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Calculate the resultant inductance of tw...

Calculate the resultant inductance of two inductor `L_(1)` and `L_(2)` when they are connected in parallel in A.C. circuit.

Text Solution

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Let `Z_(L_(1))` and `Z_(L_(2))` be the inductive reactance of two coils. Since they are connected in parallel, the resultant reactance will be,
`Z= ( Z_(L_(1)) Z_(L_(2)))/( L_(L_(1))+ Z_(L_(2)))= ( j X_(L_(1)) xx jX_(L_(2)))/(jX_(L_(1))+jX_(L_(2)))`
`:. Z = (jX_(L_(1))X_(L_(2)))/(X_(L_(1))+ X_(L_(2)))` .....(i)
If the resultant inductance is equal to L, then `Z = j omega L`.
`X_(L _(1) ) = omegaL_(1)` and `X_(L_(2)) = omega L_(2)` then
Putting this values in equation (1).
`:. j omega L = ( j omega^(2) L_(1) L_(2))/( omega L_(1) + omegaL_(2))`
`:. L = ( L_(1) L_(2))/( L_(1) + L_(2))`
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