Understanding Resonance in AC Circuits

[Toky]

Homework Statement

Hello. So, I have an AC circuit in front of me. What I need to do is pull out the formula for inductance of the inductor as L = f(C, R2, w0) and w0 =f(C, R2, L) by calculating the total impedance of the circuit. It's a voltage resonance example so it is supposed to be done by using the term Im(Z)=0. I'm not sure if I did it correctly so I'm asking for help.

Relevant Equations

XL = w0*L
XC = 1/(w0*C)

Here's my attempt at a solution.

 

[Toky]

For some reason, some of the images didn't load properly while constructing the thread, so I'm leaving the full solution in the replies.

 

[berkeman]

Well, the images are coming out small, dim and sideways. It's better to post a PDF file attachment showing the problem statement, and type your work into the forum using LaTeX. See the LaTeX Guide link in the lower left of the Edit window.

While you are learning LaTeX, can you at least post much better/bigger versions of the schematic drawings and separately post pictures of your work? Be sure to post them upright instead of sideways, and when you attach them, select "Full Size" instead of "Thumbnail". Thanks.

 

[Delta2]

Hard to "decipher" your hand writing. I can only tell you what I get. I get a quite complex expression for Im(Z), but at the end of the day the condition $Im(Z)=0$ simplifies to $$L=C(\frac{1}{C^2\omega^2}+R_2^2)$$ from which I get that $$\omega=\frac{1}{\sqrt{C(L-CR_2^2)}}$$

 

[berkeman]

Hard to "decipher" your hand writing. I can only tell you what I get. I get a quite complex expression for Im(Z), but at the end of the day the condition $Im(Z)=0$ simplifies to $$L=C(\frac{1}{C^2\omega^2}+R_2^2)$$ from which I get that $$\omega=\frac{1}{\sqrt{C(L-CR_2^2)}}$$

Yeah, I cleaned up his last page, and his final result for L looks similar to yours. Can you tell if it is the same? Also, did he work out the expression for $\omega$ ?

 

[Toky]

Well, the images are coming out small, dim and sideways. It's better to post a PDF file attachment showing the problem statement, and type your work into the forum using LaTeX. See the LaTeX Guide link in the lower left of the Edit window.

While you are learning LaTeX, can you at least post much better/bigger versions of the schematic drawings and separately post pictures of your work? Be sure to post them upright instead of sideways, and when you attach them, select "Full Size" instead of "Thumbnail". Thanks.

I apologise if I've caused any anger or inconveniences. All of my professors ask for 100% hand-written assignments, probably because I'm still only on the 1st year of college, so I never even tried writing in LaTeX, or anything similar, though it seems very convenient. I'll try to learn is ASAP, thanks for the kind advice.

 

[DaveE]

I apologise if I've caused any anger or inconveniences. All of my professors ask for 100% hand-written assignments, probably because I'm still only on the 1st year of college, so I never even tried writing in LaTeX, or anything similar, though it seems very convenient. I'll try to learn is ASAP, thanks for the kind advice.

It's really about making your work easy to read. Good handwriting and good photos are also OK.

LaTeX isn't necessary, but it is an good way to solve the readability problem. Whether it's here, in school, or in your career, you'll get better results if you make it easy for the people reviewing your work. Look at your posts and imagine that it's you that is being asked to help out; how much effort is in helping, and how much in reading the question.

It isn't unusual for me, and I suspect others, to simply not comment on a thread that I could help with because it's just too hard to figure out the information I need to assist.