Mastering LC Oscillators: Calculate Capacitance with Ease

When getting ready for the Electronics Engineering Board Exam, diving into the intricacies of LC oscillators can seem a bit daunting, right? But don’t worry! We’re going to break it down step by step and make this complex topic a piece of cake. So let’s roll up our sleeves and tackle the challenge of calculating capacitance in an LC oscillator at a frequency of 1 GHz with an inductance of 1 mH.

First off, if you've ever been curious about how oscillators work, understanding the relationship between inductance (L) and capacitance (C) is key. An LC oscillator essentially generates waveforms — think of it like a musical instrument creating a sine wave. The frequency of that wave is largely determined by the values of L and C.

Alright, so how do you actually find the capacitance needed for a specific frequency? The formula you need is:

[ f = \frac{1}{2\pi \sqrt{LC}} ]

Pretty straightforward, right? Here’s what those symbols mean:

• ( f ) is the frequency in hertz (Hz),
• ( L ) is the inductance measured in henries (H),
• ( C ) is the capacitance measured in farads (F).

Rearranging for Capacitance

To find capacitance (( C )), we can rearrange the formula:

[ C = \frac{1}{(2\pi f)^2 L} ]

Now we can substitute our known values. We're looking to build an oscillator that produces a sine wave at 1 GHz. Remember, 1 GHz is the same as ( 1 \times 10^9 ) Hz. The inductance provided is 1 mH, which is ( 1 \times 10^{-3} ) H.

Plug in the Numbers

So, we get:

[ C = \frac{1}{(2\pi (1 \times 10^9))^2 (1 \times 10^{-3})} ]

Now, let’s compute that. You might want to grab your calculator or simply run it through some handy software, because the numbers can get a little wild.

1. The denominator becomes ( (2\pi \times 1 \times 10^9)^2 \times (1 \times 10^{-3}) ).
2. After crunching those numbers, you will arrive at a capacitance of approximately ( 2.53 \times 10^{-11} ) farads, which is the equivalent of 25.3 pF (picoFarads).

What's the Bottom Line?

So, amongst the multiple-choice options, the correct answer to our original question about the required capacitance for the LC oscillator is indeed 2.53 x 10¯¹¹ pF. The others don't stack up!

Wrap-Up

Thus, not only do you understand the theory behind oscillators, but you've now also got a solid grasp of how to calculate capacitance for a given frequency and inductance. It's like having a new tool in your electronics toolkit!

Just remember, as you prep for your upcoming exams, practice makes perfect. Knowing how to apply these formulas in problem-solving is crucial. And who knows? You might even impress a few friends with your newly acquired knowledge about LC oscillators!

So, what are you waiting for? Keep that curiosity alive, and you'll not only conquer the boards but also truly understand the beautiful world of electronics.