Example 28.2: You're building a radio transmitter whose...

Example 28.2: You're building a radio transmitter whose frequency is to be determined by an $L C$ circuit. The voltage across the $L C$ circuit will be $480 \mathrm{~V}$ rms, and the maximum allowable current in the $L C$ circuit is $500 \mathrm{~mA} \mathrm{rms}$. You have available capacitors in the standard values of $1.3 \mathrm{pF}$ and $2.2 \mathrm{pF}$, and inductors of $1.3 \mu \mathrm{H}$ and $2.4 \mu \mathrm{H}$. (a) Find a pair of values for $C$ and $L$ that will keep the current under the $500-\mathrm{mA}$ maximum and will give a frequency in the FM broadcast band, which covers 88 $\mathrm{MHz}$ to $108 \mathrm{MHz}$. (b) What are the values for the corresponding frequency and rms current?

Key Concepts

LC Circuit

An LC circuit is a type of resonant circuit consisting of an inductor (L) and a capacitor (C) that can oscillate at a natural frequency. It uses the exchange of energy between the magnetic field of the inductor and the electric field of the capacitor to sustain oscillations, making it crucial in applications such as filters, oscillators, and tuners.

Resonant Frequency

The resonant frequency of an LC circuit is determined by the relationship f = 1/(2??(LC)). At this frequency, the reactive effects of the inductor and capacitor cancel each other out, leading to a condition where the circuit can either oscillate freely or achieve maximum voltage or current under driving conditions. This concept is fundamental when designing circuits that require precise frequency control.

RMS Values in AC Circuits

RMS, or Root Mean Square, values provide a measure of the effective voltage or current in AC circuits, equivalent to the DC value that delivers the same power to a load. RMS values are used extensively in circuit analysis and design to ensure that systems operate safely within desired power and current limits.

Current Limitation in Reactive Circuits

In reactive circuits involving inductors and capacitors, controlling the maximum current is critical to avoid component damage and ensure proper operation. The impedance of the circuit at the operating (resonant) frequency can be leveraged to limit the RMS current, which is essential when designing circuits that must adhere to specific current constraints.

Standard Component Selection

Electronic components such as capacitors and inductors are often available in standard, commercially available values. When designing a circuit, engineers must select the nearest standard values that satisfy the design criteria, balancing ideal calculated values with practical component availability and tolerances.

Frequency Band Specifications

When designing circuits for communication applications, such as radio transmitters, it is important to ensure that the operating frequency falls within the designated frequency band. Compliance with specific band requirements, like those of the FM broadcast band, is essential for meeting regulatory standards and ensuring proper signal transmission.

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