Given the parallel RLC circuit in Figure P7.38, (a) Derive...

Given the parallel RLC circuit in Figure P7.38, (a) Derive the expression for the resonant frequency, the half-power frequencies, the bandwidth, and the quality factor for the transfer characteristic $\mathbf{V}_{\text {out }} / \mathbf{I}_{\text {in }}$ in terms of the circuit parameters $R, L$, and $C$. (b) Compute the quantities in part (a) if $R=1 \mathrm{k} \Omega, L=10 \mathrm{mH}$, and $C=100 \mu \mathrm{F}$. Figure P7.38 can't copy

Given the parallel RLC circuit in Figure P7.38,
(a) Derive the expression for the resonant frequency, the half-power frequencies, the bandwidth, and the quality factor for the transfer characteristic $\mathbf{V}_{\text {out }} / \mathbf{I}_{\text {in }}$ in terms of the circuit parameters $R, L$, and $C$.
(b) Compute the quantities in part (a) if $R=1 \mathrm{k} \Omega, L=10 \mathrm{mH}$, and $C=100 \mu \mathrm{F}$.
Figure P7.38 can't copy

Key Concepts

Parallel RLC Circuits

A parallel RLC circuit consists of a resistor, an inductor, and a capacitor connected in parallel. This configuration is used to study how current divides among the elements and to analyze the frequency-dependent behavior of the circuit. Understanding the impedance and admittance in such circuits is crucial for determining the overall response, particularly near resonance.

Resonant Frequency

Resonant frequency is the specific frequency at which the reactive elements (the inductor and capacitor) cancel each other’s effects, resulting in a circuit that behaves as if it has only the resistive component. In RLC circuits, calculating the resonant frequency involves the inductance and capacitance values, and this frequency determines the point of peak response or minimal impedance.

Quality Factor (Q)

The quality factor, or Q factor, quantifies the sharpness of resonance in the circuit. It is defined as the ratio of the resonant frequency to the bandwidth of the circuit. A high Q indicates a narrow and sharp resonance peak, implying that the circuit stores energy efficiently relative to the rate of energy loss, while a low Q signifies broader resonance and higher damping.

Bandwidth and Half-Power Frequencies

Bandwidth is the range of frequencies over which the circuit effectively responds, typically defined between the two half-power frequencies where the output power is half of its peak value. These half-power points delineate the limits of the circuit’s effective frequency range and are directly related to the Q factor in that a higher Q corresponds to a narrower bandwidth.

Transfer Functions in RLC Circuits

The transfer function describes the relationship between the output and input of a circuit as a function of frequency. In the context of an RLC circuit, it is derived from the circuit’s impedance or admittance and is used to analyze how the circuit filters or affects particular frequency components. This analysis is fundamental when designing circuits for specific frequency characteristics, such as filters or oscillators.

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