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Question
Obtain the expression for the resonant frequency of the LCR series circuit and explain electrical resonance in an LCR series circuit.
Solution
- 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.
- The impedance (Z) of an LCR circuit is given by,
Z = `sqrt("R"^2 + (omega"L" - 1/(omega"C"))^2` - At very low frequencies, inductive reactance XL = ωL is negligible but capacitive reactance XC = `1/(omega"L")` is very high.
- As we increase the applied frequency then XL increases and XC decreases.
- At some angular frequency (ωr), XL = XC
i.e., ωrL = `1/(omega_"r""C")`
∴ `(ω_"r")^2 = 1/"LC" "or" (2pi"f"_"r")^2 = 1/"LC"`
∴ `2pi"f"_"r" = 1/sqrt("LC")`
∴ fr = `1/(2pisqrt"LC")`
Where fr is called the resonant frequency. - At this particular frequency fr, since XL = XC we get Z = `sqrt("R"^2 + 0)` = R.
This is the least value of Z. - 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.
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