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Question
In a series, LCR circuit, obtain an expression for the resonant frequency,
Solution
In a series LCR circuit,
`I=E/sqrt(R^2+(ωL-1/(ωc))^2`
From this equation, we may observe that when ω=0, I become O, 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+(ωL-1/(ωc))^2` is minimum
For this either R = 0, or ωL -` 1/(ωc) =0`
So when ωL -`1/(ωc)=0`, this is called resonance condition.
∴ At resonance,
ωL`=1/(ωc)`
⇒ `ω^2 =1/(Lc)`
⇒ `ω_r = 1/sqrt(Lc)`
⇒ `2pif_r = 1/sqrt(Lc)`
∴ `f_r` = Frequency of resonance
`=1/(2pi). 1/sqrt(Lc)`
i.e., `f_r =1/(2pi) . 1/sqrt(Lc)`
This is the required experssion for resonant frequency.
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