**Doubled Tuned Coupled Circuits:**

Figure 10.24 shows a Doubled Tuned Coupled Circuits involving two series resonant circuits.

For the circuit shown in the figure, a special case where the primary and secondary resonate at the same frequency ω_{r}, is considered here,

i.e.

The two mesh equations for the circuit are

From which

or

where A is the amplification factor given by

The maximum amplification or the maximum output voltage can be obtained by taking the first derivative of υ_{o} with respect to M, and equating it to zero.

where M_{c} is the critical value of mutual inductance. Substituting the value of M_{c }in the equation of υ_{o} we obtain the maximum output voltage as

By definition, M = K√L_{1}L_{2}, the coefficient of coupling, K at M = M_{c }is called the critical coefficient of coupling, and is given by K_{c} = M_{c}/√L_{2}L_{1}.

The critical coupling causes the secondary current to have the maximum possible value. At resonance, the maximum value of amplification is obtained by changing M; or by changing the coupling coefficient for a given value of L_{1} and L_{2}. The variation of output voltage with frequency for different coupling coefficients in Doubled Tuned Coupled Circuits is shown in Fig. 10.25.