Differential equations: Circuit RL

[masterchiefo]

Homework Statement

A simple electrical circuit consists of a voltage source E(t) = t*e^-t volts, a
resistor R = 1 and an inductor L = 1/10 H connected in series. It is assumed that I(0) = -10/81
a) The differential equation that governs the current I (t) in this circuit .
b) Find the time manually t1 where the current is maximum . Explain your answer.

Homework Equations

The Attempt at a Solution

a)
i=dq/dt
VR+VL =E
Ri+L*di/dt=E

I(t) = E(t)/1 *(1-e^-R*t/L)
I(t) = t*e^-t/1 *(1-e^-1*t/(1/10))

b) Not sure how I have to use this info I(0) = -10/81
do I do d/dt(I(t)) to find the max I?
I was also thinking of doing the limit of t to infinite

 

[vela]

Homework Statement

A simple electrical circuit consists of a voltage source E(t) = t*e^-t volts, a
resistor R = 1 and an inductor L = 1/10 H connected in series. It is assumed that I(0) = -10/81
a) The differential equation that governs the current I (t) in this circuit .
b) Find the time manually t1 where the current is maximum . Explain your answer.

Homework Equations

The Attempt at a Solution

a)
i=dq/dt
VR+VL =E
Ri+L*di/dt=E

Up to here is fine.

Now you need to solve the differential equation correctly. Do you know how to solve first-order differential equations?

I(t) = E(t)/1 *(1-e^-R*t/L)
I(t) = t*e^-t/1 *(1-e^-1*t/(1/10))

b) Not sure how I have to use this info I(0) = -10/81
do I do d/dt(I(t)) to find the max I?
I was also thinking of doing the limit of t to infinite

 

[rude man]

I(t) = E(t)/1 *(1-e^-R*t/L)
I(t) = t*e^-t/1 *(1-e^-1*t/(1/10))

where did you get these last two equations?
As vela said, you need to solve the ODE. Unfortunately, it's a bit messy. When you finally get i(t) you do indeed do di/dt = 0 etc. to find i_max and the corresponding t₁.

Not sure what the question meant by "find the time manually".

 

[masterchiefo]

where did you get these last two equations?
As vela said, you need to solve the ODE. Unfortunately, it's a bit messy. When you finally get i(t) you do indeed do di/dt = 0 etc. to find i_max and the corresponding t₁.

Not sure what the question meant by "find the time manually".

it means by hand, as we can do it pretty quick on calculator.
also the equations are from wikipedia and my note book are the same for i(t)

 

[rude man]

it means by hand, as we can do it pretty quick on calculator.
also the equations are from wikipedia and my note book are the same for i(t)

Where in wikipedia?

 

[masterchiefo]

Where in wikipedia?

https://fr.wikiversity.org/wiki/Étude_des_systèmes_électriques/Bobine_d'induction_et_circuit_RL

 

[masterchiefo]

Where in wikipedia?

also what do I do with i(o)=-10/-81? I use it to find the constant C in i(t)?

 

[rude man]

https://fr.wikiversity.org/wiki/Étude_des_systèmes_électriques/Bobine_d'induction_et_circuit_RL

Lisez bien le suivant:
"On appelle échelon de tension le passage brutal de la tension appliquée à l'ensemble {R + L} d'une valeur nulle à une valeur non nulle : on suppose qu'à t < 0, la tension du générateur est nulle et qu'à partir de t = 0 elle est égale à une constante E."

But in your case, E is not a constant, what we call a step function, commonly written here as E U(t). In your case, E = t exp(-t) which is a much more complicated voltage. Which is why you need to solve the differential equation to get i(t), then do di/dt = 0 etc. as you were previously advised to do.

 

[rude man]

also what do I do with i(o)=-10/-81? I use it to find the constant C in i(t)?

exactly right. You will find that i(0) = -10/81 was chosen for great convenience for you, not arbitrarily.