- #1
NaS4
- 19
- 0
- Homework Statement
- Learning
- Relevant Equations
- z=a+bj, angle = tan^-1(b/a) -
R = R, C = 1/jwC, L = jwL
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Okay, why don't you type out your steps here and we can see where the problem is coming from.NaS4 said:Yeah part a
But when I try to simplify it it just goes horrible wrong
I’ve seen it before. But there’s a lot happening here. And I don’t know where to startMaster1022 said:For part (a), what you have done so far looks correct. You should continue to simplify the expression (get it into a simpler fraction). I'm assuming you were asking about part a - if not, do correct me.
[EDIT]: you may want to 'rationalise' the denominators of the fractions to help reach the required form. If you are taking an AC circuit theory class, I am guessing that you have seen complex numbers before? If not, let me know and I can explain further/ point you towards some resources.
1/Z_{total} = 1/Z_{1} + 1/Z{2} in parallelMaster1022 said:Okay, why don't you type out your steps here and we can see where the problem is coming from.
Generally, we know that two parallel impedances yield ## Z_{total} = Z_1 // Z_2 = \frac{Z_1 \cdot Z_2}{Z_1 + Z_2} ## which follows directly from the formula you wrote. It might be easier to write out the final impedance that way
Lets take ## Z_1 = R + \frac{1}{j\omega C} ## and ## Z_2 = j \omega L ##. The formula then becomes:NaS4 said:I’ve seen it before. But there’s a lot happening here. And I don’t know where to start
And in series it’s Z(total) = Z(1) + Z(2)NaS4 said:1/Z_{total} = 1/Z_{1} + 1/Z{2} in parallel
It would really help to see some working. To get to the expression I have written, we can multiply the expression you have in the third line of your working by ## (R + 1/j\omega C ) ## and ## j \omega L ## (both numerator and denominator)NaS4 said:Yeah i get that. But the formula i got is not the same as the one you are using
In the beginning, ## j \omega L + \frac{1}{j\omega C} \neq j(\omega L + \frac{1}{\omega C}) ## as the j in the denominator should make it negative (i.e. ## j(\omega L - \frac{1}{\omega C}) ##)NaS4 said:Is this correct?
Ahh yes I see itMaster1022 said:In the beginning, ## j \omega L + \frac{1}{j\omega C} \neq j(\omega L + \frac{1}{\omega C}) ## as the j in the denominator should make it negative (i.e. ## j(\omega L - \frac{1}{\omega C}) ##)
Damn so many mistakes. Thx man. I’ll change it and post my next tryMaster1022 said:I think you might also be missing a term where ## j \omega L R ## multiplies the ## R ## in the first line
NaS4 said:@Master1022 How does this solution look?
Ok I’m lost now. How did you get (1 - w^2LC)Master1022 said:That looks correct to me! It might be easier to turn the ## (\omega L - \frac{1}{\omega C}) ## into a ## 1 - \omega^2 LC ## term (remembering to multiply through by ## (\omega C)^2 ## ). This looks like it will be easier to work with for the next parts of the question.
It is also useful to think about dimensions when working through these types of problems. Often seeing dimensions that match can be indicative of correct working (e.g. seeing a ## \omega ^2 LC ## or ## \omega ^ 2 RC ##)
Apologies, so you can get ## \omega^2 LC - 1 ## and then multiply that term by ## (-1)^2 = 1 ## so your denominator becomes ## (\omega C R)^2 + (\omega^2 LC - 1)^2 = (\omega C R)^2 + (1 - \omega^2 LC )^2 ##. It's just a multiplication by 1 (won't matter because the term is squared).NaS4 said:Ok I’m lost now. How did you get (1 - w^2LC)
That is true and does definitely stand out to me. However, this is what can happen in resonant circuits with very high Q-factors... I just did a quick calculation, and the number I got was slightly larger than your answer (but on the same order of magnitude). Maybe another member might notice an error that I have overlookedNaS4 said:That 11.5k V sounds abit to high
Master thanks a lot for the help man. You really are a lifesaverMaster1022 said:Also, in the expression for ## b##, I think the central term in the numerator should have the ## C ## as ## C^2 ##. However, I am not sure whether that makes much difference given the order of magnitudes. I will go back to check the original expression to ensure that your formula is exactly the same as what I got
This is an incredibly useful tool when working with this sort of complex algebra. OK, actually, useful in most of physics, really. This allows you to spot errors in equations without having to go through the whole derivation. Maybe you won't know why it's wrong, but you will know it's not correct.Master1022 said:It is also useful to think about dimensions when working through these types of problems. Often seeing dimensions that match can be indicative of correct working (e.g. seeing a ## \omega ^2 LC ## or ## \omega ^ 2 RC ##)
Thx man. I’ll look up dimensional analysisDaveE said:This is an incredibly useful tool when working with this sort of complex algebra. OK, actually, useful in most of physics, really. This allows you to spot errors in equations without having to go through the whole derivation. Maybe you won't know why it's wrong, but you will know it's not correct.
So, to elaborate on @Master1022s method:
R, 1/(ωC), ωL all have units of ohms. If I see a term in your equation like (R + L), I know without much analysis that it's wrong; that should be something like (R + ωL), or (R/ω + L). You just won't see solutions in physics where you have (ohms + volts), (force + energy), (flow + pressure), etc.; these rarely have any useful physical meaning. In the (R + L) example the units are (ohms + ohms*seconds).
BTW, you can't use this to verify correctness completely. It won't spot some errors, like the difference between (R + ωL) and (R + 1/(ωC)), or R vs. 2R.
If you want to know more look up "dimensional analysis".
It can get scary. Look for the useful bits. If they get too abstract (math, philosophy and such) that may a good place to stop, or not, if you like that stuff.NaS4 said:Thx man. I’ll look up dimensional analysis
DaveE said:It can get scary. Look for the useful bits. If they get too abstract (math, philosophy and such) that may a good place to stop, or not, if you like that stuff.
Yeah the w = 2(pi)f but since the question tells us that w is 200Master1022 said:Just a quick update, I have checked my expression:
$$ Z = \left( \frac{\omega^4 R L^2 C^2}{(1-\omega^2 LC)^2 + (\omega CR)^2} \right) + j \left( \frac{\omega L (1 - \omega^2 LC + (\omega R C)^2)}{(1-\omega^2 LC)^2 + (\omega CR)^2} \right) $$
and think it is equal to yours (thus we are either both correct... or both wrong !)
Quick question about the problem statement: are the units of ## \omega ## a typo when it says Hz? Usually we expect ## \omega ## to be in rad/s (a conversion factor of ## 2 \pi ## is needed).
Am doing the calculations in a Python notebook, so will post pictures here to show how my calculations are resulting
NaS4 said:Yeah the w = 2(pi)f but since the question tells us that w is 200