Transient Circuits lab discussion and related final practice problem

For your last lab, which is the week after Thanksgiving,  everyone will do the "Transient Circuit Analysis" lab*. (Please ignore that it says not offered in fall 2010 in your lab book.) Please do the "Transient Circuits" prelab!!!

This lab is more aligned with what we have been emphasizing in class and will allow you to see the time-dependent behavior of circuits. It is very important to notice and understand that the duration of each square wave is much longer than the characteristic time scale associated with your circuits in this lab. Thus each square wave functions as essentially a DC voltage; the beginning of each square wave "on-cycle" is essentially equivalent to the closing of a switch and the application of a DC voltage. You will look at the time-dependent response after the switch is closed.

*In this context, transient circuit analysis means analysis of the time-dependent response of the circuit to a sudden change in voltage, which we have discussed quite a bit in class and on HW.

Please feel free to post your comments and questions here.

added Friday, Dec 3

Here is a  new practice problem:

P.S. What part of the problem does not really make sense after you replace the capacitor by an inductor?  Why?

Other things you would like to learn about.

Before our class is over, I would like to ask if there any other topics or issues, related to electricity and magnetism, you would like to learn more about?  Please feel free to post comments and thoughts on that here (or to email me if you prefer).  zacksc@gmail.com

Everything you always wanted to know about oscillating circuits and more.

LCR circuits might seem daunting; I would suggest viewing them as LC circuits with a small resistor tacked on. As we learned earlier, the LC combination produces oscillations! The inductor (L) stores energy -in its magnetic (B) field- when there is any current in the circuit; while the capacitor, on the other hand, stores energy -in its charge and the associated E field-  when the current in the circuit is zero....  There is always energy somewhere, and it transfers back-and-forth, from the inductor to the capacitor, in an oscillatory fashion.

A small resistor adds a third element which transfers energy out of the circuit, e.g., into light and/or heat. We can regard the resistor as a small perturbation that gradually diminishes the energy in the circuit; with each oscillation the peak values of the charge and the current are a little less than they were the time before.

Below are some notes -calculations- which show a fundamental model equation for an LCR circuit and its solution in terms of Q(t). The main things you need to know are:

1) what the equation for Q(t) is; what its graph looks like -in detail-; and how its parameters and behavior depend on L, C and R.

2) same thing for the current, I(t). Additionally, the relationship, graphical and in equation form, between I(t) and Q(t) is important.

3) how to calculate energy from I(t) and Q(t).

4) how to calculate w, T and tau from L, C and R, and what they are and what role they play in Q, I, V and all related energies.

and additionally,

4) to have an intuitive feel for and understanding of the characteristic time scales-- tau and T -- and to understand what it means for tau to be much greater than T.

Your comments and questions are welcome!

Review: Saturday December 4, 12:30 - 2:00 PM

We'll have a review scheduled on Saturday Dec 4, at:
12:30 PM
in (our regular classroom)
Thimann 3

Homework practice problems for this week. *new 7 added on Dec 1

Thursday 1 PM: I did some minor edits.

I realize these problems may be difficult.  If you get stuck at any point, please ask questions with a comment here. Probably some other people are stuck on the same thing?  For 3a), do you know what T refers to and how to calculate it? What equation do you use for w? You can ask very simple questions. For example: what is T?, how do we calculate w and tau??, how do you calculate the current, I(t)?, etc.

(Instead of an online homework assignment this week, the class chose to instead work on some written practice/homework problems. Here they are. Feel free and encouraged to post any questions or comments here.)

PS. Even when the questions do not ask for graphs, i would strongly encourage you to graph everything you can think of, e.g., I v t, Q v t, tau v R, T v C, etc. Familiarity with graphs and their meanings is highly valued here. (I'll add that in now.)

1. In a series LRC circuit --which starts out with a charged capacitor -- which of the following statements is true:

a) the current starts out from the capacitor, then flows through the inductor and then through the resistor.

b) Current can flow either clockwise or counterclockwise, and it can change as a function of time, but at any given moment the current is the same everywhere in the circuit so it is not useful or helpful to talk about where it goes first.
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2.
a) What is an Ohm in terms of Coulombs, Volts and seconds?
b) What is a Farad in terms of Coulombs, Volts and seconds?
c) What is a Henry in terms of Coulombs, Volts and seconds?
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3. Consider a series LRC circuit with L= 1 H, R= 1 Ohm and C= 10^-2/(2*pi)^2 F*. Suppose that at t=0 the current is zero and the charge on the capacitor is 0.002 Coulombs.
(* This was 10^-6/(2*pi)^2 F earlier this week...)

a) What is tau?  What is T?  What is tau/T?

b) Write an equation for the charge on the capacitor as a function of time. (graph it)

c) At what time is the charge on the capacitor zero? What is the current in the circuit at that time? (graph it (current))

d) xc. why do you think i made C much smaller than R and L?
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4. Referring to the same circuit and initial conditions as in problem #3:
a) at the time when the charge on the capacitor has reached zero, where is the energy in the circuit and how much (energy) is there?

b) has any energy been lost? If so where did it go?

c) xc. draw a picture, a graph, which illustrates the lost energy as the area under a "curve". Describe in words what your picture represents and means.


5. Referring to the same circuit and initial conditions as in problem #3:
a)  at approximately what time, after t=0, does the current return to zero? (graph it)

b) at that time, what is the energy stored in the inductor and in the capacitor respectively. (graph the energies as a function of time.)

c) has much energy been lost? how much?  How does that compare to your result for energy lost from 4b?


6. Referring to the same circuit and initial conditions as in problem #3:
a) what is the voltage across the capacitor at t =0?

b) what is the voltage across the inductor at t = 0?

c) graph both voltages as a function of time.


7. a) Graph tau vs R.

b) Graph tau vs L.

c) Graph w vs C

d) Graph T vs C

e) extreme extra credit: graph T vs R.  what is the range and domain that makes sense to you and why?

Online homework poll

Please see the online homework poll to the right. Please feel free to comment here and suggest other questions and perspectives about the homework issue or anything related to that.

Homework 6 solutions

Homework 7: comments here

Note added: homework 7 is canceled. I will post some potentially important and hopefully interesting practice problems instead. 

HW #7 includes questions about circuits with L, C and R type elements. Please feel free to comment if some of these problems see impossible to do with what we have covered and emphasized in class. I will modify the HW later if that seems like a good idea.

In order for you to understand the nature oscillating LC circuits and LCR circuits, it is critically important for you to be quite familiar with sine and cosine functions, as well as exponential functions. 
What does cos(wt) look like from wt=0 to pi/2 ?
What does sin(wt) look like from wt=0 to pi/2 ?
Why does it seem reasonable, looking at the graphs, that the derivative of cos(wt) is -wsin(wt)?
etc...
If you can achieve that familiarity, then you will be able to visualize, understand and discuss the behavior of circuits, which will be a major point of emphasis for he final.

Consider the function: sin(wt), where w has units of 1/seconds, also known as frequency and t is time. What is its value at t=0? Can you graph this function?  What is its highest value? What is its lowest value? What is its period? These questions are best answered by looking at a graph and seeing and understanding what the function looks like -- how it goes up and down and repeats as a function of time.

The function, cos(wt), is equally important. Unlike the sine function it starts out at 1, at t=0, but just like the sine function, it oscillates up and down and repeats as a function of time. What does cos(wt) look like when it is multiplied by exp{-t/tau}...

Homework 5 solutions

Notes on oscillating circuits: solutions, phenomenology and differential equations.

Here are some notes related to the material we are covering on oscillating circuits. Your comments and questions will be greatly appreciated.

Functions and graphs that are critically important.

For the rest of this course, understanding the sine and cosine functions, their graphs, and their graphical relation to each other, will be critically important.  That will help you understand our exploration of the nature and behavior of oscillating LC  circuits.

Let's begin with a broader discussion of the essential graphs and functions which are intertwined with the physics of this course. For this course there are two types of graphs that are important. It will be essential for you to become familiar and comfortable with these graphs, and with the functions that create them. If you can achieve that familiarity, then you will probably do well in this course.

The first type, which you've already seen, are the exponential functions. These comes in two varieties:
1) a growing exponential that usually starts at zero, and grows, first at a high rate and then more slowly, to an asymptotic limiting value.
2) a decreasing exponential that begins at a finite value––not zero––and decreases exponentially to zero (or, in some cases, to a finite limiting value which is not zero).
These graphs are associated with the functions exp(-t/tau) and (1-exp(-t/tau)), respectively. If you do not feel intimately familiar with these graphs, their appearance and their functions, then you might want to review that in the near future. For the coming week, however, it is the sine and cosine functions that will be critically important, as discussed next.

The second kind of graphs that we will use extensively are the oscillating functions: sine and cosine. These will be essential for our study of oscillating circuits which begins on Monday and will be a major part of the rest of this course.

Consider the function: sin(wt), where w has units of 1/seconds, also known as frequency and t is time. What is its value at t=0? Can you graph this function?  What is its highest value? What is its lowest value? What is its period? Do you know what period means? These questions are best answered by looking at a graph and seeing and understanding what the function looks like -- how it goes up and down and repeats as a function of time.

The function, cos(wt), is equally important. Unlike the sine function it starts out at 1, at t=0, but just like the sine function, it oscillates up and down and repeats as a function of time.
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Appendix.  Advanced topics related to the graphs and discussion above:

Derivatives:
First, note that the derivative of the increasing exponential from the previous section is a decreasing exponential. One can relate that to what one sees in the graph. The increasing exponential has a positive slope which slope starts out large and decreases in magnitude as t increases. (The slope and the derivative are intimately related.)

The cosine and sine functions are similarly related. Specifically, the derivative of sin(wt) is w*cos(wt). It is also true that the derivative of cos(wt) is -w*sin(wt).

Curvature:
We say that the increasing exponential function has concave downward curvature. On the other hand, the decreasing exponential can be described as having concave upward curvature. The cosine and sine functions have more complex curvature which oscillates betweeen upward to downward.

Conundrum of the closed conducting loop.

At the bottom of page 482 there is something called “got it? 27.7”. It discusses a wire loop with three resistors symmetrically placed around the loop. The loop has an induced current, associated with a changing magnetic field; they ask about voltage differences in this system, particularly the readings of volt meters which they describe.

Has anyone looked at this? I think it's an interesting question. I am not sure that I agree with what the book says about this. Please feel free to comment here and engage in an open discussion of this potentially interesting topic. What do you think?

Oscillating circuits: HW#6, reading for this week,

For the coming week our reading will be from chapter 28. Our emphasis will be on LC circuits, 28.3. For these problems we will not be using phasor diagrams.

This week's homework assignment is due on Friday at about 1 PM.  There are not so many problems, but the problems tend to be difficult as they deal with the phenomenology of oscillating circuits. Most of the problems involve LC circuits.

There is one problem, 28.72, which is different, and is actually really difficult. I would recommend not doing 28.72. I left it in the assignment so that you could see how difficult problems become you have a resistor and capacitor together in an AC circuit. The resistor is actually the most difficult element to deal with because it dissipates energy, so within the context of the system we are looking at --the circuit-- energy is not conserved. We will come back to this in homework 7 ––next week––when we look at damped driven oscillators, in which energy is gradually lost as a function of time.

The material we cover from chapter 28 uses what we learned in Chapter 27, particularly the self inductance of the solenoid. This was our main point of emphasis from chapter 27.  I think, however, that the first section, 27.1, on induced currents is also very interesting, and can help us understand and contextualize Faraday's law (equation 27.2) more fully. Sections 27.3 and 27.4, on induction, energy, and inductors, also seem pretty interesting. I think Chapter 27 is a pretty key chapter and may be worth taking a second look at.

Midterm scores and how to prepare for the Final

The midterm was out of 104 points. Opportunities for extra credit totaled about 15 to 20 points.

Scores above 88 or 90 suggest a trajectory toward an A grade,
scores close to or above about 80 suggest a B-like trajectory,
scores below 60 suggest you should be concerned about not passing.
This is a continuous spectrum and one shouldn't place too much emphasis on magic breakpoints.
No matter what, if you do well enough on the final, you can pass the class.

If you did well on the midterm: good work. My advice is: don't slack off. The final is is very important for everyone.

I think the most important thing to get from the midterm is a sense of:
what sort of questions you will be asked, and
how to prepare for them and answer them on an exam.
Since our tests emphasize problem solving, I feel strongly that the best way to prepare is by practicing problem solving. Reading is of rather limited value because you are not being tested on retention of information, but rather on your ability to solve simple yet difficult problems.

If you read the blog posts assiduously in the week or two leading up to the final, then you will know what is going to be on the final. (One can see, looking at the blog, that the practice problems and posts on midterm preparation tell you what will be on the test and more.)  I know these problems can be very difficult; i think a good process gives you the best chance to succeed.


The final can be approached in the same way. It will focus on a few important, selected problems. From the blog posts, you will be able to discern what those will be. Even now in the class, our emphasis is on selected topics that I believe are the most important.  Those are the topics you will be tested on at the end of the quarter. Ideally the final, and the emphasis that precedes it, can enhance your learning in addition to its role for evaluation. I try to limit the number of topics we cover, so that we have possibility of some depth of coverage and understanding of the most important and memorable aspects of the course.