EE 215A Fundamentals of Electrical Engineering Lecture Notes Resistive Circuits 10/06/04. Rich Christie

Transcription

1 5A Introducton: EE 5A Fundamental of Electrcal Engneerng Lecture Note etve Crcut 0/06/04 ch Chrte The oluton of crcut wth more than two element need a lttle more theory. Start wth ome defnton: Node pont of connecton of the crcut. Wre deal wre have zero retance, are hort crcut. Can expand a note to a wre. 3 3 Can collape a wre to a node. How many node? c

2 Krchhoff' Current Law (KCL): A node jut a pont t can't tore charge. So charge flowng n to a node mut flow out of t. Flow of charge jut current, of coure. A clever way to ay th : "The um of current flowng nto a node zero." (We could alo ay the um of current flowng out of a node zero. But we ay "nto" becaue a guy named Krchhoff ad nto, and got h name on the ayng a) Krchhoff' Current Law (KCL).!

3 3 Suppoe I add a current comng out? It' jut the negatve of the current flowng n (! ) Or you can thnk of the um of current n equal to the um of current out (more natural le general) A A 6A What 3? 3()(6) A Or, nce 3 flow out, t the um of what flow n, whch 3 and 6 out 6 n plu 7 A out It good to develop the ablty to do KCL around a node for any combnaton of polarte and gn of current value. Krchhoff' Voltage Law (KVL): Start wth another defnton: Loop path that follow branche (crcut element) from node to node untl they end up back at the tartng pont. Count the loop: c

4 4 It hould be obvou that "The um of the voltage around a loop zero." It wa obvou to Krchhoff, and he got h name on another law, Krchhoff' Voltage Law, KVL. 8 V Let' go around the loop clockwe from lower left. 3V 8V 5V 0 (check) 3 V 5 V Note how polarty of voltage mportant! To deal correctly wth polarty:. Decde whch drecton you wll trace the loop.. Decde whether you wll ue the near de or the far de polarty a the gn for each term. IT DOES NOT MATTE WHICH CHOICE YOU MAKE AS LONG AS YOU APPLY IT CONSISTENTLY.

5 5 In the example above, we went clockwe and ued the far gn. Let' go clockwe and ue the near gn. We get 3V 8V 5V 0 V (ame anwer). For practce, try counterclockwe near gn and counterclockwe far gn. Intead of decdng each tme whch drecton and whch gn, make a choce now. I recommend clockwe and far gn, t eem the mot natural to me. Then ue thoe choce unle there a good reaon not to. 4 V 5 V v 3 V V What' the value of v? Wrte KVL clockwe tartng at lower left: 3 4 (5) v () 0 v V A wth KCL, we can wrte the um of the voltage around part of the loop equal to the um of the voltage around the other part. Thu, take a loop from the () termnal of v above clockwe one element, and et t equal to the voltage around the ret of the loop gong counterclockwe from the ame tartng pont to the ame endng pont: v (5) 4 3 () V Wth thee law plu Ohm' law we can olve (fnd all voltage and current) n any retve crcut. We'll pend a lot of tme learnng dfferent way to do th. Sere etor: We often fnd retor n ere. (Connected to the ame node at one end, and dfferent node at oppote end. Nothng can be connected to the mddle node.) The ame current flow through both retor (a bac tet for ere connecton).

6 6 There are two thng that are eay to do when you ee a two (or more) retor n ere. They are: v v v. Smplfy the crcut by replacng the two (or more) retor wth one ere equvalent retor.. Knowng the voltage acro the ere combnaton, fnd the voltage acro each of the retor. goe away! Note that you can't do both at the ame tme! If you combne retor, the node between them Let' do replacement frt. We want to fnd eq uch that current unchanged, for a gven voltage v. From KVL on the orgnal crcut v eq v! v v! v v v v (Ohm' Law) v 0 ( ) eq eq It hould be pretty clear that when you ee retor n ere, you can jut add them rght up. etor n ere add! What about fndng the voltage acro the retor? From above ( ) v v but from Ohm' Law v o

7 7 v v mlarly v v A voltage acro ere retor dvde among the retor n proporton to the retance of each hence the term Voltage Dvder. If there are more than two retor n ere, the um of all the retance appear n the dvor (the bottom of the fracton.) The larger retor wll have more of the voltage acro them. Example: v 4! v 6V, eq 0 Ω 0 V v 6! k! 9 V v 500!.5 k! Fnd v, power uppled by ource 500 v! 9!. 5V Power: eq 3 k! v p mw

8 8 O 9 3 ma 3 k p v 3! 9 7 mw Could alo fnd v acro each retor and p dpated by each retor. Could alo fnd p dpated n each retor ung current Could alo fnd p dpated n eq ung current Parallel etor: We often fnd retor n parallel connected to ame node at both end. Voltage acro retor the ame. Can we replace? Ye. v eq eq v v v v v eq Pragmatc: For two parallel retor, know t product over um or um over product, hmm, unt tell you whch (product over um correct). For three or more retor, ue "one over one over plu" eq 3... A you put n more parallel retor, the equvalent retance goe down. It alway le that the mallet retor n the parallel et. Note: Parallel conductance add: G G... G eq Lke voltage dvder there a current dvder. The mot common cae two retor:

9 9 v eq Note the retor you are NOT ntereted n appear n the numerator (top). Thnk about t, more current flow through the maller retor o the bgger one appear on the top. For more than two parallel retor you have to ue conductance. n G G n eq n 3... Example 6 A 0! 5! A 6! A A check! Source: Sere voltage ource mply add (KVL) Parallel current ource mply add (KCL) Parallel voltage ource of dfferent value are theoretcally mpoble. If you try th n practce, you wll hurt omethng (melt, explode or get very hot). Sere current ource of dfferent value have ame effect mpoble n theory, meltng, exploon or exce heat n practce. Ouch!

10 0 Summary: Tool you now have to attack ckt: Ohm' Law KCL KVL Sere combnaton parallel combnaton voltage dvder current dvder. Challenge to pull rght tool out of toolkt, apply to rght part of crcut, to get to anwer mot ealy (fatet, leat lkely to make an error both depend on the kll of the olver you!). When dong th n an untructured way, t called "Opportuntc crcut theory" Bt of a logc problem need to practce, practce, practce olvng crcut. Eventually t clck. There no tep by tep proce for choong the hortet correct equence of tep. (Can talk AI problem olver f you want.) There are, however, ytematc method of oluton that we'll cover n the next few lecture, nodal and meh analy.