Voltage Vector Diagram:

Resonance condition is obtained while X_L = X_C in the series circuit (it is obtained by either dropping the supply frequency while X_L would decrease and X_C would enhance or by enhancing the supply frequency making X_L to enhance and X_C to decrease).

When   X_L = X_C, Z = R + jO = R

Also,

I₀ = V/ Z = V/ R Amp      [I₀ = Current at resonance]

The p.f. (power factor) of the circuit becomes

cos φ = R/ Z = R/ R = 1

The expression of frequency of resonance may be obtained as follows :

 Let ω₀ or f₀ be the frequency at which X_L = X_C.

   i.e.

                               ω₀ L = ω_o C

or         (ω_o )² = 1/ LC   i.e. ω_o = 1/√(LC)  rad/ sec

i.e.     f₀ =  1     /2π √( LC)  Hz

Voltage vector diagram at resonance is illustrated in Figure.

Figure: Voltage Vector Diagram at Resonance (V_L = V_c)