# Math

#### hlyman

Joined Nov 29, 2005
1
Made an error on last post 429.11 is 449.11 in lesson

I am a beginner learning circuits.
I am stuck in lesson AAC Vol 2, Chapter 5, Series-Parallel R, L, & C circuits.

Rule of parallel circuits: Z=1/(1/Zr + 1/Zc)
Zr=470 [R]
Zc=1523 [L--C2]
Zr//L--C2 = 1/(1/470+1/1523.3)

My answer = 359.18

Lesson answer is 429.15 - j132.41 or 449.11 @ -17.147 deg.

I can't figure out how this is arrived at. Can anyone give me the math?

Thanks

#### peajay

Joined Dec 10, 2005
67
Well, this answer is probably a week or two too late, but I hate to go into a forum and just ask questions, I feel like I should answer a few first.

The imendance of the inductor is 90 degrees out of phase with the impendance of the resistor. Or something like that. (I just happened to read this in someone's college textbook one time, but I didn't pay a whole lot of attention as I don't care much about AC anyway.)

So 470 stays 470, but the 1523 becomes -1523j, where j is i because for some reason I guess electronics people can't tell the difference between i and I. (Actually, I guess they sound exactly the same when spoken, maybe that's why.) Impendances from inductors are multiplied by -j, impendances from capacitors are multiplied by +j. (or at least I assume that's the polarity because of the answer you gave) And because of that, it really is possible to connect a capacitor and inductor together in parallel and end up with a circuit that has infinite impendance to a given frequency, or connect them in series to make a zero impendance to that frequency. Hey, this sound like it might have uses in audio circuits, so maybe I do care. Hmm... Well, anyway...

So then the equation becomes 1 / ( 1 / 470 + 1 / -1523j ). Unfortunately I don't remember how to calculate the reciprocal of a complex number, so I'm kind of stuck there. But I do know how to calculate the absolute value of a complex number, same as a regular number, abs(x) = sqrt(x^2), and that's 0.002227, and the reciprical of that is 449.1. So that's where that comes from. (remember that j^2 = 1, and so sqrt(x^2) has the effect of removing the j term as well as making the number positive if it was negative)

Ok, looking on the internet says that we calculate the reciprocal like this: for 1 / (x + yj), the reciprocal's real part is x / ( x^2 + y^2 ) and the imaginary part is -y / ( x^2 + y^2 ).

So x is 0.002128 and y is 0.0006566 and x^2 + y^2 is 0.000004958. So the real part is 429.1 and the imaginary part is -132.4, which gives us 429.1 - 132.4j, which is basically the above. The absolute value of that will also be 449.1, as the absolute value of the reciprocal is the same as the reciprocal of the absolute value.

To figure out that angle, you take the atan2 of 429.1 - 132.4j. If you don't have an atan2 button on your calculator, then I'm not sure what to tell you there, except that atan of (-132.4 / 429.1) will give you a similar answer that's merely off by 180 degrees half the time. Maybe someone else knows how to do that correctly. I _think_ that it's off by 180 degrees when the denominator of that fraction is negative, and so you just add 180 degrees in that case, but don't take my word on that.