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A series RLC circuit has a resonant frequency = 6.00 kHz. When it is driven at a frequency = 8.00 kHz, it has an
impedance = 1000 W and a phase constant = 45°. Calculate R and L for this circuit.

Sagot :

onyebuchinnaji

The resistance (R) of the circuit is 707.1 ohms and the inductance (L) is 0.032 H.

Resistance of the circuit

For the phase constant of 45⁰, impedance is equal to the resistance of the circuit.

[tex]Z= R\sqrt{2} \\\\R = \frac{Z}{\sqrt{2} } \\\\R = \frac{1000}{\sqrt{2} } = 707.1 \ ohms[/tex]

Resonant frequency

[tex]f = \frac{1}{2\pi \sqrt{LC} } \\\\6000 = \frac{1}{2\pi \sqrt{LC} } \\\\2\pi(6000) = \frac{1}{\sqrt{LC} } \\\\\sqrt{LC} = \frac{1}{2\pi (6000)} \\\\LC = (\frac{1}{2\pi (6000)})^2\\\\LC = 7.034 \times 10^{-10} \\\\ C = \frac{7.034 \times 10^{-10} }{L} ---(1)[/tex]

At driven frequency

[tex]X_l- X_c = R\\\\\omega L - \frac{1}{\omega C} = 707.1\\\\2\pi f L - \frac{1}{2\pi fC} = 707.1\\\\2\pi (8000) L - \frac{1}{2\pi (8000) C } = 707.1\ \ --(2)\\\\[/tex]

solve 1 and 2 together

[tex]2\pi(8000) L - \frac{L}{2\pi (8000)(7.034 \times 10^{-10})} = 707.1\\\\50272L - 28279.48L = 707.1\\\\L = 0.032 \ H[/tex]

Learn more about impedance of RLC circuit here: https://brainly.com/question/372577

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