Hello,

I'm currently stuck on a conceptual part of a question. The circuit is a parallel RLC in parallel with a current source. The values are R = 16 ohms, L = 0.5H, C = 312.5 uF, iL(0) = 0.25A, and Current Source = 0.25A. The coefficients for the solution are equal to zero for v(t), t>0. How do I make sense of this? If I understand the solution correctly, the currents through the capacitor and the resistor are equal to zero for all t and iL is constant for all t. Is it even possible? Please help!

 

At DC, what does an inductor look like in steady state?

Is the current flowing in the inductor at t = 0 consistent with it being in DC steady state?

Note there is a very, very important difference between being zero for v(t), t > 0 and being zero for all t.

 

At DC, what does an inductor look like in steady state?

Is the current flowing in the inductor at t = 0 consistent with it being in DC steady state?

Note there is a very, very important difference between being zero for v(t), t > 0 and being zero for all t.

Thanks for your prompt response! The inductor acts as a wire in steady state t->infty and yes, iL is consistent with it being in DC state. So is it safe to assume that if vC(0) = 0, then vC(t) is zero for all t in overdamped circuits "if" iL(0) = DC?

 

Thanks for your prompt response! The inductor acts as a wire in steady state t->infty and yes, iL is consistent with it being in DC state. So is it safe to assume that if vC(0) = 0, then vC(t) is zero for all t in overdamped circuits "if" iL(0) = DC?

No, it really isn't. In fact, since this is a second order circuit, it really isn't sufficient to know just one initial condition, you need two. For instance, the circuit COULD be oscillating and it just happens that, at t=0, the current in the inductor is passing through a value that just happens to be the same, by coincidence, as the value it will have in steady state.

For example, it could be that, in addition to iL(0) being 0.25 A, that vC(0) = 16 V so that iR(0) = 1 A.

If you are very careful in solving the differential equation it should pop out that your solution does not have a single answer without at least one additional piece of information.

And, once again, be careful about making claims that apply "for all t"; any such claim applies to just that, all t -- be it positive, zero, or negative. You know absolutely nothing about anything in this circuit for t < 0 except that it all combined to result in the inductor having 0.25 A of current in it at t = 0.

Also, and this is something I meant to say in my original response, voltages and currents have both magnitudes and polarities. You have given no indication about what the directions of the currents in the current source and the current in the inductor are, so it is actually just as reasonable and justifiable for me, the reader, to conclude that they are in opposite directions resulting in a total net current of 0.5 A splitting between the resistor and the capacitor. This is why properly annotated schematics are so valuable and important.

 

Hello,

I'm currently stuck on a conceptual part of a question. The circuit is a parallel RLC in parallel with a current source. The values are R = 16 ohms, L = 0.5H, C = 312.5 uF, iL(0) = 0.25A, and Current Source = 0.25A. The coefficients for the solution are equal to zero for v(t), t>0. How do I make sense of this? If I understand the solution correctly, the currents through the capacitor and the resistor are equal to zero for all t and iL is constant for all t. Is it even possible? Please help!

Hello there,

What do you mean by "The coefficients for the solution are equal to zero for v(t), t>0." ??

That does not make sense because the coefficients will not be zero, even if the voltage is zero as t goes toward inf.

What i think you meant was that the VOLTAGE goes to zero as t goes toward infinity. In that case then we can probably assume that the original circuit was with a current source of 0.25 amps and inductor initial current 0 amps, and after some time the inductor current settles to equal the source (DC) current of 0.25 amps and therefore we have iL(inf)=0.25 amps and vC(inf)=0 volts.
We might also note that the initial voltage of the capacitor does not matter for the t toward infinity response, but we can look at this in more detail later. Note also even the initial current in the inductor does not matter as t goes toward infinity because it eventually assumes the entire current value from the current source.

BTW the solution is in the form v(t)=A*e^at+B*e^bt so there are no sine terms, and A and B are both non zero and a and b are both negative so v(t) tends toward zero as t tends toward infinity (and exponents at=a*t and bt=b*t).

[LATER]
Here's the solution with included initial conditions v=vC(0) and i=iL(0):
iL(t)=(e^(-160*t)*(5-v-20*i))/60-(e^(-40*t)*(20-v-80*i))/60+0.25
vC(t)=(e^(-40*t)*(20-v-80*i))/3-(e^(-160*t)*(20-4*v-80*i))/3

and we might note that everything except that constant 0.25 goes to zero at t tends toward infinity because of the exponential factors. The coefficients themselves however are not zero.

 

He was pretty explicit that the current in the inductor at t=0 is 0.25 A. Why assume otherwise?

 

He was pretty explicit that the current in the inductor at t=0 is 0.25 A. Why assume otherwise?

Hi there,

Well for one thing to show that the initial conditions dont matter for the solution at t=infinity.
We can start with either of iL=0, iL=1, iL=10, iL=2.3456, and either of vC=0, vC=1, vC=2.3456, and we still get the result iL=0.25 and vC=0 at t=infinity. That is of course because the current source is a DC current source and the circuit is over damped so all the extra energy gets dissipated as time evolves.

This result comes from the complete analysis where we include by the initial current as a variable 'i' and initial voltage as a variable 'v' and take the limit as t goes to infinity. Even though we dont specify what 'i' is or what 'v' is the result still goes to iL=0.25 and vC=0 as we let t go to infinity. I think there is a theory about this for LTI systems but i'd have to look it up. I posted the two solutions so we could see this my inspection of the equations for current and voltage anyway.

 

But what does any of that have to do with the price of kumquats in Norway? He's not asking about the voltage as t goes to infinity and he's not asking about the response when the initial current in the inductor is zero. His situation is that the initial current in the inductor at t=0 is equal to the final current in the conductor at t=∞. As a result, he is getting that the voltage across the components is zero for ALL time (greater than zero) and he is asking if this is correct and/or even possible. The answer is that, yes, it's possible -- it would just mean that the system was already in DC steady state at the moment that we decided to start the clock. But the information provided is not sufficient to require it. If he were to present his analysis I suspect we would see that he is assuming that v(t=0) = 0V even if the problem statement doesn't include that. If we could see the actual problem statement we might be able to determine what the second initial condition is.

 

From #3

The inductor acts as a wire in steady state t->infty

 

And that was in response to a question I asked him in order to guide him toward seeing that his initial inductor current happened to be equal to the DC steady state inductor current.

 

Here's the solution with included initial conditions v=vC(0) and i=iL(0):
iL(t)=(e^(-160*t)*(5-v-20*i))/60-(e^(-40*t)*(20-v-80*i))/60+0.25
vC(t)=(e^(-40*t)*(20-v-80*i))/3-(e^(-160*t)*(20-4*v-80*i))/3

and we might note that everything except that constant 0.25 goes to zero at t tends toward infinity because of the exponential factors. The coefficients themselves however are not zero.

It appears to me that the coefficients are zero if v = 0 for the directions you have chosen for iL(0) and for the .25 A current source. As WBahn pointed out in post #4, the directions of currents matter, so if the direction of iL(0) is reversed, then in that case your solution's coefficients will not be zero when v=0.

But, apparently the TS is assuming the current iL is the same magnitude and direction at t=0 and t=∞, and also apparently you made this choice in your solution, in which case the coefficients are zero when v=0.

 

The answer (based on what the TS has provided) should be expressed in terms of v(0) since no specific value was specified. That may well have been the point of the entire problem was to point out that you can't just assume unstated initial conditions.

 

The answer (based on what the TS has provided) should be expressed in terms of v(0) since no specific value was specified.

Which is what MrAl did in post #5. His solution is in terms of V(0) and iL(0), but he failed to recognize that for one set of initial conditions, the coefficients are zero.

The TS in post #3 did seem to be picking up on where you were attempting to guide him, even if he didn't get every detail:

So is it safe to assume that if vC(0) = 0, then vC(t) is zero for all t in overdamped circuits "if" iL(0) = DC?

 

The answer (based on what the TS has provided) should be expressed in terms of v(0) since no specific value was specified. That may well have been the point of the entire problem was to point out that you can't just assume unstated initial conditions.

Hi again,

What coefficients are you (we) talking about here?