Solving Series RLC Circuit Problems: Input/Output Equations & Diagram

[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.$$y(t)=v_{R}$$
2.$$Y(t)=i(t)$$
3.$$y(t)=v_{L}$$
4.$$y(t)=v_{C}$$

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

 

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

 

[SGT]

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

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

 

[Raihan]

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

 

[SGT]

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

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]

solution so far

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

 

[SGT]

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

 

[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