# Coupling cofficient of coil in Resonant Series RLC Circuit

In the above diagram the circuit is in Series Resonance. Now we are asked to find the coupling coefficient K between coils L1 and L2. For which i first calculated L(equivalent) as follows :

Leq = L1 + L2 + 2M ( Dots are in aiding position)
Leq = 12 + 3 + 2M (Assuming w = 1)
Leq = 15 + 2M (M denotes mutual inductance between two coils)


So the circuit becomes

Since The circuit is in resonace the XL should equal to Xc

  Xc = XL
-jw18 = jw(15+2M)
-18  = 15 + 2M ( w = 1)
-16.5 = M


Now, To find coupling coefficient K, we can use the follow formula

K = M / sqrt(L1*L2)
K = -16.5/ sqrt(12*3) = -2.75


Which is not possible, Since Range of k should be 0 < K < 1. Where am i wrong ? Please Help...

How did I get j27 ohms - it's a sanity check - reactance or inductance is proportional to turns squared and if coupling is 100% then the turns for L1 and L2 are perfectly coupled and can be regarded as being on the same former so, new impedance is $j(\sqrt{12} + \sqrt{3})^2$ = j27 ohms.