### Coupling

- When two coils ( or more than two coils ) are connected by common magnetic flux, they are called as coupled with each other.

### Mutual coupling

- The circuit element used to represent magnetic coupling is known as Mutual coupling.
- It is donated by symbol M and its unit is henry.
- The induced emf in the second coil due to current flows through one coil is related by

e

_{2}= M di_{1}/ dt- The dot sign indicates direction of current in the coil.
- If the current flows through both coil A and coil B, the induced emf in the two coils A and coil B is

e1 = L1 ( di

_{1}/ dt ) + M ( di_{2}/ dt )
e2 = L2 ( di

_{2}/ dt ) + M ( di_{1}/ dt )### Inductance are in series : Series aiding

- Figure B ( A ) shows two coils are connected in series such that the current enter the dot end of the coil A whereas it leaves dot end of the other coil B.
- This type of coil connection is called as series aiding connection of coil.
- The total induced emf in coil A and coil B due to self inductance and emf induced due to other coil is given by

e1 = L1 ( di /
dt ) + M ( di / dt )

= [ L1 + M ] ( di / dt ) ……..( 1 )

e2 = L2 ( di /
dt ) + M ( di / dt )

= [ L2 + M ] ( di / dt )…..….( 2 )

- From equation ( 1 ) and ( 2 )

Total
induced emf e = e1 + e2

= [ L1 + L2 + M ] ( di / dt )…..( 3 )

- Now e = L ( di / dt )……………….( 4 )
- From equation ( 3 ) and ( 4 )

L ( di / dt ) =
[ L1 + L2 + M ] ( di / dt )

L = [ L1 + L2 + 2M
]

### Series opposition

- If the coil connection is made such that the current enter the dot end from coil A and enter undotted end of coil B, it is called as series opposition connection.
- Figure B ( B ) shows series opposition connection of two coils. The axes of two inductive coils are in the same straight line.
- The coil connection is made such that mutual induced emf opposes the self inducted emf of the coils.

e1 = L1 ( di /
dt ) – M ( di / dt )

= ( L1 – M ) ( di / dt )………. ( 5 )

e2 = L2 ( di /
dt ) – M ( di / dt )

= ( L2 – M ) ( di / dt )…....… .( 6 )

- From equation ( 5 ) and ( 6 )

Total
induced emf e = e1 + e2

= [ L1 + L2 – M
] ( di / dt )……( 7 )

- Now e = L ( di / dt )…...……….( 8 )
- From equation ( 7 ) and ( 8 )

L = [ L1 + L2 –
M ]

### Two coils are Penpendicular

- If the coils axis are perpendicular to each other, there is no mutual inductance between them therefore total induced emf. Figure C shows two coils are perpendicular.

e = e1 + e2

= L1 ( di / dt )
+ L2 ( di / dt )

= [ L1 + L2 ] (
di / dt )

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