Resonance in electrical circuits : Deriving the frequency equation for series resonance

RLC circuit

The current in a series RLC circuit varies with frequency as shown in the graph

I_o is the current at the resonating frequency and I_o/√2 is the current at the lower and higher cut-off frequency and it is 0.707 of I_o value. Lower and higher cut-off frequencies are important because bandwidth is defined in between these frequencies.

Current is calculated as

At critical frequency current is

Combining these two we get

at resonance impedance of the circuit is simply R. Now writing I_o =V/R and substituting value of Z

Cancelling V from both sides and taking reciprocal on both sides

To get rid of j (equivalent i of imaginary number), write the right hand side expression in terms of it's magnitude form

square on both sides 

and simplify

Now square root both sides

Substitute in value for inductive and capacitive impedance in terms of omega,L,C

Further simplify

This is a quadratic equation in ω.

Because of +- sign there are two different quadratic equations. So considering one at a time

Now calculating roots of the above quadratic equation

Now here there are two possible roots. Since frequency cannot be negative so considering positive root.

lets call this ω1 (lower cut-off frequency)

Now considering the second quadratic equation

calculating roots

Now just for observation if you factor out R/4L form inside the squared root, we can see that the term is actually greater than R/2L term, so this term cannot be negative as will make our frequency negative which is fundamentally wrong. 

So choosing the positive root

Lets call this ω2 (higher cut-off frequency)

We can write both the roots together as