12.8 SERIES RESONANCE AND SELECTIVITY OF SERIES
RLC CIRCUIT WITH FREQUENCY VARIABLE

The impedance of a series RLC circuit is given by

where ω is any frequency.

For ω close to the resonant frequency ω₀, Equation (12.23) does not
yield very accurate results.

It is desirable in such a case to use a special relation valid only for frequencies
close to the resonant frequency. Thus, Equation (12.26) may be put as,

At resonance,

Hence,

where Q_r is the value of Q at resonance.

Hence, Equation (12.24) may be put as,

The fractional frequency variation is given by

Hence,

Hence, Equation (12.26) may be put as,

If ω is close to ω₀, then δ < 1.

Hence,

and

Hence, Equation (12.28) reduces to,

Hence, the current

Figure 12.12 shows the variation of current I with the frequency as
obtained from Equation (12.29).

At resonance,

At half-power frequencies f₁ and f₂, the magnitude of the reactive component
of impedance equals the resistance, i.e.,

At these half-power frequencies f₁ and f₂, the current

From Equation (12.29), at half-power frequencies,

In Equation (12.31), the positive sign applies for the upper half-power
frequency f₂, while the negative sign applies for the lower half-power
frequency f₁.

FIGURE 12.12   Variation of current with frequency in a series RLC current.

For upper half-power frequency f₂ from Equation (12.31),

Hence,

Similarily,

From Equations (12.32) and (12.33) we see that (f₂ − f_r) = (f_r− f₁), i.e.,
the half-power frequencies f₂ and f₁ are symmetrically disposed with respect
to the resonant frequency f_r. Also the response curve, i.e., current versus
frequency curve, is symmetrical about the resonant frequency f_r.

But Equations (12.32) and (12.33) are true only when δ << 1. This condition
is satisfied only when the circuit is highly selective, i.e., when Q_r is large.
For the low Q circuit, i.e., when R is large, the half-power frequencies f₁ and
f₂ are away from the resonant frequency f_r. The impedance Z is then given by
Equation (12.28) instead of Equation (12.29). Half-power frequencies must
then be calculated from Equation (12.28), i.e., from the following equation,

Thus, for the low selectivity circuit, for example, (f₂− f₀) ≠ (f₀ − f₁), the
response curve is not symmetrical about the resonant frequency.
Referring back to Equations (12.32) and (12.33) for the high Q circuit,
by addition of the equations, we get

Let the half-power bandwidth (f₂− f₁) be indicated by Δf. Then
selectivity

From Equation (12.36), we find for a given value of f , the bandwidth Δf
varies inversely with Q_r. The larger the value of Q_r , the lesser the bandwidth
and more selective is the circuit.

At a lower half-power frequency ω1,

Similarly, at an upper half-power frequency ω₂,

Adding Equations (12.37) and (12.38), we get

or

Hence,

But

Hence,

and we can also write