RLC Circuit question

[RedArmy]

So say I have a simple circuit like this and I need to find the differential equation for it. It's a second order since it's an RLC like I stated in the title. Therefore, what would be the steps I would go through to obtain the equation? I'm guessing I start by finding out the initial conditions (need 2 instances, one where the switch is open, and the other with it closed), and then do I just do a KCL on it and then take the derivative?

My teacher has been lacking in the finer details of the process, and I found through wikibooks the equation for the Zero-Input Response equation (only really helps in one case obviously), which helps me a little, but a explanation of some sorts would be helpful.

As always, thanks for any help you can give me

 

[Analog]

I would sub in the Laplace equivalents, then solve the circuit using KCL. To get the differential equation in the time domain, I would then take the solution back using an inverse Laplace transform.

 

[RedArmy]

Originally posted by: Analog
I would sub in the Laplace equivalents, then solve the circuit using KCL. To get the differential equation in the time domain, I would then take the solution back using an inverse Laplace transform.

I know a good deal about laplace transforms from my Diff Eq class and I understand the terminology you're using, but how would I apply that to a circuit? We've never done anything like that before but it sounds intriguing

 

[Gibson486]

Originally posted by: RedArmy

Originally posted by: Analog
I would sub in the Laplace equivalents, then solve the circuit using KCL. To get the differential equation in the time domain, I would then take the solution back using an inverse Laplace transform.

I know a good deal about laplace transforms from my Diff Eq class and I understand the terminology you're using, but how would I apply that to a circuit? We've never done anything like that before but it sounds intriguing

d/dt = s

(d/dt)^2=s^2

(d/dt)^3=s^3

you get the picture....


edit: the reason you will take the derivative is because the equation will have an integral. Solving problems with integrals and derivatives in one equation just blows. Ofcourse, if he never told you this, he may reveal this later on so you would appreciate it after you spent hours solving the equation without laplace.

 

[BrownTown]

Originally posted by: RedArmy

Originally posted by: Analog
I would sub in the Laplace equivalents, then solve the circuit using KCL. To get the differential equation in the time domain, I would then take the solution back using an inverse Laplace transform.

I know a good deal about laplace transforms from my Diff Eq class and I understand the terminology you're using, but how would I apply that to a circuit? We've never done anything like that before but it sounds intriguing

Well its pretty basic cirucits analysis, essentially instead of writing out the differential equatiosn and then solvign via the Laplace transform you just transform it directly to the s-domain and then solve using algebra. For example a capacitor goes to 1/Cs, and an inductor goes to Ls, then you just solve using Kirchoffs laws to get the transfer function. So once you learn the "trick" you will NEVER using differential equations to solve an sort of circuits problems. That having been said, since your problem is to find the differential equation it doesn't really help you to know the easy way to do it (but thats exactly whats going on here, they like showing you the absurdly hard way and then being like "oh guess what all that is worthless here the easy way to do it"). But now i'm just rambling on, so basically just look in your book and they will have the capacitor and inductor defined with differential equations, so just plug that in and solve normally.

 

[RedArmy]

Originally posted by: BrownTown

Originally posted by: RedArmy

Originally posted by: Analog
I would sub in the Laplace equivalents, then solve the circuit using KCL. To get the differential equation in the time domain, I would then take the solution back using an inverse Laplace transform.

I know a good deal about laplace transforms from my Diff Eq class and I understand the terminology you're using, but how would I apply that to a circuit? We've never done anything like that before but it sounds intriguing

Well its pretty basic cirucits analysis, essentially instead of writing out the differential equatiosn and then solvign via the Laplace transform you just transform it directly to the s-domain and then solve using algebra. For example a capacitor goes to 1/Cs, and an inductor goes to Ls, then you just solve using Kirchoffs laws to get the transfer function. So once you learn the "trick" you will NEVER using differential equations to solve an sort of circuits problems. That having been said, since your problem is to find the differential equation it doesn't really help you to know the easy way to do it (but thats exactly whats going on here, they like showing you the absurdly hard way and then being like "oh guess what all that is worthless here the easy way to do it"). But now i'm just rambling on, so basically just look in your book and they will have the capacitor and inductor defined with differential equations, so just plug that in and solve normally.

I would look in the book if we had one, it's basically more like a course book made by the teacher...so whatever he decided not to put in isn't there. I'm still having trouble grasping exactly what I should be doing since reading these different responses is making me contradict everything I do.

 

[George P Burdell]

Try googling RLC circuits... maybe some university will have online courseware on circuits.

 

[orakle]

I'm so disappointed that we didn't learn circuit analysis using Laplace. I took Laplace after circuits and kept thinking how easy circuit analysis would have been

 

[mrrman]

I dont have my books with me but that example is shown in it...its basically breaks down into 2 equations....the 1st being the capacitior and the battery(power source), the 2nd being the cap,VC,inductor and resistor...and solve for the load current

 

[RedArmy]

Originally posted by: George P Burdell
Try googling RLC circuits... maybe some university will have online courseware on circuits.

Yeah, I've been doing that for a while now...most the information is so in depth and goes off on all these different tangents that have nothing to do with what I need to know...but I'm still looking.

Originally posted by: orakle
I'm so disappointed that we didn't learn circuit analysis using Laplace. I took Laplace after circuits and kept thinking how easy circuit analysis would have been

Care to contribute any advice?

 

[orakle]

Originally posted by: RedArmy
Care to contribute any advice?

I would love to, but I don't remember anything. I just flipped through my notes on 2nd order systems and didn't find an example similar to yours, so I can't help you much. Sorry!

 

[Goosemaster]

Originally posted by: RedArmy

Originally posted by: George P Burdell
Try googling RLC circuits... maybe some university will have online courseware on circuits.

Yeah, I've been doing that for a while now...most the information is so in depth and goes off on all these different tangents that have nothing to do with what I need to know...but I'm still looking.

Originally posted by: orakle
I'm so disappointed that we didn't learn circuit analysis using Laplace. I took Laplace after circuits and kept thinking how easy circuit analysis would have been

Care to contribute any advice?

keep on googling....

all you are doing is moving into the s-domain, using algebra to solve the circuit, and converting back to the time domain. You'll discover stuff like

t-domain: xC (capacitance) = s-domain: x/c
t-domain: xL (capacitance) = s-domain: sL

Basically it comes out to rewriting, paying attention to the differences when working with current/voltage sources, and whether the circuit is in parallel/series and the associated procedures.

 

[Gibson486]

Originally posted by: RedArmy

Originally posted by: George P Burdell
Try googling RLC circuits... maybe some university will have online courseware on circuits.

Yeah, I've been doing that for a while now...most the information is so in depth and goes off on all these different tangents that have nothing to do with what I need to know...but I'm still looking.

Originally posted by: orakle
I'm so disappointed that we didn't learn circuit analysis using Laplace. I took Laplace after circuits and kept thinking how easy circuit analysis would have been

Care to contribute any advice?

it's the same exact process as DC only analysis. The only thing different if is the addition of solving differential equations.

You know for L, V=L*di/dt
and
for C, I=C*dv/dt


You will do KCL or KVL and you will modify the inductor equation to solve for I or modify the cap equation to solve for V. Either way, you will end up only modifying one of the equations and you will leave the other alone.

As a result, you end up with an integral in the equation along with derivatives. This is hard to solve. How do we fix this? We simply take the derivative (actually, it's the partial derivative) of both side to get rid of that integral. This will give double derivatives in RLC circuits, which are harder to solve, but since we know differential equations, they are easier to solve.


edit:
don't worry if you do not get it like that. It took me 2 years after I ifinished the class to understand what I was exactly doing. The paralell RLC is actually more intimidating than the series. In other words, it's a good finals question.

 

[RedArmy]

Alright, I'll see what I can do with the information you guys gave me, thanks for the help everyone!

 

[darthsidious]

Originally posted by: RedArmy

Originally posted by: George P Burdell
Try googling RLC circuits... maybe some university will have online courseware on circuits.

Yeah, I've been doing that for a while now...most the information is so in depth and goes off on all these different tangents that have nothing to do with what I need to know...but I'm still looking.

Originally posted by: orakle
I'm so disappointed that we didn't learn circuit analysis using Laplace. I took Laplace after circuits and kept thinking how easy circuit analysis would have been

Care to contribute any advice?

Goto MIT's OCW website (ocw.mit.edu), and under the Electrical Engg and Comp Sci section, you'll find the 6.002 webpage. Look at the relevbant coursenotes, especially lecture note #15. That should help.
Link for the Lazy

For now, it seems that your professor has not discussed Laplace transforms, so unless you want to try and be clever and self learn, just solve it using regular KCL + differential equations. I'll just give you a small start. Treat the voltage source + switch combined as an input, say Vi. Then, find the Output Voltage Vout as a function of Vin. This is the part where you will have to use KVL/KCL, and the relations I=Cd/dt(V) and V =L d/dt(I). make sure the final equation only has the variable of interest - this will involve you having to substitute the first derivative of current for the second derivative of voltage (or vice versa).
Once you have the diff eq., it's a matter of finding the homogenous solution (which should be in any diffeq text) and a particular solution (which should be easy to guess from the circuit config. Finally, use your initial conditions to solve for the constants

 

[esun]

What exactly are you solving for? i_L as a function of time? We also need to know the initial conditions to solve the problem. Post that information and we can help. Otherwise, it isn't clear what you want us to solve.

 

[Eli]

That's like the most simple circuit ever, and I haven't the slightest idea what the fuck is being talked about in this thread.



I've been thinking about going back to school for EE lately, but it scares me.

I guess thats not true, I get some of it.. but just realizing that its basic stuff and makes my head spin is disheartening. lol

 

[IAteYourMother]

Originally posted by: darthsidious

Originally posted by: RedArmy

Originally posted by: George P Burdell
Try googling RLC circuits... maybe some university will have online courseware on circuits.

Yeah, I've been doing that for a while now...most the information is so in depth and goes off on all these different tangents that have nothing to do with what I need to know...but I'm still looking.

Originally posted by: orakle
I'm so disappointed that we didn't learn circuit analysis using Laplace. I took Laplace after circuits and kept thinking how easy circuit analysis would have been

Care to contribute any advice?

Goto MIT's OCW website (ocw.mit.edu), and under the Electrical Engg and Comp Sci section, you'll find the 6.002 webpage. Look at the relevbant coursenotes, especially lecture note #15. That should help.
Link for the Lazy

For now, it seems that your professor has not discussed Laplace transforms, so unless you want to try and be clever and self learn, just solve it using regular KCL + differential equations. I'll just give you a small start. Treat the voltage source + switch combined as an input, say Vi. Then, find the Output Voltage Vout as a function of Vin. This is the part where you will have to use KVL/KCL, and the relations I=Cd/dt(V) and V =L d/dt(I). make sure the final equation only has the variable of interest - this will involve you having to substitute the first derivative of current for the second derivative of voltage (or vice versa).
Once you have the diff eq., it's a matter of finding the homogenous solution (which should be in any diffeq text) and a particular solution (which should be easy to guess from the circuit config. Finally, use your initial conditions to solve for the constants

i think if you just go to the 8.02 website it should have it.

Q'' + Q'(damping stuff) + Q/(LC) = 0?

 

[hypn0tik]

V = L * di/dt (The V-I relationship for an inductor) --- (1)
I = C * dV/dt (The V-I relationship for an capacitor) --- (2)
V = IR (V-I relationship for a resistor) --- (3)

Start there and apply KCL and/or KVL.

 

[Gibson486]

Originally posted by: Eli
That's like the most simple circuit ever, and I haven't the slightest idea what the fuck is being talked about in this thread.



I've been thinking about going back to school for EE lately, but it scares me.

I guess thats not true, I get some of it.. but just realizing that its basic stuff and makes my head spin is disheartening. lol

It's tough stuff....but lots of people who graduate with an EE degree still can't do it. i even know people with high gpas that can't do it. The math is just a huge hurtle. Once you really understand the math behind it, it becomes easier. At the same time, their are people who just do the math....and they forget to understand the concept. I was in that catergory during school. It took my 2 coops for my bosses to make sure i really knew these concepts....and they even expected me to not know them well out of school. It sucks, though....once I found out I really enjoyed doing it, i was not able to get a job in that field. When it comes to analog circuits, most companies will not accept less than a 3.2 GPA and in some cases a Masters.

 

[Gibson486]

Originally posted by: IAteYourMother

Originally posted by: darthsidious

Originally posted by: RedArmy

Originally posted by: George P Burdell
Try googling RLC circuits... maybe some university will have online courseware on circuits.

Yeah, I've been doing that for a while now...most the information is so in depth and goes off on all these different tangents that have nothing to do with what I need to know...but I'm still looking.

Originally posted by: orakle
I'm so disappointed that we didn't learn circuit analysis using Laplace. I took Laplace after circuits and kept thinking how easy circuit analysis would have been

Care to contribute any advice?

Goto MIT's OCW website (ocw.mit.edu), and under the Electrical Engg and Comp Sci section, you'll find the 6.002 webpage. Look at the relevbant coursenotes, especially lecture note #15. That should help.
Link for the Lazy

For now, it seems that your professor has not discussed Laplace transforms, so unless you want to try and be clever and self learn, just solve it using regular KCL + differential equations. I'll just give you a small start. Treat the voltage source + switch combined as an input, say Vi. Then, find the Output Voltage Vout as a function of Vin. This is the part where you will have to use KVL/KCL, and the relations I=Cd/dt(V) and V =L d/dt(I). make sure the final equation only has the variable of interest - this will involve you having to substitute the first derivative of current for the second derivative of voltage (or vice versa).
Once you have the diff eq., it's a matter of finding the homogenous solution (which should be in any diffeq text) and a particular solution (which should be easy to guess from the circuit config. Finally, use your initial conditions to solve for the constants

i think if you just go to the 8.02 website it should have it.

Q'' + Q'(damping stuff) + Q/(LC) = 0?

That's what it generally ends up being....but it's too general (and there is not always damping and its not always the Q'). Knowing how to derive the equation from scratch will be a life saver come finals time (and during a job interview).

 

[frostedflakes]

Well I think resistance will always be the coefficient for the first-order term. The differential equations for these RLC circuits can be related in some ways to those for spring-mass systems. In the spring-mass system, the coefficient for the x' term is the dampening constant. For an RLC circuit, the resistor performs the same function (which makes sense, it is allowing energy to leave the system in the form of heat).

The DE for series RLC is Lq''+Rq'+(1/C)q=V(t). No idea what it is for series and parallel combination you posted, though. I should probably try to derive it for practice, though. Then again, I'm sure I'll get to do plenty of this over the next few semesters when I take my circuit theory courses.

 

[TuxDave]

Is it too late to post a solution? Feel free to read only up to what you care to read before trying it out on your own.


--> Basic Equations Needed
Ic = C * Vc'
Vl = L * Il'
Vr = Il * R

And one extra one since Ic = -Il ...
Il = -C*Vc'







--> A little push...
Vc = Vl + Vr
Vc = L*Il' + R*Il









--> A full solution
Substituting Il from abote:
Vc = L*(-C*Vc')'+R*(-C*Vc')
Vc = -LC*Vc'' - RC*Vc'

And rearranging:
LC*Vc'' + RC*Vc' + Vc = 0
Vc(0) = 2
Il(0) = Vsupply/R