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
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
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
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
apaciors A d Model for a non-ideal capacior 5
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 ) ( ) (
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
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.,.06 0-4 m. The permiiviy of free space is: Ɛ 0 = 8.85 0-2 F/m. = (Ɛ 0 A)/d A= d/ Ɛ 0 =.06 0-4 / 8.85 0-2 A=.48 0 7 m 2 =.48 km 2!!!!!! 8
Example The volage across a 5-µF capacior is given below. Deermine he curren of he capacior. 9
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
Example ompue he equivalen capaciance of he following nework:
Example alculae he equivalen capaciance of he following nework: a) when he swich is open b) when he swich is closed 2
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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
Applicaion Example Simple model for invesigaing crossalk: 5
Applicaion Example Use of ground wire o reduce crossalk (simple, no oo realisic model! Why?) 6
A more accurae model: Applicaion Example 7
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
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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
Inducors N 2 A l Model for a non-ideal inducor 2
Inducors In honor of Joseph Henry (797-878), 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
Example The curren in a 0-mH inducor has he following waveform. Find he volage of he inducor. 23
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
Example Find he equivalen inducance ( T ) of he following nework: 25
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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
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
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
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
Example Which one of he following circuis is a firs-order circui? 3
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 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
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
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
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
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
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 () 0.3679V 0 2 0.353V 0 3 0.0498V 0 4 0.083V 0 5 0.0067V 0 For all pracical purposes i is ypically assumed ha he response reaches is final value afer 5. 38
Example Assuming v (0)=30V, deermine v and v x, and i o for 0 39
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
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
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
Example In he following circui, assuming i(0)=0a, calculae i() and i x (). 43
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
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( ) 0 0 0 Use he sep funcion o express he volage across he resisor in he previous example: v()= 45
Uni Sep Funcion Examples Assuming 0 is a given posiive ime, plo he following funcions: u () u( ) u ) ( 0 u( 0) u( 0 ) 46
Uni Sep Funcion Examples Wrie he funcions on he previous slide in mahemaical erms, e.g., u( ) 0 0 0 47
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
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
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
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
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
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
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
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
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
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
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
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
Example In he following circui find he energy ha is sored in he inducor and capacior, when he circui reaches seady sae. 60
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
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
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
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
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
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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
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