Finding the ODE that describes this circuit + find its transfer function

[rugerts]

Homework Statement

Find ODE that describes circuit in terms of capacitor voltage + find its transfer function (Vc/Vs)

Relevant Equations

Ohm's Law; Kirchhoff's Current Law and Voltage Law; Voltage across inductor; Current through
capacitor

As you can see, I've tried using KCL at node A to find the 2nd order ODE that describes this circuit in terms of the capacitor voltage. The problem I run into, however, is that I can't find anything to put the node voltage at A in terms of. I've tried (not shown here) doing mesh current as well, but ran into similar problems. Can anyone point me in the right direction? I'm certain that after this I'll be able to find the transfer function by applying a Laplace transform since the initial conditions are 0 to make things simple.

 

[rude man]

View attachment 240949View attachment 240950

2nd eq. for i1 is wrong (polarity).
Then substitute until you have VC = VC(VS).

 

[rugerts]

2nd eq. for i1 is wrong (polarity).
Then substitute until you have VC = VC(VS).

Does this reconcile the issue that I can't find an expression for Va? Also, could you expand on why i1 would be wrong in terms of polarity?

 

[rude man]

Does this reconcile the issue that I can't find an expression for Va?

Probably not.

Also, could you expand on why i1 would be wrong in terms of polarity?

If current flows from A to B then A is higher in voltage than B.

The way I would approach this problem is:
assign Laplace transform to all components. So L becomes Z=sL and C becomes Z=1/sC.

Then sum currents to zero at every dependent node (in your case 2). Don't use currents explicitly. For example, (V1-V2)/Z1 = (V2-V3)/Z2 + (V2-V4)/Z3 that sort of thing. This gets you your transfer function V(C)/V(S) = F(s) immediately.

To get the ODE I would take V(C) = F(s)V(S), multiply both sides by s, then go back to the time domain by s → d/dt and $s^2$ → $d^2/dt^2$.

Of course there are ways of staying in the time domain but I'm not sure how to best do that so others might help you there.

 

[NascentOxygen]

I've tried using KCL at node A to find the 2nd order ODE that describes this circuit in terms of the capacitor voltage.

You have written the equation for $i_c$, the current through the capacitor and through the inductor, so the next step is to say

$$v_A\ =\ v_c\ +\ L\cdot\dfrac{di_c}{dt}$$

 

[rugerts]

You have written the equation for $i_c$, the current through the capacitor and through the inductor, so the next step is to say

$$v_A\ =\ v_c\ +\ L\cdot\dfrac{di_c}{dt}$$

See, I thought this was the case but couldn't justify it properly to myself. By what physical law is this true might I ask? Is this a consequence of Kirchhoff's Voltage Law?

 

[NascentOxygen]

By what physical law is this true might I ask? Is this a consequence of Kirchhoff's Voltage Law?

The instantaneous voltage across multiple elements = the sum of their individual instantaneous voltages