Understanding Capacitor Transients: Solving for Initial Conditions

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The discussion focuses on understanding capacitor transients and determining initial conditions after switching in a circuit. It clarifies that while the steady-state current is zero, the initial current immediately after switching is based on the capacitor's voltage divided by resistance. The participants calculate the time constant (tau) and derive the initial current, concluding that it is 12mA. They also discuss the exponential decay of current over time, noting that after one second, the current decays to approximately 4.415mA. The conversation concludes with a better understanding of how initial conditions work in capacitor circuits.
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Homework Statement


Ok so I have attached the question I am working on.
Capacitor transient question.jpg

Basically when I have the circuit before or after switching in steady state I believe the current will equal zero. However if this is the case, how do I determine what happens immediately after switching?

Homework Equations


i(t) = A + Be-t/tau
tau = RC
(capacitor in steady state becomes an open circuit)

The Attempt at a Solution


tau = 1000x1000x10-6 = 1
i(0-) = 0 = A+B
i(0+) = 0 = B
therefore A = 0?
 
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R0CC0 said:

Homework Statement


Ok so I have attached the question I am working on. View attachment 52539
Basically when I have the circuit before or after switching in steady state I believe the current will equal zero. However if this is the case, how do I determine what happens immediately after switching?

Homework Equations


i(t) = A + Be-t/tau
tau = RC
(capacitor in steady state becomes an open circuit)

The Attempt at a Solution


tau = 1000x1000x10-6 = 1
i(0-) = 0 = A+B
i(0+) = 0 = B
therefore A = 0?

i(0+) is not zero. You have a fully charged capacitor connected to a resistor at that time. You have the correct overall equation:

i(t) = A + Be-t/tau

Now you just need to solve it given the initial condition of the charged capacitor which starts to discharge through the resistor after t=0...
 
berkeman said:
i(0+) is not zero. You have a fully charged capacitor connected to a resistor at that time.
sorry that was meant to be i(∞) = 0 = B

i(0+) is what I want to know how to work out.
 
R0CC0 said:
sorry that was meant to be i(∞) = 0 = B

i(0+) is what I want to know how to work out.

That may be true, but it's not relevant to the solution really. What is i(0+)?

Can you visualize what happens? Some initial current flows based on the initial voltage and the resistance, and that current decays according to the equation. And since you correctly calculated tau as 1 second, and they ask for something after 1 second, can you guess the answer?
 
Ignore this post I did it before I saw your last one sorry
Actually sorry I see where I went wrong that should be i(∞) = 0 = A
But that will still give me A+B=0 initially thus giving B = 0??
 
R0CC0 said:
Ignore this post I did it before I saw your last one sorry
Actually sorry I see where I went wrong that should be i(∞) = 0 = A
But that will still give me A+B=0 initially thus giving B = 0??

No. What is the initial current? It's the initial cap voltage divided by the resistance, right? In fact, the current at any time is the cap voltage divided by the resistance, right?

Use that and e0=____ (fill in the blank) to figure out the constants...
 
Ok so the initial current immediately after switching will be 12mA?
Therefore B will equal 12x10-3?
Thus i(t) = 12x10-3e-t?
 
R0CC0 said:
Ok so the initial current immediately after switching will be 12mA?
Therefore B will equal 12x10-3?
Thus i(t) = 12x10-3e-t?

You left the time constant divider out of the exponent. Even though it is "1", you should still show something there to show that the units work. So you could put a 1s in the denominator of the exponential term, for example. Other than that, it looks good. What is the answer at 1 second?
 
Ok, I'll remember that for next time.
I got i(1) = 4.415mA
 
  • #10
R0CC0 said:
Ok, I'll remember that for next time.
I got i(1) = 4.415mA

Good! And notice how after 1 second, the value has decayed to 1/e of its initial value. That's how those exponential decays work. :smile:
 
  • #11
Ahh ok that's actually really easy now that I see how it works. I never understood how the initial (0+) condition worked before.
Thanks for all the help :)
 

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