I have a homework assignment working with RLC circuit analysis. Here is what the assignment says:
1. Go to URL: www.falstad.com/circuit
2. Open up the RLC Circuit. (It should be the first one that pops up, but if not go to Circuits>Basics>LRC Circuit.)
3. Change the circuit values to build an
a. underdamped
b. critically damped and
c. overdamped circuit with natural frequency of f=1000 Hz
4. Plot the natural response in both current and voltage for each of the components R L and C of the second order system when the switch is opened (6 plots total).
5. Write an equation that matches each of the plots above.
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My first thoughts were to find values of R L and C by using the fact that, ω=2000∏ and that ζ must be <1, =1, and >1 for each of the different situations.
I also used the following: ω = 1/√[L*C]; ζ = (R/2)*√[C/L]
Using the tools at the above website and some other sources I was able to design an underdamped circuit with the following conditions:
R = 10 Ω; L = 100 mH; and C ≈ 253.3 nF
.: ζ ≈ 0.007921 < 1 and the circuit is underdamped.
I know that the general form for the solution for an underdamped circuit is:
x(t) = e^(-ζωt)*[A1 cos(ω*√[1-ζ^2]*t) + A2 sin(ω*√t[1-ζ^2]*t)]
I plug in what I got for ζ and ω but don't know how to solve for A1 and A2.
Now the next part is that I'm having trouble coming up with the critically damped and overdamped R L and C values while keeping the 1000 Hz frequency requirement.
For the critically damped circuit ζ = 1. I thought that I could keep L and C the same as the underdamped circuit (L = 100 mH; and C ≈ 253.3 nF) and use the equation ζ = (R/2)*√[C/L] to solve for a new resistance, R that will make ζ = 1. So I get the equation
1 = (R/2)*√[253.3n/100m]
Solve for R,
R = 2/√[253.3n/100m] ≈ 1256.6 Ω
I plugged the new R value in the circuit simulator and I can't tell whether or not it's a critically damped circuit by looking at the plots below. It's too small to tell, so I'm wondering if someone can show me how to check that or find a better way to go about doing it.
Provided that this new circuit is correct where:
R = 1262.39 Ω; L = 100 mH; and C ≈ 253.3 nF
.: ζ = 1 = 1 and the circuit is critically damped.
I know that the general form for the solution for a critically damped circuit is:
x(t) = B1*e^(-ζωt) + B2*t*e^(-ζωt)
Again, I plugged in ζ and ω but don't know how to solve for B1 and B2.
As far as the overdamped circuit goes... I haven't gotten anywhere because I am a bit confused on whether or not I'm on the right track with what I've done so far and want to see what you think.
I was going to talk to my prof after class, but he wasn't there today... go figure. Thanks for the help in advance.
1. Go to URL: www.falstad.com/circuit
2. Open up the RLC Circuit. (It should be the first one that pops up, but if not go to Circuits>Basics>LRC Circuit.)
3. Change the circuit values to build an
a. underdamped
b. critically damped and
c. overdamped circuit with natural frequency of f=1000 Hz
4. Plot the natural response in both current and voltage for each of the components R L and C of the second order system when the switch is opened (6 plots total).
5. Write an equation that matches each of the plots above.
----------------------------------------------------------------------
My first thoughts were to find values of R L and C by using the fact that, ω=2000∏ and that ζ must be <1, =1, and >1 for each of the different situations.
I also used the following: ω = 1/√[L*C]; ζ = (R/2)*√[C/L]
Using the tools at the above website and some other sources I was able to design an underdamped circuit with the following conditions:
R = 10 Ω; L = 100 mH; and C ≈ 253.3 nF
.: ζ ≈ 0.007921 < 1 and the circuit is underdamped.
I know that the general form for the solution for an underdamped circuit is:
x(t) = e^(-ζωt)*[A1 cos(ω*√[1-ζ^2]*t) + A2 sin(ω*√t[1-ζ^2]*t)]
I plug in what I got for ζ and ω but don't know how to solve for A1 and A2.
Now the next part is that I'm having trouble coming up with the critically damped and overdamped R L and C values while keeping the 1000 Hz frequency requirement.
For the critically damped circuit ζ = 1. I thought that I could keep L and C the same as the underdamped circuit (L = 100 mH; and C ≈ 253.3 nF) and use the equation ζ = (R/2)*√[C/L] to solve for a new resistance, R that will make ζ = 1. So I get the equation
1 = (R/2)*√[253.3n/100m]
Solve for R,
R = 2/√[253.3n/100m] ≈ 1256.6 Ω
I plugged the new R value in the circuit simulator and I can't tell whether or not it's a critically damped circuit by looking at the plots below. It's too small to tell, so I'm wondering if someone can show me how to check that or find a better way to go about doing it.
Provided that this new circuit is correct where:
R = 1262.39 Ω; L = 100 mH; and C ≈ 253.3 nF
.: ζ = 1 = 1 and the circuit is critically damped.
I know that the general form for the solution for a critically damped circuit is:
x(t) = B1*e^(-ζωt) + B2*t*e^(-ζωt)
Again, I plugged in ζ and ω but don't know how to solve for B1 and B2.
As far as the overdamped circuit goes... I haven't gotten anywhere because I am a bit confused on whether or not I'm on the right track with what I've done so far and want to see what you think.
I was going to talk to my prof after class, but he wasn't there today... go figure. Thanks for the help in advance.
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