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1、【精品文档】如有侵权，请联系网站删除，仅供学习与交流Lecture2.精品文档.Lecture 2Good morning, OK. Lets get started. We have one handout today. Thats your lecture notes. Theres some copies still outside for those who havent picked one up. In general, what I do is, in the lecture notes, I leave out large amounts of material.So, thi

2、s will enable you to keep your hands busy while Im lecturing and take down some notes and so on. So, dont assume that everything that I talk about is on here. Please follow along. OK, so as is my usual practice, let me start with a quick review of what we covered so far.So what we did primarily was

3、looked at this discipline that we call the lump matter discipline, which was very similar, very reminiscent of the point mass simplification in physics. And this discipline, this set of constraints we imposed on ourselves, allowed us to move from Maxwells equations to a very, very simple form of alg

4、ebraic equations.And specifically, the discipline took two forms. One is, we said that we will deal with elements for whom the rate of change of magnetic flux is zero outside of the elements, and for whom the rate of change of charge I want to charge inside the element was zero.So, if I took any ele

5、ment, any element that I called a lump circuit element, like a resistor or a voltage source, and I put a black box around it, then what Im saying is that the net charge inside that is going to be zero.And this is not true in general. We will see examples where, if you choose some piece of an element

6、 for example, there might be charge buildup, but net inside the, if I put a box around the entire element, I am going to assume that the rate of change of charge is going to be zero.So, what this did was it enabled us to create the lump circuit abstraction, where I could take elements, some element

7、of the sort, this could be a resistor, a voltage source, or whatever, and I could now ascribe a voltage, some voltage across an element, and also some current, i, that was going into the element.And as I go forward, when I label the voltages and currents across and through elements, Im going to be f

8、ollowing a convention. OK, the convention is that Im going to label, if I label V in the following manner, then Im going to label i for that element as a current flowing into the positive terminal.Its just a convention. By doing this, it turns out that the power consumed by the element is vi is posi

9、tive. OK, so by choosing I going in this way into the positive terminal, the power consumed by the element is going to be positive.OK, so in general of even simply following this convention, when I label voltages and currents, Ill be labeling the current into an element entering in through the plus

10、terminal. Remember, of course, if the current is going this way, lets have one amp of current flowing this way, then when I compute the current, i will come out to be negative.OK, so by making these assumptions, the assumptions of the lumped matter discipline, I said I was able to simplify my life t

11、remendously. And, in particular what it did was it allowed me to take Maxwells equations, OK, and simplify them into a very simple algebraic form, which has both a voltage law and a current law that I call Kirchhoffs voltage law, and Kirchhoffs current law.KVL simply states that if I have some circu

12、it, and if I measured the voltages in any loop in the circuit, so if I look at the voltages in any loop, then the voltages in the loop would sum to zero. OK, so I measure voltages in the loop, and they will sum to zero.Similarly, for the current, if I take a node of a circuit, if I build the circuit

13、, a node is a point in the circuit where multiple edges connect. If I take a node, then the current coming into that node, the net current coming into a node is going to be zero.OK, so if I take any node of the circuit and sum up all the currents going into that node, they will all net sum to zero.

14、So, notice what Ive done is by this discipline, by this constraint I imposed on myself, I was able to make this incredible leap from Maxwells equations to these really, really simple algebraic equations, KVL and KCL.And I promise you, going forward to the rest of 6.002, if this is all you know, you

15、can pretty much solve any circuit using these two very simple relations. Its actually really, really simple. Its all very simple algebra, OK? So, just to show you an example, let me do a little demonstration.Let me build let me build a small circuit and measure some voltages for you, and show you th

16、at the voltages, indeed, add up to zero. So, heres my little circuit. So, Im going to show you a simple circuit that looks like this, and lets go ahead and measure some voltages and currents.In terms of terminology to remember, this is called a loop. So if I start from the point C and I travel throu

17、gh the voltage source, come to the node A down through R1 and all the way down through R2 back to C, thats a loop.Similarly, this point A is a node where resistor R1 the voltage source V0, and R4 are connected. OK, just make sure your terminology is correct. So, what Ill do is Ill make some quick me

18、asurements for you, and show you that these KVL and KCL are indeed true.So, the circuits up there, could I have a volunteer? Any volunteer? All you have to do is write things on the board. Come on over. OK, so let me take some measurements, and why dont you write down what I measure on the board? Wh

19、at Ill do is, let me borrow another piece of chalk here.What Ill do is focus on this loop here, and focus on this node and make some measurements. All right, so you see the circuit up there. OK, so I get 3 volts for the voltage from C to A. so why dont you write down 3 volts? OK, so the next one is

20、-1.6.And so that will be, Im doing AB, V_AB. OK, and then let me do the last one. It is -1.37. The measurements, I guess, have been this way. So, whats written is V_AC. But its OK for now. Dont worry about it.So, well, thank you. I appreciate your help here. OK, so within the bonds of experimental e

21、rror, noticed that if I add up these three voltages, they nicely sum up to zero. OK, next let me focus on this node here.And at this node, let me go ahead and measure some currents. What Ill do now is change to an AC voltage so that I can go ahead and measure the current without breaking my circuit.

22、 OK, this time around, youll get to see the measurements that Im taking as well.So, what I have here, I guess you can see it this way. What I have here is three wires that I have pulled out from D. And this is the node D, OK? So, I have three wires coming into the node D just to make it a little bit

23、 easier for me to measure stuff.OK, so everybody keep your fingers crossed so I dont look like a fool here. I hope this works out. So, you roughly get, whats that, 10 mV. OK, so its about 10 mV peak to peak out there, and lets say that if the waveform raises on the left-hand side, its positive.So, i

24、ts positive 10 mV. And another positive 10 mV, so thats 20 mV. And this time, its a negative, roughly 20, I guess, -20. So, Im getting, in terms of currents, I have a -10, -10, Im sorry, positive 10, positive 10, and a -20 that adds up to zero.But more interestingly, I can show you the same thing by

25、 holding this current measuring probe directly across the node. And, notice that the net current that is entering into this node here is zero.OK, so that should just show you that KCL does indeed hold in practice, and it is not just a figment of our imaginations. So, before I go on, I wanted to poin

26、t one other thing out. Notice that Ive written down two assumptions of the lumped matter discipline, OK? There is a total assumption of the lump matter discipline, and that assumption is, in spirit, at least, shared by the point mass simplification in physics as well.Can someone tell me what that as

27、sumption is? A total assumption, which I did not mention, which you can read in your notes in section 8.2 in the appendix, whats a total assumption that is shared in spirit with the point mass simplification? Anybody? A total assumption to be made here is that in all the signals that we will study i

28、n this course, weve made the assumption that the signal speeds of interest, transition speeds, and so on, are much slower than the speed of light.OK, that my signal transition speeds of interest are much slower than the speed of light. Remember, the laws of motion, the discrete laws of motion break

29、down if your objects begin moving at the speed of light.OK, the same token here, our lump circuit abstraction breaks down if we approach the speed of light. And there are follow on courses that talk about waveguides and other distributed analysis techniques that deal with signals that travel close t

30、o speeds of light.OK, so with that, let me go on to talking about method one of circuit analysis. This is called the basic KVL KCL method. So just based on those two simple algebraic relations, I can analyze very interesting and complicated circuits.The method goes as follows. So, lets say our goal

31、is, given a circuit like this, our goal is to solve it. OK, in this course, we will do two kinds of things: analysis and synthesis. Analysis says, given a circuit, OK, what can you tell me about the circuit? OK, so well solve existing circuits for all the voltages and currents, voltages across eleme

32、nts, and currents through those elements.Synthesis says, given a function, I may ask you to go and build circuits. OK, so for analysis here, we can apply this method that I want to show you. And the idea here is that, given a circuit like this, let us figure out all the voltages and currents that ar

33、e a function of the way these elements are connected.So, the basic KVL and KCL method has the following steps. The first step is to write down the element VI relationships. OK, right down the element VI relationships for all the elements. The second step is write KCL for all the nodes, and the third

34、 step is to write KVL for all the loops in the circuit.Thats it. Just go ahead and write down element rules, KVL, and KCL, and then go ahead and solve the circuit. So, what well do, well do an example, of course. But, just as a refresher, weve looked at a bunch of elements so far, and for the resist

35、or, the element relation says that V is pi R, where R is the resistance of the element here.For a voltage source, V is equal to V nought. Thats the element relationship. And for a current source, the element is the relation is, i is simply the current flowing through the element. OK, so these are so

36、me of the simple element rules for the devices that the current source, voltage source, and the resistor.So lets go ahead and solve this simple circuit. And what Ill do is go ahead and solve the circuit for you. OK, if you turn to page five of your notes, Im going to go ahead and edit the circuit he

37、re.You can scribble the values on your notes on page five. OK, so as a first step of my KVL KCL method, I need to write down all my element VI relationships. So, before I do that, let me go ahead and label all the voltages and currents that are unknowns in the circuit.So, let me label the voltages a

38、nd currents associated with the voltage source as here. Notice, I continue to follow this convention where whenever I label voltages and currents for an element, I will show the current going into the positive terminal of the element variable, OK, after element variable voltage.So here, I have V nou

39、ght and I nought. Let me pause here for five seconds and show you a point of confusion that happens sometimes. Often times, people confuse between what is called the variable that is associated with the element versus the element value.OK, notice that here, capital V nought is the voltage that this

40、voltage source provides, while this name here, v nought, is simply a variable that weve used to label the voltage across that element.So, similarly, I can label v1 as the voltage across the resistor, and i1 is the current flowing through the resistor. So this method of labeling, where I follow the c

41、onvention, that the current flows into the positive terminal is called the associated variables discipline.I was trying to use the word discipline in situations where you have a choice, OK, and of a variety of possible choices, you pick one as the convention. OK, so here, as a convention, we use the

42、 associated variables discipline, and use that method to consistently label the unknown voltages and currents in our circuits.OK, so let me continue the labeling here, v4, i4, i3, v3 here, and v2 and i2, v5, and i5. I think thats it. So, Ive gone ahead and labeled all my unknowns. So each of these v

43、oltages and currents are the voltages and currents associated with each of the elements.And my goal is to solve for these. OK, so in terms of our solution here, lets follow the method that I outlined for you. So, as the first step I am simply going to go ahead and write down all the element VI relat

44、ionships.OK, so as a first step, Im going to go ahead and write down all the VI relationships. So, can someone yell out for me the VI relationship for the voltage source? OK, good. So, v0 is capital V nought, that is that the variable V nought is simply equal to the voltage, v0.Similarly, I can writ

45、e the others. v1 is i1, R1. v2 is i2, R2, and so on. OK, and I have one, two, three, four, five, six elements. So, I will get six such equations. Step two, Im going to go ahead and write KCL for the nodes in my system.So, let me start with node A. So, for node A, let me take as positive the currents

46、 going out of the node. So, I get i nought flowing out, plus i1 flowing out, plus i4 flowing out, and they must sum to zero for node A.Then, I can go ahead and do the other nodes, lets say, for example, I do node B. For node B, I have i2 going out. Thats positive, i3, and i1 is coming in, so I get -

47、i1 equals zero. OK, so I have one, two, three, four, I have four nodes.OK, so I would get four equations. It turns out that the fourth equation is not independent. You can derive it from the others. So, I get three independent equations out of this. I can then write KVL.And for KVL, I just go down m

48、y loops here. And let me go through this first loop here in this manner. OK, and a simple trick that I use, you have to be incredibly careful when you go through this in keeping your minuses and pluses correct.Otherwise you can get hopelessly muddled. Once you label it, you need to be sure that you

49、get all your minuses and pluses correct. So, for KVL, what Id like to do is, lets say I start at C, and from C Im going to go to A.For A I go to B, and from B Im going to come back to C. OK, thats how I traverse my loop. And, the trick that Im going to follow is, as my finger walks through that loop, Im going to label the voltage as the first sign that I see for that voltage.OK,