Tuesday, August 11, 2009

Resonance - the tight-rope walk, blindfolded !

I've no idea why, but for some reason I've started with resonance, and the resonant frequency of a circuit. It's like trying to learn to walk by doing a tight-rope walk blindfolded between to skyscrapers.

And although I think I've learned the formula fairly quickly - f equals 1 over two by pi by the square root of LC, where L = inductance in Henrys and C is capacitance in Farads, I struggled since yesterday to get the correct answer in one of the questions on the sample paper.

After much perseverance, scratching of heads and pushing calculator buttons, I finally worked out the correct answer. I'll tell you how and where I went wrong . . .

But first, I have to pop up to Tony's to collect a book, so I'll continue later.

Right, I'm back. Me and Tony had a good old chinwag. Took me two hours to collect the book, and he only lives 5 minutes away !!

Anyhow, here's the question in the Sample Paper which I attempted to answer:

In the circuit below, given that L = 100 μH, C = 150 nF, what is the resonant frequency of the circuit? This is question 9 in the sample paper.

OK, so when I originally set about to answer it, I got an answer which wasn't any of the four options outlined below:

A [ ] 410.94kHz
B [ ] 41.094 kHz
C [ ] 4.1094 kHz
D [ ] 0.41094 kHz

Right, so we start with the inductance (L) which is 100 μH. One μH (micro henry) is .000001 Henrys. So that's .0001 H. The capacitance (C) is 150 nF or nano Farads. One nano Farad nF is .000000001, so 150 nF is .00000015 F. Remember I gave you the Micro, Nano, Pica equivalents in the section on capacitors? OK, if you can't then just remember the number of decimal places - micro (μ) = .000001 (6 places), nano (n) = .000000001 (9 places) and pica (p) = .000000000001 (12 places).

So the first thing we need to do is multiply LxC. .0001 x .00000015 is 0.000000000015 or otherwise, as the scientific calculator gives it, 1.5 x 10 to the power of minus 11.

Now we need the square root of 0.000000000015 which is .000003872.

Now, I know what you're thinking. Probably the same thing I was thinking. "This looks like it's going wrong. How the heck can I be on the right track with such tiny fractions of numbers?" Yes, I very much doubted where I was going with this. But stick with me.

The next stage is to find the value of two by pi. Now the value of pi given by the scientific calculator is 3.141592654, and this is where things started to go wrong for me. I'll explain why in a minute. Multiplying this pi value by 2 gives 6.283185307. This is further multiplied by .000003872 (square root of L x C). Still with me? Hope so! That gives .000024328. And one over .000024328 gives 41104. That's the frequency (f) in Hertz. 1,000 Hz is 1 kHz. So 41.104kHz is the answer But our answer is not given in the list:

A [ ] 410.94kHz
B [ ] 41.094 kHz
C [ ] 4.1094 kHz
D [ ] 0.41094 kHz

Answer B - 41.094 kHz looks close, but were' 10 Hz out. How come? I racked my brains over this one for an hour this evening - even had my wife Ann helping out. Turns out that it's so long since I did formulae or any sort of complex maths that I haven't a clue how to do all this stuff. So I let the scientific calculator do the work.

After getting LxC = .000003872, I multiplied by pi then by 2, which gives .000024334 (remember previously we got .000024328 - well those six billionths make all the difference!!) One over .000024334 gives 41.093.6

41093.6 rounded up is 41094 Hertz. That's 41.094 kHz. So we now know the true answer is B. It took a good while but we got there. Phew! Questions welcome if you are stuck on any of this. Remember, I'm doing this for the first time too, so if you're stuck I can only promise encouragement !

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