Transient circuit analysis problem with a dependent source

[Michael_0039]

Homework Statement

Find the Vo of the Transient circuit

Relevant Equations

nil

Hi all,

I tried to solve this, but I can't find the R(equal). I use the following method.

The circuit:

For t<0 the switch is closed so the iL(0-)=4 A

For t>0 it opens and the iL(∞)=0 A

Now, I want to find the value τ=L/R(equal)

Then I will use the: iL(t) = iL(∞) + ( iL(0-) - iL(∞) )* ( e^(-(t/τ) ) )

And finally VL(t) = L * [ ∂iL(t) / ∂t ] which is the Vo

What do you suggest ?Thanks.

 

[Tom.G]

Been a long time since I've used a simulator but I'm assuming the device in parallel with the inductor is a Voltage Controlled Current Source (VCCS) with 4Vo being its transfer function.

If that is the case, what resistance value would yield a current of 4 times the applied voltage?

 

[Michael_0039]

Thanks for your answer!

I think I solved it¨, this is my try:

 

[alan123hk]

There seems to be contradiction, I am confused.

On the one hand, the inductor current should maintain 4A at the moment the switch is turned off.

However, on the other hand, if the voltage of the inductor is initially zero, the VCCS should force the inductor current to zero immediately after the switch is turned off.

Where is my inference wrong?

 

[cnh1995]

However, on the other hand, if the voltage of the inductor is initially zero, the VCCS should force the inductor current to zero immediately after the switch is turned off.

Hint: Voltage across an inductor can change instantaneously but current through the inductor can't change instantaneoulsy.

 

[scottdave]

After the switch is opened, the current throughout the right-hand loop is all the same. Use this fact to set up the differential equation, and integrate to solve for I(t).

The expression will be an exponential, so the exponent will tell you what the time constant is.

 

[Michael_0039]

@scottdave Thanks for your answer.

What about my post #3 is that a correct process? For this kind of problems I use those steps in post #3. I never tried something else.

 

[alan123hk]

Hint: Voltage across an inductor can change instantaneously but current through the inductor can't change instantaneoulsy.

Thanks for your hint, it is true that the voltage across inductor can change instantaneously, but current flowing through inductor cannot change instantaneously.

I am only worried that there may be ambiguous situation in this particular circuit.

I mean that I = Io * e ^(-t /(4L))= 4 * e ^(-50t) may be just a possible answer, and there may be another possible answer.

Since the voltage across the inductor is zero, the current source should be zero when the switch is opened, so why can't the voltage across the inductor become infinite because the current source can be considered as infinite resistance?

It also means that the current flowing through the inductor disappears in an extremely short period of time in this situation.

 

[cnh1995]

Since the voltage across the inductor is zero,

What is the voltage across the inductor the moment you open the switch?

 

[alan123hk]

What is the voltage across the inductor the moment you open the switch?

Thank you for your valuable reply.
After re-study the circuit, I agree that I = 4 * e ^(-50t) is the only answer.

 

[cnh1995]

Thank you for your valuable reply.
After re-study the circuit, I agree that I = 4 * e ^(-50t) is the only answer.

I haven't verified the calculations, but there is no contradiction in this circuit. The moment you open the switch, the inductor voltage instantaneously increases to whatever is required to maintain the 4A current.

 

[scottdave]

Note which direction I is flowing through the inductor. How will it flow through the dependent source? What will the polarity of Vo be?