Question

Obtain the expression for the resonant frequency of the LCR series circuit and explain electrical resonance in an LCR series circuit.

Answer 1

1. A circuit in which inductance L, capacitance C, and resistance R are connected in series and the circuit admits maximum current corresponding to a given frequency of AC, is called a series resonance circuit.
   
   
2. The impedance (Z) of an LCR circuit is given by,
   Z = `sqrt("R"^2 + (omega"L" - 1/(omega"C"))^2`
3. At very low frequencies, inductive reactance X_L = ωL is negligible but capacitive reactance X_C = `1/(omega"L")` is very high. 
4. As we increase the applied frequency then X_L increases and X_C decreases. 
5. At some angular frequency (ω_r), X_L = X_C  
   i.e., ω_rL = `1/(omega_"r""C")`
   ∴ `(ω_"r")^2 = 1/"LC" "or"  (2pi"f"_"r")^2 = 1/"LC"`
   ∴ `2pi"f"_"r" = 1/sqrt("LC")`
   ∴ f_r = `1/(2pisqrt"LC")`
   Where f_r is called the resonant frequency.
6. At this particular frequency f_r, since X_L = X_C we get Z = `sqrt("R"^2 + 0)` = R.
   This is the least value of Z.
7. Thus, when the impedance of an LCR circuit is minimum, the circuit is said to be purely resistive, current and voltage are in phase, and hence the current `"i"_0 = "e"_0/"Z" = "e"_0/"R"` is maximum. This condition of the LCR circuit is called resonance condition and this frequency is called series resonant frequency.