:

M L L M L L

Leq 221

221


31/41

Parallel opposing of inductors:

M L L

M L L Leq

221

221


32/41

Single tuned coupled circuit:

11 L j R


33/41

Loop 1:

v MI j L j R I

v MI j I L j R I

2111

21111


34/41

Loop 2:

01

0

2221

1222

22

C L j R I MI j

MI j I L jC

jI R I


35/41

02

1

22

1 v I I

jX R M j M j R

C L X

122


36/41

22221

22221

220

22221

2

22221

1

2

)(

1)(

)(

0

M jX R RC MV

c j M jX R R MV j

C j I

C j

I v

M jX R R

MV j I

M jX R R

M j

v R

I


37/41

Amplification factor

When the secondary is tuned, X 2 =0

A resonance

222210

)( M jX R RC M

V V A

C L 1

2

22

021 M R RC

M


38/41

21Max0

0

21

20

2122

021Resonance0

2

R M

output,maximumFor

V

R RC V V

R

M M

R RC

V

M R RC

MV

o


39/41

DOUBLE TUNED COUPLED CIRCUIT:


40/41

222

1

C L X

111

1C

L X Let

0

V

I

I

jXR M j

M j jXR

2

1

22

11


41/41

222211

21

2 ))((

0

M jX R jX R

M j

V jX R

I

2222112

))(( M jX R jX R

MV j I

]))([( 2222112

2

2220

M jX R jX RC MV

C j I

X I V C

Coupled Circuits

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