A stereo receiver is tuned to 98 mHz on the FM band. The tuning knob controls a variable capacit

step 1 Hello friends here it is given. FM radio station having the broadcast at frequency 100 megahertz

A stereo receiver is tuned to 98 mHz on the FM band. The...

A stereo receiver is tuned to $98 \mathrm{mHz}$ on the FM band. The tuning knob controls a variable capacitor in a parallel resonant circuit. If the inductance of the circuit is $2 \mu \mathrm{H}$ and the $Q$ is 100, determine the values of $C$ and $G$.

Key Concepts

Resonant Frequency in LC Circuits

This concept refers to the natural frequency at which an LC (inductor?capacitor) circuit oscillates. It is given by the formula ?? = 1/?(LC) (or f? = 1/(2??(LC)) when using frequency), which directly relates the inductance and capacitance of the circuit. This expression is fundamental for tuning circuits such as those found in radio receivers, where adjusting the capacitor alters the resonant frequency.

Quality Factor (Q) in Resonant Circuits

The quality factor, Q, quantifies the selectivity or sharpness of the resonance. In resonant circuits, a high Q indicates a narrow bandwidth and low energy losses relative to the stored energy, while a low Q represents a broad resonance with high losses. For a parallel resonant circuit, Q can be expressed in terms of the circuit's resistance (or its reciprocal, conductance), inductance, and capacitance, thereby linking the stored energy to the dissipative elements.

Determining Component Values in Parallel Resonant Circuits

By combining the resonant frequency formula with the definition of the quality factor, one can determine unknown components such as the capacitance needed for a desired resonant frequency and the conductance (the inverse of the effective resistance) required to achieve a specified Q. This method of calculating component values is essential for the design and tuning of circuits in applications like radio receivers, where precision and stability are critical.

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