APAITANE AND INDUTANE Inroduces wo passve, energy sorng devces: apacors and Inducors LEARNING GOALS APAITORS Sore energy n her elecrc feld (elecrosac energy) Model as crcu elemen INDUTORS Sore energy n her magnec feld Model as crcu elemen APAITOR AND INDUTOR OMBINATIONS Seres/parallel combnaons of elemens R OP-AMP IRUITS Inegraon and dfferenaon crcus
apacors-apacance The capacor s a passve elemen and follows he passve sgn convenon LEARNING BY DOING apacors only sore and release ELETROSTATI energy. They do no creae Lnear capacor crcu represenaon dv d
Q V apacance Law If he volage vares he charge vares and here s a dsplacemen curren One can also express he volage across n erms of he curren V ( ) Q ( x) dx Inegral form of apacance law Or one can express he curren hrough n erms of he volage across dq d dv d Dfferenal form of apacance law The mahemacal mplcaon of he negral form s... V ( ) V ( ); Volage across a capacor MUST be connuous Implcaons of dfferenal form?? V ons 0 D or seady sae behavor A capacor n seady sae acs as an OPEN IRUIT
APAITOR AS IRUIT ELEMENT v dvc d v ( x) dx v v 0 0 0 v ( 0 ) ( x) dx ( x) dx 0 ( x) dx 0 The fac ha he volage s defned hrough an negral has mporan mplcaons... v v c ( O ) R R v R R R Ohm s Law R LEARNING EXAMPLE 5μF DETERMINE THE URRENT dv d 60mA 6 4 V 5 0 [ F ] 0mA 3 6 0 s 0 elsewhere
dvc d dvc p v d d p ( ) v d w APAITOR AS ENERGY STORAGE DEVIE Insananeous power p v If s mnus nfny we alk abou energy sored a me. v (, ) v ( ) v ( ) Energy s he negral of power w (, ) p ( x) dx p v ( x) dx p q dq q d d q c d If boh lms are nfny hen we alk abou he oal energy sored. w W (, ) q ( ) q ( )
5μF LEARNING EXAMPLE w w Energy sored n 0-6 msec (0,6) v (6) v (0) 6 (0,6) 5*0 [ F ]*(4) [ V q harge sored a 3msec q ( 3) v (3) (3) 5*0 6 [ F]*[ V ] 60μ ] oal energy sored?... oal charge sored?... If charge s n oulombs and capacance n Farads hen he energy s n.
SAMPLE PROBLEM v() v 30sn (0π ) μ F WHAT VARIABLES AN BE OMPUTED? E(/ 40) q v ( (/0) Energy sored a a gven me E( ) v harge sored a a gven me ) urren hrough he capacor () 6 π *0 [ F]*30 sn 6 q *0 [ ]*sn( π )[ V ] 0 dv 6 (/0) *0 *30*0π cos( π ) d p v ( W Elecrc power suppled o capacor a a gven me ) Energy sored over a gven me nerval w(, ) v ( ) v ( ) J J A
A TIME VARYING MAGNETI FLUX INDUES A VOLTAGE v L dφ d Inducon law FOR A LINEAR INDUTOR THE FLUX IS PROPORTIONAL TO THE URRENT LL d DIFFERENTIAL FORM L vl L OF INDUTION LAW φ Inducors-Inducance d THE PROPORTIONALITY ONSTANT, L, IS ALLED THE INDUTANE OF THE OMPONENT LEARNING by Dong INDUTANE IS MEASURED IN UNITS OF henry (H). DIMENSIONALLY HENRY Vol Amp sec INDUTORS STORE ELETROMAGNETI ENERGY. THEY MAY SUPPLY STORED ENERGY BAK TO THE IRUIT BUT THEY ANNOT REATE ENERGY. THEY MUST ABIDE BY THE PASSIVE SIGN ONVENTION Follow passve sgn convenon
A drec consequence of negral form v L dl L d L( ) vl ( x) dx L L L ( L Dfferenal form of nducon law 0 ) vl ( x) dx; L A drec consequence of dfferenal forml ons. vl 0 w(, ) LL( ) LL( ) w ( ) L L L( ) Energy sored a me Mus be non-negave. Passve elemen!!! 0 Inegral form of nducon law ( ) ( ); urren MUST be connuous L Power and Energy sored dl d pl vl L W pl L L L ( ) d d L d wl(, ) LL( x) dx J urren n Amps, Inducance n Henrys d yeld energy n Joules Energy sored on he nerval an be posve or negave 0 D (seady sae) behavor
LEARNING EXAMPLE FIND THE TOTLA ENERGY STORED IN THE IRUIT In seady sae nducors ac as shor crcus and capacors ac as open crcus W V W LI L L V V 9 A A @ A: 3A 0 9 6 VA 8 [ V ] 5 I 3A I I.A L L L V 96I V 6.V L 6 V V 0.8V A 6 3 I L VA.8A 9
L v v
IDEAL AND PRATIAL ELEMENTS () () () () v() v() v() v() IDEAL ELEMENTS dv d v L d d APAITOR/INDUTOR MODELS INLUDING LEAKAGE RESISTANE v( ) dv R d leak MODEL FOR LEAKY APAITOR d v Rleak ( ) L d MODEL FOR LEAKY INDUTORS
SERIES APAITORS s Seres ombnaon of wo capacors 6μF μf 3 S μ F NOTIE SIMILARITY WITH RESITORS IN PARALLEL
LEARNING EXAMPLE DETERMINE EQUIVALENT APAITANE AND THE INITIAL VOLTAGE μ F μ F 3 6 OR WE AN REDUE TWO AT A TIME V 4V V ALGEBRAI SUM OF INITIAL VOLTAGES POLARITY IS DITATED BY THE REFERENE DIRETION FOR THE VOLTAGE
LEARNING EXAMPLE Two uncharged capacors are conneced as shown. Fnd he unknown capacance V 8V - 4V 30μ F FIND 8V SAME URRENT. ONNETED FOR THE SAME TIME PERIOD SAME HARGE ON BOTH APAITORS Q ( 30μ F )(8V ) 40μ Q V Q ( μf )(6V ) 7μ 7μ 8V 4μ F
PARALLEL APAITORS dv k d k () LEARNING EXAMPLE P 4 6 3 5μF
SERIES INDUTORS v d Lk d k LEARNING EXAMPLE d v LS d Leq 7H
PARALLEL INDUTORS () LEARNING EXAMPLE ( N 0 ) j ( 0) j INDUTORS OMBINE LIKE RESISTORS APAITORS OMBINE LIKE ONDUTANES 4mH mh ( 0 ) 3A 6A A A
R OPERATIONAL AMPLIFIER IRUITS INTRODUES TWO VERY IMPORTANT PRATIAL IRUITS BASED ON OPERATIONAL AMPLIFIERS THE IDEAL OP-AMP IDEAL RO 0, R, A R R O 0 v O A ( v v ) A
R OPERATIONAL AMPLIFIER IRUITS -THE INTEGRATOR v 0 v IDEAL OP-AMP ASSUMPTIONS v 0 ( A ) ( R )
R OPERATIONAL AMPLIFIER IRUITS - THE DIFFERENTIATOR R KVL v 0 KL@ v v : IDEAL OP-AMP ASSUMPTIONS v 0 ( A ) ( R ) R v ( x) dx d dv R d d R v O 0 DIFFERENTIATE replace n erms of v o ( v R dvo dv R vo R d d IF R OULD BE SET TO ZERO WE WOULD HAVE AN IDEAL DIFFERENTIATOR. IN PRATIE AN IDEAL DIFFERENTIATOR AMPLIFIES ELETRI NOISE AND DOES NOT OPERATE. THE RESISTOR INTRODUES A FILTERING ATION. ITS VALUE IS KEPT AS SMALL AS POSSIBLE TO APPROXIMATE A DIFFERENTIATOR o )