1. ## Circuit Differential Equation

Hi all,

My friend and I have been trying to solve this differential equation, however we did not manage to get very far! The questions states that we need to:

Find the currents $I_1$ and $I_2$ in the circuits shown when $R = 2.5$ Ohms, $L=1$ Henry, $C= 0.04$ Farad, and $E(t) = 169sin(t)$ Volts. Assume that there are no currents in the circuits at the initial instant of time, i.e. $I_1(0) = 0$ and $I_2(0) = 0$.

Am I correct in assuming that we need to use Kirchoff's second law to begin, or am I on the wrong track to begin with? And if so, would i go about finding what each of the voltages are of each component with respect to current? Sorry if that part did not make sense, I do not have much experience at all with electrical circuits at all.

We did a question in class which was a series circuit but this parallel circuit has thrown me right out!

Cotty

2. ## Re: Circuit Differential Equation

Your source voltage is sinusoidal so you can just treat this as an ordinary DC circuit problem using the complex impedance of your circuit elements.

$\begin{pmatrix}Z_R + Z_L &-Z_R \\ -Z_R & Z_R+Z_C\end{pmatrix}\begin{pmatrix}I_1 \\ I_2\end{pmatrix}=\begin{pmatrix}E \\ 0\end{pmatrix}$

You should have learned what the complex impedances are for circuit elements.

$Z_R=R$

$Z_C=\dfrac 1 {\jmath \omega C}$

$Z_L=\jmath \omega L$

You should be able to finish from here.

3. ## Re: Circuit Differential Equation

Hi romsek, thanks for getting back to me! Thank you for taking the time to write this out for me, however I cannot remember seeing anything like this is my classes! We have been dealing with differential equations, and from what I can gather I think the background knowledge related to circuits (for questions like this) is assumed knowledge. Are we able to set this up in the form of an ODE? We dealt with a simple series circuit before, and ended up coming out with a differential equation of:

$\frac{d^2I}{dt^2}+\frac{R}{L}\frac{dI}{dt}+\frac{ 1}{LC} = 0$

By using Kirchoff's Voltage Law.

As i said before though, thank you for taking the time to type that up It seems you help me quite a bit in these forums!!

4. ## Re: Circuit Differential Equation

Originally Posted by Cotty
Hi romsek, thanks for getting back to me! Thank you for taking the time to write this out for me, however I cannot remember seeing anything like this is my classes! We have been dealing with differential equations, and from what I can gather I think the background knowledge related to circuits (for questions like this) is assumed knowledge. Are we able to set this up in the form of an ODE? We dealt with a simple series circuit before, and ended up coming out with a differential equation of:

$\frac{d^2I}{dt^2}+\frac{R}{L}\frac{dI}{dt}+\frac{ 1}{LC} = 0$

By using Kirchoff's Voltage Law.

As i said before though, thank you for taking the time to type that up It seems you help me quite a bit in these forums!!
ok, we'll start from the beginning. You need two bits of info to attack this problem

1) kirchoff's circuit law for voltage - i.e. the sum of the voltages around a circuit loop is 0.
2) the relations between voltage and current for the 3 circuit elements

Looking at (2) first we have

a) $V(t)=I(t) R$

b) $V(t)=\dfrac 1 C \int I(\tau)~d\tau$

c) $V(t) = L \dfrac{dI}{dt}(t)=L I^\prime(t)$

Now looking at (1) for the first loop we have

$E(t)=I_1(t)R + L I_1^\prime(t) - I_2(t) R$

for the 2nd

$0=I_2(t) R + \dfrac 1 C \int I_2(\tau)~d\tau - I_1(t) R$

You can differentiate eq 2 to obtain

$0=I_2^\prime R + \dfrac 1 C I_2(t) - I_1^\prime(t)$

see if you can take it from here.

5. ## Re: Circuit Differential Equation

So I have had a few different attempts so far and then had a sleep on it but haven't gotten very far. Just to clarify, should I be trying to get 1 of these equations in terms of only $I_1$ or $I_2$ + their derivatives?