Simple RLC Circuit problem

Raihan

Please help me to solve this RLC circuit problem. I am completely confused.If you give me the direct answer it would be much appreciated.
For the series RLC circuit in Figure, find the input/output
difference equation for

1.[tex] y(t)=v_{R} [/tex]
2.[tex] Y(t)=i(t) [/tex]
3.[tex] y(t)=v_{L} [/tex]
4.[tex] y(t)=v_{C} [/tex]

I have attached the Circuit diagram in a .jpg file.

Attached Thumbnails

 

berkeman

You must show your own work in order for us to help you (PF homework forum rules). Would KCL or KVL be the best way to start?

Raihan

Hey first I tried taking the KVL around the loop
something like
[tex] -x(t) + v_c(t) + v_L (t) + v_R (t) = 0 [/tex]----(1)
replaced v_L(t) with first order Ldi_L(t)/dt and make an equation for
[tex] v_C(t) [/tex]
and then as its in series I tried to write a function for
[tex] i_L(t) = \frac {v_R (t)} R [/tex]------(2)
and for [ tex ] v_R(t)/R=C \frac {dv_c(t)} {dt} [/tex]----(3)
Then tried sub (3) in (1)
and got
[tex] v_C(t) = x(t) - \frac {L} {R} dv_R(t)/dt - v + R(t)[/tex]----(4)
and then tried sub it i eqn 3. and didnt come up with a satisfactory result.
Please help.
Thanks

SGT

In series circuits you should always use [tex]v_C[/tex] as the independent variable (and [tex]i_L[/tex] in parallel circuits).
Since the current is the same for all elements, write [tex]v_L[/tex] and [tex]v_R[/tex] as functions of the current. Finally write the current as a function of [tex]v_C[/tex].

Raihan

Thank you very much for your info SGT, would you please help little bit more.

SGT

Make the substitutions I suggested in your equation 1. More help will only be provided after you show some work.

Raihan

I tried And I am not going anywhere. please help

SGT

Post what you have done and I will give you more hints.

Raihan

heres what I got so far.. please help after this point..
thanks

Attached Thumbnails

 

SGT

In the second equation don't use the integral term. Keep it as [tex]V_C(t)[/tex].
In the two other terms replace i by [tex]C\frac{dV_C}{dt}[/tex]. You get a second order equation in [tex]V_C[/tex]

Raihan

Would you please not mind to show me please. I have tried this so far. please help after this.
thanks

SGT

The rules of the forum are that you must do your work. We only give hints. Rewrite the second equation with the suggestions I made and post it here.

serienumerica

The easiest way to solve any RCL circuit with an input vs(t) is by a difference equation.

Let curr= (q1-q0)/dt


q2=2.*q1-q0 + dt**2*( -q1/(L*c) -(R/L)*curr +vs(t-dt) ).

Then everything else follows ,

Vc(t) = q2/C , VL = L * ( q2-2*q1+q0)/dt^2 , VR = R*(q2-q1)/dt
SEE http://www.geocities.com/serienumerica/RCLfree.doc