How do I correctly position A1 and A2 in overdamped parallel RLC circuit?

[sevens]

For some reason i can't get my A1 and A2 in the correct spots. I get the correct value of these two, but if someone could point out where I've gone wrong, i would apreciate it.

V(0) = 125v
i(0) = 0A

value of the inductor is 1.25h
value of the resistor is 2 Ohms
value of the capacitor is 50mf ( yes i know huge... but its theory, not practice)

the final answer is to find the voltage across the capacitor

Thanks any advice will help.

 

[doodle]

Your answer looks good to me.

 

[berkeman]

Why do you say \omega_c = \frac{1}{2} Cv^2 ?

What are the units of this \omega ?

 

[sevens]

The units of W are Jouls, its energy stored in the capacitor, and i use that formula to find the initial voltage in the capacitor. The answer i got is V(t) and then under that i wrote what the answer should be "actual answer" i keep getting A1 and A2 in the wrong place for several of my circuit analysis. Am i missing some tiny detail.?

 

[berkeman]

Oh, you meant W like work or energy, not \omega like angular frequency. I get that now.

i use that formula to find the initial voltage in the capacitor

But you're given v(0) = 250V as part of the problem statement. And the "final" voltage across the capacitor will be zero. What exactly does the question ask for? It wants the capacitor voltage when?

 

[sevens]

constants in wrong spot

Yes its true that final voltage across the capacitor will be zero given enough time. The question wants an equation that can be defined in terms of t. So if you take the found equation for example t =2 sec the capacitor will be zero. however if i take t=.0001s there will be some voltage left in the capacitor.


Originaly i was told how much energy was stored in the capacitor which is why i used Wc to find initial Voltage.

overal the problem is that my A1 and my A2 are with the wrong positions. they are the correct values themselves but if you see where i wrote "actual answer" that's the position they should be in.

Thanks for the help
this problem will crack sooner or later

 

[SGT]

How did you get i_R(0) = 0?
i_R(0) = \frac{V_C(0)}{R}.

 

[berkeman]

He just said i(0)=0. I think it's just one of the initial conditions, like if you were closing a switch at t=0 to connect the cap into the rest of the circuit.

 

[SGT]

He just said i(0)=0. I think it's just one of the initial conditions, like if you were closing a switch at t=0 to connect the cap into the rest of the circuit.

Since the initial energy in the inductor is 0, it follows i_L(0) = 0.
The resistor is in parallel with the capacitor, which has an initial voltage of 125 V. So, the initial current in the resistor is \frac{125}{R}A.

 

[berkeman]

Since the initial energy in the inductor is 0, it follows i_L(0) = 0.
The resistor is in parallel with the capacitor, which has an initial voltage of 125 V. So, the initial current in the resistor is \frac{125}{R}A.

You've worked with initial conditions in differential equations before, right? As usually stated, the current in the resistor would be zero at t=0-, and the value you state at t=0+. Standard stuff.

 

[doodle]

Hmm... SGT is right. If Vc(0+) = 125V, then Ir(0+) = 125/2 = 62.5A since current across the resistor can change suddenly. This means that Ic(0+) = -Ir(0+) = -62.5A which then gives the actual answer.

 

[SGT]

You've worked with initial conditions in differential equations before, right? As usually stated, the current in the resistor would be zero at t=0-, and the value you state at t=0+. Standard stuff.

When you replace initial conditions in the general solution, you use the conditions at t = 0+. The conditions at t = 0- are irrelevant, except in the case of energy storage elements, like capacitors and inductors (or masses and sprigs). For those elements the conditions at t = 0- and t = 0+ are the same, except in the case of an impulsional excitation.

 

[acceler8]

i hate it when experiments don't work the way they should

soooooooooo frustrating!

xxxx Gareth