In a series resonant circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), the half-power frequencies are the frequencies at which the power dissipated in the circuit is half of the maximum power dissipated. These frequencies are also known as the bandwidth frequencies and are important in understanding the behavior of resonant circuits.
The resonant frequency (fâ) of the circuit, where the impedance is purely resistive (minimum), can be calculated using the formula:
fâ = 1 / (2 * Ď * â(L * C))
Where:
fâ is the resonant frequency in Hertz (Hz)
Ď is the mathematical constant pi (approximately 3.14159)
L is the inductance of the coil in Henrys (H)
C is the capacitance of the capacitor in Farads (F)
For a series resonant circuit, the impedance is given by:
Z = R + j(Xl - Xc)
Where:
Z is the total impedance
R is the resistance in Ohms (Ί)
j is the imaginary unit (â(-1))
Xl is the inductive reactance in Ohms (Ί)
Xc is the capacitive reactance in Ohms (Ί)
At the resonant frequency (fâ), the inductive reactance (Xl) is equal to the capacitive reactance (Xc), and the total impedance (Z) is purely resistive (minimum).
Now, to find the half-power frequencies, you need to consider the points on the frequency response curve where the power dissipated is half of the maximum power dissipated. At these frequencies, the impedance will be the square root of two times the resistance (R) since power is proportional to the square of the current, which in turn is inversely proportional to impedance.
So, the expression for the half-power frequencies (fâ and fâ) can be given as:
fâ = fâ * (1 - 1 / â2)
fâ = fâ * (1 + 1 / â2)
Where:
fâ is the resonant frequency
fâ is the lower half-power frequency
fâ is the upper half-power frequency
â2 is the square root of 2 (approximately 1.414)
These expressions give you the frequencies at which the power dissipated in the RLC series resonant circuit is half of the maximum power dissipated. The bandwidth of the circuit is given by the difference between fâ and fâ:
Bandwidth = fâ - fâ
Understanding these expressions helps in designing and analyzing resonant circuits for various applications.