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प्रश्न
In a series, LCR circuit, obtain an expression for the resonant frequency.
Derive an expression for resonant frequency of series resonant circuit.
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उत्तर
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)`
⇒ ωr = `1/sqrt(L c)`
⇒ 2 π fr = `1/sqrt(L c)`
∴ Frequency of resonance (fr) = `1/(2 pi) * 1/sqrt(L c)`
i.e., fr = `1/(2 pi) * 1/sqrt(L c)`
This is the required expression for resonant frequency.
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