How can I solve the following problem: "Show that the voltage Vc (t) across the capacitor as a function of time goes as 4.125(1 - exp(-t/0.1375)) volts. When the steady state has been reached, what is the power dissipated in each resistor and the energy stored in the capacitor?"
Circuit diagram:
formatting link
I know how to do this for a normal RC circuit, without the 22kOhm resistor in parallel around the capacitor. I think I need to somehow use Kirchhoff laws to figure this out, but I could really use a hand getting started.
There is only one node in this circuit that is not fixed. Let's call this node A and the node voltage va. Apply KCL at this node: -(6V - va)/10k + va/22k + 20uF* dva/dt = 0
One way to look at it is that the cap is being fed from a voltage source that has a series resistance of 10k || 22k, and a voltage that is equal to the voltage divider voltage.
Notice that 6*22/(22+32) = 4.125, and that (10k||22k)*20e-6 = 0.1375.
If your textbook has mentioned Thevenin's theorem, that is simply the more formal way to do this. It states that if you have a two-terminal passive element in a circuit, you can find another circuit that consists of the device, a single resistor, and a single current source, all in series. The value of the voltage source is the thevenin equivalent voltage, and the value of the resistor is the thevenin equivalent resistance. The method to do this is trivial:
1) Remove the component you are analyzing from the schematic (in this case C), and determine the voltage between the place where the leads previously connected. In this case, it is V = 6 * (22/(10+22)). That is the thevenin equivalent voltage.
2) Now, in that modified circuit, replace the voltage source with a wire, and compute the resistance between those same two points. In this case, it is 10k || 22k = 6.875k. This is the thevenin equivalent resistance.
Your circuit is then equivalent to
4.125V ------[6.875k ohms]------[20uF]-----GND
from the point of view of the cap, which is far easier to analyze.
Regarding the resistors, when the cap is in its 'steady state', it effectively disappears from the circuit. So, use the equation for power given voltage and resistance: (6-4.125)^2/10000 and 4.126^2/22000.
--
Regards,
Bob Monsen
Nature does not at once disclose all Her mysteries. - Lucius Seneca (Roman
philosopher)
Whoa.... is this a homework problem? The only person who should be doing your homework is you, but I don't mind helping.
Break the problem down.
The 10k and 22k resistors form a voltage divider. What is the final voltage across the capacitor? (Pretend the capacitor isn't there.)
When the current is first applied (switch not shown) what was the voltage across the capacitor?
At switch on (t = 0), the capacitor is effectively a short circuit. At t = infinity, it is effectively an open circuit.
6 * 10/22 = ? Looks like your 4.125 isn't quite right. Maybe that's the problem.
What is exp(-t) when t is large? You can use a calculator for that. Hint: On Microsoft Windows calculator, use Inv (the check box) ln (the natural logarithm function)
Yep... My apologies. I realised it right after I posted, so I'll have to eat it. I 'fess up, I made a boo-boo.My granddaughter called as I was about to re-read what I'd written, so I posted in haste.
The equation is ..
Of course. I didn't engage brain before opening my big mouth, so call me an idiot (this time). :-)
Look up Thevenin's theorem. It says that your circuit can be replaced by one with a series resistor equal to the two of them in parallel and a battery voltage that is reduced by the voltage divider effect. Old man had the right idea but Thevenin's is simpler.
ElectronDepot website is not affiliated with any of the manufacturers or service providers discussed here.
All logos and trade names are the property of their respective owners.