Electronics Engineering (ELEX) Board Practice Exam

To build an LC oscillator producing a sine wave of frequency 1 GHz with an inductance of 1 mH, what is the required capacitance?

1. 5 pF

2. 10 pF

3. 2.53 x 10¯¹¹ pF

4. 1 x 10¯¹⁰ pF

The correct answer is: 2.53 x 10¯¹¹ pF

The frequency of an LC oscillator can be calculated using the formula for the resonant frequency of an LC circuit, given by: \[ f = \frac{1}{2\pi \sqrt{LC}} \] where \( f \) is the frequency in hertz, \( L \) is the inductance in henries, and \( C \) is the capacitance in farads. To find the required capacitance for a given frequency and inductance, we can rearrange the formula to solve for \( C \): \[ C = \frac{1}{(2\pi f)^2 L} \] Plugging in the values: - Frequency \( f = 1 \, \text{GHz} = 1 \times 10^9 \, \text{Hz} \) - Inductance \( L = 1 \, \text{mH} = 1 \times 10^{-3} \, \text{H} \) Now substituting these values into the capacitance formula: \[ C = \frac{1}{(2\pi (1 \times 10^9))^2 (1 \times 10^{-3})} \] Calculating \( (