RLC circuit with very large Resistance

[gonzalesdp]

Homework Statement

Consider an RLC circuit with very large Resistance.
a) when t = 0, q(0) = 0
b) When t = T_L q(T_L) = 2Q(1+e)^-1cosh(T_L/T_C)
Where T_L and T_C are the inductive and capacitive time constants, respectively.

Show that the charge as a function of time is given by:

q(t) = Q(1+e)^-1e^-t/TC + Q(1+e^-1)^-1e^-t/TLe^t/TC

Homework Equations

There is a hint that you might need to use the binomial theorem.

The Attempt at a Solution

I'm not sure where to begin with this one.

 

[vela]

Start by writing down the differential equation that the charge q satisfies, or if you already have it, the general solution to that equation.

 

[gonzalesdp]

Thanks, I have gotten that far since I posted.

I have the diff eq as

L dq^2/dt +R dq/dt +q/c = 0

I'm solving using the general form ay" + by' +cy = 0

the auxiliary equation is

am^2 + bm +c = 0

From what I understand is there are three cases to this solution:
1: Distinct real roots when b^2 - 4ac > 0
Solution: y =c_1exp(xm_1) +c_2exp(xm_2)
2: Reapeated Real roots when b^2 - 4ac = 0
Solution: y = c_1exp(xm) +xc_2exp(xm)
3: conjugate complex roots when b^2 -4ac < 0
Solution: y = exp(alphax)[c_1cos(Bx) + c_2sin(Bx)
alpha = -b/2a
B = (1/2a)(4ac-b^2)^1/2
Usually the third case is the one to use, since circuits are only useful with small resistances.
However since one of the conditions is a "very large resistance". I've been trying to use the first two.

 

[vela]

What you want to do is use the quadratic equation to write down what the roots are, and then you want to use the fact that R is large to approximate the radical. That's where the hint comes in.

 

[rwisz]

I would approach this problem by first defining the RLC circuit and its components. The RLC circuit consists of a resistor (R), inductor (L), and capacitor (C) connected in series. The resistance (R) is a measure of how much the circuit impedes the flow of current, while the inductance (L) and capacitance (C) are measures of the circuit's ability to store energy in the form of magnetic and electric fields, respectively.

Next, I would consider the given initial conditions (a) and (b) and use them to write out the equations for the charge (q) at t = 0 and t = TL. From there, I would use the given equations and the binomial theorem to manipulate and simplify the expressions for q(0) and q(TL) into the desired form.

Finally, I would use my knowledge of the behavior of an RLC circuit with large resistance to explain how the resulting charge function relates to the circuit's response. In an RLC circuit with very large resistance, the current will decrease rapidly due to the high resistance, resulting in a slower discharge of the capacitor. This means that the charge will decrease slowly over time, as shown in the resulting charge function.

In conclusion, the charge function q(t) = Q(1+e)-1e-t/TC + Q(1+e-1)-1e-t/TLet/TC is a result of the initial conditions and the behavior of an RLC circuit with very large resistance. The binomial theorem is used to simplify the expressions, and the resulting function shows how the charge decreases over time in this type of circuit.