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In a series RCL circuit, the generator is set to a frequency that is not the resonant frequency. This non-resonant frequency is such that the ratio of the inductive reactance to the capacitive reactance of the circuit is observed to be 5.68. The resonant frequency is 240 Hz. What is the frequency of the generator?

Answer :

Answer:

Explanation:

Resonant frequency is 240

4π² x 240² = 1 / LC

230400π² = 1 / LC

Let the required frequency = n

inductive reactance = 2 πn L

capacitative reactance = 1 / 2 π n C

inductive reactance / capacitative reactance

= 4π² x n ² x LC = 5.68

4π² x n ² = 1 / LC x 5.68

= 230400π² x 5.68

4n ²= 230400 x 5.68

n ²= 57600 x 5.68

n ² = 327168

n = 572 approx

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Rewritten by : Barada

The generator's frequency in this RLC circuit, where the ratio of inductive to capacitive reactance is 5.68 and the resonant frequency is 240 Hz, is approximately 571 Hz.

To solve this problem, we need to use the relationship between inductive and capacitive reactance in an RLC circuit. Given that the ratio of the inductive reactance (XL) to the capacitive reactance (XC) is 5.68,

  • X_L / X_C = 5.68

Reactance is frequency-dependent, with the following formulas for inductive and capacitive reactance:

  • X_L = 2πfL
  • X_C = 1 / (2πfC)

Given the resonant frequency as 240 Hz, we start by calculating the resonance reactances:

  • At resonance: X_L = X_C, so 2π(240)L = 1 / (2π(240)C)

Using the non-resonant frequency, we use the reactance ratio:

  • 2πfL / (1 / (2πfC)) = 5.68

Rearranging gives us:

  • f² = 5.68 / (2π)²(LC)

Since 240 Hz is the resonant frequency, substituting f:

  • f² = 5.68 * (240)²

Simplifying yields:

  • f = 240 * √(5.68)
  • f ≈ 571 Hz

Therefore, the generator frequency is approximately 571 Hz.