Find current through inductor that parallels a resistor

[Color_of_Cyan]

Homework Statement

The switch in the circuit has been closed for a long time before it is opened at t = 0.


Find:

a. I_L(t) for t > 0

b. i₀(t) for t > 0

c. V₀(t) for t > 0

Homework Equations

equivalent resistance, equivalent current, equivalent voltage

voltage division, current division,

V through inductor = L*(dI/dt)

Thevenin / Norton procedures

The Attempt at a Solution

So at the beginning I changed the above to this and simplified the resistors to the right of the inductor to 10Ω with 10 parallel 40 which is 8 and then added to 2 in series to get 10:




Then I changed the current source to the Thevenin equivalent voltage to get this:





But since the switched was closed for a long time and the current through the inductor wasn't changing, the voltage through the inductor is 0.

But what I DO NOT know now...


Does this mean that there's ALSO NO voltage through the 10Ω resistor which now parallels it?


This would mean that there's only voltage across the 0.1Ω resistor which is 2V across. Therefore the current across the 0.1Ω would be

2/0.1 = 20A

So would all the current go across the inductor and not the 10Ω?


If so, then the current through the inductor at the beginning with the switch closed for a long time would be 20A at t=0 .

So then with the switch open the equivalent resistance in the loop left over is 10Ω and so the total V is

10Ω * I +L(dI/dt) = 0;

10Ω*I = -2H*(dI/dt)

(-10Ω/2H)dt = dI/I

integrating both sides gets:

-5t + c = ln I

then if t = 0 then I = 20A. Substituting this initial condition gets C = 3 so

-5t + 3 = ln I; so


e^{(-5t + 3)} = e^{(ln I)}


So I_Lt = 20e^(-5t)A. And this is the total current for the rest of the loop in the circuit with the switch opened too.

Using current division between the 10Ω and 40Ω gets

I = 4e^(-5t)A through the 40Ω

Since i₀ is going in the opposition direction from the diagram it would be

i₀t = -4e^(-5t)A

then V = IR so

V₀t = 4e^(-5t)A * 40Ω

= 80e^(-5t)V





Did I do all of this right or did I go wrong somewhere? Thank you.

 

[gneill]

You've got the right results for I_L(t) and i_o(t). Check your math for V_o(t); what's -4 x 40?

 

[Color_of_Cyan]

Thanks again gneill.

Rushed the end of my post there though so V₀t is actually

-160e^(-5t)V

So the drop is negative because the current goes in the direction from - to + across V₀,

and would be positive if it was + to - instead, right?

 

[gneill]

Thanks again gneill.

Rushed the end of my post there though so V₀t is actually

-160e^(-5t)V

So the drop is negative because the current goes in the direction from - to + across V₀,

and would be positive if it was + to - instead, right?

Yup. Looks good.

 

[rezeew]

Your calculations and approach seem to be correct. The key concept in this problem is that the switch has been closed for a long time and therefore the current through the inductor is constant. This means that the voltage across the inductor is 0, as you correctly stated. This also means that there is no voltage drop across the 10Ω resistor, since it is in parallel with the inductor. Therefore, all of the voltage drop in the circuit must occur across the 0.1Ω resistor, resulting in a current of 20A.

Your calculations for the current through the inductor and the total current in the circuit after the switch is opened are correct. Similarly, your calculations for the current and voltage across the 40Ω resistor are also correct. Overall, your approach and calculations seem to be correct and you have successfully found the current through the inductor and the other components in the circuit. Well done!