12.9 SELECTIVITY OR SERIES RLC CIRCUIT WITH A
CAPACITANCE VARIABLE

In certain cases, we feed to a series RLC circuit a fixed frequency signal and
it is required to vary the capacitance C to obtain the condition of resonance.
In such a case, as C varies, the response current in the circuit varies as shown
in Figure 12.13. The maximum response is reached when the value of C is
such that the circuit resonates at the frequency of the applied signal.

FIGURE 12.13   Response of a series RLC circuit with a C variable.

Let C₀ be the value of C at resonance and let C₁ and C₂ be the values
of capacitance C at which the magnitude of reactance
equals the resistance R.

Then

and

At resonance

From Equations (12.40) and (12.41), we get

However, if the circuit is highly selective, then

Hence,

and

For a highly selective circuit, ΔC₁ and ΔC₂ are small and almost equal.
Then

and

Substituting the value of C₁C₂ from Equation (12.45) into Equation
(12.43), we get

Combining Equations (12.46) and (12.42), we get

or

or

(C₂− C₁) gives the total variation in C in moving from one half-power
condition to the other. The quantity thus gives the selectivity of the
series-tuned circuit with C variables and this equals , as may be seen from
Equation (12.47).

Figure 12.14 gives curves showing the variation of current I, voltage V_l
across inductor L, and voltage V_c across capacitor C as C is varied. Voltage
V_l(=ωL · I) varies in the same manner as current I and becomes maximum
at resonance, i.e., when C = C₀. At resonance = V_l= V_c, i.e.,

FIGURE 12.14   Variation of current and voltage V_l and V_c with a variation of C.