Derive an expression for resonant frequency of series circuit containing inductor, capacitor and resistor.

Explanation

Series LCR circuit is said to be in resonance when current through the circuit is maximum. In a series LCR circuit current amplitude is given by
`i_(m)=(V_(m))/(Z)`
`i_(m)=(V_(m))/(sqrt(R^(2)+(X_( C )-X_(L))^(2)))`
where `X_( C )=(1)/(omegaC)` and `X_(L)=omega_(L)`
If frequency is varied, at particular angular frequency `omega_(0)` the condition
`X_( C )=X_(L)` is achieved, this condition is called resonance, `(1)/(omega_(o)C)=omega_(0)L`
`:." "omega_(0)=(1)/(sqrt(LC))`
`2pi v_(o)=(1)/(sqrt(LC))" "(because omega_(0)=2pi v_(0))`
`v_(0)=(1)/(2pi sqrt(LC))` is called resonant frequency