6.01: Introduction to EECS I Lecture 7 March 15, 2011

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1 6.0: Introducton to EECS I Lecture 7 March 5, : Introducton to EECS I Crcuts The Crcut Abstracton Crcuts represent systems as connectons of elements through whch currents (through arables) flow and across whch oltages (across arables) deelop. March 5, 20 The Crcut Abstracton We can represent the flashlght as a oltage source (battery) connected to a resstor (lght bulb). The Crcut Abstracton We can represent the flow of water by a crcut. r (t) r r o h(t) r o (t) P = ρgh The oltage source generates a oltage across the resstor and a current through the resstor. Flow of water nto and out of tank are represented as through arables r and r o, respectely. Hydraulc pressure at bottom of tank s represented by the across arable P = ρgh. The Crcut Abstracton Crcuts are mportant for two ery dfferent reasons: as physcal systems power (from generators and transformers to power lnes) electroncs (from cell phones to computers) as models of complex systems neurons bran cardoascular system hearng The Crcut Abstracton Crcuts are bass of enormously successful semconductor ndustry Dual-Core Itanum 2 Pentum 4 Pentum III Pentum II Pentum Itanum 2 Itanum,000,000,000 00,000,000 0,000,000,000,000 00, , , year # of transstors What desgn prncples enable deelopment of such complex systems?

2 6.0: Introducton to EECS I Lecture 7 March 5, 20 The Crcut Abstracton Crcuts as models of complex systems: myelnated neuron. The Crcut Abstracton Crcuts represent systems as connectons of elements through whch currents (through arables) flow and across whch oltages (across arables) deelop. h(t) r (t) r o (t) r r o P = ρgh The prmtes are the elements: sources, capactors, and resstors. The rules of combnaton are the rules that goern flow of current (through arable) and deelopment of oltage (across arable). Analyzng Crcuts: Elements Analyzng Smple Crcuts We wll start wth the smplest elements: resstors and sources R = R 0 = 0 I 0 = I 0 Analyzng smple crcuts s straghtforward. Example : Ω The oltage source determnes the oltage across the resstor, =, so the current through the resstor s = /R = / = A. Example 2: A Ω The current source determnes the current through the resstor, = A, so the oltage across the resstor s = R = =. What s the current through the resstor below? Analyzng More Complex Crcuts More complex crcuts can be analyzed by systematcally applyng Krchhoff s oltage law (KL) and Krchhoff s current law (KCL). Ω A. A 2. 2A 3. 0A 4. cannot determne 5. none of the aboe 2

3 6.0: Introducton to EECS I Lecture 7 March 5, 20 Analyzng Crcuts: KL KL: The sum of the oltages around any closed path s zero. Analyzng Crcuts: KL KL: The sum of the oltages around any closed path s zero. = = Example: 2 4 = 0 or equalently = 2 4. How many other KL relatons are there? How many KL equatons can be wrtten for ths crcut? Analyzng Crcuts: KL Planar crcuts can be characterzed by ther nner loops. KL equatons for the nner loops are ndependent. = = 0 A : 2 4 = 0 A 2 B C B : = 0 C : = 0 Analyzng Crcuts: KL KL: Summary All possble KL equatons for planar crcuts can be generated by combnatons of the nner loops. The sum of the oltages around any closed path s zero. One KL equaton can be wrtten for eery closed path n a crcut. Sets of KL equatons are not necessarly lnearly ndependent. = AB KCL equatons for the nner loops of planar crcuts are lnearly ndependent. A : 2 4 = 0 B : = 0 AB : = = 0 3

4 6.0: Introducton to EECS I Lecture 7 March 5, 20 Krchhoff s Current Law The flow of electrcal current s analogous to the flow of ncompressble flud (e.g., water). Krchhoff s Current Law The net flow of electrcal current nto (or out of) a node s zero Current flows nto a node and two currents 2 and 3 flow out: = 2 3 Here, there are two nodes, each ndcated by a dot. The net current out of the top node must be zero: 2 3 = 0. Krchhoff s Current Law Electrcal currents cannot accumulate n elements, so current that flows nto a crcut element must also flow out. How many lnearly ndependent KCL equatons can be wrtten for the followng crcut? = 4 2 = 5 3 = Snce 2 3 = 0 t follows that = 0. How many dstnct KCL relatons can be wrtten for ths crcut? Analyzng Crcuts: KCL The net current out of any closed surface (whch can contan multple nodes) s zero node : 2 3 = 0 node 2: = 0 nodes 2: = = 0 4

5 6.0: Introducton to EECS I Lecture 7 March 5, 20 Analyzng Crcuts: KCL The net current out of any closed surface (whch can contan multple nodes) s zero KCL: Summary The sum of the currents out of any node s zero. One KCL equaton can be wrtten for eery closed surface (whch contan one or more nodes) n a crcut. Sets of KCL equatons are not necessarly lnearly ndependent. KCL equatons for eery prmte node except one (ground) are lnearly ndependent. nodes 2: = 0 node 3: = 0 nodes 23: = 4 5 = 0 Net current out of nodes 23 = net current nto bottom node! KL, KCL, and Consttute Equatons Crcuts can be analyzed by combnng all lnearly ndependent KL equatons, all lnearly ndependent KCL equatons, and one consttute equaton for each element. = KL, KCL, and Consttute Equatons Unfortunately, there are a lot of equatons and unknowns. = unknowns:, 2, 3, 4, 5, 6,, 2, 3, 4, 5 and 6. 2 equatons: 3 KL 3 KCL 5 for resstors for source Ths crcut s characterzed by 2 equatons n 2 unknowns! Node oltages The node method s one (of many) ways to systematcally reduce the number of crcut equatons and unknowns. label all nodes except one: ground (gnd) 0 olts wrte KCL for each node whose oltage s not known Loop Currents The loop current method s another way to systematcally reduce the number of crcut equatons and unknowns. label all the loop currents wrte KL for each loop 0 0 R 2 R 3 R 6 e e 2 R 4 R 5 KCL at e : e 0 R 2 e e 2 R 6 e R 4 = 0 KCL at e 2 : e 2 0 R 3 e 2 e R 6 e 2 R 5 = 0 0 a R 2 R 3 b R 6 R 4 R 5 c loop a: 0 R 2 ( a b ) R 4 ( a c ) = 0 loop b: R 2 ( b a ) R 3 ( b ) R 6 ( b c ) = 0 loop c: R 4 ( c a ) R 6 ( c b ) R 5 ( c ) = 0 gnd sole (here just 3 equatons and 3 unknowns) sole (here just 2 equatons and 2 unknowns) 5

6 6.0: Introducton to EECS I Lecture 7 March 5, 20 Analyzng Crcuts: Summary We hae seen three (of many) methods for analyzng crcuts. Each one s based on a dfferent set of arables: currents and oltages for each element node oltages loop currents Each requres the use of all consttute equatons. Each prodes a systematc way of dentfyng the requred set of KL and/or KCL equatons. Determne the current I n the crcut below. 5 I 3 Ω 2 Ω 0 A. A A 3. A 4. 5 A 5. none of the aboe Common Patterns Crcuts can be smplfed when two or more elements behae as a sngle element. Seres Combnatons The seres combnaton of two resstors s equalent to a sngle resstor whose resstance s the sum of the two orgnal resstances. A one-port s a crcut that can be represented as a sngle element. R R 2 R s A one-port has two termnals. Current enters one termnal () and exts the other (), producng a oltage () across the termnals. = R R 2 R s = R R 2 = R s The resstance of a seres combnaton s always larger than ether of the orgnal resstances. Parallel Combnatons The parallel combnaton of two resstors s equalent to a sngle resstor whose conductance (/resstance) s the sum of the two orgnal conductances. What s the equalent resstance of the followng one-port. R R 2 R p 2 = R R 2 = R p = = R R 2 R p = R p R R 2 R R 2 R = R R 2 R R 2 R R R 2 2 The resstance of a parallel combnaton s always smaller than ether of the orgnal resstances. 6

7 6.0: Introducton to EECS I Lecture 7 March 5, 20 oltage Dder Resstors n seres act as oltage dders. I Current Dder Resstors n parallel act as current dders. I I I 2 R R R 2 R 2 2 = (R R 2 ) I I = R R 2 = R I = R R R 2 I = = R R 2 I = R R 2 R 2 I = I R R R R R 2 R R 2 I 2 = = R R 2 I = R R 2 R I = I R 2 R 2 R 2 R R 2 R R 2 2 = R 2 I = R 2 R R 2 Summary Ω 3Ω Crcuts represent systems as connectons of elements through whch currents (through arables) flow and across whch oltages (across arables) deelop. 5 Whch of the followng s true?. o < o < o < o 2 5. o > 2 2Ω 6Ω o We hae seen three (of many) methods for analyzng crcuts. Each one s based on a dfferent set of arables: currents and oltages for each element node oltages loop currents We can smplfy analyss by recognzng common patterns: seres and parallel combnatons oltage and current dders 7

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