Proof of an expression concerning an RLC circuit

[batshwa]

Hi,

How can you prove that, in a damped RLC circuit, the logarithmic decrement equals the period times the damping factor:

$$\delta=T\zeta$$

(I'm using $$\delta=\ln\frac{x(t_n)}{x(t_n+T)}\quad \zeta=\frac{R}{2L}$$ )

Thanks in advance!

 

[berkeman]

What equation do you get for v(t) for a parallel RLC circuit that has initial conditions v(0) = Vo and I(0) = 0 in the inductor?

 

[batshwa]

I don't quite understand... but it seems to be my fault -- I forgot to mention that I'm talking about a series RLC circuit, not a parallel one.

 

[berkeman]

Okay, but my question still applies, just with different initial conditions. I'm trying to see what equation you have for the ringout response of the RLC circuit. I'm not familiar with the equations you wrote above, so I'm hoping that once you write the equation for the response of the RLC circuit, I'll be able to map that familiar equation onto what you are asking above. (Or maybe you will see the answer to your question too, once you write the equation for how the RLC circuit works...)

 

[batshwa]

I'm sorry, but I'm a total newbie regarding electric oscillations and RLC stuff. I don't quite underastand what equations you have in mind. I don't think you meant the differential equation of a damped oscillation, did you?
I hope there is no misunderstanding: I believe the expression above is generally valid (which is why I'm a bit troubled by your mentioning initial conditions).

Excuse any awkwardness my ignorance in physics is causing :) Can you recommend me any special (introductory) documents about RLC circuits on internet (apart from the Wikipedia article, which is somewhat unclear for me)?

Thanks a lot!

 

[berkeman]

I'm sorry, but I'm a total newbie regarding electric oscillations and RLC stuff. I don't quite underastand what equations you have in mind. I don't think you meant the differential equation of a damped oscillation, did you?

No worries. Yes, I was referring to the differential equation for the voltage (parallel RLC ) or current (series RLC) as a function of time. I'm not familiar with the term you mentioned in your first post ("logarithmic decrement"), but I'd imagine it is one way of expressing the damping factor of the circuit. I was hoping that by seeing how you wrote the equations for the V(t) or I(t) for the circuit, I'd be able to infer what the logarithmic decrement was and maybe answer your question.

I googled rlc circuit tutorial damped, and got lots of great hits. Try that google to see if some of those web pages help out. -Mike-

 

[batshwa]

Right, this is the differential equation of my RLC circuit:

$$L\ddot q+R\dot q+\frac 1Cq=0$$

Its solution is the following:

$$q(t)=\hat q\:e^{-\zeta t}\sin\left(\omega t+\varphi_0\right)$$

 

[batshwa]

where $$\hat q$$ symbolizes the maximum charge (or peak charge?).

I'll continue searching on Google, but until now I have found that expression I've posted here only once, unfortunately. It doesn't seem to be current.

 

[berkeman]

You're getting closer, but yes, for the series RLC circuit, solve for the series current. You get an integral term, a differential term, and a linear term, then differentiate to get the DiffEq to solve. You should see some things that you can manipulate to answer your original question...