6.01: Introduction to EECS 1 Week 6 October 15, 2009

Transcription

1 6.0: ntroducton to EECS Week 6 October 5, : ntroducton to EECS Crcuts The Crcut Abstracton Crcuts represent systems as connectons of component through whch currents (through arables) flow and across whch oltages (across arables) deelop. Week 6 October 5, 2009 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 the bass of our enormously successful semconductor ndustry Dual-Core tanum 2 Pentum 4 Pentum Pentum Pentum tanum 2 tanum,000,000,000 00,000,000 0,000,000,000,000 00, , , year # of transstors The Crcut Abstracton Crcuts as models of complex systems: myelnated neuron. Model of Myelnated Nere Fber What s a Crcut? Crcuts are connectons of components through whch currents (through arables) flow and across whch oltages (across arables) deelop. _ nternode Node of Raner nternode Node of Raner nternode Dendrtc tree nternode Node of Raner Cell body Myelnated fber Axonal tree Fgure by MT OpenCourseWare.

2 6.0: ntroducton to EECS Week 6 October 5, 2009 Rules Goernng Flow Rule : Currents flow n loops. Example: flow of electrcal current through a flashlght Rules Goernng Flow Rule 2: Lke the flow of water, the flow of electrcal current (charged partcles) s ncompressble. Example: flow of water through a branchng pont 2 3 When the swtch s closed, electrcal current flows through the loop. The same amount of current flows nto the bulb (top path) and out of the bulb (bottom path). What comes n must go out. Here = 2 3. Krchoff s Current Law: the sum of the currents nto a node s zero. Rules Goernng Flow n electrcal crcuts, we represent current flow by arrows on lnes representng connectons (wres). Nodes Nodes are represented n crcut dagrams by lnes that connect crcut components. 2 The followng crcut has three components, each represented wth a box. = The dot represents a node whch represents a connecton of two or more segments. There are two nodes, each ndcated by a dot. The net current nto or out of each of these nodes s zero. Therefore 2 3 = 0. What s a Crcut? Crcuts are connectons of components through whch currents (through arables) flow and across whch oltages (across arables) deelop. Rules Goernng oltages oltages accumulate n loops. Example: the seres combnaton of two.5 batteres supples Krchoff s oltage Law: the sum of the oltages around a closed loop s zero. 2

3 6.0: ntroducton to EECS Week 6 October 5, 2009 Alternate Representaton: Node oltages Node oltages represent the oltage between each node n a crcut and an arbtrarly selected ground. e e e gnd Node oltages and component oltages are dfferent but equalent representatons of oltage. component oltages represent the oltages across components. node oltages represent the oltages n a crcut. Rules Goernng Components Each component s represented by a relatonshp between the oltage across the component to the current through the component. Examples: R = R 0 = 0 (regardless of ) 0 = 0 (regardless of ) Node-oltage-and-Component-Current (NCC) Method Combnng KCL, node oltages, and component equatons leads to the NCC method for solng crcuts: Assgn node oltage arables to eery node except ground (whose oltage s arbtrarly taken as zero). Assgn component current arables to eery component n the crcut. Wrte one consttute relaton for each component n terms of the component current arable and the component oltage, whch s the dfference between the node oltages at ts termnals. Express KCL at each node except ground n terms of the component currents. Sole the resultng equatons. Analyzng Smple Crcuts Analyzng smple crcuts s straghtforward. Ω The oltage source determnes the oltage across the resstor, =, so the current through the resstor s = /R = / = A. 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? Ω A Common Patterns There are a number of common patterns that facltate desgn and analyss: seres resstances parallel resstances oltage dders current dders. A 2. 2A 3. 0A 4. cannot determne 5. none of the aboe 3

4 6.0: ntroducton to EECS Week 6 October 5, 2009 Seres Combnatons The seres combnaton of two resstors s equalent to a sngle resstor whose resstance s the sum of the two orgnal resstances. R R s 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. R R p = R = R s = R = R p R s = R The resstance of a seres combnaton s always larger than ether of the orgnal resstances. = = R R p = R p R R R 2 R = R R R R 2 The resstance of a parallel combnaton s always smaller than ether of the orgnal resstances. oltage Dder Resstors n seres act as oltage dders. What s the equalent resstance of the followng crcut. R = R R = R = R 2 = = R Current Dder Resstors n parallel act as current dders. Fnd o. 2 = (R ) R 2 8 Ω 2 Ω 6 Ω o = R = R R = R R R = 2 = = R = R R = R R R 4

5 6.0: ntroducton to EECS Week 6 October 5, 2009 Loadng Addng (or changng the alue of) a component alters all of the oltages and currents n a crcut (except n degenerate cases). Consder dentcal lght bulbs connected n seres across a battery. What happens f we add thrd lght bulb? # #2 #3 Because the same current passes through both lght bulbs, they are equally brght. Closng the swtch wll make. bulb brghter 2. bulb 2 dmmer 3.. and bulbs, 2, & 3 equally brght 5. none of the aboe Loadng Loadng dd not occur n LT systems. Loadng Q: What s dfferent about a crcut? Example: addng H 2 has no effect on Y Y X H Y X H H 2 Y = H X regardless of H 2. Z L A: A new component generally alters the currents at the nodes to whch t connects. Bufferng Summary Effects of loadng can be dmnshed or elmnated wth a buffer. An deal buffer s an amplfer that senses the oltage at ts nput wthout drawng any current, and sets ts output oltage equal to the measured nput oltage. x L =0 y = x Crcuts represent systems as connectons of components through whch currents (through arables) flow and across whch oltages (across arables) deelop. There are a number of common patterns that facltate desgn and analyss: seres resstances parallel resstances oltage dders current dders Buffers elmnate loadng and thereby smplfy desgn and analyss. We wll dscuss how to use op-amps to make buffers n next lecture. 5

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