Analyzing the LCR Circuit: Oscillatory Currents

[albega]

Homework Statement

I have the LCR circuit attached below.

At time t=0 the capacitor is uncharged and the switch is closed. By solving an appropriate
differential equation, show that the current through the resistor is oscillatory provided
L<4CR². By considering the boundary conditions at t=0 and as t→∞, sketch the
form of this current as a function of time.

Homework Equations

V=IR, V=LdI/dt, V=Q/C.

The Attempt at a Solution

So the first bit is pretty simple, giving a DE of
LCRd²I/dt²+LdI/dt+RI=V₀.

Solving for the transient complentary functions using the quadratic formula gives the required condition for an oscillatory current.

I can solve the DE to obtain
I=exp(-t/2CR)(Asinβt+Bcosβt)+(V₀/R), where β=[√(4LCR²-L²)]/2LCR and we are assuming oscillatory solutions do exist as the question wants.

I have the initial condition Q=0 for the capacitor when t=0, so then I=0. Then B=-V₀/R. However the condition as t→∞ is problematic. I expect I→V_o/R (the steady-state solution) as t→∞. If I let t tend to infinity in my solution, the CF vanishes, so it doesn't allow me to implement any sort of condition. I can only think of this meaning I can let A be zero but I don't think that would be ok.

Any clues would be great, thanks!

 

[SammyS]

Homework Statement

I have the LCR circuit attached below.

At time t=0 the capacitor is uncharged and the switch is closed. By solving an appropriate
differential equation, show that the current through the resistor is oscillatory provided
L<4CR². By considering the boundary conditions at t=0 and as t→∞, sketch the
form of this current as a function of time.

Homework Equations

V=IR, V=LdI/dt, V=Q/C.

The Attempt at a Solution

So the first bit is pretty simple, giving a DE of
LCRd²I/dt²+LdI/dt+RI=V₀.

Solving for the transient complentary functions using the quadratic formula gives the required condition for an oscillatory current.

I can solve the DE to obtain
I=exp(-t/2CR)(Asinβt+Bcosβt)+(V₀/R), where β=[√(4LCR²-L²)]/2LCR and we are assuming oscillatory solutions do exist as the question wants.

I have the initial condition Q=0 for the capacitor when t=0, so then I=0. Then B=-V₀/R. However the condition as t→∞ is problematic. I expect I→V_o/R (the steady-state solution) as t→∞.

What is $\displaystyle\ \lim_{t\,\to\,\infty}e^{-t/(2RC)} \ ?$

If I let t tend to infinity in my solution, the CF vanishes, so it doesn't allow me to implement any sort of condition. I can only think of this meaning I can let A be zero but I don't think that would be ok.

Any clues would be great, thanks!

 

[albega]

What is $\displaystyle\ \lim_{t\,\to\,\infty}e^{-t/(2RC)} \ ?$

Zero... As I said, that causes the CF, the transient response, to vanish.

However this isn't useful in finding the arbitrary constant A because it all just disappears.

 

[SammyS]

I = 0 initially because of the nature of inductors, not because Q = 0.

Q = 0 implies that at t = 0, the voltage drop across the capacitor is zero.

Take the derivative of your solution and apply a boundary condition to that at t = 0. That should give what A is.

 

[albega]

I = 0 initially because of the nature of inductors, not because Q = 0.

Q = 0 implies that at t = 0, the voltage drop across the capacitor is zero.

Take the derivative of your solution and apply a boundary condition to that at t = 0. That should give what A is.

I don't see how Q=0 at t=0 implies anything about dI/dt at t=0...

 

[ehild]

The voltage across the resistor and capacitor is U=Vo-LdI/dt. At t=0 it is zero.
Note that the problem asks the current through the resistor, which is U/R.
Oscillatory means also damped oscillation.

ehild

 

[SammyS]

I don't see how Q=0 at t=0 implies anything about dI/dt at t=0...

It wouldn't except that the inductor is involved.

Initially, t = 0, the voltage drop across the resistor/capacitor is zero, so you know that the voltage drop across the inductor is V₀. But you also know how the voltage drop across the inductor is related to the current. Right?

 

[albega]

It wouldn't except that the inductor is involved.

Initially, t = 0, the voltage drop across the resistor/capacitor is zero, so you know that the voltage drop across the inductor is V₀. But you also know how the voltage drop across the inductor is related to the current. Right?

Yes, that gives dI_total/dt=V₀/L at t=0. How can I relate that to I through the resistor? Also I notice we're dealing with t=0 here. What about t→∞ as the question suggests?

 

[SammyS]

Yes, that gives dI_total/dt=V₀/L at t=0. How can I relate that to I through the resistor? Also I notice we're dealing with t=0 here. What about t→∞ as the question suggests?

Last question first:
Presumably the initial solution to the differential equation contained an additive constant, which was set to V₀/R because of the behavior of I(t) as t→∞ .


How to find A:
You have an expression for dI/dt at t = 0: dI/dt=V₀/L .

That's why I suggested evaluating the first derivative of your solution for I(t) at t = 0.

Take the derivative of $\displaystyle \ e^{-t/(2RC)}\left(A\sin(\beta t)+B\cos(\beta t) \right) +\frac{V_0}{R} \,,\$ evaluate that at t = 0 & set it equal to V₀/L

 

[albega]

Last question first:
Presumably the initial solution to the differential equation contained an additive constant, which was set to V₀/R because of the behavior of I(t) as t→∞ .


How to find A:
You have an expression for dI/dt at t = 0: dI/dt=V₀/L .

That's why I suggested evaluating the first derivative of your solution for I(t) at t = 0.

Take the derivative of $\displaystyle \ e^{-t/(2RC)}\left(A\sin(\beta t)+B\cos(\beta t) \right) +\frac{V_0}{R} \,,\$ evaluate that at t = 0 & set it equal to V₀/L

Right, I'm fine with all that, but surely I have an expression for dI_tot/dt at t=0 and not dI/dt, because the current through the resistor is not the same as that through the inductor.

 

[SammyS]

Right, I'm fine with all that, but surely I have an expression for dI_tot/dt at t=0 and not dI/dt, because the current through the resistor is not the same as that through the inductor.

What is I(t) ? Current through the source? ... Current through the inductor ? ... Current through the resistor/capacitor combination ?

Yes to all three.


Although it doesn't matter for the issue at hand: At t = 0, the voltage drop across the resistor is zero. Therefore, no current flows through the resistor.

 

[albega]

What is I(t) ? Current through the source? ... Current through the inductor ? ... Current through the resistor/capacitor combination ?

Yes to all three.


Although it doesn't matter for the issue at hand: At t = 0, the voltage drop across the resistor is zero. Therefore, no current flows through the resistor.

Is there a mathematical way to show dI_tot/dt=dI/dt though? I can't find a way which is a little annoying.

 

[ehild]

Is there a mathematical way to show dI_tot/dt=dI/dt though? I can't find a way which is a little annoying.

It is not true. You solved the differential equation for the total current I.

The current through the resistor is i = U/R where U is the voltage across the resistor.

The voltages add along the loop (Kirchhoff's loop rule) so the voltage across the inductor + voltage across the capacitor= voltage of the source. U_L+U=U₀.
The voltage across the inductor is proportional to the time derivative of the current. U_L=LdI/dt.

So LdI/dt + U = U₀ →U=U₀-LdI/dt and the current through the resistor is i=U/R. ehild

 

[albega]

It is not true. You solved the differential equation for the total current I.

The current through the resistor is i = U/R where U is the voltage across the resistor.

The voltages add along the loop (Kirchhoff's loop rule) so the voltage across the inductor + voltage across the capacitor= voltage of the source. U_L+U=U₀.
The voltage across the inductor is proportional to the time derivative of the current. U_L=LdI/dt.

So LdI/dt + U = U₀ →U=U₀-LdI/dt and the current through the resistor is i=U/R.


ehild

I solved it for the current through the resistor only.

Here:
Let the current through the inductor be I1+I2, that through the resistor I2 and that through the capacitor I1.

Going around the resistor capacitor loop,
Q1/C-I2R=0 and differentiating this gives I1=CRdI2/dt.
Going around the cell, inductor and capacitor loop,
V0-Ld(I1+I2)/dt-Q1/C=0.
V0-LdI1/dt-LdI2/dt-Q1/C=0

Substituting for I1 we have

LCRd²I2/dt²+LdI2/dt²+RI2=V0.

Then solve for the complementary function, assuming L<4CR² for underdamped solutions. Then solve for the PI.

I obtain the general solution I2=exp(-t/2CR)(Asinβt+Bcosβt)+(V0/R), where β=[√(4LCR2-L2)]/2LCR.

 

[ehild]

Sorry, I did not recognize it was the equation for the resistor current in the OP.
It is still valid that LdI(total)/dt+RI₂=Vo.
To find the other constant, A, use that the current through the inductor (the total current) is zero at t=0.
I(total)=I₁+I₂, and I₁=dQ/dt=d(UC)/dt=RCdI₂/dt.


ehild

 

[SammyS]

Sorry, I did not recognize it was the equation for the resistor current in the OP.
It is still valid that LdI(total)/dt+RI₂=Vo.
To find the other constant, A, use that the current through the inductor (the total current) is zero at t=0.
I(total)=I₁+I₂, and I₁=dQ/dt=d(UC)/dt=RCdI₂/dt.

ehild

Thanks ehild for clearing up this issue!


@albega,
I apologize for misreading the question. You're in excellent hands now, with my friend ehild.

And of course, you were correct in stating that Q=0 @ t=0 → I₂ = 0 at t=0 !

Placing subscripts on the currents sure does help clarify things!

 

[albega]

Sorry, I did not recognize it was the equation for the resistor current in the OP.
It is still valid that LdI(total)/dt+RI₂=Vo.
To find the other constant, A, use that the current through the inductor (the total current) is zero at t=0.
I(total)=I₁+I₂, and I₁=dQ/dt=d(UC)/dt=RCdI₂/dt.


ehild

So my initial conditions are I₂=dI₂/dt=0 at t=0.

If so, thanks for clearing that up!

 

[albega]

Thanks ehild for clearing up this issue!


@albega,
I apologize for misreading the question. You're in excellent hands now, with my friend ehild.

And of course, you were correct in stating that Q=0 @ t=0 → I₂ = 0 at t=0 !

Placing subscripts on the currents sure does help clarify things!

Ah sorry, I should have made it clearer by using subscripts from the start! That will teach me not to take shortcuts in the future...

 

[ehild]

So my initial conditions are I₂=dI₂/dt=0 at t=0.

If so, thanks for clearing that up!

Yes, we figured it out at the end The problem text was confusing: t→∞ helped to find the particular solution.


ehild