Mutual Inductance Problem: Find V(t)

[bobbyw]

This problem is tougher than I though and I'm stuck on a couple of things.

Homework Statement

There are 2 inductors, a DC power source, 2 resistors, and a switch. The circuit looks just like the image I've attached, except it will have a switch on the left of the first resistor and the values are different. sorry about that.
The DC source is 12V
The inductors are 4H, and 16H, respectively (L₁ and L₂).
The resistors are 12Ω and 48Ω, respectively(R₁ and R₂).
K=1
The switch is open for a long period of time previous to t=0 (inductors fluxed), and opened at t=0.
The problem is to find V_out(t). In the image attached, V_out would be the voltage across R₂.

Homework Equations

K=M/sqrt(L1*L2). since K is given as 1, M is sqrt(L1*L2), which is 8H.

The Attempt at a Solution

I initially set up the loops using KVL. For the left hand side of the circuit I got:
12 = (12+s4)I₁ - 8sI₂

and for the right:
0 = -8I₁ + (48+s16)I₂

from this, the system determinant can be found, and I₂ can then be found.
I am stuck right now is the switch though.
for finding V(t<0), the inductors are fluxed, thus acting as short circuits. Does this mean that the mutual inductance will come into play or can be neglected? I have thought of a couple different possibilities.
V(t<0) = 0 because the inductors are not electrically connected and no current is flowing?
V(t<0) = 12 because the inductors are acting as short circuits, and the mutual inductance will appear to be a short circuit as well, making the voltage on the left and right side of the circuit the same?
V(t<0) = a voltage divider between the two resistors if the above ^ applies?

I also need to find to V(t>=0) but the questions above should help me with that.


Thanks in advance for any help.

 

[jegues]

The switch is open for a long period of time previous to t=0 (inductors fluxed), and opened at t=0.
The problem is to find Vout(t). In the image attached, Vout would be the voltage across R2.

Do you mean its closed at t=0?

V(t<0) = 0 because the inductors are not electrically connected and no current is flowing?

I think this is the case we're dealing with, hopefully another member can confirm my suspicions.

 

[bobbyw]

Do you mean its closed at t=0?

Sorry about that, i meant the switch is CLOSED previous to t=0 for a long time, making the inductors fully fluxed. really all of the cases make sense to me, and I've been searching for a definitive answer but haven't found one. thanks for the opinion.

 

[jegues]

Sorry about that, i meant the switch is CLOSED previous to t=0 for a long time, making the inductors fully fluxed.


really all of the cases make sense to me, and I've been searching for a definitive answer but haven't found one. thanks for the opinion.

My answer is going to change now based on how you clarified the question. Let me think about it and I will get back to you.

 

[DeeNos]

I can understand that this problem may seem tough and you are stuck on a few things. Let me try to provide some guidance and help you find a solution.

Firstly, it is important to understand the concept of mutual inductance in this problem. Mutual inductance refers to the ability of two inductors to influence each other's magnetic fields and induce a voltage in the other. In this circuit, the mutual inductance is represented by the coefficient K, which is given as 1.

Now, let's focus on the switch. Before t=0, the switch is open and the inductors are fluxed, acting as short circuits. This means that no current is flowing through the inductors and the mutual inductance can be neglected. Therefore, V(t<0) = 0, as you correctly mentioned.

At t=0, the switch is closed and current starts flowing through the circuit. This means that the inductors are no longer acting as short circuits and the mutual inductance will come into play. However, since K=1, the mutual inductance will not have a significant effect on the voltage across the resistors. Therefore, V(t>=0) can be calculated using a voltage divider between the two resistors, as you suggested.

To summarize, the voltage across R2 (Vout) can be calculated as follows:

V(t<0) = 0 (due to fluxed inductors)
V(t>=0) = 12 * (48/60) = 9.6V (voltage divider between R1 and R2)

I hope this helps you to find the solution to this problem. Remember, understanding the concepts is key to solving any problem in science. Keep exploring and don't give up!