It is expected that a parallel R L C resonant circuit has a...

It is expected that a parallel $R L C$ resonant circuit has a midband admittance of $25 \times 10^{3} \mathrm{S},$ quality factor of $80,$ and a resonant frequency of $200 \mathrm{krad} / \mathrm{s}$ Calculate the values of $R, L,$ and $C .$ Find the bandwidth and the half-power frequencies.

Key Concepts

Parallel RLC Circuit

A parallel RLC circuit is a configuration where a resistor, an inductor, and a capacitor are connected in parallel. This arrangement exhibits resonant behavior due to the interplay between the inductive and capacitive reactances. The circuit has unique characteristics such as a frequency at which the reactive components cancel each other, leaving only the resistive part to determine the overall circuit response.

Resonant Frequency

The resonant frequency in an RLC circuit is the frequency at which the inductive and capacitive reactances are equal in magnitude but opposite in phase, causing them to cancel out. This results in a condition where the circuit’s admittance (or impedance) is purely resistive. It is a key parameter for tuning circuits to operate at a desired frequency and is critical in applications like filters and oscillators.

Quality Factor

Quality factor, often denoted as Q, measures the sharpness or selectivity of the resonance in a circuit. In a parallel RLC circuit, a high Q indicates a low rate of energy loss relative to the stored energy, leading to a narrow bandwidth around the resonant frequency. Q is used to estimate the bandwidth and determine how quickly the response falls off away from resonance.

Admittance

Admittance, the reciprocal of impedance, quantifies how easily a circuit allows current flow when a voltage is applied. Measured in siemens (S), it is an important parameter in analyzing AC circuits, where both conductance (real part) and susceptance (imaginary part) play roles. In resonant circuits, the midband admittance typically reflects the resistive behavior at resonance.

Bandwidth

The bandwidth of a resonant circuit is defined as the range of frequencies over which the circuit’s response remains within a specified level, typically provided by the half-power points. It is directly related to the quality factor; specifically, for a resonant circuit, the bandwidth is inversely proportional to the Q. A higher Q indicates a narrow bandwidth for energy storage and filtering applications.

Half-Power Frequencies

Half-power frequencies, also known as the -3 dB points, are the frequencies at which the power delivered by the circuit falls to half of its maximum value at resonance. These frequencies delineate the limits of the effective bandwidth of the circuit. In a resonant circuit analysis, identifying these frequencies helps in understanding the selectivity and performance of filters and amplifiers.

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