Question

How do you calculate the resonant frequency and bandwidth of RLC circuits?

Answer 1

To calculate the resonant frequency and bandwidth of RLC circuits, you first need to understand the basics of RLC circuits. An RLC circuit is an electrical circuit that consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series or parallel. The resonant frequency is the frequency at which the circuit exhibits maximum impedance, and the bandwidth is the range of frequencies around the resonant frequency within which the circuit's impedance remains relatively high.

To calculate the resonant frequency and bandwidth, follow these steps:

Resonant Frequency (fr):
The resonant frequency is the frequency at which the reactance of the inductor (XL) equals the reactance of the capacitor (XC). At resonance, these reactances cancel each other out, and only the resistance remains.

The resonant frequency (fr) is given by the formula:
fr = 1 / (2 * π * √(L * C))

where:
fr = Resonant frequency in Hertz (Hz)
π (pi) ≈ 3.14159
L = Inductance in Henrys (H)
C = Capacitance in Farads (F)

Bandwidth (BW):
The bandwidth is the difference between the two frequencies at which the impedance of the circuit is √2 (approximately 1.414) times the value at the resonant frequency (fr). These frequencies are known as the half-power frequencies.

The bandwidth (BW) is given by the formula:
BW = f2 - f1

where:
BW = Bandwidth in Hertz (Hz)
f1 = Lower half-power frequency in Hertz (Hz)
f2 = Upper half-power frequency in Hertz (Hz)

To find the half-power frequencies, you need to calculate the impedances at these frequencies and solve for f1 and f2:

At f1 and f2, the impedance Z is given by:
Z = √(R^2 + (XL - XC)^2)

Now, set Z equal to √2 times the impedance at the resonant frequency (fr):
√2 * Z(fr) = √(R^2 + (XL(fr) - XC(fr))^2)

Solve the above equation for XL(fr) and XC(fr). Then use the formulas for XL and XC at the resonant frequency:

XL(fr) = 2 * π * fr * L
XC(fr) = 1 / (2 * π * fr * C)

Substitute these values back into the equation for Z to find the half-power frequencies (f1 and f2).

Once you have calculated the resonant frequency (fr) and the bandwidth (BW), you can better understand how the RLC circuit behaves at different frequencies and design circuits accordingly.