17 January 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 )


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 )