## 17 January 2018

### 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
e2 = M di1 / dt
• 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 ( di1 / dt ) + M ( di2 / dt )
e2 = L2 ( di2 / dt ) + M ( di1 / 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 )