2 Electric Circuits Concepts Durham

Transcription

1 Chaper 3 - Mehods Chaper 3 - Mehods nroducion Elecrical laws Definiions Kirchhoff Faraday Conservaion Power Complee Fields Laplace dc Branches, Nodes, Loops Superposiion Topology definiions Topology Theorem Kirchhoff laws Circui Laws Circui nalysis Mehods Nodal - node volage Mesh - loop curren Problems...

2 2 Elecric Circuis Conceps Durham 3. nroducion One of he firs requiremens for analysis is mahemaical ools, echniques, and ransforms. Nex a fundamenal undersanding of elecrical measuremens and calculaions is necessary. Then he consrucion of passive elemens o define impedance can be accomplished. Now, circui performance can be analyzed. Numerous mehods are available. These pracices are developed from he elecromagneic energy equaion and he conservaion of energy []. W pq Σ W Elecrical laws The conceps embedded in he very fundamenal node form of he elecromagneic energy equaion are saggering Definiions Firs, he definiion of volage, curren, and frequency is conained in he expression. How many more relaionships can be found? Consider jus a few. Elecrical energy is volage muliplied by charge. Magneic energy is curren muliplied by magneic pole flux. These relaionships were used o describe he energy sorage in impedance elemens. W W Elecric Magneic vq ip Kirchhoff Now he plo hickens even furher. Circui analysis is ofen described using wo laws developed by he Prussian mahemaician and physicis, Gusav Rober Kirchhoff in 854. Boh hese laws are imbedded in he very simple elecromagneic energy definiion. Firs, apply he consrain of conservaion o he relaionship. This ses he sum of he energy equal o zero. Nex, hold one erm consan. Then, he sum of he changing erm is zero. Kirchhoff s curren law (KCL) can be saed succincly: When he magneic flux is consan, he sum of he curren a a node is zero. Σ W 0 pq Σ 0 Σ 0 p k Similarly, Kirchhoff s volage law (KVL) can be saed: When he charge is consan, he sum of he volage around a loop is zero.

3 Chaper 3 Mehods 3 pq Σ 0 Σ V 0 q k Faraday Faraday s law is also imbedded in he elecromagneic energy correlaion. saes ha he rae of change of he magneic flux or pole srengh is equal o he induced volage. This is he definiion of volage. V induced p q k Conservaion When conservaion of energy is applied o he elecromagneic energy expression, an enire paradigm is developed. Since charge, flux, and ime canno conver o he oher, conservaion applies o each iem individually. Conservaion of charge saes he sum of he charge is zero. Charge is discree and is an inegral muliple of he charge on an elecron or proon. -9 q Coulombs The oal charge is always balanced. proon is 836 imes heavier han an elecron. Neverheless, i has exacly he same charge. Conservaion of magneic pole srengh saes he sum of he magneic flux is zero. Magneic poles always exis in a balanced pair wih a norh and souh poles. Conservaion of ime and frequency implies ha she sum of frequency is zero. lernaively, ime is conserved. Conservaion of ime and frequency is direcly relaed o Planck. The number of waves or cycles, w, is a discree number, jus like charge, q. w W h h Planck's consan wdescree numbers These relaionships are implici in he elecromagneic expression when he consrain of conservaion of energy is imposed Power Power is he energy over ime. Power imposes anoher ime on he elecromagneic energy. W p r pq r vi Σ W 0 pq Σ 0 Σ q p 0 k Σ p q 0 k Σ 0 pq k Specrum Frequency Direc curren 0 C power 60 Hz Sound khz M radio MHz FM radio 00 MHz UHF TV 500 MHz Cell phone 800 MHz Cell phone 2.9 GHz Saellie radio 2.2 GHz Wireless LN 2.4 GHz Microwave 2.5 GHz Radar 5.0 GHz nfrared Visible Ulraviole X-ray Gamma ray ll objecs have an elecromagneic (radio) frequency. PGE 3

4 4 Elecric Circuis Conceps Durham Complee Oher han Ohm s Law, which provides he definiion of impedance, all elecrical laws and relaionships are conained in he elecromagneic energy relaionship. Z V The complee suie of relaionships used for circui analysis is rooed in one, very simple, elegan equaion. The developmen and applicaion of he conceps can mos ofen be done wih mahemaics no more complex han algebra. This opens he undersanding of elecromagneic science o an enire new level of applicaion Fields noher class of analysis embraces elecromagneic energy dispersed hrough a medium, raher han focused a a poin or node. The invesigaion requires knowing he locaion in space. pansion o he field form of elecromagneic energy includes disances and direcions. EXMPLES V s N H Siuaion: Figure Find: Use KCL o obains he equaion for 2 Σ Siuaion: 0 0, 2 Find \ Siuaion: Figure Find: Use KVL o find he volage beween he H-N erminals. Σ V 0 V V + V 0 N S H V V V V S H N HN 0 2 s σ + jω jω σ 3.3 Laplace dc The Laplace ransform is defined in erms of a real, sabiliy facor and he orhogonal frequency. n a large class of problems, he frequency is zero. This is a direc curren or sep funcion. s σ + jω s σ

5 Chaper 3 Mehods 5 Frequency is dependen on he sorage elemens, capaciors and inducors. f eiher of hese is missing, hen here is no a frequency componen o he Laplace. f here is only one sorage elemen, hen he response will be an exponenial decay. f here are no sorage elemens, hen he impedance is only resisance. Then, he response will be a sep funcion wih he magniude deermined by he source and resisance. 2π ω α e α cosω Z R+ sl+ sc f () e α Wrie Ohm s Law where curren is he response and volage is he source. V() s Zs () () s V() s s () Z() s Consider a nework wih only resisance and a sep or dc volage supply or source. Find he curren in Laplace form. V s () s R Now ake he inverse Laplace o find he curren in he ime domain. V s () R s V i () R 0.37 e For he special case of a resisance nework and a direc curren, sep source, hen he ime domain response is simply he magniudes of he Laplace coefficiens. Therefore, here is no loss of accuracy o simply wrie he equaions wihou using he sep operaor, /s, and hen finding he inverse Laplace. α f() 3.4 Branches, Nodes, Loops circui is defined as a nework or combinaion of sources and impedances. The sources can be eiher volage or curren. The impedances can be resisors, inducors, and capaciors. n addiion acive elemens consising of elecronic devices can be included Superposiion The unknowns can be volage, curren, or impedance a any locaion wihin he circui. The law of superposiion allows he invesigaion o be broken down ino smaller pieces. The sysem is he sum of he behavior of he individual componens acing separaely. PGE 5

6 6 Elecric Circuis Conceps Durham For he purposes of circui analysis, all impedances will be drawn as resisors and labeled in Ohms. Conversion can be made from he impedance back o he resisors, inducors, and capaciors Topology definiions branch is a single elemen such as a source or an impedance. branch is a wo erminal elemen. The nework figure has hree impedances and wo sources o give five branches. node is he poin of connecion of wo or more branches. single node resuls if a wire wihou impedance connecs wo nodes. The nework figure has four nodes labeled a, b, c and g. The g node is a reference and is ofen conneced o ground. 6V a 2Ω 0Ω g b 4Ω 8V c loop is a closed pah in a circui. loop sars a a node, passes hrough oher nodes and reurns o he saring node wihou passing hrough any node more han one ime. loop is someimes called a mesh or a pane as in window. n independen loop conains a leas one branch ha is no par of any oher loop. The figure has wo independen loops Topology Theorem The heorem of nework opology relaes he componens. The number of branches (b) is he sum of he loops (l) plus he nodes (n) minus. b l+ n Series elemens are conneced so hey exclusively share one node and carry he same curren. Parallel elemens are conneced o he same wo nodes and have he same volage across hem. Oher connecions exis where more han wo elemens are conneced o a node. The combinaions require he applicaion of more sophisicaed echniques for analysis. The remainder of he chaper is devoed o hese echniques. EXMPLES 3.2- Siuaion: Figure Find: he number of branches, nodes, and loops. Nodes4 Loop Branches2 sources + 2 impedance 4 Check: bl+n OK

7 Chaper 3 Mehods Kirchhoff laws 3.5. Circui Laws Two circui laws are used in all circui analysis. These circui laws adhere o he Conservaion of Energy There is nohing new under he sun; or, more radiionally, he sum of he energy in a closed sysem is zero. Kirchhoff s Curren Law (KCL) relaes o currens enering a node. KCL saes ha he sum of currens enering a node is equal o zero (0). Σ 0 i n i + i + i + i By convenion, if curren eners a node, i is considered negaive. Curren leaving a node is considered posiive. Kirchhoff s Volage Law (KVL) relaes o volages in a circui loop. KVL saes ha he sum of volages in a loop is equal o zero (0). Σ v 0 n v + v + v + v 0 R L 2 2 ( ) V V R+ sl By convenion, if curren goes ino of volage source, hen he volage is considered posiive. f curren goes ino + of volage source, hen he volage is considered negaive. f curren goes ino he + of an impedance, hen i is a volage drop. f curren goes ino he of an impedance, hen i is a volage rise Circui nalysis Circui analysis is obviously a deailed process ha requires many seps. Neverheless, here is a general approaches ha applies o any resoluion of any circui problem. Five rules aid in any circui analysis. ) ssume a consisen group of currens & volages for each impedance elemen (R, L, C). Skech hese currens & volages. 2) Conservaion: Wrie equaions using Kirchhoff s Law (KCL or KVL). 3) Subsiue elemen definiions (Ohm s Law) for each impedance. 4) Combine equaions in erms of unknowns, eiher curren or volage. 5) Solve simulaneous equaions for unknown volage or curren using Cramer s rule or eliminaion. 6) f necessary, calculae any oher desired value from he unknown. Circui analysis has a source (exernal forcing funcion for volage or curren) and impedance elemens (opposiion). The answer o a circui analysis problem is he volage and curren across an impedance elemen. PGE 7

8 8 Elecric Circuis Conceps Durham Occasionally he answer is a derived value such as power or energy. However, ha requires he volage and curren. Once he curren and volage are known, hen any impedance ha is unknown can be found. EXMPLES Siuaion: The curren diagram above wih -, 2 2, 3 3. Find: Siuaion: Loop diagram above wih R0 Ohms and L0 mhy. V 24 and V 2 2. Find: Volage V 2. V V V V Find: (s) V2 s () s sl+ R 2 2 s 0 s s s+ 0 ( ) Now expand ino parial fracions k f k s () 3 s + s + 0 Muliply hrough and equae numeraors. 2 3 f ( ) ( kf k ) s 2 0 k s+ 0 + k s k Deermine coefficiens. k f.2 k k.2 f f Wrie (s) from he parial fracion expansion..2.2 () s 3 s s + 0 Find: i() i ().2.2 Find: ime consan. c α 000 Find: Final value. i( ) e 3 0 sec

9 Chaper 3 Mehods Mehods Three mehods of solving circui analysis are commonly used. These are direcly relaed o and derived from Kirchhoff s Law.. Kirchhoff s Law a. Loop curren uses KVL and is called mesh analysis. b. Node volage uses KCL and is called nodal analysis. 2. Equivalen impedances are combinaions of KVL and KCL o reduce he problem o simpler srucure. a. Series / parallel combinaion b. Volage / curren dividers c. Dela-wye conversion 3. Equivalen source is based on Ohm s Law a. Thevenin equivalen volage source b. Noron equivalen curren source 3.6. Nodal - node volage The nodal mehod is a parallel approach o he problem. Nodal analysis has broad applicaions ouside of elecrical engineering, since i only requires informaion a specific locaions or nodes and general informaion beween he nodes. Nodal analysis has been adoped by peroleum engineers o describe he performance of underground reservoirs of oil and gas. The applicaion of he general rules for circui analysis yield specific seps for nodal analysis. ) ssign node volage a each connecion. (Ground ref 0V) 2) ssume a branch curren direcion (eg. always leaves node). 3) Wrie KCL a each node. 4) Subsiue impedance elemens (Ohm s law) for curren. 5) Solve for unknown volage a each node. 6) The impedances are known. Once he volage is found a each node, hen he curren in each branch can be deermined. ample Find he curren in he 0Ω resisor of he circui. Soluion: There is only one unique node ha is no in a branch. n addiion here is he reference node.. The nodes are labeled and he volage a he node is from he node o he reference. 2. ssume all curren leaves node. 3. Wrie KCL a node b PGE 9

10 0 Elecric Circuis Conceps Durham i i i KCL node b 2Ω 0Ω 4Ω 0 4. pply Ohm s Law o he currens. Vb Va Vb 6 i2 Ω Ohm s Law 2 2 i 0Ω i 4Ω Vb 0 0 Vb Vc Vb Solve for unknown volage a nodes. Vb 6 Vb 0 Vb Subsiue 60 0V V 2V 0 Solve equaion 7V 00 V b b b b b 5.88v 6. Now calculae desired value i 0Ω Vb v Ω Mesh - loop curren The mesh or loop mehod is a series approach o he problem. Mesh analysis has applicaion where here are less loops han independen nodes. The applicaion of he general rules for circui analysis yield specific seps for mesh analysis. ) ssign curren loops o he circui using a window-pane mehod. nclude all impedances and sources. 2) ssume he volage polariy for he impedance elemens (+ ino Z) 3) Wrie KVL around each loop. 4) Subsiue impedance elemens (Ohm s law) for he volage across he Z. 5) Solve for unknown curren in he desired branch. 6) The impedances are known. Once he volage is found a each node, hen he curren in each branch can be deermined. ample Find he curren in he 0Ω resisor of he circui. Soluion:. Draw currens in each window pane. 2. ssume he volage polariy for he impedance elemens (+ ino Z) 3. Wrie he KVL around each loop.

11 Chaper 3 Mehods 4. pply Ohm s Law o he volages. For illusraion purposes, seps 3 and 4 are combined below. 2Ω 4Ω KVL Lef Loop V + V + V KVL lef loop 6V 2Ω 0Ω 0 6V i 0Ω i 2 8V V2 Ω 2i Ohm s law V 0 0( i Ω i 2) 6 + 2i + 0( i i ) 0 Combine 2 KVL Righ Loop V V + V KVL righ loop 8V 0Ω 4Ω 0 V4 Ω 4i 2 Ohm s law V 0 0( i Ω i2 ) 8 + 0( i i ) + 4i 0 Combine Solve for unknown curren in each loop. 2i 0i 6 2 0i + 4i 8 2 Se up using Cramer s Rule. i i Solve equaions simulaneously 8 4 (6)(4) ( 0)( 8) (2)(4) ( 0)( 0) Now calculae desired value i0 i i Cramer s Rule Se up a marix for he impedance, unknowns, and RHS forcing funcion. f solving for i n subsiue RHS for i n in numeraor. Use marix of coefficiens in denominaor. Use deerminans o manipulae using crossmuliplicaion 3.8 Problems Pracice Problem -2 (Old Syle) STUTON: saring circui is needed ha will limi he saring curren in a dc moor o wo and one half ime (2.5pu) he normal full load curren. The swiches S and S 2 in he saring circui shown below are o close sequenially when he curren has dropped o normal full load curren (pu) Boh swiches are open when he main breaker S M is closed. PGE

12 2 Elecric Circuis Conceps Durham SOLUTON: On Closing of S m V V 2.5pu R+ R2 + R 0.4 pu R + R + R 2 Wih V.0, V RT R+ R2 + R 0.4 pu 2.5 pu V E V Ea + R R When drops o.0 pu V E + (0.4 pu) E V pu a Close S and raises o 2.5pu, a ha insance E 0.6pu V Ea 0.6 RT R + R2 0.6 pu 2.5 When again drops o.0 pu E V R * pu T Close S 2 and raises o 2.5pu, a ha insance E 0.84pu V Ea 0.84 RT R pu 2.5 When again drops o.0 pu E V RT * pu R pu R + R 0.6 pu R pu 2 2 R + R + R 0.4 pu R 0.24 pu 2