A series RCL circuit contains a resistor of value R, an inductor of value L, and

Question

A series RCL circuit contains a resistor of value R, an inductor of value L, and a capacitor of value C, and operates at frequency f_1, such that one finds the square of the impedence is Z_1^2, the inductive reactance is XL=2ohm and the capacitive reactance is Xc= 6ohm. If the frequency is doubled to f_2=2f_1, and the resistor is removed from the cicruit leaving a series LC circuit, one then finds the square of the impedance, Z_2^2 is 24ohm^2 less than Z_1^2 (i.e. Z_2^2 = Z_1^2 - 24ohm^2). What is the value of the resistor, R?

The answers is 3ohms, but need help getting there

Explanation / Answer

Z_1 = R + j(XL1 - XC1)

Z_2 = j(XL2 - XC2)

With frequency f1, we have XL1 = 2 ohms and XC1 = 6 ohms

By increasing the frequency i.e. doubling the frequency inductive reactance will get doubled and capactive reactance will be halved

Hence XL2 = 2*2 = 4 ohms , XC2 = 6/2 = 3 ohms

Z_2 = j(4 - 3) = j

Z_1^2 = 24 + Z^2 = 25

hence we get

R^2 + |j(2-6)|^2 = 25

R^2 + 16 = 25

R^2 = 9 ohms

Hence we get R = 3 ohms

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