Sections begin this week. Cancelled Sections: Th Labs begin this week. Attend your only second lab slot this week.

Transcription

1 Announcements Sectons begn ths week Cancelled Sectons: Th 122. Labs begn ths week. Attend your only second lab slot ths week. Cancelled labs: ThF 25. Please check your Lab secton. Homework #1 onlne Due next Monday at 6pm EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 1 Revew Fnd v ab, v ca, v cb a 2 V c 1 V v cd b v bd d Note that the labelng conventon has nothng to do wth whether or not v s postve or negatve. EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 2 1

2 Lecture #2 OUTLINE Crcut Elements Crcut element IV characterstcs Constructon of a crcut model Krchhoff s laws a closer look Readng (Chapter 1, begn Ch. 2) EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 3 Crcut Elements 5 deal basc crcut elements: voltage source current source resstor nductor capactor actve elements, capable of generatng electrc energy passve elements, ncapable of generatng electrc energy Many practcal systems can be modeled wth just sources and resstors The basc analytcal technques for solvng crcuts wth nductors and capactors are smlar to those for resstve crcuts EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 4 2

3 Electrcal Sources An electrcal source s a devce that s capable of convertng nonelectrc energy to electrc energy and vce versa. Examples: battery: chemcal electrc dynamo (generator/motor): mechancal electrc (Ex. gasolnepowered generator, Bonnevlle dam) à Electrcal sources can ether delver or absorb power EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 5 Ideal Voltage Source Crcut element that mantans a prescrbed voltage across ts termnals, regardless of the current flowng n those termnals. Voltage s known, but current s determned by the crcut to whch the source s connected. The voltage can be ether ndependent or dependent on a voltage or current elsewhere n the crcut, and can be constant or tmevaryng. Devce symbols: v s _ vs =m v x _ vs =r x _ ndependent voltagecontrolled currentcontrolled EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 6 3

4 Ideal Current Source Crcut element that mantans a prescrbed current through ts termnals, regardless of the voltage across those termnals. Current s known, but voltage s determned by the crcut to whch the source s connected. The current can be ether ndependent or dependent on a voltage or current elsewhere n the crcut, and can be constant or tmevaryng. Devce symbols: s s =a v x s =b x ndependent voltagecontrolled currentcontrolled EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 7 Electrcal Resstance Resstance: the rato of voltage drop and current. The crcut element used to model ths behavor s the resstor. Crcut symbol: R Unts: Volts per Ampere ohms (W) The current flowng n the resstor s proportonal to the voltage across the resstor: v = R (Ohm s Law) Georg Smon Ohm where v = voltage (V), = current (A), and R = resstance (W) EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 8 4

5 Electrcal Conductance Conductance s the recprocal of resstance. Symbol: G Unts: semens (S) Example: Consder an 8 W resstor. What s ts conductance? Werner von Semens EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 9 Short Crcut and Open Crcut Wre ( short crcut ): R = 0 à no voltage dfference exsts (all ponts on the wre are at the same potental) Current can flow, as determned by the crcut Ar ( open crcut ): R = à no current flows Voltage dfference can exst, as determned by the crcut EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 10 5

6 Current vs. Voltage (IV) Characterstc Voltage sources, current sources, and resstors can be descrbed by plottng the current () as a functon of the voltage (v) v _ Passve? Actve? EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 11 IV Characterstc of an Ideal Voltage Source a V ab _ b _ v s =0 V s >0 1. Plot the IV characterstc for v s > 0. For what values of does the source absorb power? For what values of does the source release power? Repeat V s >0 (1) àfor <0 v s release < 0. power; >0 absorb power What s the IV characterstc for an deal wre? v EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 12 6

7 IV Characterstc of an Ideal Voltage Source a V ab _ b _ v s V s <0 v 2. Plot the IV characterstc for v s < 0. For what values of does the source absorb power? For what values of does the source release power? V s <0 à >0 release power; <0 absorb power EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 13 IV Characterstc of an Ideal Current Source v _ s v 1. Plot the IV characterstc for s > 0. For what values of v does the source absorb power? For what values of v does the source release power? V>0 absorb power; V<0 release power EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 14 7

8 IV Characterstc of Ideal Resstor a v _ R v b 1. Plot the IV characterstc for R = 1 kw. What s the slope? EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 15 Lumped Element Crcut Modelng (Model = representaton of a real system whch smplfes analyss) In crcut analyss, mportant characterstcs are grouped together n lumps (separate crcut elements) connected by perfect conductors ( wres ) An electrcal system can be modeled by an electrc crcut (combnaton of paths, each contanng 1 or more crcut elements). EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 16 8

9 Constructon of a Crcut Model The electrcal behavor of each physcal component s of prmary nterest. We need to account for undesred as well as desred electrcal effects. Smplfyng assumptons should be made wherever reasonable. EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 17 IV Characterstc of a real Voltage Source a _ v s V ab _ R v s v b 1. What s the IV characterstc for an real current source? 2. What s the IV characterstc for an deal wre? EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 18 9

10 Termnology: Nodes and Branches Node: A pont where two or more crcut elements are connected Branch: A path that connects two nodes EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 19 Notaton: Node and Branch Voltages Use one node as the reference (the common or ground node) label t wth a symbol The voltage drop from node x to the reference node s called the node voltage v x. The voltage across a crcut element s defned as the dfference between the node voltages at ts termnals v 1 Example: v a _ a _ c v s R 1 EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 20 b R 2 v b _ ß REFERENCE NODE 10

11 Crcut Loops A loop s formed by tracng a closed path n a crcut through selected basc crcut elements wthout passng through any ntermedate node more than once. Below are marked loops 1,2 and 3: v a 3 v 2 1 v b v 3 2 v c EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 21 Krchhoff s Laws Krchhoff s Current Law (KCL): The algebrac sum of all the currents enterng any node n a crcut equals zero. Krchhoff s Voltage Law (KVL): The algebrac sum of all the voltages around any loop n a crcut equals zero. Gustav Robert Krchhoff EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 22 11

12 Usng Krchhoff s Current Law (KCL) Consder a node connectng several branches: Use reference drectons to determne whether currents are enterng or leavng the node wth no concern about actual current drectons EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 23 Formulatons of Krchhoff s Current Law (Charge stored n node s zero.) Formulaton 1: Sum of currents enterng node = sum of currents leavng node Formulaton 2: Algebrac sum of currents enterng node = 0 Currents leavng are ncluded wth a mnus sgn. Formulaton 3: Algebrac sum of currents leavng node = 0 Currents enterng are ncluded wth a mnus sgn. EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 24 12

13 A Major Implcaton of KCL KCL tells us that all of the elements n a sngle branch carry the same current. We say these elements are connected n seres. Current enterng node = Current leavng node 1 = 2 EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 25 KCL Example 5 ma 10 ma 15 ma Currents enterng the node: Currents leavng the node: KCL: EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 26 13

14 Generalzaton of KCL The sum of currents enterng/leavng a closed surface s zero. Crcut branches can be nsde ths surface,.e. the surface can enclose more than one node! 2 3 Ths could be a bg chunk of a crcut, e.g. a black box 1 4 EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 27 Generalzed KCL Examples 50 ma 5mA 2mA EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 28 14

15 Examples Are these nterconnectons permssble? Ths crcut connecton s permssble. Ths s because the current sources can sustan any voltage across; Hence ths s permssble. Ths crcut connecton s NOT permssble. It volates the KCL. EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 29 Usng Krchhoff s Voltage Law (KVL) Consder a branch whch forms part of a loop: loop v 1 _ loop v 2 Movng from to We add V 1 Movng from to We subtract V 1 Use reference polartes to determne whether a voltage s dropped No concern about actual voltage polartes EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 30 15

16 Formulatons of Krchhoff s Voltage Law (Conservaton of energy) Formulaton 1: Sum of voltage drops around loop = sum of voltage rses around loop Formulaton 2: Algebrac sum of voltage drops around loop = 0 Voltage rses are ncluded wth a mnus sgn. (Handy trck: Look at the frst sgn you encounter on each element when tracng the loop.) Formulaton 3: Algebrac sum of voltage rses around loop = 0 Voltage drops are ncluded wth a mnus sgn. EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 31 A Major Implcaton of KVL KVL tells us that any set of elements whch are connected at both ends carry the same voltage. We say these elements are connected n parallel. v a_ v b _ Applyng KVL n the clockwse drecton, startng at the top: v b v a = 0 è v b = v a EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 32 16

17 KVL Example Three closed paths: v a a v 2 b v 3 c 3 1 v b 2 v c Path 1: Path 2: Path 3: EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 33 An Underlyng Assumpton of KVL No tmevaryng magnetc flux through the loop Otherwse, there would be an nduced voltage (Faraday s Law) Note: Antennas are desgned to pck up electromagnetc waves; regular crcuts often do so undesrably. B v (t) Avod these loops! How do we deal wth antennas (EECS 117A)? v(t) Include a voltage source as the crcut representaton of the nduced voltage or nose. (Use a lumped model rather than a dstrbuted (wave) model.) EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 34 17

18 Tps for KCL, KVL Fnd one formulaton whch works for you and stck wth t. I suggest: KCL Algebrac sum of currents leavng node = 0 Currents enterng are ncluded wth a mnus sgn. KVL Algebrac sum of voltage drops around loop = 0 Voltage rses are ncluded wth a mnus sgn. EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 35 Equvalent Resstance Arrangements of several resstors can be lumped nto one equvalent resstance: I I R 1 R 2 R 3 R 4 V º R EQ V EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 36 18

19 Resstors n Seres Consder a crcut wth multple resstors connected n seres. Fnd ther equvalent resstance. V SS I R 1 R 2 R 3 R 4 KCL tells us that the same current (I) flows through every resstor KVL tells us: Equvalent resstance of seres resstors s the sum of resstances EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 37 Voltage Dvder I = V SS / (R 1 R 2 R 3 R 4 ) I R 1 V 1 V SS R 2 R 3 V 3 R 4 EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 38 19

20 When can the Voltage Dvder Formula be Used? I I V SS R 1 R 2 R 3 V 2 V SS R 1 R 2 R 3 V 2 R 4 R 4 R 5 R V 2 2 = V SS R R R R Correct, f nothng else s connected to nodes R V 2 2 V SS R 1 R 2 R R 3 4 Why? What s V 2? EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 39 Resstors n Parallel Consder a crcut wth two resstors connected n parallel. Fnd ther equvalent resstance. x KVL tells us that the same voltage s dropped across each resstor I 1 I 2 KCL tells us: I SS R 1 R 2 Equvalent conductance of parallel resstors s the sum of conductances EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 40 20

21 Current Dvder x V x = I 1 R 1 = I 2 R 2 I 1 I 2 I SS R 1 R 2 EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 41 Generalzed Current Dvder Formula Consder a current dvder crcut wth >2 resstors n parallel: I V I 1 I 2 I 3 R 1 R 2 R 3 I V = æ 1 ö æ 1 ö æ 1 ö ç ç ç è R1 ø è R2 ø è R3 ø I 3 = V R 3 é = Iê ë1/r 1 1/R 1/R 3 2 1/R 3 ù ú û EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 42 21

22 Example: Power Absorbed by a Resstor p = v = ( R ) = 2 R p = v = v ( v/r ) = v 2 /R Note that p > 0 always, for a resstor à a resstor Example: dsspates electrc energy a) Calculate the voltage v g and current a. b) Determne the power dsspated n the 80W resstor. EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 43 Summary An deal voltage source mantans a prescrbed voltage regardless of the current n the devce. An deal current source mantans a prescrbed current regardless of the voltage across the devce. A resstor constrans ts voltage and current to be proportonal to each other: v = R (Ohm s law) EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 44 22

23 Summary (cont d) Krchhoff s current law (KCL) states that the algebrac sum of all currents at any node n a crcut equals zero. Krchhoff s voltage law (KVL) states that the algebrac sum of all voltages around any closed path n a crcut equals zero. Resstors n Seres Voltage Dvder Conductances n Parallel Current Dvder EE40 Summer 2006: Lecture 2 Instructor: Octavan Florescu 45 23