# RCL circuit help

Discussion in 'Homework Help' started by micachu, Nov 5, 2014.

1. ### micachu Thread Starter New Member

Oct 13, 2014
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Hi I'm suppose to analyze the circuit shown below

and for preliminary calculations I'm suppose to find what value of R would give me under-damped and over-damped.

For lab we were assigned to use 6.8k ohm and a 560 ohm resistor for R. I used pspice to simulate the circuit. I got under-damped for 6.8k ohm for R and over-damped for 560 ohm for R.

For the preliminary calculations, my first step was to find the second order.
(LC)i'' + (L/R)i' + i = f(t)/R
Givens: L = 68mH, C=.01uF

After substituting the values for L and C my characteristic equation was
m^2 + [(1E8)/R]m + 1.47E9 = 0

a = 1
b = (1E8)/R
c = 1.47E9

b^2-4ac > 0 is over-damped
b^2-4ac < 0 is under-damped

[(1E8)/R]^2 - 4(1)(1.47E9)

I'm not sure if I'm doing this correctly. I just assumed that if I were to isolate R from b^2-4ac and get something like R > # or R < #, then I found the resistant values that I can use that would give me under-damped or over-damped. So I got R > 1304, R<1304 which is way off from the pspice simulation. Any ideas? Thanks in advance.

2. ### shteii01 AAC Fanatic!

Feb 19, 2010
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Why do you say it is way off from the pspice simulation?

3. ### WBahn Moderator

Mar 31, 2012
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I'll echo the question. You simulated it at 6.8 kΩ and got underdamped and you simulated it at 560 Ω and got overdamped. You then calculated that anything over 1.3 kΩ should be underdamped and anything under that should be overdamped. What's the problem? Have you run sims at around 1.3 kΩ?

4. ### JoeJester AAC Fanatic!

Apr 26, 2005
3,397
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I'd say post your simulation graphs at the three values of R1.

5. ### MrAl Distinguished Member

Jun 17, 2014
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Hi,

Maybe your result is not accurate enough (1304) as the more accurate result is 1303.84048 and that's still approximate.
If you use 1304-0.1 ohm and 1304+0.1 ohm for example you will not see the change, you need to use a larger change if you estimate at 1304, like 10 ohms. So try 1304+10 and 1304-10 ohms and see what you get.

6. ### WBahn Moderator

Mar 31, 2012
18,085
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1304 Ω is barely more than 0.01% away from the true value. In most sims you would be hard pressed to discern the difference with even a 1% change (13 Ω).

7. ### JoeJester AAC Fanatic!

Apr 26, 2005
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If I didn't know I stepped the resistor the values, I would have been hard pressed to recognize the differences.

So if there was only one trace and not three ... I would have assumed it to be "close" enough.

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8. ### WBahn Moderator

Mar 31, 2012
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My guess is that you would want to run a set of sims, say 11, sweeping the value of R in 5% or even 10% increments about the 1.3 kΩ value.

9. ### micachu Thread Starter New Member

Oct 13, 2014
15
0
I'm sorry I'm really confused. I'm not sure if I'm doing the calculations correct at all, but that is how I understood the problem. The problem asked "what range of values of R correspond to over,under, critically damped cases" for the circuit I posted above for preliminary calculations. What I got was,

(LC)i(t)''+(L/R)i(t)'+i(t) = (v(t)/R) for my second order
given: L=68mH, C=.01uF, R=?
anything over 1.3kΩ is overdamped
anything under 1.3kΩ is underdamped

Then when we constructed the circuit and were asked to use 6.8kΩ and 1.5kΩ for R. (Sorry, I forgot to mention that instead of using 560Ω, my professor made us use 1.5kΩ instead). I uploaded a file showing what we got using pspice and oscilloscope.

I apologize if I don't make that much sense >_<
I appreciate all the help so far

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10. ### WBahn Moderator

Mar 31, 2012
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And so what's the problem? It appears that 6.8 kΩ is very underdamped and that 1.5 kΩ is significantly less underdamped, but still underdampled. That is very consistent with 1.3 kΩ being critically damped. Keep in mind that you are probably not restricted to ONLY trying the values that your instructor told you to use. Try other values and see what happens! Try 1.3 kΩ. Try 1 kΩ, try 560 Ω.

11. ### micachu Thread Starter New Member

Oct 13, 2014
15
0
The problem is im really confused lol. Sorry can you explain how it is consistent with the 1.3k? For my understanding 6.8kΩ > 1.3kΩ so if a value that is greater than 1.3k, it should be overdamped? 1.5k,1.3k,1k, and 560 looks very similar to each other. Sorry for the trouble >.<

12. ### WBahn Moderator

Mar 31, 2012
18,085
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Why do you think that a value greater than 1.3 kΩ should be overdamped.

What you did was set up an equality for the case of it being critically damped and then guess which side of that was overdamped and which side was underdamped.

Don't guess!

Set up the inequality specifically for the case of the circuit being overdamped.

Also, ask if your conclusions make sense. What if the resistor value was zero? Would the LC part of the circuit be able to ring? What if the resistor value was so large that it effectively isolated the LC from the source (allowing just a bit of energy to sneak in)? Would the LC part of the circuit stop ringing?

13. ### JoeJester AAC Fanatic!

Apr 26, 2005
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In the underdamped pictures, it appears you have your oscilloscope triggered on a positive edge, so you will notice that the first positive going cycle on the simulation is the same amplitude as the oscilloscope.

Run your simulation with a 750 Hz square wave.

Then you can trigger on the positive edge of the square wave to start the oscilloscope.

On edit ... added simulation setup and virtual oscilloscope ....

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• ###### damped-oscilloscope.png
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Last edited: Nov 6, 2014
14. ### MikeML AAC Fanatic!

Oct 2, 2009
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Based on this, I'd say that 1.2K and 1.3K are overdamped, and 1.4K and 1.5K are underdamped...

15. ### WBahn Moderator

Mar 31, 2012
18,085
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I think I'd agree with that. Can you run some sims between 1.3 kΩ and 1.4 kΩ to narrow it down more? I suspect we will find that the 1304 Ω value is right on the money. I don't have any sim software installed on my home machine anymore and I'm not interested in chewing up what little space I have left on this drive by installing any.

16. ### MikeML AAC Fanatic!

Oct 2, 2009
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Zooming way in, and varying R1 in steps of 10 Ohms from 1300 Ohms to 1400 Ohms, 1300, 1310, and 1320 are overdamped, 1330 is critical (Yellow trace), and >=1340 undershoot:

17. ### WBahn Moderator

Mar 31, 2012
18,085
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Thanks. So the question is what accounts for the very small difference (0.02%) between your sim results and the theoretical calculations? I'm willing to bet that it is in your simulator, given all of the various quantization and time-step errors that exist in simulators. Heck, I suspect that just the choice of integration method might account for it. But they have definitely served their purpose in showing that 1.3 kΩ is right at the critical value for the resistor and that it is overdamped for smaller R and underdamped for larger R (just as would be expected).