Understanding LRC Theory Intuitively

[bobthenormal]

Hello, I am having some difficulty that I thought some of you might be able to help with.

In the LRC circuit models, the phasor method is used to find the instantaneous values of various properties. I've been having a lot of trouble picturing how this theory actually works in reality.

I apologize if the questions seem too easy, I've been struggling with the physical concepts and working through them, so I might be missing something obvious just from the fatigue.

Usually the situations I don't understand have to do with the inductance part of the circuit. Inductors seem to get treated more mathematically than physically.

For example, a problem with an AC voltage source (Vmax*Sin(omega*t)), an inductor and a resistor, all in parallel.

When I think about how this would work out, I end up going in circles. Here's why:

1) The voltage in the circuit doesn't seem to be constant. It would be instantaneously, but as soon as things begin to move, the EMF induced by the inductor will cause the effective voltage to be lower, even across the resistor, correct?

2) Shouldn't the current in the circuit skyrocket to infinity?

3) The inductor provides a back-EMF, but how long does that back-emf last? It's not really clear from formulas how quickly a magnetic field, once created, re-establishes a current.

4) Shouldn't the inductor cause a spike in current on the 2nd quarter-cycle of the AC input voltage source? That is, in the first quarter-cycle, the voltage is increasing, which presumably is met with a large back-EMF created by the inductor... when it reaches the voltage maximum and the 2nd quarter-cycle begins, shouldn't the current be at a maximum? The input voltage is peaking, so the inductor should be just leveling off because there is no longer (for a short time) any attempt to change the rate of the current.

5) After the situation in (4), I would expect the current to fall slowly or flatten off? Because, as the voltage falls, the current should also be trying to fall, but that fall is resisted by the inductor providing a now forward-EMF, effectively increasing the voltage difference (which I would expect to look like an almost flat line, the inductor and the input voltage working together to provide a constant current as long as the inductor can maintain it).

Being able to picture this has really been holding me back... and it's not the phasor diagrams or math that I'm not getting, so please don't derive the inductor's current from Kirchoff's loop rule. I know you can derive the 90 degree lag by math, I'm trying to get a feel for why THAT graph is not the one I intuitively expect.

--Bob

 

[uart]

Being able to picture this has really been holding me back... and it's not the phasor diagrams or math that I'm not getting, so please don't derive the inductor's current from Kirchoff's loop rule. I know you can derive the 90 degree lag by math, I'm trying to get a feel for why THAT graph is not the one I intuitively expect.

But at the moment your "intuition" seems to be really holding you back as you seem to have a pretty muddled view of how this stuff works.

So can I just ask if you currently have any experience with calculus (derivatives) because a proper time domain description (involving derivatives) might just set you on the right path to understanding this better.

 

[f95toli]

The "phasor methods" is really just Fourier analysis in disguise.
As uart has already pointed out there is nothing stopping you from writing down the correct ODEs instead and solving them some other way; although you will probably find that solving them by using a Fourier transform (or a Laplace transform, they are pretty much equivalent in this case) is usually the quickest and easiest method; there are however cases where it is more conventient to look at the ODEs in the time domain.

 

[bobthenormal]

Heh, yes I'm familiar with Calculus... derive and integrate away if you think it can help to explain it in an intuitive way. (No offense but I'm skeptical, anyone that has jumped to the math to explain it so far has had no intuitive grasp of what it means. But do try, and I hope I'm wrong.)

 

[bobthenormal]

Again, I'm not trying to solve the equations. I know the equations. I can give you the "right answer."

What is PHYSICALLY happening that causes the phases to be SHIFTED the way they are, is my question. It doesn't line up with what I expect to physically happen.

 

[f95toli]

And the suggestion was to write down the ODEs instead of using the phasor methods.
Note that when you are using phasors you are really working in the frequency domain, meaning the formulas aren't very useful if you are interested if is happening as a function of time (which, since you are asking about lag, you are).

Now, if you do take the time to write down the ODEs you will see that inductors are essentially integrating elements meaning what happens in them depends on their "history", this is why it takes time for them to "react" and is causing the phase shift. Now, if you want to understand the reason for that you need the full EM theory (Maxwell's equations) for real inductors, this is presumably beyond the scope of a course in circuit theory.
Asking "why something is REALLY happening" is -as is usually the case in physics- a rabbit hole.

Do you have access to some software that can be used to solve ODEs as a function of time? If you do I would suggest you spend some time solving the ODEs for your circuit different initial conditions etc looking at the instantaneous values for voltage or current, this is -in my experience- the best way to get an intuitive grasp of what is happening.
Or, alternatively, you can just download a free circuit simulator such as LTSpice (see http://www.linear.com/designtools/software/ ,very useful piece of software). The advantage here is that you can "probe" different parts of the circuit and see voltage, current etc.
Since it is such a simple circuit you are dealing with you can of course get the same information from the formulas but using a numerical simulator leaves you more time to think about the physics (since you don't have to worry about the math)

 

[bobthenormal]

I was asking for a more in-depth answer, and this isn't circuits, it's general physics. I don't think it's a "rabbit hole" to ask for the mechanism that results in induction. I KNOW the formula works and that it is an integral solution (because Maxwell's equation contains a flux derivative) but that doesn't actually telll me anything. In FACT, it is even less intuitive because, according to that, it should be when there is a large change in CURRENT that causes induced EMF, but that current itself is AFFECTED by the inductance! The equation has infinite solutions.

I said it multiple times, solving the equations is not an intuitive way of understanding what is happening IN THIS SITUATION - because of the above problem of the current determining the inductance, and vice-versa.

I'll check out the software when I get a chance to get home and hopefully it will help, but I'm a bit skeptical if it just prints the sin graphs (I can picture a very simple sin function just fine without having to draw it, as I'm sure most people can, thanks).

I appreciate the effort you're putting forward, while it is frustrating that I have to repeat that I've gone through all the usual steps, it is still very helpful (and of course not necessary for you to waste your time if I refuse to be helped) so we'll see what playing with the software does. (I'm also going to do a few experiments today if I have time, with real inductors and a function generator... see if I can't get a better grasp of what effect the inductance has on this circuit.)

--Bob

 

[bobthenormal]

Well, it turns out the problem IS more complicated than the usual circuit theory suggests.

Luckily, a professor was able to point me in the right direction, and with a little sleuthing I found a great explanation.

The reason it doesn't make much sense is that the phasor form assumes a steady-state solution of the problem, while I was tending to think of it as beginning from a stop.

The solution for starting the circuit are much harder, usually upper-division level class study according to this professor.

Anyway, here is a site that has a nice illustration of why the current (I) waveform is different during the transient "beginning" stages of a simple LR circuit. It doesn't cover the circuit in parallel, which I imagine is a bit harder, but I'll keep looking for a way to solve it. http://www.csupomona.edu/~apfelzer/demos/transient/rl/14-rlsin.html

--Bob