FE Review Basic Circuits. William Hageman

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1 FE eview Basic Circuis William Hageman -8-04

2 FE opics General FE 4. Elecriciy, Power, and Magneism 7 A. Elecrical fundamenals (e.g., charge, curren, volage, resisance, power, energy) B. Curren and volage laws (Kirchhoff, Ohm) C. DC circuis D. Equivalen circuis (series, parallel, Noron s heorem, Thevenin sheorem) E. Capaciance and inducance F. AC circuis (e.g., real and imaginary componens, complex numbers, power facor, reacance and impedance) G. Measuring devices (e.g., volmeer, ammeer, wameer) Indusrial FE Engineering Sciences 5 8 A. Work, energy, and power B. Maerial properies and selecion C. Charge, energy, curren, volage, and power Mechanical FE Elecriciy and Magneism 3 5 A. Charge, curren, volage, power, and energy B. Curren and volage laws (Kirchhoff, Ohm) C. Equivalen circuis (series, parallel) D. AC circuis E. Moors and generaors Chemical FE Engineering Sciences 4 6 A. Applicaions of vecor analysis (e.g., saics) B. Basic dynamics (e.g., fricion, force, mass, acceleraion, momenum) C. Work, energy, and power (as applied o paricles or rigid bodies) D. Elecriciy and curren and volage laws (e.g., charge, energy, curren, volage, power, Kirchhoff, Ohm)

3 Topics o Be Covered DC Circuis and Analysis DC Volage DC Curren Power esisance esisiviy Ohms Law Series and Parallel Combinaions of esisors Volage and Curren Dividers Capaciors Inducors Kirchhoff s Laws Simple esisive Circui Solving Dela-Wye Transforms Superposiion Source Transforms and Thevenin/Noron Equivalen Circuis Maximum Power Transfer C and LC Transiens AC Circuis AC Waveforms Sine and Cosine elaions MS Values Phasor Transforms Phase Angles Impedance Ohm s Law for AC circuis Complex Power esonance Transformers

4 DC Circuis and Analysis

5 Definiion of Volage Separaing elecric charge requires energy. The poenial energy per uni charge creaed by charge separaion is VOLTAGE. I is ofen referred o as elecrical poenial. where v volage in vols, w energy in joules q charge in coulombs I is a Δquaniy i.e. measured a one poin relaive o anoher mos ypically relaive o he elecrical poenial of he Earh (ground).

6 Definiion of Curren Flowing charges creae elecrical effecs. The rae of charge flow is called CUENT where i curren in amperes, q charge in coulombs ime in seconds We ypically hink of curren flowing hrough wires in our circuis much like waer hrough pipes

7 Elecrical Power The power supplied or dissipaed in a circui elemen can be found from he following relaion: P iv A negaive power denoes power being supplied by he elemen, posiive power denoes power being dissipaed by he elemen

8 esisance esisance is he abiliy of a circui elemen o impede curren flow. I is denoed by. esisance is measured in Ohms (Ω). Energy is dissipaed in a resisor and convered ino hea.

9 esisiviy esisiviy (ρ) is a maerial propery ha governs resisance. The resisiviy of a maerial along wih is physical geomery deermine is resisance. I is defined in erms of unis of (Ohm * meers) l l ρ A A

10 Ohm s Law Ohm s Law provides a relaionship beween he volage dropped across a resisive elemen and he curren flowing hrough i. v i By convenion, he ohm s law volage drops in he direcion of curren flow

11 Power dissipaed in a esisor ecall ha: Power v*i Using Ohm s Law, he power dissipaed in a resisor can be re-wrien muliple ways: P (i*)*i i P v*(v/) v / These expressions hold rue regardless of sign convenion Power is always dissipaed in a resisor.

12 Combinaions of esisors Separae esisors in a circui can be combined o creae a simplified equivalen resisance esisors in Series When resisors are conneced in series (end o end) hey can simply be added eq

13 Combinaions of esisors esisors in Parallel When resisors are conneced in Parallel (side o side) hey add in he inverse eq

14 Example FE Problem If eq 0.8Ω, wha is? a) Ω b) 0Ω c) 8Ω d) 8Ω

15 Solving DC Circuis We have covered volage and curren (he wo parameers we wish o find hroughou an elecrical circui) as well as basic circui elemens. This allows us o begin o consruc and model basic resisive circuis We will need some basic circui heorems in order o solve hese circuis.

16 Circui Terminology

17 Pracice How Many Nodes? How Many Essenial Nodes? How Many Branches? How Many Essenial Branches? How Many Meshes?

18 Kirchoff scurren Law Kirchoff s Curren Law (KCL) The algebraic sum of all he currens a any node in a circui equals zero This means ha he ne curren enering a node is equal o he ne curren exiing he node no charge is sored in he node. To use KCL you mus assign a sign o use for all curren enering he node, and an opposie sign o all curren exiing he node. All currens enering and exiing he node are hen summed o zero.

19 Kirchoff svolage Law Kirchoff s Volage Law (KVL) The algebraic sum of all he volage drops around any closed pah in a circui is equal o zero A closed pah in a circui is a pah ha akes you from a saring node back o ha node wihou passing hrough any inermediae node more han once To use KVL, simply find a closed pah and sum he volage drops you encouner and se he sum equal o zero

20 Sysemaic Approach By immediaely idenifying he essenial nodes and branches of a circui, we can immediaely go abou sysemaically solving he circui. Using KVL and KCL on essenial nodes and branches is a perfecly good and valid echnique for circui solving However, here are new variables called node volages and mesh currens ha le us furher reduce he number of equaions needed for solving a circui

21 Node-Volage Mehod s Seps in Node-Volage Mehod Make a nea layou of your circui wih no branches crossing over Mark he essenial nodes Pick one of he essenial nodes o be your reference node and mark i. (ypically he node wih he mos branches) Define he node volages (volages from reference node o oher essenial nodes)

22 Node-Volage Mehod Coninued Wih he node volages labeled in your circui you are ready o generae your node-volage equaions This is done by wriing each of he essenial branch currens leaving he essenial nodes in erms of he node volages and hen summing hem o zero in accordance wih Kirchoff scurren law.

23 Example of N-V mehod () () (3) There are hree branch currens coming ou of node : The curren () hrough he Ωresisor and 0V source The curren () hrough he 5Ω resisor The curren (3) hrough he Ω resisor These can be wrien in erms of he node volages, v and v. v 0 V v v v Ω 5 Ω Ω 0

24 Finishing Example This same process can be used o produce he N-V equaion for node. v v v 0 These wo equaions have wo unknowns, v and v, hus he wo node volages can be calculaed and he branch currens deermined from hem. v 9.09V 0 v 0.9V

25 Exercise Find he wo node volage equaions ha would allow us o find v and v.

26 Soluion v v ma k 6k 6k v v 4mA 6k 6k 6k 3 v v V v v 4V Node wih v: Node wih v: v ma k v v 6k 0 v v v 4mA 0 6k 6k v v 4 ( v 4) v V 3 4v 60V v 5V 3v 30 V v 6V

27 Example FE Exam Problem Find Vo in he circui. a) 3.33 V b) 8.5 V c) 9.33 V d).5 V

28 Addiional Techniques

29 Volage Divider The volage divider is a simple way o find he volage across wo resisors in series. v v s v v s

30 Curren Divider The curren divider allow you o find he curren flowing hrough one of wo parallel resisors i i s i i s

31 Dela-WyeTransforms There is anoher simplificaion ha can be made o make solving resisive circuis easier This is he Dela-Wye Transform

32 Transform Equaions c b a b a c b a a c c b a c b c b a Dela o Y Y o Dela

33 Superposiion All linear sysems obey he principle of superposiion. This means ha whenever a linear sysem is driven by muliple sources, he oal response of he sysem is he sum of he responses o each individual source This principle is used in all areas of engineering and physics and is of course useful in Circui Theory

34 Wha does his mean for solving circuis? The value of volage or curren relaing o any elemen in a circui can be hough of as he sum of he volages and currens resuling on ha elemen from each of he independen sources in he circui individually. v v v v v v i i i i ' i ' i '' '' i i 3 4 i i 3 4 ' i 3 ' i 4 '' '' v v v v ' v ' v '' ''

35 Uses for superposiion Superposiion is highly useful if you have a circui ha you have already solved ha is alered by adding a new source of some ype. All you need o do is deacivae all he original sources and find he values resuling from your new source The rue alered values in your circui are jus he original values plus wha resuls from he new source Also very useful when analyzing muliple signals in a sysem. You find found he oupu for each signal individually and hen sum he resuls o find ha full soluion.

36 Example Wih only 4V source: ' 5 5 v o 4V 4V 4. 8V Wih only 40mA source: '' 5(00) v o 40mA ( 5//00) 40mA 0. V ' '' v v v V o o o

37 Source Transformaions Anoher mehod like Dela-Wyeand Parallel/Series combinaions o simplify circuis Can swich beween equivalen sources: a volage source wih series resisor or curren source wih parallel resisor

38 How o Transform Each of he wo equivalen sources mus produce he same curren hrough any load conneced across he source. L L I load vs L I load is (curren divider) L vs L i s L v s i s O i s vs

39 Source Transform Example Wha is Power of 6V source? i i (9.V-6V)/6 0.85A P i*v 0.85A*6V 4.95W

40 Example FE Exam Problem Find Iin he circui. a) 8 A b) -4 A c) 0 A d) 4 A

41 TheveninEquivalen Circuis

42 Terminal Behavior We are ofen ineresed in looking a he behavior of an elecrical circui a i s oupu erminals. If a load is hooked o a circui, how does he circui behave relaive o he load? If he circui is made of linear elemens, he enire circui can be simplified ino a Thevenin (or Noron) Equivalen circui.

43 Jusificaion We can consider he enire circui a black box wih wo oupu erminals. The behavior a hese erminals can be modeled by a volage source and a series resisor.

44 Finding he TheveninVolage If we look a he erminal wih no load aached (open circui oupu), he volage beween he oupu erminals is V Th. Wih an open circui, no curren flows This means no volage drop across Th Hence V ab V Th

45 Finding Thevinenesisance Now les look a wha happens when he oupu is shor circuied i sc If we can find he shor circui curren and have already found V Th, hen we can easily calculae Th i sc Th V Th Th V i Th sc

46 Power Transfer We are ofen concerned wih maximizing he amoun of power we ransfer from one circui o anoher In Power sysems he primary concern is efficienly ransferring power o he load. Losses mus be kep o a minimum. In communicaion sysems we ofen wan o ransfer daa from a ransmier o a deecor. Opimizing he power ha is ransferred o a receiver or oher load circui is highly imporan

47 Maximum Power Transfer We have some nework of sources and linear elemens and will connec his nework o a load L. We wan o maximize power in he load. The power ransfer is maximized when L is equal o TH

48 Example FE Problem Deermine he maximum power ha can be delivered o he load resisor L in he circui shown. a) mw b) 0 mw c) 4 mw d) 8 mw

49 L and C Circuis

50 Inducors Inducors are magneic coils he produce an induced volage proporional o he rae of change of curren in he coil Curren and Volage elaions v( ) i( ) L di d vdτ i L 0 ( ) 0 Power as a funcion of ime Energy Sored in he inducor P( ) w Li Li di d

51 The Capacior A capacior is a circui componen represened by he leer C Capaciance is measured in Farads Typical values are in he pf o μf range A F capacior is exremely large The symbol for a capacior is shown below: Sign Convenion

52 Capacior Curren and Volage elaions The curren and volage relaions for a capacior are opposie o hose of he inducor Inducor: Capacior: v( ) i( ) L L di d τ i( ) v( ) idτ v( ) vd 0 0 i( ) C C dv d 0 0

53 Power and Energy elaions Similarly, he power and energy sorage relaionships for a capacior are reversed in erms of volage and curren from hose of he inducor: Inducor Capacior Power as a funcion of ime Sored Energy P( ) w Li Li di d P( ) w Cv Cv dv d

54 Series and Parallel Combinaions of Inducors and Capaciors Inducors add in series much like resisors: v L di d v L di d v3 L3 di d v ( L L L ) v v v3 3 di d L eq di d L L L L L... eq 3 4 L n

55 Parallel Inducors ( ) i vd L i 0 0 τ ( ) i vd L i 0 0 τ ( ) i vd L i τ 3 i i i i ( ) ( ) ( ) i i i vd L L L i τ n eq L L L L L L ) (... ) ( ) ( ) ( i i i i n

56 Combinaions of Capaciors Capaciors add when in parallel: i i i 3 i i i i3 i C dv d i C dv d i3 C3 dv d i ( C C C ) i i i3 3 dv d C eq dv d C C C... eq C n

57 Series Combinaions of Capaciors ( ) v id C v 0 0 τ v n v v v... ( ) ( ) ( ) v v v id C C C v n n τ eq C n C C C... ) (... ) ( ) ( ) ( v v v v n Inducors add in he inverse when in parallel, jus like resisors ( ) v id C v 0 0 τ ( ) n n n v id C v 0 0 τ C n v n

58 C and L Circui Transiens Inducors and capaciors can be charged or discharged hrough a resisive elemen Discharge Charge

59 General Form of C and L esponse All off hese simple C and L circuis have he same form of response: X ( ) X final ( X X ) iniial final e ( ) τ 0 X can be curren or volage in eiher he inducor or capacior Tau is he ime consan: τc for C circui τl/ for L circui

60 Example FE problem Wha is he equivalen inducance of he circui? a) 9.5 mh b).5 mh c) 6.5 mh d) 3.5 mh

61 Example Problem # Wha is he volage across he 0 mhinducor expressed as a funcion of ime? a) b) c) d)

62 AC Circuis and Analysis

63 AC Waveforms AC waveforms are sinusoidal and can be described by eiher a sine or cosine v ( ) V m cos ( ω φ) V m Ampliude ω angular frequency (equal o πf or π/t) φ phase consan

64 Sine-Cosine elaions The following are useful relaionships for AC circuis: π cos( ω) sin( ω ) π sin( ω) cos( ω ) π sin( ω ) π cos( ω )

65 MS Values AC sinusoids have an average of zero, so we use MS (roo of he mean squared funcion) o ge a measure for effecive average value. V rms T 0 T ( V ( ) ) m cos ω φ 0 d V rms V m

66 Phasorsand PhasorTransform We use Euler s Ideniy o express sinusoidal funcions in he complex number domain. ± θ e j cosθ ± j sinθ Euler s Ideniy v V m cos( ω φ) V m { j( ω φ )} { j ω j e V e e φ } m The Phasor epresenaion or Phasor Transform of a Sinusoidal waveform: V jφ { cos( ω φ) } V e P V m m

67 Impedance Impedance (Z) is he complex analog o simple resisance. V ZI Ohm s Law for AC Circuis Circui Elemen Impedance eacance esisor - Inducor jωl ωl Capacior -j/ωc -/ωc eacance is he imaginary par of he Impedance

68 Kirchhoff s Laws wih AC circuis All of Kirchhoff s Laws sill hold wih he phasor ransform Jus sum complex currens and volages in eiher he polar or recangular form Impedances simply sum when in series and sum in he inverse when in parallel (jus like resisors)

69 FE exam problem Wha is he curren Io in he circui? a) b) c) d)

70 Transformers Two coils are wound on a single core o ensure magneic coupling

71 Ideal Transformer For an ideal ransformer he volages and currens hrough each winding are relaed as shown below: V V I N IN N N N number of winding in coil N number of winding in coil V volage across coil V volage across coil I Curren hrough coil I Curren hrough coil

72 Volage Polariy in an Ideal Transformer The four circuis below show he do convenion and he relaive volage polariy and direcion of curren Volage ha is posiive a a doed erminal resuls in a volage ha is posiive a he oher doed erminal Curren flowing ino a doed erminal should resul in curren flowing ou of he oher doed erminal

73 AC POWE Insananeous Power v()*i() PPcos(ω)Qsin(ω) PAverage Power Qeacive Power V I m m V I m m cos( φ φ ) v sin( φ φ ) v i i

74 Complex Power We have discussed eal Power (P) and eacive Power (Q) These can be hough of as he real and imaginary componens of complex power (S) S P jq

75 Apparen Power When ploed in he complex plane, Complex Power can be hough of as he resul of he addiion of perpendicular P and Q. θ ( i φ v φ ) The magniude of he complex power S is known as Apparen Power

76 Meaning of Apparen Power Apparen Power is he amoun of Power ha mus be generaed in order o supply a given amoun of average power If he circui has reacive elemens (which any power nework will, such as power lines, generaor windings, ec.) There will be a reacive elemen o he oal power. If Q >0, hen he amoun of power ha mus be supplied o drive a circui ha is dissipaing a cerain amoun of real power (P) is greaer han P. Q can be balanced by compensaing inducance wih capaciance, making he power facor [cos(φ v -φ i )] Wih proper balancing he amoun of apparen power needed o provide a given average power o a load can be minimized

77 eal Power: Expressions for Power P eacive Power: Complex Power: P I rms V rms I rms Q I rms V rms Q cos( φ φ ) V rms * S I rms V rms S I rms I Χ rms Z v X sin( φ φ ) V V v rms rms Z * Χ i i X

78 Maximizing Power Delivered o a Load emember from resisive circuis ha he maximum power delivered from a circui o a load resisor occurred when he load resisance was equal o he Thevenin esisance of he driving circui.

79 Maximizing Power Transfer wih complex circuis Wih circuis conaining reacive elemens, maximum power is ransferred when he load impedance is equal o he complex conjugae of he Thevenin Equivalen Impedance * Z L Z Th P max 4 V Th L

80 Explanaion Look a a complex impedance in series wih i s complex conjugae: Z Z Z Th * Th Th A A Z * Th jb jb A jb A The reacive elemens will cancel, causing he oal power facor o equal. Each elemen has he same resisance resuling in maximum power ransfer ino he resisive par of he load (max real power) jb A

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