12.10 SELECTIVITY OF SERIES RLC CIRCUIT WITH
AN L VARIABLE

In certain cases, we feed to a series RLC circuit, a fixed frequency signal, and
it is required to vary the inductance L to obtain the condition of resonance.

In such a case, as L is varied, the current varies as shown in Figure 12.15. The
maximum response is obtained when the value of L is such that the circuit
resonates at the frequency of the applied signal.

Let L_r be the value of L at resonance and let L₁ and L₂ be the values of
L at which the magnitude of reactance equals the resistance R.

Then

and

FIGURE 12.15   Response of a series RLC circuit with an L variable.

At resonance,

Subtracting Equation (12.48) from Equation (12.49), we get

or

(L₂ − L₁) gives the total variation in L in moving from one-half power
condition to the other. The quantitythus gives the selectivity of the
series RLC circuit with the L variable and this equals , as may be seen
from Equation (12.51).

Figure 12.16 gives curves showing a variation of current I, voltage V₁
across inductor L, and voltage Vc across capacitor C as L is varied. Voltage
V_c, being equal to , varies in the same manner as current I and becomes
maximum at resonance, i.e., when L = L_r.

At resonance,

FIGURE 12.16   Variation of current I and voltages V_c and V_l with a variation of L.