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 5.53. The resonant frequency is 220 Hz. What is the frequency of the generator?
Xl/Xc=2PIfl*2PI*f*C= (2PI f)^2 LC
but 220=1/2pi sqrt LC
or LC=1/(2PI*220)^2
so we now have
5.53=(2PIf)^2/(2PI*220)^2 = f^2/220^2
solve for f.
To find the frequency of the generator in a series RCL circuit, we need to use the given information about the ratio of inductive reactance to capacitive reactance.
In a series RCL circuit, the total impedance is given by:
Z = √((R^2 + (Xl - Xc)^2))
Where:
Z is the total impedance
R is the resistance
Xl is the inductive reactance
Xc is the capacitive reactance
At resonance, the inductive reactance and capacitive reactance cancel each other out, so Xl - Xc = 0.
Given that the resonant frequency is 220 Hz, we can calculate the inductive reactance (Xl) and the capacitive reactance (Xc) using the following formulas:
Xl = 2πfL
Xc = 1 / (2πfC)
Where:
f is the frequency
L is the inductance
C is the capacitance
Now, let's substitute the known values and solve for the ratio of inductive reactance to capacitive reactance:
5.53 = Xl / Xc
Substituting the formulas for Xl and Xc:
5.53 = (2πfL) / (1 / (2πfC))
Simplifying the equation:
5.53 = 4π^2f^2LC
Now, we have an equation in terms of f (frequency), L (inductance), and C (capacitance).
Since the resonant frequency is given as 220 Hz, we can substitute this value and solve for the unknowns:
5.53 = 4π^2(220)^2LC
Simplifying the equation further and rearranging for LC:
LC = (5.53 / (4π^2(220)^2))
Now, you will need to know the values of L and C in order to calculate the LC product. Once you have those values, substitute them and solve for LC.
Once you have the value of LC, you can substitute it back into the earlier equation in terms of f, L, and C to calculate the frequency of the generator:
5.53 = 4π^2f^2LC
Solving for f:
f^2 = (5.53 / (4π^2LC))
Taking the square root:
f = √(5.53 / (4π^2LC))
Substituting the LC value you calculated earlier, you can compute the frequency (f) of the generator.