A series RCL circuit is constructed with an AC generator who

A series RCL circuit is constructed with an AC generator whose frequency can be adjusted without changing its rms voltage. The circuit\'s resonant frequency is between 100 Hz and 500 Hz. The resistor\'s resistance is chosen to be smaller than the inductive and capacitive reactances at the resonant frequency (R < XC = XL when f = f0). The generator is initially set to a frequency f1 that\'s at least 50 Hz below f0, and the impedance, rms current, and average power output are measured. Then the generator is set to a frequency f2 that\'s as far above f0 as f1 was below it, and the same measurements are repeated. (For example, if f0 = 400 Hz, the first frequency might be f1 = f0 - 80 Hz = 320 Hz, and the second frequency would then be f2 = f0 + 80 Hz = 480Hz.) Question: given that f1 and f2 are equally \"distant\" from the resonant frequency f0, where the average power output peaks, are the circuit\'s impedance, rms current, and average power output the same at both frequencies?

Solution

Here f1 and f2, are said to be half power frequencies whose magnitude of power at this frequencies is exactly the Half the magnitude at resonant frequency.

Moreover resonant frequency fo=square root of (f1f2)

Magnitude at half power is given by

P/2= VI/2

=(V/ sqt2) (I/sqt 2)

At f2-f1 is known as Bandwidth. Between which signal is transferred.

MOREOVER RESONANT FREQUENCY FO = GEOMENTRIC MEAN OF F1F2.