Calculating Time Constant and Effective Resistance for a Parallel RC Circuit

[galoisien]

I am trying obtain some equations for an RC circuit that has a resistor R1 after a battery, which then splits into a capacitor C and a resistor R2 (the circuit closes after this point).

When I try to seek help online, the resources are quite unhelpful. I cannot seem for example, obtain any formulas for the time constant for a capacitor connected in parallel to a resistor. When I try to look up the effective resistance of a capacitor I get resources that talk about alternating current circuits!

I know that capacitors start at zero resistance in the beginning and increase in resistance later. I cannot for the life of me however seem to plot the effective resistance (reactance) of a capacitor.

At tau/2, what is the resistance of a capacitor? How would I obtain RC constants?

 

[vela]

The reactance of a capacitor is a function of frequency. That's why you're finding all the discussions involving AC circuits. A step-function input, which is what I assume you have here as you're talking about a time constant, is comprised of an infinite combination of frequencies; there is no single number describing the reactance of a capacitor.

One thing you can do is use Thevenin and Norton equivalents to simplify the circuit to a basic RC circuit with only one resistor. If you're not familiar with these, just write down the loop equations, i.e. apply Kirchoff's laws, and solve them. It's pretty straightforward.

 

[galoisien]

There has to be a value describing the resistance of a capacitor in a DC circuit, because if it's connected in parallel to a resistor, that resistance describes how much current is flowing through the resistor (as opposed to charging the capacitor).

I'm trying to analyse it as a current divider. The lack of any formulas is really annoying! What is the time-dependent resistance of a capacitor?

"If you're not familiar with these, just write down the loop equations, i.e. apply Kirchoff's laws, and solve them. It's pretty straightforward."

I can't figure out the time constants for this circuit... I need values as a function of time.

 

[galoisien]

Okay every time they talk about Kirchoff's law for an RC circuit, I only see the example for a single resistor in series with a capacitor -- much too simple

OR multiple capacitors with multiple batteries and inductors and everything with a whole mess of equations -- much too complex.

where are some resources where I can get help for a two-resistor, one-capacitor, two-loop, one-battery circuit? I don't know where to start to set up a differential equation.

 

[vela]

There has to be a value describing the resistance of a capacitor in a DC circuit, because if it's connected in parallel to a resistor, that resistance describes how much current is flowing through the resistor (as opposed to charging the capacitor).

I'm trying to analyse it as a current divider. The lack of any formulas is really annoying! What is the time-dependent resistance of a capacitor?

The problem is, your approach won't work because there is no such thing as a "time-dependent resistance of a capacitor."

The basic relationship between the voltage v across a capacitor and the current i into the capacitor C is

$$i = C\frac{dv}{dt}$$

"If you're not familiar with these, just write down the loop equations, i.e. apply Kirchoff's laws, and solve them. It's pretty straightforward."

I can't figure out the time constants for this circuit... I need values as a function of time.

 

[galoisien]

If I am trying to figure out how much current will pass through a capacitor given a resistor in parallel for some particular point in time.

 

[vela]

I know, and I've already told you how to analyze this type of problem. If you're just going to ignore the suggestion, there's not much more I can say.