Question

The frequency of output signal of LC oscillator circuit is \(100 \mathrm{~Hz}\) with capacitance value \(0.1 \mu \mathrm{F}\). If value of capacitance is taken as \(0.2 \mu \mathrm{F}\), the frequency of output signal (A) decreases by \((1 / \sqrt{2})\) (B) increases by \((1 / \sqrt{2})\) (C) decreases by \((1 / 2)\) (D) increases by \((1 / 2)\)

Short answer

The frequency of the output signal decreases by \((1 / \sqrt{2})\) when the capacitance is doubled to $0.2 \mu \mathrm{F}$.

Step-by-step solution

Write down the given information

The given information is: Initial frequency, \(f_{1} = 100 \text{ Hz}\) Initial capacitance, \(C_{1} = 0.1 \ \mu \text{F}\) New capacitance, \(C_{2} = 0.2 \ \mu \text{F}\) We do not know the value of the inductance 'L', but since it does not change in this problem, we can express the change in frequency as a ratio between the two capacitance values.

Frequency formula for LC circuit

The formula for the resonant frequency of an LC circuit is given by: \[f = \frac{1}{2 \pi \sqrt{LC}}\] In our case, the inductance 'L' is constant, so we can write the initial and new frequencies as: \[f_{1} = \frac{1}{2 \pi \sqrt{LC_{1}}}\] \[f_{2} = \frac{1}{2 \pi \sqrt{LC_{2}}}\]

Find the ratio of the frequencies

Divide the equation for \(f_{1}\) by the equation for \(f_{2}\): \[\frac{f_{1}}{f_{2}} = \frac{\frac{1}{2 \pi \sqrt{LC_{1}}}}{\frac{1}{2 \pi \sqrt{LC_{2}}}} = \frac{\sqrt{LC_{2}}}{\sqrt{LC_{1}}}\] Since \(L\) is constant, it cancels out from both numerator and denominator: \[\frac{f_{1}}{f_{2}} = \frac{\sqrt{C_{2}}}{\sqrt{C_{1}}}\] Now, plug in the given values for \(C_{1}\) and \(C_{2}\) and solve for the ratio: \[\frac{f_{1}}{f_{2}} = \frac{\sqrt{0.2 \ \mu \text{F}}}{\sqrt{0.1 \ \mu \text{F}}} = \sqrt{2}\]

Match the result with the given options

The ratio of the frequencies is \(\sqrt{2}\), which means that when the capacitance is doubled, the frequency decreases by a factor of \(1/\sqrt{2}\). Thus, the correct answer is: (A) decreases by \((1 / \sqrt{2})\)

Key concepts

Capacitance

Capacitance is a key characteristic of capacitors, which are components that store electrical energy in an electric field. The unit of capacitance is the farad (F). In simpler terms, it measures a capacitor's ability to store charge. The relationship between charge (Q), voltage (V), and capacitance (C) is given by the formula:
\( Q = C imes V \).

• The larger the capacitance, the more charge a capacitor can hold at a given voltage.
• Capacitors can affect the frequency in LC circuits, as increasing or decreasing capacitance alters the system's resonant frequency.

In the context of an LC circuit, changing the capacitance directly influences how quickly the circuit can oscillate. This concept is crucial for designing circuits like oscillators, which depend on precise frequency control.

Inductance

Inductance involves a coil that stores energy in a magnetic field when current flows through it. It's measured in henrys (H). The inductance of a coil tells us how effective it is at generating a magnetic field. This property is fundamental in LC circuits.
In LC circuits, inductance works with capacitance to determine the natural frequency of oscillation. As the current oscillates in the circuit, energy shifts between the capacitor (electric field) and the inductor (magnetic field), creating oscillations at a frequency called the resonant frequency.

• An increase in inductance results in a decrease in the resonant frequency.
• A decrease in inductance leads to an increase in the resonant frequency.

Understanding inductance helps in tuning circuits to the desired frequency by considering both the role of inductors and the balance with capacitance.

Resonant Frequency Formula

The resonant frequency of an LC circuit determines how fast the circuit oscillates. It's given by the formula:
\[ f = \frac{1}{2 \pi \sqrt{LC}} \] This expression shows how the circuit's frequency depends on both the inductance (L) and the capacitance (C).

• The term \(2 \pi\) represents the circular nature of oscillation in the circuit.
• \(\sqrt{LC}\) indicates the interplay between inductance and capacitance in regulating oscillation speeds.

Using this formula, we can understand how changes in either L or C affect the frequency. Decreasing capacitance or inductance increases frequency, while increasing them decreases frequency. This relationship is vital for designing circuits with specific oscillating properties.

Frequency Change in LC Circuit

Understanding how frequency changes in an LC circuit is essential for applications like signal processing and radio transmission. When either capacitance or inductance is altered, the frequency shifts in predictable ways.
From the formula \( f = \frac{1}{2 \pi \sqrt{LC}} \), we learn:

• If capacitance is increased, the frequency decreases because the circuit takes longer to charge and discharge.
• If capacitance is decreased, the frequency increases as the circuit charges and discharges more quickly.

For example, in the given exercise, doubling the capacitance from \(0.1 \mu \text{F}\) to \(0.2 \mu \text{F}\) results in a frequency decrease by a factor of \(1/\sqrt{2}\). This occurs because the increased capacitance slows the energy transfer between the electric and magnetic fields. This predictable change helps engineers and technicians optimize circuits for desired functions.