12 Kirchhoff s Rules, Terminal Voltage

Transcription

1 12 Kichhoff s Rules, Teminal Voltage Thee ae two cicuit-analysis laws that ae so simple that you may conside them statements of the obvious and yet so poweful as to facilitate the analysis of cicuits of geat complexity. The laws ae known as Kichhoff s Laws. The fist one, known both as Kichhoff s Voltage Law and The Loop Rule states that, stating on a conducto 1, if you dag the tip of you finge aound any loop in the cicuit back to the oiginal conducto, the sum of the voltage changes expeienced by you fingetip will be zeo. (To avoid electocution, please think of the finge dagging in an actual cicuit as a thought expeiment.) Kichhoff s Voltage Law (a.k.a. the Loop Rule) To convey the idea behind Kichhoff s Voltage Law, I povide an analogy. Imagine that you ae exploing a six-stoy mansion that has 20 staicases. Suppose that you stat out on the fist floo. s you wande aound the mansion, you sometimes go up stais and sometimes go down stais. ach time you go up stais, you expeience a positive change in you elevation. ach time you go down stais, you expeience a negative change in you elevation. No matte how convoluted the path of you exploations might be, if you again find youself on the fist floo of the mansion, you can est assued that the algebaic sum of all you elevation changes is zeo. To elate the analogy to a cicuit, it is best to view the cicuit as a bunch of conductos connected by cicuit elements (athe than the othe way aound as we usually view a cicuit). ach conducto in the cicuit is at a diffeent value of electic potential (just as each floo in the mansion is at a diffeent value of elevation). You stat with you fingetip on a paticula conducto in the cicuit, analogous to stating on a paticula floo of the mansion. The conducto is at a paticula potential. You pobably don t know the value of that potential any moe than you know the elevation that the fist floo of the mansion is above sea level. You don t need that infomation. Now, as you dag you finge aound the loop, as long as you stay on the same conducto, you fingetip will stay at the same potential. But, as you dag you fingetip fom that conducto, though a cicuit element, to the next conducto on you path, the potential of you fingetip will change by an amount equal to the voltage acoss the cicuit element (the potential diffeence between the two conductos). This is analogous to climbing o descending a flight of stais and expeiencing a change in elevation equal to the elevation diffeence between the two floos. If you dag you fingetip aound the cicuit in a loop, back to the oiginal conducto, you finge is again at the potential of that conducto. s such, the sum of the changes in electic potential expeienced by you finge on its tavesal of the loop must be zeo. This is analogous to stating that if you stat on one floo of the mansion, and, afte wandeing though the mansion, up and 1 Cicuits consist of cicuit elements and wies, I am calling the wies conductos. Moe specifically, a conducto in a cicuit is any wie segment, togethe will all othe wie segments connected diectly to the wie segment (with no intevening cicuit elements). 90

2 down staicases, you end up on the same floo of the mansion, you total elevation change is zeo. In dagging you finge aound a closed loop of a cicuit (in any diection you want, egadless of the cuent diection) and adding each of the voltage changes to a unning total, the citical issue is the algebaic sign of each voltage change. In the following example we show the steps that you need to take to get those signs ight, and to pove to the eade of you solution that they ae coect. xample Find the cuent though each of the esistos in the following cicuit. 222 Ω 560 Ω 15 volts 27 volts 18 volts Befoe we get stated, let s define some names fo the given quantities: ach two-teminal cicuit element has one teminal that is at a highe potential than the othe teminal. The next thing we want to do is to label each highe potential teminal with a and each lowe-potential teminal with a. We stat with the seats of MF. They ae tivial. By definition, the longe paallel line segment, in the symbol used to depict a seat of MF, is at the highe potential. 91

3 Next we define a cuent vaiable fo each leg of the cicuit. leg of the cicuit extends fom a point in the cicuit whee thee o moe wies ae joined (called a junction) to the next junction. ll the cicuit elements in any one leg of the cicuit ae in seies with each othe, so, they all have the same cuent though them. Note: In defining you cuent vaiables, the diection in which you daw the aow in a paticula leg of the cicuit, is just a guess. Don t spend a lot of time on you guess. It doesn t matte. If the cuent is actually in the diection opposite that in which you aow points, you will simply get a negative value fo the cuent vaiable. The eade of you solution is esponsible fo looking at you diagam to see how you have defined the cuent diection and fo intepeting the algebaic sign of the cuent value accodingly. Now, by definition, the cuent is the diection in which positive chage caies ae flowing. The chage caies lose electic potential enegy when they go though a esisto, so, they go fom a highe-potential conducto, to a lowe-potential conducto when they go though a esisto. That means that the end of the esisto at which the cuent entes the esisto is the highe potential teminal (), and, the end at which the cuent exits the esisto is the lowepotential teminal () of the esisto. 92

4 Now let s define some vaiable names fo the esisto voltages: V R1 V R2 Note that the and signs on the esistos ae impotant pats of ou definitions of V R1 and V R2. If, fo instance, we calculate V R1 to have a positive value, then, that means that the left (as we view it) end of V R1 is at a highe potential than the ight end (as indicated in ou diagam). If V R1 tuns out to be negative, that means that the left end of is actually at a lowe potential than the ight end. We do not have to do any moe wok if V R1 tuns out to be negative. It is incumbent upon the eade of ou solution to look at ou cicuit diagam to see what the algebaic sign of ou value fo V R1 means. With all the cicuit-element teminals labeled fo highe potential o fo lowe potential, we ae now eady to apply the Loop Rule. I m going to daw two loops with aowheads. The loop that one daws is not supposed to be a vague indicato of diection but a specific statement that says, Stat at this point in the cicuit. Go aound this loop in this diection, and, end at this point in the cicuit. lso, the stating point and the ending point should be the same. In paticula, they must be on the same conducto. (Neve stat the loop on a cicuit element.) In the following diagam ae the two loops, one labeled 1O and the othe labeled 2 O. 93

5 V R1 1 2 V R2 Now we wite KVL 1O to tell the eade that we ae applying the Loop Rule (Kichhoff s Voltage Law) using loop 1O, and tanscibe the loop equation fom the cicuit diagam: KVL 1O V 1 V R1 = 0 The equation is obtained by dagging you fingetip aound the exact loop indicated and ecoding the voltage changes expeienced by you fingetip, and then, emembeing to wite = 0. Stating at the point on the cicuit closest to the tail of the loop 1 aow, as we dag ou finge aound the loop, we fist tavese the seat of MF,. In tavesing we go fom lowe potential () to highe potential (). That means that the finge expeiences a positive change in potential, hence, entes the equation with a positive sign. Next we come to esisto. In tavesing we go fom highe potential () to lowe potential (). That s a negative change in potential. Hence, V R1 entes ou loop equation with a negative sign. s we continue ou way about the loop we come to the seat of MF and go fom lowe potential () to highe potential () as we tavese it. Thus, entes the loop equation with a positive sign. Finally, we aive back at the stating point. That means that it is time to wite = 0. We tanscibe the second loop equation in the same fashion: KVL 2 O V R2 = 0 With these two equations in hand, and knowing that V R1 = and V R2 =, the solution to the example poblem is staightfowad. (We leave it as an execise fo the eade.) It is now time to move on to Kichhoff s othe law. 94

6 Kichhoff s Cuent Law (a.k.a. the Junction Rule) Kichhoff s junction ule is a simple statement of the fact that chage does not pile up at a junction. (Recall that a junction is a point in a cicuit whee thee o moe wies ae joined togethe.) I m going to state it two ways and ask you to pick the one you pefe and use that one. One way of stating it is to say that the net cuent into a junction is zeo. Check out the cicuit fom the example poblem: V R1 V R2 In this copy of the diagam of that cicuit, I put a dot at the junction at which I wish to apply Kichhoff s Cuent Law, and, I labeled that junction. Note that thee ae thee legs of the cicuit attached to junction. In one of them, cuent flows towad the junction. In anothe, cuent flows towad the junction. In the thid leg, cuent flows away fom the junction. cuent away fom the junction counts as the negative of that value of cuent, towad the junction. So, applying Kichhoff s Cuent Law in the fom, The net cuent into any junction is zeo, to junction yields: KCL = 0 Note the negative sign in font of. cuent of into junction is the same thing as a cuent of out of that junction, which is exactly what we have. The othe way of stating Kichhoff s Cuent Law is, The cuent into a junction is equal to the cuent out of that junction. In this fom, in applying Kichhoff s Cuent Law to junction in the cicuit above, one would wite: KCL = Obviously, the two esults ae the same. 95

7 Teminal Voltage Moe Realistic Model fo a o DC lectical Powe Souce Ou model fo a battey up to this point has been a seat of MF. I said that a seat of MF can be consideed to be an ideal battey. This model fo a battey is good as long as the battey is faily new and unused and the cuent though it is small. Small compaed to what? How small? Well, small enough so that the voltage acoss the battey when it is in the cicuit is about the same as it is when it is not in any cicuit. How close to being the same? That depends on how accuate you want you esults to be. The voltage acoss a battey deceases when you connect the battey in a cicuit. If it deceases by five pecent and you calculate values based on the voltage acoss the battey when it is in no cicuit, you esults will pobably be about 5% off. bette model fo a battey is an ideal seat of MF in seies with a esisto. battey behaves vey much as if it consisted of a seat of MF in seies with a esisto, but, you can neve sepaate the seat of MF fom the esisto, and if you open up a battey you will neve find a esisto in thee. Think of a battey as a black box containing a seat of MF and a esisto. The esisto in this model is called the intenal esistance of the battey. Lowe-Potential () Teminal Highe-Potential () Teminal Seat of MF Intenal Resistance of the The point at which the seat of MF is connected to the intenal esistance of the battey is inaccessible. The potential diffeence between the teminals of the battey is called the teminal voltage of the battey. When the battey is not pat of a cicuit, the teminal voltage is equal to the MF. You can deduce this fom the fact that when the battey is not pat of a cicuit, thee can be no cuent though the esisto. If thee is no cuent though the esisto than the two teminals of the esisto must be at one and the same value of electic potential. Thus, in the diagam above, the ight end of the esisto is at the same potential as the high-potential teminal of the seat of MF. Now, let s put the battey in a cicuit: 96

8 B I R I ve indicated the two points and B on the cicuit fo communication puposes. The teminal voltage is the voltage fom to B (V B ). If you tace the cicuit, with you fingetip, fom to B, the teminal voltage (how much highe the potential is at B than it is at ) is just the sum of the voltage changes you finge expeiences along the path. (Note that this time, we ae not going all the way aound a loop. We do not wind up on the same conducto upon which we stated. So, the sum of the voltage changes fom to B is not zeo.) To sum the voltage changes fom to B, I will mak the teminals of the components between and B with fo highe potential and fo lowe potential. Fist the seat of MF: That s tivial. The shote side of the MF symbol is the lowe potential () side and the longe side is the highe potential () side. B I R Now, fo the intenal esistance of the battey: The end of the intenal esistance that the cuent entes is the highe-potential () end, and, the end that it exits is the lowe-potential () end. 97

9 V B I R Note that I have also defined, in the peceding diagam, the vaiable V fo the voltage acoss the intenal esistance of the battey. Remembe, to get the teminal voltage V B of the battey, all we have to do is to sum the potential changes that ou fingetip would expeience if we wee to dag it fom to B in the cicuit. (This is definitely a thought expeiment because we can t get ou fingetip inside the battey.) V B = V V B = I Note that, in the second line, I used the definition of esistance (V=I R) in the fom V = I, to eplace V with I. We have been consistent, in this book, with the convention that a double subscipt such as B can be ead to B meaning, in the case at hand, that V B is the sum of the potential changes fom to B (athe than the othe way aound), in othe wods, that V B is how much highe the electic potential at point B is than the electic potential at point. Still, thee ae some books out thee that take V B (all by itself) to mean the voltage of with espect to B (which is the negative of what we mean by it). So, fo folks that may have used a diffeent convention than you use, it is a good idea to diagammatically define exactly what you mean by V B. Putting a voltmete, labeled to indicate that it eads V B, and labeled to indicate which teminal is its teminal and which is its teminal is a good way to do this. V B I V B R 98