EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits
|
|
- Alfred Richards
- 6 years ago
- Views:
Transcription
1 EEE25 ircui Analysis I Se 4: apaciors, Inducors, and Firs-Order inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca
2 Overview Passive elemens ha we have seen so far: resisors. We will look ino wo oher ypes of passive componens, namely capaciors and inducors. We have already seen differen mehods o analyze circuis conaining sources and resisive elemens. We will examine circuis ha conain wo differen ypes of passive elemens namely resisors and one (equivalen) capacior (R circuis) or resisors and one (equivalen) inducor (R circuis) Similar o circuis whose passive elemens are all resisive, one can analyze R or R circuis by applying KV and/or K. We will see wheher he analysis of R or R circuis is any differen! Noe: Some of he figures in his slide se are aken from (R. Decarlo and P.-M. in, inear ircui Analysis, 2 nd Ediion, 200, Oxford Universiy Press) and (.K. Alexander and M.N.O Sadiku, Fundamenals of Elecric ircuis, 4 h Ediion, 2008, McGraw Hill) 2
3 Reading Maerial hapers 6 and 7 of he exbook Secion 6.: apaciors Secion 6.2: Inducors Secion 6.3: apacior and Inducor ombinaions Secion 6.5: Applicaion Examples Secion 7.2: Firs-Order ircuis Reading assignmen: Review Secion 7.4: Applicaion Examples (7.2, 7.3, and 7.4) 3
4 apaciors A capacior is a circui componen ha consiss of wo conducive plae separaed by an insulaor (or dielecric). apaciors sore charge and he amoun of charge sored on he capacior is direcly proporional o he volage across he capacior. The consan of proporionaliy is he capaciance of he capacior. Tha is: ( ) v ( ) apacior sores energy in is elecric field. q 4
5 apaciors A d Model for a non-ideal capacior 5
6 6 apaciors In honor of Michael Faraday (79-867), an English chemis and physicis, he uni of capaciance is named Farad (F). The volage-curren relaionship of he capacior is: why? ) ( ) ( ) ( ) ( ) ( ) ( 0 0 v d i v d i q v or d dv i ) ( ) (
7 Noe ha; apaciors A capacior acs as an open circui when conneced o a D volage source A capacior impede he abrup change of is volage The insananeous power absorbed by he capacior is: p dv ( ) ( ) i ( ) v ( ) v ( ) d and he oal sored energy in he capacior is: W ( ) p ( ) d v ( ) dv v 2 ( ) Have we assumed anyhing in wriing he above equaion?! ( ) 2 7
8 Example alculae he area of simple parallel plae F capacior. Assume ha he plaes are separaed by air wih a disance of he hickness one shee of paper, i.e., m. The permiiviy of free space is: Ɛ 0 = F/m. = (Ɛ 0 A)/d A= d/ Ɛ 0 = / A= m 2 =.48 km 2!!!!!! 8
9 Example The volage across a 5-µF capacior is given below. Deermine he curren of he capacior. 9
10 Series and Parallel apaciors The equivalen capaciance of series-conneced capaciors is he reciprocal of he sum of he reciprocals of he individual capaciances. Why? 2 eq 2 n n The equivalen capaciance of parallel capaciors is he sum of he individual capaciances. Why? eq 2 n 2 n 0
11 Example ompue he equivalen capaciance of he following nework:
12 Example alculae he equivalen capaciance of he following nework: a) when he swich is open b) when he swich is closed 2
13 Board Noes 3
14 Applicaion Example In inegraed circuis, wires carrying high-speed signals are closely spaced as shown by he following micrograph. As a resul, a signal on one conducor can myseriously appear on a differen conducor. This phenomenon is called crossalk. e us examine his condiion and propose some mehods for reducing i. 4
15 Applicaion Example Simple model for invesigaing crossalk: 5
16 Applicaion Example Use of ground wire o reduce crossalk (simple, no oo realisic model! Why?) 6
17 A more accurae model: Applicaion Example 7
18 Design Example We have all undoubedly experienced a loss of elecrical power in our office or our home. When his happens, even for a second, we ypically find ha we have o rese all of our digial alarm clocks. e's assume ha such a clock's inernal digial hardware requires a curren of ma a a ypical volage level of 3.0 V, bu he hardware will funcion properly down o 2.4 V. Under hese assumpions, we wish o design a circui ha will hold he volage level for a shor duraion, for example, second. 8
19 Board Noes 9
20 Inducors An inducor is ypically a coil of conducing wire. Inducor sores energy in is magneic field. If curren passes hrough an inducor he volage across he inducor is direcly proporional o he ime rae of change of he curren: v ( ) di( ) d The consan of proporionaliy is he inducance of he inducor. i ( ) v( ) d or i( ) v( ) d i( 0) 0 20
21 Inducors N 2 A l Model for a non-ideal inducor 2
22 Inducors In honor of Joseph Henry ( ), an American physicis, he uni of inducance is named Henry (H). Noe ha: An inducor acs like a shor circui o D curren. Inducor impede insananeous changes of is curren. Insananeous power delivered o he inducor is: p The oal sored energy is: W ( ) di( ) ( ) v( ) i( ) i( ) d p ( ) d (Have we assumed anyhing in wriing he above equaion?!) i ( ) di ( ) 2 i 2 ( ) 22
23 Example The curren in a 0-mH inducor has he following waveform. Find he volage of he inducor. 23
24 Series and Parallel Inducors The equivalen inducance of series-conneced inducors is he sum of he individual inducances. Why? eq 2 n 2 n The equivalen inducance of parallel inducors is he reciprocal of he sum of he reciprocals of he individual inducances. Why? eq 2 n 2 n 24
25 Example Find he equivalen inducance ( T ) of he following nework: 25
26 26
27 Firs-Order ircuis Applying KV and/or K o purely resisive circuis resuls in algebraic equaions. Applying hese laws o R and R circuis resuls in differenial equaions. In general, differenial equaions are a bi more difficul o solve compared o algebraic equaions! If here is only one or jus one in he circui he resuling differenial equaion is of he firs order (and i is linear). A circui ha is characerized by a firs-order differenial equaion is called a firs-order circui. 27
28 Wha Do We Mean By Equivalen apacior? The equivalen capaciance of series-conneced capaciors is he reciprocal of he sum of he reciprocals of he individual capaciances. Why? 2 eq 2 n n The equivalen capaciance of parallel capaciors is he sum of he individual capaciances. Why? eq 2 n 2 n 28
29 Wha Do We Mean by Equivalen Inducor? The equivalen inducance of series-conneced inducors is he sum of he individual inducances. Why? eq 2 n 2 n The equivalen inducance of parallel inducors is he reciprocal of he sum of he reciprocals of he individual inducances. Why? eq 2 n 2 n 29
30 Firs-Order ircuis So in an R circui if we have more han one capacior, however, we can combine he capaciors (series and/or parallel combinaion) and represen hem wih one equivalen capacior, we sill have a firs-order circui. The same is rue for R circuis, ha is if we can combine all he inducors and represen hem wih one equivalen circui hen we sill have a firs-order circui In such circuis we can find he Thevenin (or Noron) equivalen circui seen by he equivalen capacior (or Inducor) and hen solve he circui. e s sar wih he circuis ha have no source! 30
31 Example Which one of he following circuis is a firs-order circui? 3
32 Source-Free or Zero-Inpu Firs-Order ircui Recall ha in general if here is only one (equivalen) inducor or capacior in he circui one can model he circui seen by he inducor or capacior by is Thevenin equivalen circui. In he case of source-free circui (no independen source in he circui) he Thevenin equivalen circui will be.. a resisor. 32
33 33 Source-Free or Zero-Inpu Firs-Order ircui ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( i R d di i d di R d di R R v R v i i i R R R ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( v R d dv R v d dv R v R v i i i R R R R
34 Source-Free Firs-Order R ircui e s assume ha we know he charge or equivalenly he volage across he capacior a ime 0. Tha is: ( 0) V v 0 Recall: i dv ( ) v ( ) dv ( ) ( ) ir( ) 0 0 v ( ) d R d R e s ry o solve his differenial equaion. Because of he simple form of his equaion we can re-arrange he erm as: dv v ( ) ( ) R d 34
35 Source-Free Firs-Order R ircui Before going any furher can you ell he unis for R from he equaion: dv ( ) d v ( ) R Now le s go furher! and inegrae boh sides of he equaion from 0 o : ' 0 ln v dv v ( ') 0 ( ') ( ') ' 0 v ( ) ln V ' 0 ' R d' R ' 0 v R ( ) V e 0 R 35
36 Source-Free Firs-Order R ircui The volage response of an source-free firs-order R circui is an exponenial decay from is iniial volage value: v () V 0 v ( ) V 0 e R 0.368V 0 R The ime ha is required for he response o decay by a facor of /e (36.8% or by engineering approximaion! 37%) of is iniial value is called ime consan of he circui and is ypically denoed by. 5 36
37 Source-Free Firs-Order R ircui Philosophical quesion: When here is no source in he circui, how come we have such a response? Wha is he response due o? In general, he response of a source-free circui which is due o he iniial energy sored in he sorage elemens (in his case ) and no due o exernal sources is called naural response. In firs order R (and R) circuis naural response is a decaying exponenial. To find he naural response of a firs-order R circui we need wo pieces of informaion: Iniial volage across he capacior The ime consan =R 37
38 Source-Free Firs-Order R ircui Time consan of he circui gives us an indicaion of how rapidly he response decays, in oher words how fas is he response. R e s calculae he naural response v ( ) V0e for imes equal differen muliples of he ime consan: v () V V V V V 0 For all pracical purposes i is ypically assumed ha he response reaches is final value afer 5. 38
39 Example Assuming v (0)=30V, deermine v and v x, and i o for 0 39
40 Source-Free Firs-Order R Example In he following circui, find he volage across he capacior for 0. Assume ha v(0)=0v. =s + =0.F R =0 v() R 2 =0-40
41 Source-Free Firs-Order R ircui e s assume ha we know he iniial curren in he inducor a ime 0. Tha is: ( 0) I i 0 Afer a bi of equaion wriing! we have: di( ) d R i( ) i( ) I0 Wha is he ime consan of his circui? To find he naural response of a firs-order R circui we need wo pieces of informaion: Iniial curren hrough he inducor The ime consan =/R e R 4
42 Source-Free Firs-Order R Example In he following circui, find he curren hrough he inducor for 0. Assume ha i(0)=a. + v R 2 = v i() =mh - 42
43 Example In he following circui, assuming i(0)=0a, calculae i() and i x (). 43
44 Uni Sep Funcion Sep funcion is a very useful funcion o model he signals in he circuis ha have swiches. Example: In he following circui, find he volage across he resisor R for - <<. =0 5V R + V() - 44
45 Uni Sep Funcion To model abrup changes in a volage or curren one can use a uni sep funcion. The uni sep funcion is defined as follows: u( ) Use he sep funcion o express he volage across he resisor in he previous example: v()= 45
46 Uni Sep Funcion Examples Assuming 0 is a given posiive ime, plo he following funcions: u () u( ) u ) ( 0 u( 0) u( 0 ) 46
47 Uni Sep Funcion Examples Wrie he funcions on he previous slide in mahemaical erms, e.g., u( )
48 Recall: Differenial Equaions In general, he differenial equaion ha model a firs-order R or R circui wih a source ha is swiched in a 0, is of he form: dx( ) x( ) f ( 0 ) u( 0), x( 0) x0 d valid for 0, where x() is he volage or curren of ineres and x 0 is he iniial condiion a ime 0 and f() is a funcion of he source (or force funcion). For noaion simpliciy and wihou loss of generaliy, le s assume 0 = 0, hen, he equaion can be wrien as: dx( ) x( ) f ( ), x(0) x0 d Noe ha his is a special ype of differenial equaions! (Wha is so special abou i?) 48
49 Recall: Differenial Equaions Many echniques for solving his ype of differenial equaions exis. The fundumenal heorem of differenial equaions saes ha if x p () is a soluion of dx( ) x( ) f ( ) d and x h () is a soluion o he homogeneous equaion dx( ) x( ) 0 d hen x( ) Kxh ( ) xp( ), where K is a consan is a soluion o he original differenial equaion. x p () is called he paricular soluion or forced response. x h () is called he homogeneous soluion or naural response (also called complemenary soluion). 49
50 Recall: Differenial Equaions If we only have D sources in he circui, hen f ( ) F where F is a consan. an you find x p () and x h ()? 50
51 Recall: Differenial Equaions In general, he soluion o: is of he form of: dx( ) d x( ) x( ) K e K 2 F is called he naural frequency of he circui! or / is called he ime consan of he circui. Recall, ha he firs erm in he above expression is called naural response (is due o sored energy or iniial condiion) and he second erm is called forced response (is due o independen sources). How do we find K and K 2? 5
52 D or Sep-Response of Firs-Order ircuis When a D source in an R or R circui is suddenly applied (for example by urning on a swich), he volage or curren source can be modeled using he source and a swich (using a sep funcion!). The response of he circui o such a sudden change (when he exciaion is a sep funcion) is called he sep response of he circui. In general he D or sep response of a firs-order circui saisfies a differenial equaion of he following form (assuming ha he sep is applied a = 0): dx( ) ax( ) Fu( ), x(0 ) x0 d Do you know wha do we mean by and why we are using i? 0 52
53 D or Sep-Response of Firs-Order ircuis Using dx( ) he soluion o ax( ) Fu( ), d is of he form: x(0 ) x 0 Noe ha: x( ) K x( ) lim e F x( ) F K x(0 ) x( ) Thus, he response of a firs-order circui has he following form: x(0 ) x( ) e x( ) x( ) The sep response of any volage or curren in a firs-order circui has he above form. 53
54 D or Sep-Response of Firs-Order ircuis If he sep is applied a = 0 (or he swich changes is posiion a = 0 ), given he iniial condiion a = 0 + hen he general form of he soluion is of he form: x( ) ( 0 ) 0 x( ) x( ) e x( ) 54
55 D or Sep-Response of Firs-Order ircuis For example, in a firs-order R circui he sep response of he curren hrough he inducor is of he form: i ( 0 ) 0 i ( ) i ( ) e R i ( ) ( ) / and in a firs-order R circui he sep response of he volage across he capacior is of he form: v ( ) ( 0 ) 0 v ( ) v ( ) e R v ( ) These equaions are very useful! and in general for a sep response of any firs-order circui we have: any volage or curren ( Iniial value Final value) e elapsedime imeconsan Final value The iniial value can be found using he iniial condiion of he circui. 55
56 D or Sep-Response of Firs-Order ircuis The complee response can be divided ino wo porions: omplee Response Transien Response emporary par due o sored energy Seady - Sae Response permanen par due o independen sources The ransien response is he circui s emporary response ha will die ou wih ime. The seady-sae response is he porion of he response ha remains afer he ransien response has died ou (behavior of he circui a long ime afer he exernal exciaion is applied). 56
57 D or Sep-Response of Firs-Order ircuis Wha are he ransien and seady-sae porions of he following response: v ( ) To find he complee response of a firs-order circui we need o find iniial value, final value, and ime consan of he circui: Iniial value can be found using he iniial condiion. Time consan can be found by finding he Thevenin equivalen resisance seen across he capacior (or inducor) How abou he final value. ( 0 ) 0 v ( ) v ( ) e R v ( ) 57
58 D or Sep-Response of Firs-Order ircuis ouple of ineresing poins (ricks) ha are only valid for calculaing final values of D sep-response: A capacior acs like a open circui long ime afer he exernal exciaion is applied. an you inuiively jusify his saemen? An inducor acs like a shor circui long ime afer he exernal exciaion is applied. Why? 58
59 Seady-Sae Response oosely speaking, he behaviour of he circui a long ime afer an exciaion is applied o he circui is called seady-sae response. For example, if in a circui a swich is opened (or closed) he response of he circui o his exciaion long ime afer he swich is opened (or closed) is referred o as seady-sae response. If we only have D sources in he circui, a seady sae capaciors ac like open circui and inducors ac like a shor circui. 59
60 Example In he following circui find he energy ha is sored in he inducor and capacior, when he circui reaches seady sae. 60
61 Example In he following circui, he swich has been in posiion A for a long ime and hen a =0, he swich moves o posiion B. Find he energy sored in he capacior jus before he swich moves. Also, wha is he energy sored in he capacior a long ime afer swich is moved o B, i.e., =.. 6
62 Example In he following circui, he swich has been closed for a long ime and a =0 he swich is opened. Wha is he energy sored in he inducor jus before he swich is opened? Wha is he energy sored in he inducor a long ime afer he swich is opened. i.e., =. 62
63 Example Find v() for >0 in he following circui. Assume he swich has been open for a long ime before i is closed a =0. R =3 =0s 2V =0.F + v() R 2 =6 0.5A - 63
64 Example Find i() for >0 in he following circui. Assume he swich has been open for a long ime before i is closed a =0. R =3 =0s 2u(-)+24u() V 0.6mH i() R2=6 0.5A 64
65 Example In he following circui, assume he swich has been open for a long ime before being closed a ime 0. Find v 0 () for >0 65
66 Noes 66
67 Example In he following circui, assume he swich has been open for a long ime before being closed a ime 0. Find i 0 () for >0 67
68 Noes 68
EEEB113 CIRCUIT ANALYSIS I
9/14/29 1 EEEB113 CICUIT ANALYSIS I Chaper 7 Firs-Order Circuis Maerials from Fundamenals of Elecric Circuis 4e, Alexander Sadiku, McGraw-Hill Companies, Inc. 2 Firs-Order Circuis -Chaper 7 7.2 The Source-Free
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationCHAPTER 6: FIRST-ORDER CIRCUITS
EEE5: CI CUI T THEOY CHAPTE 6: FIST-ODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions
More informationBasic Circuit Elements Professor J R Lucas November 2001
Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive and passive elemens. Some Definiions/explanaions
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More informationFirst Order RC and RL Transient Circuits
Firs Order R and RL Transien ircuis Objecives To inroduce he ransiens phenomena. To analyze sep and naural responses of firs order R circuis. To analyze sep and naural responses of firs order RL circuis.
More informationINDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
More informationHomework-8(1) P8.3-1, 3, 8, 10, 17, 21, 24, 28,29 P8.4-1, 2, 5
Homework-8() P8.3-, 3, 8, 0, 7, 2, 24, 28,29 P8.4-, 2, 5 Secion 8.3: The Response of a Firs Order Circui o a Consan Inpu P 8.3- The circui shown in Figure P 8.3- is a seady sae before he swich closes a
More informationdv 7. Voltage-current relationship can be obtained by integrating both sides of i = C :
EECE202 NETWORK ANALYSIS I Dr. Charles J. Kim Class Noe 22: Capaciors, Inducors, and Op Amp Circuis A. Capaciors. A capacior is a passive elemen designed o sored energy in is elecric field. 2. A capacior
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationChapter 4 AC Network Analysis
haper 4 A Nework Analysis Jaesung Jang apaciance Inducance and Inducion Time-Varying Signals Sinusoidal Signals Reference: David K. heng, Field and Wave Elecromagneics. Energy Sorage ircui Elemens Energy
More informationLab 10: RC, RL, and RLC Circuits
Lab 10: RC, RL, and RLC Circuis In his experimen, we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors. We will sudy he way volages and currens change in
More informationEE100 Lab 3 Experiment Guide: RC Circuits
I. Inroducion EE100 Lab 3 Experimen Guide: A. apaciors A capacior is a passive elecronic componen ha sores energy in he form of an elecrosaic field. The uni of capaciance is he farad (coulomb/vol). Pracical
More informationLecture 13 RC/RL Circuits, Time Dependent Op Amp Circuits
Lecure 13 RC/RL Circuis, Time Dependen Op Amp Circuis RL Circuis The seps involved in solving simple circuis conaining dc sources, resisances, and one energy-sorage elemen (inducance or capaciance) are:
More informationEECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits
EECE25 Circuit Analysis I Set 4: Capacitors, Inductors, and First-Order Linear Circuits Shahriar Mirabbasi Department of Electrical and Computer Engineering University of British Columbia shahriar@ece.ubc.ca
More informationECE 2100 Circuit Analysis
ECE 1 Circui Analysis Lesson 35 Chaper 8: Second Order Circuis Daniel M. Liynski, Ph.D. ECE 1 Circui Analysis Lesson 3-34 Chaper 7: Firs Order Circuis (Naural response RC & RL circuis, Singulariy funcions,
More informationChapter 10 INDUCTANCE Recommended Problems:
Chaper 0 NDUCTANCE Recommended Problems: 3,5,7,9,5,6,7,8,9,,,3,6,7,9,3,35,47,48,5,5,69, 7,7. Self nducance Consider he circui shown in he Figure. When he swich is closed, he curren, and so he magneic field,
More informationDesigning Information Devices and Systems I Spring 2019 Lecture Notes Note 17
EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive
More informationCapacitors & Inductors
apaciors & Inducors EEE5 Elecric ircuis Anawach Sangswang Dep. of Elecrical Engineering KMUTT Elecric Field Elecric flux densiy Elecric field srengh E Elecric flux lines always exend from a posiively charged
More informationDirect Current Circuits. February 19, 2014 Physics for Scientists & Engineers 2, Chapter 26 1
Direc Curren Circuis February 19, 2014 Physics for Scieniss & Engineers 2, Chaper 26 1 Ammeers and Volmeers! A device used o measure curren is called an ammeer! A device used o measure poenial difference
More informationChapter 16: Summary. Instructor: Jean-François MILLITHALER.
Chaper 16: Summary Insrucor: Jean-François MILLITHALER hp://faculy.uml.edu/jeanfrancois_millihaler/funelec/spring2017 Slide 1 Curren & Charge Elecric curren is he ime rae of change of charge, measured
More information7. Capacitors and Inductors
7. Capaciors and Inducors 7. The Capacior The ideal capacior is a passive elemen wih circui symbol The curren-volage relaion is i=c dv where v and i saisfy he convenions for a passive elemen The capacior
More informationR.#W.#Erickson# Department#of#Electrical,#Computer,#and#Energy#Engineering# University#of#Colorado,#Boulder#
.#W.#Erickson# Deparmen#of#Elecrical,#Compuer,#and#Energy#Engineering# Universiy#of#Colorado,#Boulder# Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance,
More informationUniversity of Cyprus Biomedical Imaging and Applied Optics. Appendix. DC Circuits Capacitors and Inductors AC Circuits Operational Amplifiers
Universiy of Cyprus Biomedical Imaging and Applied Opics Appendix DC Circuis Capaciors and Inducors AC Circuis Operaional Amplifiers Circui Elemens An elecrical circui consiss of circui elemens such as
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationECE 2100 Circuit Analysis
ECE 1 Circui Analysis Lesson 37 Chaper 8: Second Order Circuis Discuss Exam Daniel M. Liynski, Ph.D. Exam CH 1-4: On Exam 1; Basis for work CH 5: Operaional Amplifiers CH 6: Capaciors and Inducor CH 7-8:
More informationElectrical Circuits. 1. Circuit Laws. Tools Used in Lab 13 Series Circuits Damped Vibrations: Energy Van der Pol Circuit
V() R L C 513 Elecrical Circuis Tools Used in Lab 13 Series Circuis Damped Vibraions: Energy Van der Pol Circui A series circui wih an inducor, resisor, and capacior can be represened by Lq + Rq + 1, a
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationVoltage/current relationship Stored Energy. RL / RC circuits Steady State / Transient response Natural / Step response
Review Capaciors/Inducors Volage/curren relaionship Sored Energy s Order Circuis RL / RC circuis Seady Sae / Transien response Naural / Sep response EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu Lecure
More information(b) (a) (d) (c) (e) Figure 10-N1. (f) Solution:
Example: The inpu o each of he circuis shown in Figure 10-N1 is he volage source volage. The oupu of each circui is he curren i( ). Deermine he oupu of each of he circuis. (a) (b) (c) (d) (e) Figure 10-N1
More information2.9 Modeling: Electric Circuits
SE. 2.9 Modeling: Elecric ircuis 93 2.9 Modeling: Elecric ircuis Designing good models is a ask he compuer canno do. Hence seing up models has become an imporan ask in modern applied mahemaics. The bes
More information2.4 Cuk converter example
2.4 Cuk converer example C 1 Cuk converer, wih ideal swich i 1 i v 1 2 1 2 C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode C 1 i 1 i v 1 2 Q 1 D 1 C 2 v 2 28 Analysis sraegy This converer
More informationLabQuest 24. Capacitors
Capaciors LabQues 24 The charge q on a capacior s plae is proporional o he poenial difference V across he capacior. We express his wih q V = C where C is a proporionaliy consan known as he capaciance.
More informationPhys1112: DC and RC circuits
Name: Group Members: Dae: TA s Name: Phys1112: DC and RC circuis Objecives: 1. To undersand curren and volage characerisics of a DC RC discharging circui. 2. To undersand he effec of he RC ime consan.
More informationChapter 2: Principles of steady-state converter analysis
Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer
More information( ) = Q 0. ( ) R = R dq. ( t) = I t
ircuis onceps The addiion of a simple capacior o a circui of resisors allows wo relaed phenomena o occur The observaion ha he ime-dependence of a complex waveform is alered by he circui is referred o as
More informationAC Circuits AC Circuit with only R AC circuit with only L AC circuit with only C AC circuit with LRC phasors Resonance Transformers
A ircuis A ircui wih only A circui wih only A circui wih only A circui wih phasors esonance Transformers Phys 435: hap 31, Pg 1 A ircuis New Topic Phys : hap. 6, Pg Physics Moivaion as ime we discovered
More informationInductor Energy Storage
School of Compuer Science and Elecrical Engineering 5/5/ nducor Energy Sorage Boh capaciors and inducors are energy sorage devices They do no dissipae energy like a resisor, bu sore and reurn i o he circui
More informationEE202 Circuit Theory II , Spring. Dr. Yılmaz KALKAN & Dr. Atilla DÖNÜK
EE202 Circui Theory II 2018 2019, Spring Dr. Yılmaz KALKAN & Dr. Ailla DÖNÜK 1. Basic Conceps (Chaper 1 of Nilsson - 3 Hrs.) Inroducion, Curren and Volage, Power and Energy 2. Basic Laws (Chaper 2&3 of
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationES 250 Practice Final Exam
ES 50 Pracice Final Exam. Given ha v 8 V, a Deermine he values of v o : 0 Ω, v o. V 0 Firs, v o 8. V 0 + 0 Nex, 8 40 40 0 40 0 400 400 ib i 0 40 + 40 + 40 40 40 + + ( ) 480 + 5 + 40 + 8 400 400( 0) 000
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More information- If one knows that a magnetic field has a symmetry, one may calculate the magnitude of B by use of Ampere s law: The integral of scalar product
11.1 APPCATON OF AMPEE S AW N SYMMETC MAGNETC FEDS - f one knows ha a magneic field has a symmery, one may calculae he magniude of by use of Ampere s law: The inegral of scalar produc Closed _ pah * d
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationLAB 5: Computer Simulation of RLC Circuit Response using PSpice
--3LabManualLab5.doc LAB 5: ompuer imulaion of RL ircui Response using Ppice PURPOE To use a compuer simulaion program (Ppice) o invesigae he response of an RL series circui o: (a) a sinusoidal exciaion.
More information8.022 (E&M) Lecture 9
8.0 (E&M) Lecure 9 Topics: circuis Thevenin s heorem Las ime Elecromoive force: How does a baery work and is inernal resisance How o solve simple circuis: Kirchhoff s firs rule: a any node, sum of he currens
More informationdv i= C. dt 1. Assuming the passive sign convention, (a) i = 0 (dc) (b) (220)( 9)(16.2) t t Engineering Circuit Analysis 8 th Edition
. Assuming he passive sign convenion, dv i= C. d (a) i = (dc) 9 9 (b) (22)( 9)(6.2) i= e = 32.8e A 9 3 (c) i (22 = )(8 )(.) sin. = 7.6sin. pa 9 (d) i= (22 )(9)(.8) cos.8 = 58.4 cos.8 na 2. (a) C = 3 pf,
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationEE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu 1. Measuring Voltages and Currents
Announemens HW # Due oday a 6pm. HW # posed online oday and due nex Tuesday a 6pm. Due o sheduling onflis wih some sudens, lasses will resume normally his week and nex. Miderm enaively 7/. EE4 Summer 5:
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationCapacitors. C d. An electrical component which stores charge. parallel plate capacitor. Scale in cm
apaciors An elecrical componen which sores charge E 2 2 d A 2 parallel plae capacior Scale in cm Leyden Jars I was invened independenly by German cleric Ewald Georg von Kleis on Ocober 745 and by Duch
More informationi L = VT L (16.34) 918a i D v OUT i L v C V - S 1 FIGURE A switched power supply circuit with diode and a switch.
16.4.3 A SWITHED POWER SUPPY USINGA DIODE In his example, we will analyze he behavior of he diodebased swiched power supply circui shown in Figure 16.15. Noice ha his circui is similar o ha in Figure 12.41,
More informationLearning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure
Lab 4: Synchronous Sae Machine Design Summary: Design and implemen synchronous sae machine circuis and es hem wih simulaions in Cadence Viruoso. Learning Objecives: Pracice designing and simulaing digial
More informationHomework: See website. Table of Contents
Dr. Friz Wilhelm page of 4 C:\physics\3 lecure\ch3 Inducance C circuis.docx; P /5/8 S: 5/4/9 9:39: AM Homework: See websie. Table of Conens: 3. Self-inducance in a circui, 3. -Circuis, 4 3.a Charging he
More informationPhysics 1402: Lecture 22 Today s Agenda
Physics 142: ecure 22 Today s Agenda Announcemens: R - RV - R circuis Homework 6: due nex Wednesday Inducion / A curren Inducion Self-Inducance, R ircuis X X X X X X X X X long solenoid Energy and energy
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More informationReal Analog Chapter 6: Energy Storage Elements
1300 Henley C. Pullman, WA 99163 509.334.6306 www.sore.digilen.com 6 Inroducion and Chaper Objecives So far, we have considered circuis ha have been governed by algebraic relaions. These circuis have,
More informationElectromagnetic Induction: The creation of an electric current by a changing magnetic field.
Inducion 1. Inducion 1. Observaions 2. Flux 1. Inducion Elecromagneic Inducion: The creaion of an elecric curren by a changing magneic field. M. Faraday was he firs o really invesigae his phenomenon o
More informationHall effect. Formulae :- 1) Hall coefficient RH = cm / Coulumb. 2) Magnetic induction BY 2
Page of 6 all effec Aim :- ) To deermine he all coefficien (R ) ) To measure he unknown magneic field (B ) and o compare i wih ha measured by he Gaussmeer (B ). Apparaus :- ) Gauss meer wih probe ) Elecromagne
More information6.01: Introduction to EECS I Lecture 8 March 29, 2011
6.01: Inroducion o EES I Lecure 8 March 29, 2011 6.01: Inroducion o EES I Op-Amps Las Time: The ircui Absracion ircuis represen sysems as connecions of elemens hrough which currens (hrough variables) flow
More informationSub Module 2.6. Measurement of transient temperature
Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure,
More information3. Alternating Current
3. Alernaing Curren TOPCS Definiion and nroducion AC Generaor Componens of AC Circuis Series LRC Circuis Power in AC Circuis Transformers & AC Transmission nroducion o AC The elecric power ou of a home
More information5.2. The Natural Logarithm. Solution
5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,
More informationPhysics 1502: Lecture 20 Today s Agenda
Physics 152: Lecure 2 Today s Agenda Announcemens: Chap.27 & 28 Homework 6: Friday nducion Faraday's Law ds N S v S N v 1 A Loop Moving Through a Magneic Field ε() =? F() =? Φ() =? Schemaic Diagram of
More informationSection 3.8, Mechanical and Electrical Vibrations
Secion 3.8, Mechanical and Elecrical Vibraions Mechanical Unis in he U.S. Cusomary and Meric Sysems Disance Mass Time Force g (Earh) Uni U.S. Cusomary MKS Sysem CGS Sysem fee f slugs seconds sec pounds
More informationProblem Set #1. i z. the complex propagation constant. For the characteristic impedance:
Problem Se # Problem : a) Using phasor noaion, calculae he volage and curren waves on a ransmission line by solving he wave equaion Assume ha R, L,, G are all non-zero and independen of frequency From
More information8.022 (E&M) Lecture 16
8. (E&M) ecure 16 Topics: Inducors in circuis circuis circuis circuis as ime Our second lecure on elecromagneic inducance 3 ways of creaing emf using Faraday s law: hange area of circui S() hange angle
More informationPhysics for Scientists & Engineers 2
Direc Curren Physics for Scieniss & Engineers 2 Spring Semeser 2005 Lecure 16 This week we will sudy charges in moion Elecric charge moving from one region o anoher is called elecric curren Curren is all
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationEE202 Circuit Theory II
EE202 Circui Theory II 2017-2018, Spring Dr. Yılmaz KALKAN I. Inroducion & eview of Fir Order Circui (Chaper 7 of Nilon - 3 Hr. Inroducion, C and L Circui, Naural and Sep epone of Serie and Parallel L/C
More informationWall. x(t) f(t) x(t = 0) = x 0, t=0. which describes the motion of the mass in absence of any external forcing.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES 7 Mechanics applicaions of second-order ODEs Second-order linear ODEs wih consan coefficiens arise in many physical applicaions. One physical sysems whose behaviour
More informationName: Total Points: Multiple choice questions [120 points]
Name: Toal Poins: (Las) (Firs) Muliple choice quesions [1 poins] Answer all of he following quesions. Read each quesion carefully. Fill he correc bubble on your scanron shee. Each correc answer is worh
More informationfirst-order circuit Complete response can be regarded as the superposition of zero-input response and zero-state response.
Experimen 4:he Sdies of ransiional processes of 1. Prpose firs-order circi a) Use he oscilloscope o observe he ransiional processes of firs-order circi. b) Use he oscilloscope o measre he ime consan of
More informationChapter 9 Sinusoidal Steady State Analysis
Chaper 9 Sinusoidal Seady Sae Analysis 9.-9. The Sinusoidal Source and Response 9.3 The Phasor 9.4 pedances of Passive Eleens 9.5-9.9 Circui Analysis Techniques in he Frequency Doain 9.0-9. The Transforer
More informationElectrical and current self-induction
Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml mende_fedor@mail.ru Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of
More information) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationExperimental Buck Converter
Experimenal Buck Converer Inpu Filer Cap MOSFET Schoky Diode Inducor Conroller Block Proecion Conroller ASIC Experimenal Synchronous Buck Converer SoC Buck Converer Basic Sysem S 1 u D 1 r r C C R R X
More informationV L. DT s D T s t. Figure 1: Buck-boost converter: inductor current i(t) in the continuous conduction mode.
ECE 445 Analysis and Design of Power Elecronic Circuis Problem Se 7 Soluions Problem PS7.1 Erickson, Problem 5.1 Soluion (a) Firs, recall he operaion of he buck-boos converer in he coninuous conducion
More informationSection 2.2 Charge and Current 2.6 b) The current direction is designated as the direction of the movement of positive charges.
Chaper Soluions Secion. Inroducion. Curren source. Volage source. esisor.4 Capacior.5 Inducor Secion. Charge and Curren.6 b) The curren direcion is designaed as he direcion of he movemen of posiive charges..7
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationMEMS 0031 Electric Circuits
MEMS 0031 Elecric Circuis Chaper 1 Circui variables Chaper/Lecure Learning Objecives A he end of his lecure and chaper, you should able o: Represen he curren and volage of an elecric circui elemen, paying
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationLecture Outline. Introduction Transmission Line Equations Transmission Line Wave Equations 8/10/2018. EE 4347 Applied Electromagnetics.
8/10/018 Course Insrucor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@uep.edu EE 4347 Applied Elecromagneics Topic 4a Transmission Line Equaions Transmission These Line noes
More informationStructural Dynamics and Earthquake Engineering
Srucural Dynamics and Earhquae Engineering Course 1 Inroducion. Single degree of freedom sysems: Equaions of moion, problem saemen, soluion mehods. Course noes are available for download a hp://www.c.up.ro/users/aurelsraan/
More informationIE1206 Embedded Electronics
IE06 Embee Elecronics Le Le3 Le4 Le Ex Ex PI-block Documenaion, Seriecom Pulse sensors I, U, R, P, series an parallel K LAB Pulse sensors, Menu program Sar of programing ask Kirchhoffs laws Noe analysis
More informationPhysical Limitations of Logic Gates Week 10a
Physical Limiaions of Logic Gaes Week 10a In a compuer we ll have circuis of logic gaes o perform specific funcions Compuer Daapah: Boolean algebraic funcions using binary variables Symbolic represenaion
More informationThe problem with linear regulators
he problem wih linear regulaors i in P in = i in V REF R a i ref i q i C v CE P o = i o i B ie P = v i o o in R 1 R 2 i o i f η = P o P in iref is small ( 0). iq (quiescen curren) is small (probably).
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationIE1206 Embedded Electronics
E06 Embedded Elecronics Le Le3 Le4 Le Ex Ex P-block Documenaion, Seriecom Pulse sensors,, R, P, serial and parallel K LAB Pulse sensors, Menu program Sar of programing ask Kirchhoffs laws Node analysis
More informationEE 101 Electrical Engineering. vrect
EE Elecrical Engineering ac heory 3. Alernaing urren heory he advanage of he alernaing waveform for elecric power is ha i can be sepped up or sepped down in poenial easily for ransmission and uilisaion.
More informationLinear Circuit Elements
1/25/2011 inear ircui Elemens.doc 1/6 inear ircui Elemens Mos microwave devices can be described or modeled in erms of he hree sandard circui elemens: 1. ESISTANE () 2. INDUTANE () 3. APAITANE () For he
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationIntroduction to AC Power, RMS RMS. ECE 2210 AC Power p1. Use RMS in power calculations. AC Power P =? DC Power P =. V I = R =. I 2 R. V p.
ECE MS I DC Power P I = Inroducion o AC Power, MS I AC Power P =? A Solp //9, // // correced p4 '4 v( ) = p cos( ω ) v( ) p( ) Couldn' we define an "effecive" volage ha would allow us o use he same relaionships
More informationCAPACITANCE AND INDUCTANCE
APAITANE AND INDUTANE Inroduces wo passive, energy soring devices: apaciors and Inducors APAITORS Sore energy in heir elecric field (elecrosaic energy) Model as circui elemen INDUTORS Sore energy in heir
More informationTimer 555. Digital Electronics
Timer 555 Digial Elecronics This presenaion will Inroduce he 555 Timer. 555 Timer Derive he characerisic equaions for he charging and discharging of a capacior. Presen he equaions for period, frequency,
More information