CAPACITANCE AND INDUCTANCE

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1 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 magneic field Model as circui elemen APAITOR AND INDUTOR OMBINATIONS Series/parallel combinaions of elemens

2 APAITORS Firs of he energy sorage devices o be discussed Typical apaciors Basic parallel-plaes capacior IRUIT REPRESENTATION NOTIE USE OF PASSIVE SIGN ONVENTION

3 A = ε d ε Dielecric consan of maerial in gap PATE SIZE FOR EQUIVAENT AIR-GAP APAITOR A 8 55F = A = m Normal values of capaciance are small. Microfarads is common. For inegraed circuis nano or pico farads are no unusual

4 Basic capaciance law Q = f V ) ( inear capaciors obey oulomb s law Q = V is called he APAITANE of he device and has unis of charge volage One Farad(F)is he capaciance of a device ha can sore one oulomb of charge a one Vol. Farad = oulomb Vol inear capacior circui represenaion EXAMPE V = Volage across a capacior of micro Farads holding 0m of charge Q = 3 *0 0*0 = apaciance in Farads, charge in oulombs resul in volage in Vols apaciors can be dangerous!!! V

5 apaciors only sore and release EETROSTATI energy. They do no creae The capacior is a passive elemen and follows he passive sign convenion inear capacior circui represenaion dv i = d

6 Q = V apaciance aw If he volage varies he charge varies and here is a displacemen curren One can also express he volage across in erms of he curren V ( ) = Q = i ( x) dx Inegral form of apaciance law Or one can express he curren hrough in erms of he volage across i dq d = = dv d Differenial form of apaciance law The mahemaical implicaion of he inegral form is... V ( ) = V ( + ); Volage across a capacior MUST be coninuous Implicaions of differenial form?? V = ons i = 0 D or seady sae behavior A capacior in seady sae acs as an OPEN IRUIT

7 APAITOR AS IRUIT EEMENT EARNING EXAMPE i i + v dvc d = v = i ( x) dx = + v v 0 = 0 0 = v ( 0 i ) + ( x) dx + ( x) dx 0 i ( x) dx 0 The fac ha he volage is defined hrough an inegral has imporan implicaions... i i v v c ( O ) R R = v R = Ri R Ohm s aw R = 5μF DETERMINE THE URRENT dv i = d 60mA 6 4 V i = 5 0 [ F ] = 0mA s i = 0 elsewhere

8 dvc i = d dvc p = v d d p ( ) = v d w APAITOR AS ENERGY STORAGE DEVIE Insananeous power p v i = If is minus infiniy we alk abou energy sored a ime. i + v (, ) = v ( ) v ( ) Energy is he inegral of power w (, ) = p ( x) dx p v = i ( x) dx = p q dq q d = d = q c d If boh limis are infiniy hen we alk abou he oal energy sored. w W (, ) = q ( ) q ( )

9 = 5μF w w Energy sored in 0-6 msec (0,6) = v (6) v 6 (0,6) = 5*0 [ F]*(6) (0) [ V ] EXAMPE q harge sored a 3msec q ( 3) = v (3) (3) = 5*0 6 [ F]*[ V ] = 60μ

10 = 4μ F. FIND THE VOTAGE v( 0) = 0 v( ) = v(0) + i( x) dx; 0 > 0 v( ) = v() + i( x) dx; > 0 v( ) = [ V < 4ms ]

11 INDUTORS NOTIE USE OF PASSIVE SIGN ONVENTION Flux lines may exend beyond inducor creaing sray inducance effecs ircui represenaion for an inducor A TIME VARYING FUX REATES A OUNTER EMF AND AUSES A VOTAGE TO APPEAR AT THE TERMINAS OF THE DEVIE

12 A TIME VARYING MAGNETI FUX INDUES A VOTAGE v dφ d = Inducion law FOR A INEAR INDUTOR THE FUX IS PROPORTIONA TO THE URRENT φ = i v = di d DIFFERENTIA FORM OF INDUTION AW THE PROPORTIONAITY ONSTANT,, IS AED THE INDUTANE OF THE OMPONENT INDUTANE IS MEASURED IN UNITS OF henry (H). DIMENSIONAY HENRY = Vol Amp sec INDUTORS STORE EETROMAGNETI ENERGY. THEY MAY SUPPY STORED ENERGY BAK TO THE IRUIT BUT THEY ANNOT REATE ENERGY. THEY MUST ABIDE BY THE PASSIVE SIGN ONVENTION Follow passive sign convenion

13 A direc consequence of inegral form v = di d i = v ( x) dx i = i ( i Differenial form of inducion law 0 ) + v ( x) dx; A direc consequence of differenial formi = ons. v = 0 w ( p v i 0 Inegral form of inducion law ( ) = i ( + ); urren MUST be coninuous Power and Energy sored di d = = i ( ) d d = W p i, ) = d d i ( x) dx w(, ) = i( ) i( ) w ( ) i = ( ) J urren in Amps, Inducance in Henrys yield energy in Joules Energy sored on he inerval an be posiive or negaive 0 D (seady sae) behavior

14 EXAMPE FIND THE TOTA ENERGY STORED IN THE IRUIT In seady sae inducors ac as shor circuis and capaciors ac as open circuis W = V W = I V V 9 A A: 3A+ + = VA = 8 [ V ] 5 I + 3A= I I =.A V = 9 6I V = 6.V 6 V = V = 0.8V A 6+ 3 I VA = =.8A 9

15 EXAMPE =0mH. FIND THE VOTAGE A m = = s A s di v = d m = 0 A s THE DERIVATIVE OF A STRAIGHT INE IS ITS SOPE di d 0( A/ s) 0 ms = 0( A/ s) < 4ms 0 elsewhere di ) 0( A/ s) d v( ) = = 0 0 H ( = 3 V = 00mV ENERGY STORED BETWEEN AND 4 ms w(4,) w(4,) = = 0 0.5*0*0 i (4) i () 3 (0*0 3 THE VAUE IS NEGATIVE BEAUSE THE INDUTOR IS SUPPYING ENERGY PREVIOUSY STORED ) J

16 APAITOR SPEIFIATIONS APAITANE IN RANGE STANDARD VAUES p F 50mF STANDARD 6.3V 500V APAITOR RATINGS STANDARD TOERANE ± 5 %, ± 0%, ± 0% INDUTANE RANGES IN STANDARD VAUES STANDARD INDUTOR SPEIFIATIONS ma A nh 00mH INDUTOR RATINGS STANDARD ± 5 %, ± 0% TOERANE

17 v i i v

18 IDEA AND PRATIA EEMENTS i () i() i() + i() + v() + v() + v() v() IDEA EEMENTS dv di i = v = d d APAITOR/INDUTOR MODES INUDING EAKAGE RESISTANE v( ) dv i = + R d leak MODE FOR EAKY APAITOR di v = Rleak i( ) + d MODE FOR EAKY INDUTORS

19 SERIES APAITORS = s + Series ombinaion of wo capaciors 6μF μf 3 = S μ F NOTIE SIMIARITY WITH RESITORS IN PARAE

20 PARAE APAITORS i dv k d k = i() P = = 5μF

21 SERIES INDUTORS v di k d k = di v = S d eq = 7H

22 PARAE INDUTORS i() i( = N 0 ) i j ( 0) j= INDUTORS OMBINE IKE RESISTORS APAITORS OMBINE IKE ONDUTANES 4mH mh i( 0 ) = 3A 6A + A = A

23

24 EARNING EXAMPE FIP HIP MOUNTING I WITH WIREBONDS TO THE OUTSIDE GOA: REDUE INDUTANE IN THE WIRING AND REDUE THE GROUND BOUNE EFFET A SIMPE MODE AN BE USED TO DESRIBE GROUND BOUNE

25 MODEING THE GROUND BOUNE EFFET V dig ball d GB = ball 0. nh m = A s IF A GATES IN A HIP ARE ONNETED TO A SINGE GROUND THE URRENT AN BE QUITE HIGH AND THE BOUNE MAY BEOME UNAEPTABE USE SEVERA GROUND ONNETIONS (BAS) AND AOATE A FRATION OF THE GATES TO EAH BA

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CAPACITANCE AND INDUCTANCE APAITANE AND INDUTANE Inroduces wo passive, energy soring devices: apaciors and Inducors EARNING GOAS APAITORS Sore energy in heir elecric field (elecrosaic energy) Model as circui elemen INDUTORS Sore

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