How Long Does It Take for a High-Resistance RLC Circuit to Lose Half Its Energy?

[Nivlac2425]

Homework Statement

Consider a simple RLC circuit. If R >> (4L)/C , compute approximately how long it takes for the circuit to lose half of the initial energy stored in the system.

(There is no circuit drawing given; the problem assumes a general RLC circuit)

Homework Equations

The Attempt at a Solution

I have already worked out the differential equation that is associated with RLC circuits, and got an expression for

ω=[iR±sqrt(4L/C - R²)]/(2L)

and since R >> (4L)/C ,

ω= 0, (iR)/L

We consider the non-trivial case, ω= (iR)/L
Then it turns out that

Q=A_oexp(-Rt/L) , and

dQ/dt = I = -R/L A_oexp(-Rt/L)

From these, I solved for the total energy stored in the RLC system (capacitor energy + inductor energy) and got:

E_tot= A_o² [(L+CR²)/2LC] exp(-2Rt/L)

Then at this point, I'm sort of stuck and not sure where to go. Although I can derive the average power in a cycle by:

P= E_tot/t_diss = A_o²(R/L)[(L+CR²)/2LC]exp(-2Rt/L)
where t_diss is the energy dissipation time, equal to L/R

From here, I can integrate the power expression to come up with a relation between the energy and initial energy,

E= (E_o/2)exp(-2t/t_diss)

And then, finally attempting to answer the question, I set E=E_o/2 and get t=0.

Somehow this doesn't make much sense to me and makes me believe that I did something very wrong in my work above. In my class lecture notes, a similar derivation was done assuming that the resistance was small (R << (4L)/C), but now the resistance is huge.

From inspection, I can see that the current through the circuit must be very small and that perhaps the capacitor is discharging very slowly as well. But what does this mean about the energy dissipation? Assuming that my derivation above is correct (it is probably not), the circuit can only store half of the total energy predicted from the E_tot expression.

Can anyone make some sense of this?
Thanks!

 

[ehild]

The Attempt at a Solution

I have already worked out the differential equation that is associated with RLC circuits, and got an expression for

ω=[iR±sqrt(4L/C - R²)]/(2L)

and since R >> (4L)/C ,

ω= 0, (iR)/L

We consider the non-trivial case, ω= (iR)/L
Then it turns out that

Q=A_oexp(-Rt/L) , :

You lost a factor 2 from the denominator of ω

ehild

 

[Nivlac2425]

You lost a factor 2 from the denominator of ω

ehild

Thanks for the reply,
I believe it was

ω=(iR±iR)/2L, (after using R>>(4L)/C

ω=2iR/2L

ω=iR/L

Correct me if I'm wrong with this

Thanks!

 

[ehild]

Q=A_oexp(-Rt/L) , and

dQ/dt = I = -R/L A_oexp(-Rt/L)

From these, I solved for the total energy stored in the RLC system (capacitor energy + inductor energy) and got:

E_tot= A_o² [(L+CR²)/2LC] exp(-2Rt/L)

Then at this point, I'm sort of stuck and not sure where to go.

You are right, Q(t)=A_oexp(-Rt/L)
The formula you got for the total energy at time t is also correct:

E_tot= A_o² [(L+CR²)/2LC] exp(-2Rt/L). The initial stored energy is E_tot(0)=A_o² isn't it? The question is at what time becomes the energy half of the initial value.

ehild

 

[Nivlac2425]

You are right, Q(t)=A_oexp(-Rt/L)
The formula you got for the total energy at time t is also correct:

E_tot= A_o² [(L+CR²)/2LC] exp(-2Rt/L). The initial stored energy is E_tot(0)=A_o² isn't it? The question is at what time becomes the energy half of the initial value.

ehild

At t=0, E_tot(0)= A_o² [(L+CR²)/2LC] ?
I thought this would be true since exp(-2Rt/L) is the exponential decay term and the "coefficient" would be the initial value. Setting t=0 also makes the exp term become 1.

Assuming this is true, I could set E_tot= E_o/2 where E_o is the intial energy, and solve for time t.

I would get

A_o²/2 [(L+CR²)/2LC] = A_o² [(L+CR²)/2LC] exp(-2Rt/L)

and t= (L/2R)ln(2) , or t= (t_diss/2) ln(2) when t_diss=L/R

Comparing this to the solution for small resistance, it takes a shorter amount of time to drain the energy to half when we have a large resistance. This somehow makes more sense than my previous solution