# How to derive equation from RLC circuit?

• Engineering
Nat3

## Homework Statement

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

Derive a linear differential equation for the above LTI system.

## Homework Equations

$i_C=C\frac{dV_C(t)}{dt}$

$V_L=L\frac{di_L(t)}{dt}$

## The Attempt at a Solution

Using KVL, I can get the following equation:

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

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

$ay'' + by' + cy = g(x)$

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

$(aD^2+bD+c)y = g(x)$

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

Any advice on how to proceed?

## Answers and Replies

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

• 1 person
Nat3
Well, I thought about differentiating it to get rid of the integral, which results in:

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

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

Mentor
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 RB as well as RA. So RB 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)*RB.

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

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

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

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

$$\Delta V = V_{out} - V_{in}$$

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 RB as well as RA. So RB 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)*RB.

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 $ay′′+by′+cy=g(x)$, I'm just not sure how to get there :(

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

edit: gotchya, missed that...

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