Using Complex Impedances in these RLC Circuit Calculations

[DaveE]

Thx man. I’ll 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.

 

[etotheipi]

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.

Good place to start is Orodruin's article:
https://www.physicsforums.com/insights/learn-the-basics-of-dimensional-analysis/

 

[Master1022]

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]

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

Yeah the w = 2(pi)f but since the question tells us that w is 200

 

[Master1022]

Yeah the w = 2(pi)f but since the question tells us that w is 200

Ah okay, that explains the discrepancy I had earlier. After setting $\omega = 200$, I did get the same value of the gain and the final voltage as you (give or take some rounding errors) - the computer output $11551.24...$ (V), so this does agree with your result.

Hope that is of some help.

 

[NaS4]

Ah okay, that explains the discrepancy I had earlier. After setting $\omega = 200$, I did get the same value of the gain and the final voltage as you (give or take some rounding errors) - the computer output $11551.24...$ (V), so this does agree with your result.

Hope that is of some help.

Master1022 I got nothing but respect for you man. Really makes me happy to see people like you helping out people like me

 

[DaveE]

Quick question about the problem statement: are the units of ω a typo when it says Hz? Usually we expect ω to be in rad/s (a conversion factor of 2π is needed).

It's very common for people to be sloppy about that. I guess they assume you'll know about the 2π correction.

 

[Master1022]

Master1022 I got nothing but respect for you man. Really makes me happy to see people like you helping out people like me

Happy to help, hopefully that answer is correct. One other sanity check is to see whether you can find a formula for the damping ratio of an RLC circuit with this setup and see whether it is very small (which is the only way such a large value of the gain is possible). I did have a quick glance on wikipedia, but I cannot see any similar variation. Regardless of the numerical values of the question, you did get the overall method correct (finding the transfer function and then calculating the gain).

I am going to ask @DaveE as I think he is more knowledgeable: can this topology of the RLC circuit be re-arranged into another form (eg. from the list on wiki here) such we can use the same formula for the damping ratio $\zeta$?

 

[DaveE]

The canonical form for this particular LCR impedance is:

Z(s) = sL⋅(1+sRC)/(1+sRC+s²LC), where s=jω (sorry, I have habits that are hard to break!)

Then if you define some common features: ω=1/sqrt(LC), Zo=sqrt(L/C), Q=1/(2ζ)=Zo/R

The quadratic term looks like 1+(1/Q)(s/ω)+(s/ω)².

This looks very different from the homework problem which wants a solution as a+ib. Hence a lot of algebra ensues.

The "factored pole-zero" or "ratio of factored polynominals" form above is how we really do this in EE/Controls world. I gives much better insight into the dynamics of this circuit. It's also easier to derive, IMO. Also notice that the polynomials are dimensionless, except the leading sL term, which is ohms, of course.

edit: all of my ω's above (except s=jω) should be ω_o, it's not that ω, it's a constant.

 

[Master1022]

Thanks for the reply @DaveE !

The canonical form for this particular LCR impedance is:

Z(s) = sL⋅(1+sRC)/(1+sRC+s²LC), where s=jω (sorry, I have habits that are hard to break!)

Agreed! (I also prefer working in the $s$-domain)

Then if you define some common features: ω=1/sqrt(LC), Zo=sqrt(L/C), Q=1/(2ζ)=Zo/R

The quadratic term looks like 1+(1/Q)(s/ω)+(s/ω)².

Yes, that is correct. This leads to
$$ \zeta = \frac{R}{2} \sqrt \frac{C}{L} = \frac{10^4}{2} \sqrt{\frac{5 \cdot 10^{-6}}{20}} = 2.5 $$
which is odd because that doesn't suggest gain on the order of $10^3$...

Hmm, perhaps I ought to have another look over my algebra tomorrow to see if I come across any errors.

Thanks once again.