In a series RCL circuit the generator is set to a frequency that is not the resonant frequency. This nonresonant frequency is such that the ratio of the inductive reactance to the capacitive reactance of the circuit is observed to be 4.91. The resonant frequency is 240 Hz. What is the frequency of the generator?

Question

In a series RCL circuit the generator is set to a frequency that is not the resonant frequency. This nonresonant frequency is such that the ratio of the inductive reactance to the capacitive reactance of the circuit is observed to be 4.91. The resonant frequency is 240 Hz. What is the frequency of the generator?

Answer 1

To find the frequency of the generator in a series RCL circuit, we can use the formula for the impedance of the circuit:

Z = sqrt((R^2) + (Xl - Xc)^2)

Where:
Z is the impedance of the circuit
R is the resistance
Xl is the inductive reactance
Xc is the capacitive reactance

First, let's determine the inductive and capacitive reactances at the resonant frequency. The inductive reactance (Xl) can be calculated using the formula:

Xl = 2πfL

Where:
f is the frequency
L is the inductance of the circuit

Given f = 240 Hz, we need to find L.

Next, let's calculate the capacitive reactance (Xc) using the formula:

Xc = 1 / (2πfC)

Where:
C is the capacitance of the circuit

To find C, we need Xc. We know that the ratio of Xl to Xc at the non-resonant frequency is 4.91, so we can set up the equation:

Xl / Xc = 4.91

Now we have two equations:
Xl = 2πfL
Xc = 1 / (2πfC)

We can solve these equations simultaneously to find L and C.

Once we know L, we can substitute its value back into the equation for Xl to find Xl at the non-resonant frequency.

Finally, we can find the impedance (Z) at the non-resonant frequency using the equation:

Z = sqrt((R^2) + (Xl - Xc)^2)

Since we already know the value of Xl - Xc (from the given ratio), we can rearrange the equation to solve for R:

R = sqrt(Z^2 - (Xl - Xc)^2)

Now we have the resistance R at the non-resonant frequency.

To find the frequency of the generator, we can use the impedance Z and resistance R at the non-resonant frequency in the formula for impedance:

Z = sqrt((R^2) + (Xl - Xc)^2)

Rearranging the equation, we get:

Z^2 - R^2 = (Xl - Xc)^2

We can now substitute the known values of Z, R, Xl, and Xc and solve for f:

f = sqrt((Z^2 - R^2) / (2π))^2

Solving this equation will give us the frequency of the generator.