apaiane and Induane OUTINE apaiors apaior volage, urren, power, energy Induors eure 9, 9/9/5 Reading Hambley haper 3 (A) EE4 Fall 5 eure 9, Slide The apaior Two onduors (a,b) separaed by an insulaor: differene in poenial V ab > equal & opposie harge Q on onduors Q V ab (sored harge in erms of volage) where is he apaiane of he sruure, posiive () harge is on he onduor a higher poenial Parallel-plae apaior: area of he plaes A (m ) separaion beween plaes d (m) dieleri permiiviy of insulaor ε (F/m) Aε F(F) > apaiane d EE4 Fall 5 eure 9, Slide
EE4 Fall 5 eure 9, Slide 3 apaior Symbol: or Unis: Farads (oulombs/vol) (ypial range of values: pf o µf; for superapaiors up o a few F!) urren-volage relaionship: i dq d dv d v d d i Elerolyi (polarized) apaior v Noes: If (geomery) is unhanging: i dv /d Q (and v ) mus be a oninuous funion of ime Seady-sae (I and V onsan) > i > Open irui EE4 Fall 5 eure 9, Slide 4
Volage in Terms of urren Q( ) i ( ) d Q() v ( ) Q() i ( ) d i ( ) d v () Uses: apaiors are used o sore energy for amera flashbulbs, in filers ha separae various frequeny signals, and hey appear as undesired parasii elemens in iruis where hey usually degrade irui performane EE4 Fall 5 eure 9, Slide 5 Sored Energy APAITORS STORE EETRI ENERGY You migh hink he energy sored on a apaior is QV V, whih has he dimension of Joules. Bu during harging, he average volage aross he apaior was only half he final value of V for a linear apaior. QV V Thus, energy is. Example: A pf apaiane harged o 5 Vols has ½(5V) (pf).5 pj (A 5F superapaior harged o 5 vols sores 63 J; if i disharged a a onsan rae in ms energy is disharged a a 63 kw rae!) EE4 Fall 5 eure 9, Slide 6 3
A more rigorous derivaion Heads Up: This derivaion holds independen of he irui! i v w Final Iniial v i v V d v v V Final Iniial dq v V d v d v V Final Iniial dq v V w v v V Final Iniial dv V Final V Iniial EE4 Fall 5 eure 9, Slide 7 Example: urren, Power & Energy for a apaior v (V) v( ) i( ) d v() τ τ v() µf i() 3 4 5 i (µa) dv i d 3 4 5 (µs) v and q mus be oninuous funions of ime; however, i an be disoninuous. (µs) Noe: In seady sae (d operaion), ime derivaives are zero is an open irui EE4 Fall 5 eure 9, Slide 8 4
p (W) 3 4 5 (µs) v() µf i() p vi w (J) 3 4 5 (µs) w pd τ v EE4 Fall 5 eure 9, Slide 9 apaiors in Series and Parallel v () v () i() i() eq v()v ()v () Series eq Noe: For apaiors in Parallel, he volage is he same on eah and he harges and hene apaianes add. EE4 Fall 5 eure 9, Slide 5
EE4 Fall 5 apaiive Volage Divider Q: Suppose he volage applied aross a series ombinaion of apaiors is hanged by v. How will his affe he volage aross eah individual apaior? v v Q v Q Q -Q Q v v Q Q v () v Q Q Q v eure 9, Slide v v v Noe ha no ne harge an an be inrodued o his node. Therefore, Q Q v v v v Noe: apaiors in series have he same inremenal harge. Appliaion Example: MEMS Aeleromeer o deploy he airbag in a vehile ollision apaiive MEMS posiion sensor used o measure aeleraion (by measuring fore on a proof mass) MEMS miro- elero-mehanial sysems g g FIXED OUTER PATES EE4 Fall 5 eure 9, Slide 6
Sensing he Differenial apaiane Begin wih apaianes elerially disharged Fixed elerodes are hen harged o V s and V s Movable elerode (proof mass) is hen harged o V o irui model V s V s V o V o V V o s Referene Volage Vs (V s εa εa g g g g εa εa g g g g Volage Division ) g g ons Volage V is proporional o he displaemen V s EE4 Fall 5 eure 9, Slide 3 Symbol: Induor Unis: Henrys (Vols seond / Ampere) (ypial range of values: µh o H) urren in erms of volage: Noes: i mus be a oninuous funion of ime Seady-sae (I and v onsan) > v > Shor irui EE4 Fall 5 dflux di v ( ) d d i ( ) v ( τ ) dτ i( eure 9, Slide 4 ) i v 7
Sored Energy INDUTORS STORE MAGNETI ENERGY onsider an induor having an iniial urren i( ) i p( ) v( ) i( ) w( ) w( ) p( τ ) dτ i i EE4 Fall 5 eure 9, Slide 5 Induors in Series and Parallel ommon urren ommon VOlage EE4 Fall 5 eure 9, Slide 6 8
EE4 Fall 5 apaior dv i d w v v anno hange insananeously i an hange insananeously Do no shor-irui a harged apaior (-> infinie urren!) n n ap. s in series: n ap. s in parallel: eq eq Summary i n i i i eure 9, Slide 7 Induor i anno hange insananeously v an hange insananeously Do no open-irui an induor wih urren (-> infinie volage!) n ind. s in series: di v d w i n ind. s in parallel: eq eq n i n i i i v - Muual Induane i M i v v di di M d d di di M d d Example: Transformer % flux linkage N urns N urns v / v N /N v - EE4 Fall 5 eure 9, Slide 8 9
Poenial Plos for a Single Resisor and Two Resisors in Series (Poenial is Ploed Verially) Arrows represen volage drops EE4 Fall 5 eure 9, Slide 9 Poenial Plo for Two Resisors in Parallel Arrows represen volage drops EE4 Fall 5 eure 9, Slide