# [SOLVED]Circuit DE

#### dwsmith

##### Well-known member
Determine the differential equation relating $$v_i(t)$$ and $$v_0(t)$$ for the RLC circuit in the figure.

Would this just be
$v_i(t) = 3i + \frac{di}{dt} + 2\int i(t)dt$
but $$v_0 = 2\int i(t)dt$$. Do I need write it as $$v_0$$ or as $$2\int i(t)dt$$?

#### chisigma

##### Well-known member
Determine the differential equation relating $$v_i(t)$$ and $$v_0(t)$$ for the RLC circuit in the figure.
View attachment 2100
Would this just be
$v_i(t) = 3i + \frac{di}{dt} + 2\int i(t)dt$
but $$v_0 = 2\int i(t)dt$$. Do I need write it as $$v_0$$ or as $$2\int i(t)dt$$?
Correct is...

$\displaystyle v_{i} (t) = 3\ i + \frac{d i}{d t} + 2\ \int_{- \infty}^{t} i(\tau)\ d \tau$

Kind regards

$\chi$ $\sigma$

#### dwsmith

##### Well-known member
Correct is...

$\displaystyle v_{i} (t) = 3\ i + \frac{d i}{d t} + 2\ \int_{- \infty}^{t} i(\tau)\ d \tau$

Kind regards

$\chi$ $\sigma$
Why do you have bounds of $$(-\infty, t)$$?

#### chisigma

##### Well-known member
Why do you have bounds of $$(-\infty, t)$$?
... the voltage across a capacitor is proportional to the amount of charge You have pumped on it. If You don't like the term $\infty$ it is possible to write $\displaystyle v_{C} (t) = v_{c} (0) + \frac{1}{c}\ \int_{0}^{t} i(\tau)\ d \tau$...

Kind regards

$\chi$ $\sigma$

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#### dwsmith

##### Well-known member
... the voltage across a capacitor is proportional to the amount of charge You have pumped on it. If You don't like the term $\infty$ it is possible to write $\displaystyle v_{C} (t) = v_{c} (0) + \frac{1}{c}\ \int_{0}^{t} i(\tau)\ d \tau$...

Kind regards

$\chi$ $\sigma$
Is it possible to solve for $$v_0(t)$$ using a Laplace transform?

#### Ackbach

##### Indicium Physicus
Staff member
Is it possible to solve for $$v_0(t)$$ using a Laplace transform?
Yes. In fact, electrical engineers will typically write down circuit elements on the diagram with their Laplace-transformed expressions, so that the very first equation they write down is the LT'ed equation.

#### dwsmith

##### Well-known member
Yes. In fact, electrical engineers will typically write down circuit elements on the diagram with their Laplace-transformed expressions, so that the very first equation they write down is the LT'ed equation.
I don't understand. Can you demonstrate what you mean?

Suppose $$v_i(t) = e^{-3t}\mathcal{U}(t)$$. Then how can we get back $$v_0$$ from a Laplace transform?

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#### dwsmith

##### Well-known member
If I take the Laplace transform of
$e^{-3t}\mathcal{U}(t) = 3i + \frac{di}{dt} + 2\int_{-\infty}^ti(\tau)d\tau,$
I get
$i(t) = -\frac{3}{2}e^{-3t} + 2e^{-2t} - \frac{1}{2}e^{-t}.$
How do I get to $$v_0(t)$$ for $$t > 0$$?

#### Ackbach

##### Indicium Physicus
Staff member
I don't understand. Can you demonstrate what you mean?
Sorry for the delay. This is called $s$-domain circuit analysis. You have the $s$-domain impedances (kinda like what you have in steady-state sinusoidal analysis with phasors):
\begin{align*}
Z_{R}&=R \\
Z_{L}&=Ls \\
Z_{C}&= \frac{1}{Cs}.
\end{align*}
These come straight from the V-I characteristics of resistors, capacitors, and inductors: just LT those relationships, and compare with $V=IZ$, Ohm's Law, to get the $s$-domain impedances (these are defined with the initial conditions zeroed out).

There are tricks for dealing with initial conditions as well - you usually insert a voltage or current source to handle that. Here is a good summary of the method.

#### dwsmith

##### Well-known member
Sorry for the delay. This is called $s$-domain circuit analysis. You have the $s$-domain impedances (kinda like what you have in steady-state sinusoidal analysis with phasors):
\begin{align*}
Z_{R}&=R \\
Z_{L}&=Ls \\
Z_{C}&= \frac{1}{Cs}.
\end{align*}
These come straight from the V-I characteristics of resistors, capacitors, and inductors: just LT those relationships, and compare with $V=IZ$, Ohm's Law, to get the $s$-domain impedances (these are defined with the initial conditions zeroed out).

There are tricks for dealing with initial conditions as well - you usually insert a voltage or current source to handle that. Here is a good summary of the method.
I had the solutions in my office which I retrieved yesterday. It has that
$i(t) = \frac{1}{2}\frac{dv_0(t)}{dt}$
where does this identity come from? I know we have that $$\dot{v}_0(t) = 2$$ but how does that equate to $$2i(t)$$?

Edit 1:

I also worked out $$V_0(s)$$ and obtained:

If $$v_i(t) = e^{-3t}\mathcal{U}(t)$$, then
\begin{align*}
e^{-3t}\mathcal{U}(t) &= \frac{3}{2}\frac{dv_0(t)}{dt} +
\frac{1}{2}\frac{d^2v_0(t)}{dt^2} + v_0(t)\\
\frac{1}{s + 3} &= \frac{3}{2}(sV_0(s) - v_0(0)) + \frac{1}{2}(s^2V_0(s) -
sv_0(0) - v_0'(0))\\
\frac{2}{s + 3} &= 3sV_0(s) - 1 + s^2V_0(s) - s - 2\\
V_0(s)(s^2 + 3s) &= \frac{2}{s + 3} + s + 3\\
V_0(s) &= \frac{2}{(s + 3)(s^2 + 3s)} + \frac{s + 3}{s^2 + 3s}\\
&= \frac{2 + (s + 3)^2}{s(s + 3)^2}\\
&= \frac{2 + s^2 + 6s + 9}{s(s + 3)^2}\\
&= \frac{s^2 + 6s + 11}{s(s + 3)^2}
\end{align*}
but the solution says that answer is
$\frac{2(s^2 + 5s + 7)}{(s + 1)(s + s)(s + 3)}.$
Is the solution wrong? I don't see how we can get that.

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#### Ackbach

##### Indicium Physicus
Staff member
I had the solutions in my office which I retrieved yesterday. It has that
$i(t) = \frac{1}{2}\frac{dv_0(t)}{dt}$
where does this identity come from?
This comes from the $VI$ characteristic of a capacitor. For any capacitor of capacitance $C$, it is true that $i=C dV/dt$. Since you have a series $RLC$ circuit, all the current goes through the capacitor. Since $C=1/2$ in your case, that's how you get $i=(1/2) dv_0/dt$.

I know we have that $$\dot{v}_0(t) = 2$$ but how does that equate to $$2i(t)$$?

Edit 1:

I also worked out $$V_0(s)$$ and obtained:

If $$v_i(t) = e^{-3t}\mathcal{U}(t)$$, then
\begin{align*}
e^{-3t}\mathcal{U}(t) &= \frac{3}{2}\frac{dv_0(t)}{dt} +
\frac{1}{2}\frac{d^2v_0(t)}{dt^2} + v_0(t)\\
\frac{1}{s + 3} &= \frac{3}{2}(sV_0(s) - v_0(0)) + \frac{1}{2}(s^2V_0(s) -
sv_0(0) - v_0'(0))\\ \end{align*}
Shouldn't this be
$$\frac{1}{s + 3} = \frac{3}{2}(sV_0(s) - v_0(0)) + \frac{1}{2}(s^2V_0(s) - sv_0(0) - v_0'(0)) \color{red}{+V_0(s)}?$$