Analyzing source free RLC circuit

[dwn]

Homework Statement

Attached an image with the exact question.

Homework Equations

α = 1/2RC
ω = 1/√(LC) α > ω
i(t) = Ae^st + Be^ct

The Attempt at a Solution

(a) First I found ω and α ---- ω = 1.4*10^5 1.4*10^5 < 1/2RC → R > 357.1429

Is it safe to assume that R is 358 Ω...? Is this how R is found?

(b) i(t) = Ae^st + Be^ct
s = -α - √(α²-ω²)
c = -α + √(α²-ω²)

I have not started this yet, because I want to make sure R is correct.

 

[gneill]

Homework Statement

Attached an image with the exact question.

Homework Equations

α = 1/2RC
ω = 1/√(LC) α > ω
i(t) = Ae^st + Be^ct

The Attempt at a Solution

(a) First I found ω and α ---- ω = 1.4*10^5 1.4*10^5 < 1/2RC → R > 357.1429

Is it safe to assume that R is 358 Ω...? Is this how R is found?

You should keep a few more decimal places for intermediate values, otherwise truncation and roundoff errors can creep into your significant figures in multi-step calculations. When I keep a few more decimals for ω_o I get a slightly smaller value for R.

Your method is okay, but you should make it clear that for critical damping the damping factor $\zeta = \frac{\alpha}{\omega_o}$ is unity (1).

 

[dwn]

I took your advice and recalculated using a calculator instead of MatLab which truncated it to result shown above. I did get a more accurate calculation, so is it safe to assume that R = 355.56? (actual was 353.55339).

Thanks for damping factor, I must have missed that in the text...I didn't realize we can check our result using α/ω

 

[gneill]

I took your advice and recalculated using a calculator instead of MatLab which truncated it to result shown above. I did get a more accurate calculation, so is it safe to assume that R = 355.56? (actual was 353.55339).

Thanks for damping factor, I must have missed that in the text...I didn't realize we can check our result using α/ω

With the expressions for $\zeta$, $\omega_o$, and $\alpha$ you could find an expression for R in terms of L & C alone (assuming critical damping). In fact, with the given values of L and C being such nice numbers you could even show (with a little work) that
$$R = \frac{1000}{2 \sqrt{2}} \Omega$$
But that's probably going a bit overboard for what you need. If you were solving the rest of the problem via Laplace Transforms then it might make the transform prettier

But by your approach you should be able to hit the "actual" value bang on if you keep enough digits through all the intermediate steps.

 

[dwn]

I will have to save the Laplace Transforms for another time, when I'm not on the clock, so to speak. haha.

As for part b, I'm not comfortable with the results, or at least the first step to solving part b. I need to calculate for s and c using the equations previously stated. However, the result I am getting seems a bit off.

s = -0.2828422 - √(0.28284222² - 141421.356232²) = 141421.073388
c = -0.2828422 + √(0.28284222² - 141421.356232²) = 141421.639072

What should follow:
1.

A + B = i[SUB]L[/SUB](0) = 0.1A

2.

Ae[SUP]st[/SUP] + Be[SUP]ct[/SUP] = V[SUB]L[/SUB](0) = L∂i/∂t

3.

L∂i/∂t = 5*10[SUP]-3[/SUP](sAe[SUP]st[/SUP] + cBe[SUP]ct[/SUP])

4.

V(0) = -400 V = 5*10[SUP]-3[/SUP](sA + cB)

Then solve number 1 and 4, for A and B using the elimination method. Except I'm not sure how these two relate since the units don't match -- A/s and the other just A.

I'm sorry if there is any confusion..hopefully this all makes sense, if it does not, I will do my best to clean it up.

 

[gneill]

Okay, one problem. For the critically damped case the form of the solution is not $A e^{s t} + B e^{c t}$.

Take a look at the Wikipedia entry for the RLC circuit, and in particular the section on Critically Damped Response

 

[dwn]

Argh! I forgot they varied depending on damping.

v = e[SUP]-αt[/SUP](At + B)

In which case, V(0) = -400 V = B

Does everything else look alright?
Also, can you explain to me how i_max differs from i? I found the equation:

(V[SUB]0[/SUB]/e)√(C/L)

Does this typically happen within a reasonable amount of time, something intuitive?

Thanks for your help with this. Really appreciate it!

(Do you just learn to deal with and accept the "speed bumps"? Frustrating)

 

[dwn]

What type of damping circuit do ee's generally prefer? I imagine there can be a case made for each, but I just finished reading an article that talked about the effects of overdamping and that it kills the peak current in the circuit...is this done to prevent a circuit from blowing? Where would underdamping be beneficial?

Just trying to get a practical understanding of the circuit damping. Thanks!

 

[gneill]

Argh! I forgot they varied depending on damping.

v = e[SUP]-αt[/SUP](At + B)

In which case, V(0) = -400 V = B

Does everything else look alright?

That's the right idea. Find your values for A and α too.

Also, can you explain to me how i_max differs from i? I found the equation:

(V[SUB]0[/SUB]/e)√(C/L)

Does this typically happen within a reasonable amount of time, something intuitive?

Offhand I don't recognize that formula. I suppose it might be a solution to finding the extrema of the current function, but I wouldn't know without actually solving the problem myself. So, write the equation for the current w.r.t. time and find the extrema...

The current is a function of time and it will have maximum and minimum values. The magnitudes of the two are not equal, and one will be positive and the other negative. So the question is asking you to find the greatest current magnitude as well as the greatest positive valued current.

Note that you can always check your thinking/results and "see" the voltage and current curves using a simulator like LTSpice.

The time for the output (current in this case) to settle to its final value is determined by the constant in the exponential. That would be your α, for which a simple formula is available for the parallel RLC circuit. The rule of thumb is that after 5 time constants ($\tau = 1/\alpha$) all the exciting stuff is over and done with. Critical damping achieves the minimal settling time without "ringing" (oscillations about the final value).

Thanks for your help with this. Really appreciate it!

(Do you just learn to deal with and accept the "speed bumps"? Frustrating)

Yup. That's life