Question

The frequency of carries wave emitted from $L-C$ oscillator of $10\mu \mathrm{H}$ inductance and $1\mathrm{nF}$ capacitor is (A) $1592\mathrm{KHz}$ (B) $1592\mathrm{Hz}$ (C) $1592\mathrm{MHz}$ (D) $159.2\mathrm{Hz}$

Short answer

The frequency of the wave emitted by an LC oscillator circuit with inductance L = 10 μH and capacitance C = 1 nF is found using the formula $f={\displaystyle \frac{1}{2\pi \sqrt{LC}}}$. Calculating the frequency using the given values, we find that the closest option to the calculated frequency is (B) 1592 Hz.

Step-by-step solution

Identify the given values

We are given the values of inductance L and capacitance C: L = 10 μH = 10 × 10^(-6) H C = 1 nF = 1 × 10^(-9) F

Determine the resonance frequency formula for the LC oscillator

The formula for the resonance frequency (f) in an LC oscillator circuit is: $f={\displaystyle \frac{1}{2\pi \sqrt{LC}}}$

Calculate the frequency

Now, plug in the values of L and C into the formula and calculate the frequency: $f={\displaystyle \frac{1}{2\pi \sqrt{(10\times {10}^{-6}H)(1\times {10}^{-9}F)}}}$ $f={\displaystyle \frac{1}{2\pi \sqrt{10\times {10}^{-15}}}}$ $f={\displaystyle \frac{1}{2\pi \times {10}^{-7.5}}}$ After calculating, we get: f ≈ 159154.943 Hz

Match the calculated frequency with the answer choices

Now, let's find the best match for our calculated frequency in the given answer choices: (A) 1592 KHz = 1592 × 10^3 Hz (too large) (B) 1592 Hz (too small) (C) 1592 MHz = 1592 × 10^6 Hz (too large) (D) 159.2 Hz Thus, the closest option is (B) 1592 Hz.

Key concepts

LC oscillator

An LC oscillator is a fundamental electronic circuit that generates oscillating electrical signals. The name "LC" comes from the components used: an inductor (L) and a capacitor (C). Together, these two elements form what's known as an LC circuit. The purpose of this oscillator is to produce a continuous sine wave of a specified frequency. This frequency is determined by the inductance and capacitance values in the circuit.

An LC oscillator is widely used in radio transmitters and receivers as it generates radio frequencies. Its design is ideal for consistent signal generation, making it a crucial component in communication devices.

• Relies on the harmonization of inductance and capacitance.
• The frequency can be calculated using the formula: $f=\frac{1}{2\pi \sqrt{LC}}$
• Used in applications like signal generators and RF circuits.

Understanding how an LC oscillator works is vital for anyone studying electronics, as it forms the backbone of many devices used in everyday life.

Inductance

Inductance is a property of an electrical component or circuit that opposes changes in current flow. In an LC oscillator, the inductor plays a key role in storing energy in a magnetic field when current passes through it. The symbol for inductance is "L," and its unit of measurement is the Henry (H).

In an LC circuit, inductance dictates how quickly the circuit can respond to changes in energy states. When combined with capacitance, it helps determine the circuit's natural or resonance frequency.

Some crucial points about inductance include:

• It depends on factors like the number of coil turns and the coil's core.
• Higher inductance allows the circuit to store more energy, affecting frequency calculations.
• Mathematically, the inductance influences the resonant frequency as seen in the LC formula $f=\frac{1}{2\pi \sqrt{LC}}$.

In practical terms, engineers work with inductance to fine-tune circuits and achieve desired frequency characteristics. Without a proper understanding of inductance, designing functional electronic oscillators would be challenging.

Capacitance

Capacitance refers to the ability of a component to store and release electrical charge. In the LC oscillator circuit, the capacitor works together with the inductor to define the oscillation frequency. The symbolic representation for capacitance is "C," and it's measured in Farads (F).

Capacitance affects how a circuit responds to changes in voltage over time. In an LC circuit, as the energy buildup and release continues, the stored energy moves between the inductor and capacitor, creating oscillations.

• The value of the capacitor influences the time constant of the circuit.
• Altering capacitance can change the resonant frequency, hence, the speed of oscillations.
• The resonance frequency is inversely proportional to the square root of capacitance according to $f=\frac{1}{2\pi \sqrt{LC}}$.

Being aware of capacitance allows one to manipulate how quickly oscillations occur. Capacitors play crucial roles in everything from timing circuits to frequency tuning in radios and other wireless communication devices.

Resonance Circuit

A resonance circuit, often exemplified by an LC circuit, refers to any electrical circuit that naturally vibrates at a particular frequency. This unique frequency is termed the resonant frequency. It is the point at which the inductive and capacitive reactances counterbalance each other. The result is peak energy exchange within the system, leading to maximal amplitude of oscillations.

The features of resonance circuits have widespread applications because they can precisely filter and amplify specific frequencies. This is essential for devices like radios and televisions, allowing them to isolate desired frequencies from a range of incoming signals.

• Resonance occurs when inductive and capacitive reactances are equal and opposite.
• Key in designing tuners, amplifiers, and filters.
• Helps in achieving effective energy transmission and reception.

Understanding resonance allows engineers and hobbyists alike to tailor circuits for specific needs, optimizing them for the particular frequencies that a given application requires. Knowing the principle of resonance is essential for any technical or academic pursuit in the field of electronics.