How to derive equation from RLC circuit?

[Nat3]

Homework Statement

http://imageshack.com/a/img580/682/z3mt.jpg

Derive a linear differential equation for the above LTI system.

Homework Equations

[itex]i_C=C\frac{dV_C(t)}{dt}[/itex]

[itex]V_L=L\frac{di_L(t)}{dt}[/itex]

The Attempt at a Solution

Using KVL, I can get the following equation:

[itex]V_{in}(t)=L\frac{di(t)_L}{dt}+i(t)R_A+\int\frac{i(t)}{C}dt+V_o(t)[/itex]

However, I don't know where to go from here. All of the differential equations describing LTI systems in my textbook look like:

[itex]ay'' + by' + cy = g(x)[/itex]

Or something similar to that, and then we factor out the y to get something like

[itex](aD^2+bD+c)y = g(x)[/itex]

Then we factor what's in the parenthesis to find the characteristic roots.

Any advice on how to proceed?

 

[Pythagorean]

so what operation could you do to that whole equation to get it into a form you feel comfortable with?

 

[Nat3]

Well, I thought about differentiating it to get rid of the integral, which results in:

[itex]\frac{dV_{in}(t)}{dt}=L\frac{d^2i(t)}{dt^2}+\frac{di(t)}{dt}R_A+\frac{i(t)}{C}+\frac{dV_{o}(t)}{dt}[/itex]

But then I don't know where to go from there.. I think it's the [itex]V_{in}[/itex] and [itex]V_o[/itex] terms that are getting me tripped up.

 

[gneill]

Is the objective to find a D.E. that describes Vo(t) when there's a driving function of Vin(t)? Note that the current i(t) depends on R_B as well as R_A. So R_B needs to appear in your equation.

If you find the D.E. for i(t) given Vin(t) (ignoring Vo for the moment), then you can convert it to a D.E. for Vo(t) easily enough since Vo(t) = i(t)*R_B.

 

[Pythagorean]

Well, I thought about differentiating it to get rid of the integral, which results in:

[itex]\frac{dV_{in}(t)}{dt}=L\frac{d^2i(t)}{dt^2}+\frac{di(t)}{dt}R_A+\frac{i(t)}{C}+\frac{dV_{o}(t)}{dt}[/itex]

But then I don't know where to go from there.. I think it's the [itex]V_{in}[/itex] and [itex]V_o[/itex] terms that are getting me tripped up.

That's what I would have done. I don't know the convention in circuit analysis, but:

[tex]\Delta V = V_{out} - V_{in}[/tex]

And then you're measuring the potential difference.

 

[Nat3]

Is the objective to find a D.E. that describes Vo(t) when there's a driving function of Vin(t)? Note that the current i(t) depends on R_B as well as R_A. So R_B needs to appear in your equation.

If you find the D.E. for i(t) given Vin(t) (ignoring Vo for the moment), then you can convert it to a D.E. for Vo(t) easily enough since Vo(t) = i(t)*R_B.

The instructions say to derive a linear differential equation describing the circuit I posted, where the equation should be expressed in terms of the differentiation operator D and the circuit parameters (L, RA, RB, etc.)

The next problem is to find the characteristic roots and modes of the system, so I'm pretty sure that I need to get the equation in the standard form of [itex]ay′′+by′+cy=g(x)[/itex], I'm just not sure how to get there :(

 

[Pythagorean]

I think you have it already. You're not asked to solve it right?

edit: gotchya, missed that...