Question

In a series, LCR circuit, obtain an expression for the resonant frequency,

Derive an expression for resonant frequency of series resonant circuit.

Answer 1

In a series LCR circuit,

I = `E/sqrt(R^2+(omega L - 1/(omega c))^2`

From this equation, we may observe that when ω = 0, I becomes 0, and again when ω = ∞, I = 0.

∴ This implies there must be a value of ω at which I is maximum.

So when I is maximum, `sqrt(R^2 + (omega L - 1/(omega c))^2` is minimum.

For this either R = 0, or `omega L - 1/(omega c)` = 0

So when `omega L -1/(omega c)` = 0,  this is called the resonance condition.

∴ At resonance,

ωL = `1/(omega c)`

⇒ ω² = `1/(L c)`

⇒ ω_r  = `1/sqrt(L c)`

⇒ 2 π f_r = `1/sqrt(L c)`

∴ Frequency of resonance (f_r) = `1/(2 pi) * 1/sqrt(L c)`

i.e., f_r = `1/(2 pi) * 1/sqrt(L c)`

This is the required expression for resonant frequency.