17/01/2018

Concept of Mutual coupling and Inductances in Series Connection

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





Inductances are in series : Series aiding

  • Figure 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 )
Now 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 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 )
Now 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 ]
If the coils axis are perpendicular to each other, there is no mutual inductance between them therefore total induced emf
e  = e1 + e2
   = L1 ( di / dt ) + L2 ( di / dt )
   = [ L1 + L2 ] ( di / dt )