Question

A parallel RLC resonant circuit has a resistance R=2kΩ and half-power frequencies of 86kHz and 90kHz. Determine: (a) the capacitance C; (b) the inductance L; (c) the resonant frequency; (d) the bandwidth; and (e) the quality factor.

Answer 1

To find the values for a parallel RLC resonant circuit, calculate the resonant frequency, bandwidth, and quality factor using the provided half-power frequencies. Subsequently, use these values to solve for the inductance and capacitance.

Step-by-step explanation:

To solve for the values requested in the RLC resonant circuit, we follow these steps:

Calculate resonant frequency (fr) using the formula fr = ½(f1 + f2), where f1 and f2 are the half-power frequencies.

Calculate the bandwidth (Δf) as the difference between f2 and f1.

Calculate the quality factor (Q) using the formula Q = fr / Δf.

Using Q and R, calculate the inductance (L) with L = R / (2πfrQ).

Finally, calculate the capacitance (C) using the resonant frequency and L with C = 1 / (4π²fr²L).

Let's substitute the given values:

Resonant frequency(fr) = ½(86000 Hz + 90000 Hz) = ½(176000 Hz) = 88000 Hz (88 kHz).

Bandwidth(Δf) = 90000 Hz - 86000 Hz = 4000 Hz (4 kHz).

Quality factor(Q) = 88000 Hz / 4000 Hz = 22.

Inductance(L) = 2000 Ω / (2π * 88000 Hz * 22) = (approximately) 146.45 μH.

Capacitance(C) = 1 / (4π² * (88000 Hz)² * 146.45 μH) = (approximately) 2.06 nF.