PHY a.c

In an LCR series circuit, the total impedance across the circuit is given by, Z

= √[R^2 + (XL - XC)^2]

where XL is the inductive impedance, XL = ωL = 2πfL

XC is the capacitance impedance, XC = 1 / ωC = 1 / (2πfC)

So, Z can be expressed as

= √[R^2 + (2πfL - 1 / (2πfC))^2]


Now, as the impedance across the circuit are the same at f = 25 Hz and 225 Hz

Therefore, we have:

√[R^2 + (2π(25)L - 1 / (2π(25)C))^2]

= √[R^2 + (2π(225)L - 1 / (2π(225)C))^2]
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2π(25)L - 1 / (2π(25)C) = 2π(225)L - 1 / (2π(225)C)

or 2π(25)L - 1 / (2π(25)C) = -[2π(225)L - 1 / (2π(225)C)]

2π(225)L - 2π(25)L = 1 / (2π(225)C) - 1 / (2π(25)C)

or 2π(225)L + 2π(25)L = 1 / (2π(225)C) + 1 / (2π(25)C)

400πL = -4/(225πC) (rejected)

or 500πL = 1/(45πC)
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So, LC = 1/(22 500π^2)

22 500 = 1/(π^2LC)

150 = 1/π√(LC)

75 = 1/ 2π√(LC)


Now, as resonant frequency is given by f = 1/ 2π√(LC)

Therefore, we know that the resonant frequency of the circuit is f = 75 Hz.