EE105 - Fall 2006 Microelectronic Devices and Circuits
|
|
- Ferdinand Caldwell
- 5 years ago
- Views:
Transcription
1 EE15 - Fll 6 Microelectroic evice Circuit Prof. J M. Rbey (@eec Lecture 4: Ccitor P Juctio Overview Lt lecture iffuio curret Overview of IC fbrictio roce Review of electrottic Thi lecture Ccitce Juctio 1
2 Amiitrtivi Aother Mke-u Lecture Moy t 4m (treme O LECTURE O TUESAY 3 IC MIM Ccitor Bottom Plte To Plte Bottom Plte Cotct Thi Oie Q C By formig thi oie metl (or olyilico lte, ccitor i forme Cotct re me to to bottom lte Pritic ccitce eit betwee bottom lte ubtrte 4
3 Review of Ccitor E l Et o E to Q Q E S EA A t Q E S For iel metl, ll chrge mut be t urfce Gu lw: Surfce itegrl of electric fiel over cloe urfce equl chrge iie volume o Q E S Q C C A t o 5 Ccitor Q- Reltio Q y Q(y y Q C Totl chrge i lierly relte to voltge Chrge eity i elt fuctio t urfce (for erfect metl 6 3
4 A o-lier Ccitor Q y Q(y y Q f ( We ll oo meet ccitor tht hve o-lier Q- reltiohi If lte re ot iel metl, the chrge eity c eetrte ito urfce 7 Wht the Ccitce? For o-lier ccitor, we hve Q f ( C We c t ietify ccitce Imgie we ly mll igl o to of voltge: Q f ( v f ( f ( v Cott chrge The icremetl chrge i therefore: Q Q q f ( f ( v 8 4
5 Smll Sigl Ccitce Brek the equtio for totl chrge ito two term: Q Q q Cott Chrge Icremetl Chrge f ( f ( v f ( q v C v f ( C 9 Emle of o-lier Ccitor et lecture we ll ee tht for P uctio, the chrge i fuctio of the revere : Q ( q 1 b oltge Acro P Juctio Chrge At Sie of Juctio Cott Smll igl ccitce: Q C ( q b 1 1 b C 1 b 1 5
6 P Juctio (ioe 11 Crrier Cocetrtio Potetil I therml equilibrium, there re o eterl fiel we thu eect the electro hole curret eitie to be zero: J q μe q μ oe q kt o o o kt q o th 1 6
7 Crrier Cocetrtio Potetil ( We hve equtio reltig the otetil to the crrier cocetrtio kt q If we itegrte the bove equtio we hve ( ( ( l ( We efie the otetil referece to be itriic Si: th ( ( o th i 13 Crrier Cocetrtio eru Potetil The crrier cocetrtio i thu fuctio of otetil e ( / th ( i Check tht for zero otetil, we hve itriic crrier cocetrtio (referece. If we o imilr clcultio for hole, we rrive t imilr equtio e ( / th ( i ote tht the lw of m ctio i uhel ( / ( / ( ( i e th th e i 14 7
8 The oig Chge Potetil ue to the log ture of the otetil, the otetil chge lierly for eoetil icree i oig: ( ( ( ( th l 6m l 6m ( l1 log 1 ( ( 1 i ( ( 6m log 1 1 ( ( 6m log 1 1 i Quick clcultio i: For -tye cocetrtio of 1 16 cm -3, the otetil i -36 m -tye mteril hve oitive otetil with reect to itriic Si 15 P Juctio: Overview 16 8
9 P Juctio: Overview 17 P Juctio: Overview Preet i mot IC tructure 18 9
10 P Juctio: Overview The mot imortt evice i uctio betwee -tye regio -tye regio Whe the uctio i firt forme, ue to the cocetrtio griet, mole chrge trfer er uctio Electro leve -tye regio hole leve -tye regio Thee mole crrier become miority crrier i ew regio (c t eetrte fr ue to recomtio ue to chrge trfer, voltge ifferece occur betwee regio Thi crete fiel t the uctio tht cue rift curret to ooe the iffuio curret I therml equilibrium, rift curret iffuio mut blce -tye A -tye 19 P Juctio Curret Coier the P uctio i therml equilibrium Agi, the curret hve to be zero, o we hve o J qμe q o qμe q o kt 1 E μ q o kt 1 E μ q 1
11 P Juctio Fiel -tye -tye A ( E J iff i i E J iff Tritio Regio 1 Totl Chrge i Tritio Regio To olve for the electric fiel, we ee to write ow the chrge eity i the tritio regio: ρ ( q( I the -ie of the uctio, there re very few electro oly ccetor: ρ ( q( Sice the hole cocetrtio i ecreig o the - ie, the et chrge i egtive: ρ > ( < < < 11
12 Chrge o -Sie Alogou to the -ie, the chrge o the -ie i give by: ρ( q( < < The et chrge here i oitive ice: ρ ( > > i E J iff Tritio Regio 3 Ect Solutio for Fiel Give the bove roimtio, we ow hve ereio for the chrge eity q( ie ρ( q( ( / th ( / th ie o < < < < We lo hve the followig reult from electrottic E ρ ( otice tht the otetil er o both ie of the equtio ifficult roblem to olve A much imler wy to olve the roblem 4 1
13 eletio Aroimtio Let ume tht the tritio regio i comletely elete of free crrier (oly immole ot eit The the chrge eity i give by q ρ( q o < < < < The olutio for electric fiel i ow ey E E ρ ( ρ( ' ' E( ( Fiel zero outie tritio regio 5 eletio Aroimtio ( Sice chrge eity i cott ρ( ' q E ( ' ( o If we trt from the -ie we get the followig reult ρ( ' q E( ' E( ( E( Fiel zero outie tritio regio q E( ( 6 13
14 Plot of Fiel I eletio Regio -tye A eletio Regio -tye q E ( ( o q E( ( E-Fiel zero outie of eletio regio ote the ymmetricl eletio with Which regio h higher oig? Sloe of E-Fiel lrger i -regio. Why? Pek E-Fiel t uctio. Why cotiuou? 7 Cotiuity of E-Fiel Acro Juctio Recll tht E-fiel iverge o chrge. For heet chrge t the iterfce, the E-fiel coul be icotiuou I our ce, the eletio regio i oly oulte by bckgrou eity of fie chrge o the E-Fiel i cotiuou Wht oe thi imly? q q E ( E ( o o q q o o Totl fie chrge i -regio equl fie chrge i - regio! Somewht obviou reult. 8 14
15 Potetil Acro Juctio From our erlier clcultio we kow tht the otetil i the -regio i higher th -regio The otetil h to moothly tritio form high to low i croig the uctio Phyiclly, the otetil ifferece i ue to the chrge trfer tht occur ue to the cocetrtio griet Let itegrte the fiel to get the otetil: q ( ( o ( ' q ' ( ' o o ' 9 Potetil Acro Juctio We rrive t otetil o -ie (rbolic q o ( ( o itegrl o -ie q ( Potetil mut be cotiuou t iterfce (fiel fiite t iterfce q q ( ( ( 3 15
16 Solve for eletio Legth We hve two equtio two ukow. We re filly i oitio to olve for the eletio eth q q (1 q q o o ( o q o q > 31 Sity Check oe the bove equtio mke ee? Let y we oe oe ie very highly. The hyiclly we eect the eletio regio with for the hevily oe ie to roch zero: lim q lim q q Etire eletio with roe cro -regio 3 16
17 Totl eletio With The um of the eletio with i the ce chrge regio 1 1 X q Thi regio i eetilly elete of ll mole chrge ue to high electric fiel, crrier move cro regio t velocity turte ee X 1 q μ E 1 4 1μ 1 cm 33 Hve we ivete bttery? C we hre the P uctio tur it ito bttery? A A th l l th l i i i? umericl emle: 15 A 1 1 6m l 6m log 1 15 i 6m 34 17
18 Cotct Potetil The cotct betwee P uctio crete otetil ifferece Likewie, the cotct betwee two iimilr metl crete otetil ifferece (roortiol to the ifferece betwee the work fuctio Whe metl emicouctor uctio i forme, cotct otetil form well If we hort P uctio, the um of the voltge rou the loo mut be zero: m m m m ( m m 35 P Juctio Ccitor Uer therml equilibrium, the P uctio oe ot rw y (much curret But otice tht P uctio tore chrge i the ce chrge regio (tritio regio Sice the evice i torig chrge, it ctig like ccitor Poitive chrge i tore i the -regio, egtive chrge i i the -regio: q q o o 36 18
19 19 37 Revere Bie P Juctio Wht he if we revere- the P uctio? Sice o curret i flowig, the etire revere e otetil i roe cro the tritio regio To ccommote the etr otetil, the chrge i thee regio mut icree If o curret i flowig, the oly wy for the chrge to icree i to grow (hrik the eletio regio < 38 oltge eeece of eletio With C reo the mth but i the e we relize tht the equtio re the me ecet we relce the built-i otetil with the effective revere : q X 1 1 ( ( ( ( q 1 ( ( q 1 ( ( X X 1 (
20 Chrge eru Bi A we icree the revere, the eletio regio grow to ccommote more chrge QJ ( q ( q 1 Chrge i ot lier fuctio of voltge Thi i o-lier ccitor We c efie mll igl ccitce for mll igl by brekig u the chrge ito two term Q J ( v QJ ( q( v 39 erivtio of Smll Sigl Ccitce From lt lecture we fou C C Q ( J Q v QJ ( v L Q ( q 1 q C C 1 1 otice tht q q q C q R 4
21 1 41 Phyicl Iterrettio of eletio C otice tht the ereio o the right-h-ie i ut the eletio with i therml equilibrium Thi look like rllel lte ccitor! q C X q C ( ( X C 4 A rible Ccitor (rctor Ccitce vrie veru : Alictio: Rio Tuer C C
22 iffuio Reitor -tye iffuio Regio Oie P-tye Si Subtrte Reitor i ccitively ioltio from ubtrte Mut Revere Bi P Juctio! 43
EE105 - Fall 2005 Microelectronic Devices and Circuits
EE5 - Fll 5 Microelectroic evice ircuit Lecture 4 P Juctio Lecture Mteril Lt lecture I mufcturig roce iffuio curret Review of electrottic I ccitor Thi lecture P uctio rrier ocetrtio Potetil I therml equilibrium,
More informationLecture 27: PN Junctions
EECS 15 Srig 5, Lecture 7 Lecture 7: P Juctio Prof. ikej ertmet of EECS EECS 15 Fll 3, Lecture 7 Prof. A. ikej iffuio iffuio occur whe there exit cocetrtio griet I the figure below, imgie tht we fill the
More informationPN Junction Outline. Gate. Lec. 5. Gate electrode. Drain electrode. Source electrode. Source (n + ) Drain (n + ) Gate oxide. p-type Si substrate
P Juctio Outlie Gte electroe Source electroe Dri electroe Gte Source + Dri + Gte oie -tye Si ubtrte -chel MOSFET MOS Deletio regio P Juctio 0 Cotct otetil Built-i otetil Diffuio - + J iff =J rift Electric
More informationChapter 5: The pn Junction
Chter 5: The Juctio oequilibrium ece crrier i emicouctor Crrier geertio recombitio Mthemticl lyi of ece crrier Ambiolr trort The juctio Bic tructure of the juctio Zero lie bi evere lie bi o-uiformly oe
More informationCapacitors and PN Junctions. Lecture 8: Prof. Niknejad. Department of EECS University of California, Berkeley. EECS 105 Fall 2003, Lecture 8
CS 15 Fall 23, Lecture 8 Lecture 8: Capacitor ad PN Juctio Prof. Nikejad Lecture Outlie Review of lectrotatic IC MIM Capacitor No-Liear Capacitor PN Juctio Thermal quilibrium lectrotatic Review 1 lectric
More informationLecture 4: Heterojunction pn-diode
Lecture 4: Heterojuctio -ioe 16-1-5 Lecture 4, High See Devices 16 1 Lecture 4: Heterojuctio P-ioe + heterojuctio uer equilibrium + heterojuctio uer exterl s Gre heterojuctios + gre heterojuctio: Curret
More informationLecture 10: PN Junction & MOS Capacitors
Lecture 10: P Junction & MOS Cpcitors Prof. iknej eprtment of EECS Lecture Outline Review: P Junctions Therml Equilibrium P Junctions with Reverse Bis (3.3-3.6 MOS Cpcitors (3.7-3.9: Accumultion, epletion,
More informationChapter #3 EEE Subsea Control and Communication Systems
EEE 87 Chter #3 EEE 87 Sube Cotrol d Commuictio Sytem Cloed loo ytem Stedy tte error PID cotrol Other cotroller Chter 3 /3 EEE 87 Itroductio The geerl form for CL ytem: C R ', where ' c ' H or Oe Loo (OL)
More informationRemarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
More information4-4 E-field Calculations using Coulomb s Law
1/11/5 ection_4_4_e-field_clcultion_uing_coulomb_lw_empty.doc 1/1 4-4 E-field Clcultion uing Coulomb Lw Reding Aignment: pp. 9-98 Specificlly: 1. HO: The Uniform, Infinite Line Chrge. HO: The Uniform Dik
More informationKOREA UNIVERSITY. 5. I D -V D Relationship
KOREA UERSTY 5. - Reltioshi 1 Betwee oit A d B, it is the ohmic regio of the JFET. t is the regio where the voltge d curret reltioshi follows ohm's lw. At oit B, the dri curret is t mximum for S = coditio
More informationSpherical refracting surface. Here, the outgoing rays are on the opposite side of the surface from the Incoming rays.
Sphericl refrctig urfce Here, the outgoig ry re o the oppoite ide of the urfce from the Icomig ry. The oject i t P. Icomig ry PB d PV form imge t P. All prxil ry from P which trike the phericl urfce will
More informationELEC 372 LECTURE NOTES, WEEK 6 Dr. Amir G. Aghdam Concordia University
ELEC 37 LECTURE NOTES, WEE 6 Dr mir G ghdm Cocordi Uiverity Prt of thee ote re dpted from the mteril i the followig referece: Moder Cotrol Sytem by Richrd C Dorf d Robert H Bihop, Pretice Hll Feedbck Cotrol
More informationFig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg
ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced
More informationTopic 4 Fourier Series. Today
Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationArea, Volume, Rotations, Newton s Method
Are, Volume, Rottio, Newto Method Are etwee curve d the i A ( ) d Are etwee curve d the y i A ( y) yd yc Are etwee curve A ( ) g( ) d where ( ) i the "top" d g( ) i the "ottom" yd Are etwee curve A ( y)
More informationz line a) Draw the single phase equivalent circuit. b) Calculate I BC.
ECE 2260 F 08 HW 7 prob 4 solutio EX: V gyb' b' b B V gyc' c' c C = 101 0 V = 1 + j0.2 Ω V gyb' = 101 120 V = 6 + j0. Ω V gyc' = 101 +120 V z LΔ = 9 j1.5 Ω ) Drw the sigle phse equivlet circuit. b) Clculte
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationChapter #5 EEE Control Systems
Sprig EEE Chpter #5 EEE Cotrol Sytem Deig Bed o Root Locu Chpter / Sprig EEE Deig Bed Root Locu Led Cotrol (equivlet to PD cotrol) Ued whe the tedy tte propertie of the ytem re ok but there i poor performce,
More informationDiode in electronic circuits. (+) (-) i D
iode i electroic circuits Symbolic reresetatio of a iode i circuits ode Cathode () (-) i ideal diode coducts the curret oly i oe directio rrow shows directio of the curret i circuit Positive olarity of
More informationCalculus II Homework: The Integral Test and Estimation of Sums Page 1
Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the
More informationAll the Laplace Transform you will encounter has the following form: Rational function X(s)
EE G Note: Chpter Itructor: Cheug Pge - - Iverio of Rtiol Fuctio All the Lplce Trform you will ecouter h the followig form: m m m m e τ 0...... Rtiol fuctio Dely Why? Rtiol fuctio come out turlly from
More information8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
8.1 Arc Legth Wht is the legth of curve? How c we pproximte it? We could do it followig the ptter we ve used efore Use sequece of icresigly short segmets to pproximte the curve: As the segmets get smller
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationChapter #2 EEE Subsea Control and Communication Systems
EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio
More informationLecture 2. Dopant Compensation
Lecture 2 OUTLINE Bac Semicoductor Phycs (cot d) (cotd) Carrier ad uo PN uctio iodes Electrostatics Caacitace Readig: Chater 2.1 2.2 EE105 Srig 2008 Lecture 1, 2, Slide 1 Prof. Wu, UC Berkeley oat Comesatio
More informationParticle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise
Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationmoment = m! x, where x is the length of the moment arm.
th 1206 Clculus Sec. 6.7: omets d Ceters of ss I. Fiite sses A. Oe Dimesiol Cses 1. Itroductio Recll the differece etwee ss d Weight.. ss is the mout of "stuff" (mtter) tht mkes up oject.. Weight is mesure
More informationUNIT #5 SEQUENCES AND SERIES COMMON CORE ALGEBRA II
Awer Key Nme: Dte: UNIT # SEQUENCES AND SERIES COMMON CORE ALGEBRA II Prt I Quetio. For equece defied by f? () () 08 6 6 f d f f, which of the followig i the vlue of f f f f f f 0 6 6 08 (). I the viul
More informationFirst assignment of MP-206
irt igmet of MP- er to quetio - 7- Norml tre log { : MP Priipl tree: I MP II MP III MP Priipl iretio: { I { II { III Iitill uppoe tht i tre tte eribe i the referee tem ' i the me tre tte but eribe i other
More informationLecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More informationCHAPTER 3 NETWORK ADMITTANCE AND IMPEDANCE MATRICES
CHAPTER NETWORK ADTTANCE AND PEDANCE ATRCES As we hve see i Chter tht ower system etwor c e coverted ito equivlet imedce digrm. This digrm forms the sis of ower flow (or lod flow) studies d short circuit
More informationChap8 - Freq 1. Frequency Response
Chp8 - Freq Frequecy Repoe Chp8 - Freq Aged Prelimirie Firt order ytem Frequecy repoe Low-p filter Secod order ytem Clicl olutio Frequecy repoe Higher order ytem Chp8 - Freq 3 Frequecy repoe Stedy-tte
More informationWe will look for series solutions to (1) around (at most) regular singular points, which without
ENM 511 J. L. Baai April, 1 Frobeiu Solutio to a d order ODE ear a regular igular poit Coider the ODE y 16 + P16 y 16 + Q1616 y (1) We will look for erie olutio to (1) aroud (at mot) regular igular poit,
More information2.Decision Theory of Dependence
.Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationis completely general whenever you have waves from two sources interfering. 2
MAKNG SENSE OF THE EQUATON SHEET terferece & Diffrctio NTERFERENCE r1 r d si. Equtio for pth legth differece. r1 r is completely geerl. Use si oly whe the two sources re fr wy from the observtio poit.
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probbility d Stochstic Processes: A Friedly Itroductio for Electricl d Computer Egieers Roy D. Ytes d Dvid J. Goodm Problem Solutios : Ytes d Goodm,4..4 4..4 4..7 4.4. 4.4. 4..6 4.6.8 4.6.9 4.7.4 4.7.
More information0 dx C. k dx kx C. e dx e C. u C. sec u. sec. u u 1. Ch 05 Summary Sheet Basic Integration Rules = Antiderivatives Pattern Recognition Related Rules:
Ch 05 Smmry Sheet Bic Itegrtio Rle = Atierivtive Ptter Recogitio Relte Rle: ( ) kf ( ) kf ( ) k f ( ) Cott oly! ( ) g ( ) f ( ) g ( ) f ( ) g( ) f ( ) g( ) Bic Atierivtive: C 0 0 C k k k k C 1 1 1 Mlt.
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationOrthogonal functions - Function Approximation
Orthogol uctios - Fuctio Approimtio - he Problem - Fourier Series - Chebyshev Polyomils he Problem we re tryig to pproimte uctio by other uctio g which cosists o sum over orthogol uctios Φ weighted by
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationState space systems analysis
State pace ytem aalyi Repreetatio of a ytem i tate-pace (tate-pace model of a ytem To itroduce the tate pace formalim let u tart with a eample i which the ytem i dicuio i a imple electrical circuit with
More informationMathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Nme : Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits c e prite give to the istructor or emile to the istructor t jmes@richl.eu.
More informationEE Control Systems LECTURE 8
Coyright F.L. Lewi 999 All right reerved Udted: Sundy, Ferury, 999 EE 44 - Control Sytem LECTURE 8 REALIZATION AND CANONICAL FORMS A liner time-invrint (LTI) ytem cn e rereented in mny wy, including: differentil
More informationSection 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x
Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to:
More informationDegenerate and Non degenerate semiconductors. Nondegenerate semiconductor Degenerate and nondegenerate semiconductors
4.3.4 Degeerte egeerte semicuctrs Degeerte egeerte semicuctrs Smll mut f t tms (imurity tms) itercti betwee t tms Discrete, iterctig eergy stte. F t the bg F F r ccetr egeerte semicuctr 4.3.4 Degeerte
More information, so the state may be taken to be l S ÅÅÅÅ
hw4-7.b: 4/6/04:::9:34 Hoework: 4-7 ü Skuri :, G,, 7, 8 ü. bel the eigefuctio of the qure well by S, with =,,. The correpodig wve fuctio re x = px i Iitilly, the prticle i kow to be t x =, o the tte y
More information5. Fundamental fracture mechanics
5. Fudmetl frcture mechic Stre cocetrtio Frcture New free urfce formtio Force Mteril Eviromet Frcture mechic Ect crck growth drivig force? Reitce to frcture of mteril? Iititio Growth Stre cocetrtio Notch
More informationOverview of Silicon p-n Junctions
Overview of Silico - Juctios r. avid W. Graham West irgiia Uiversity Lae eartmet of omuter Sciece ad Electrical Egieerig 9 avid W. Graham 1 - Juctios (iodes) - Juctios (iodes) Fudametal semicoductor device
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationNumerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials
Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,
More informationLecture 10: P-N Diodes. Announcements
EECS 15 Sprig 4, Lecture 1 Lecture 1: P-N Diodes EECS 15 Sprig 4, Lecture 1 Aoucemets The Thursday lab sectio will be moved a hour later startig this week, so that the TA s ca atted lecture i aother class
More informationNumerical Integration
Numericl tegrtio Newto-Cotes Numericl tegrtio Scheme Replce complicted uctio or tulted dt with some pproimtig uctio tht is esy to itegrte d d 3-7 Roerto Muscedere The itegrtio o some uctios c e very esy
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationPhys-272 Lecture 25. Geometric Optics Lenses
Phy-7 Lecture 5 Geometric Optic Lee h h Rerctio o Sphericl Surce θ β φ φ α θ + + ; θ θ θ θ i i ( )φ β α + δ φ δ β δ α + R h h h t ; t ; t R h h h φ β α ; ; R + Rerctio o Sphericl Surce R + Mgiictio θ θ
More informationMathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits shoul e emile to the istructor t jmes@richl.eu. Type your me t the top
More informationThe limit comparison test
Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does
More informationIntroduction to Modern Control Theory
Itroductio to Moder Cotrol Theory MM : Itroductio to Stte-Spce Method MM : Cotrol Deig for Full Stte Feedck MM 3: Etitor Deig MM 4: Itroductio of the Referece Iput MM 5: Itegrl Cotrol d Rout Trckig //4
More informationEVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More informationMASSACHUSETTS INSTITUTE of TECHNOLOGY Department of Mechanical Engineering 2.71/ OPTICS - - Spring Term, 2014
.7/.70 Optic, Spri 04, Solutio for Quiz MASSACHUSETTS INSTITUTE of TECHNOLOGY Deprtmet of Mechicl Eieeri.7/.70 OPTICS - - Spri Term, 04 Solutio for Quiz Iued Wed. 03//04 Problem. The ive opticl ytem i
More informationFourier Series. Topic 4 Fourier Series. sin. sin. Fourier Series. Fourier Series. Fourier Series. sin. b n. a n. sin
Topic Fourier Series si Fourier Series Music is more th just pitch mplitue it s lso out timre. The richess o sou or ote prouce y musicl istrumet is escrie i terms o sum o umer o istict requecies clle hrmoics.
More informationDiscrete Mathematics I Tutorial 12
Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece
More informationCalculus Summary Sheet
Clculus Summry Sheet Limits Trigoometric Limits: siθ lim θ 0 θ = 1, lim 1 cosθ = 0 θ 0 θ Squeeze Theorem: If f(x) g(x) h(x) if lim f(x) = lim h(x) = L, the lim g(x) = L x x x Ietermite Forms: 0 0,,, 0,
More informationChapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Dr. Hamid R. Rabiee Fall 2013
Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Dr. Hmid R. Rbiee Fll 03 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Outlie Itroductio to the -Trsform Properties
More information10. 3 The Integral and Comparison Test, Estimating Sums
0. The Itegrl d Comriso Test, Estimtig Sums I geerl, it is hrd to fid the ect sum of series. We were le to ccomlish this for geometric series d for telescoig series sice i ech of those cses we could fid
More informationUniversity of California at Berkeley College of Engineering Dept. of Electrical Engineering and Computer Sciences.
Uversty of Clfor t Berkeley College of Egeerg et. of Electrcl Egeerg Comuter Sceces EE 5 Mterm I Srg 6 Prof. Mg C. u Feb. 3, 6 Gueles Close book otes. Oe-ge formto sheet llowe. There re some useful formuls
More informationVectors. Vectors in Plane ( 2
Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector
More informationLecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem
Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem Merge Sort MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA,
More informationDefinition Integral. over[ ab, ] the sum of the form. 2. Definite Integral
Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ
More informationMATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
More informationG x, x E x E x E x E x. a a a a. is some matrix element. For a general single photon state. ), applying the operators.
Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Proble Set # Quetio With G x, x E x E x E x E x G pqr p q r where G pqr i oe trix eleet For geerl igle
More information10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
More informationCHAPTER 7 SYMMETRICAL COMPONENTS AND REPRESENTATION OF FAULTED NETWORKS
HAPTER 7 SMMETRAL OMPOETS AD REPRESETATO OF FAULTED ETWORKS A uled three-phe yte e reolved ito three led yte i the iuoidl tedy tte. Thi ethod of reolvig uled yte ito three led phor yte h ee propoed y.
More informationLecture 9: Diffusion, Electrostatics review, and Capacitors. Context
EECS 5 Sprig 4, Lecture 9 Lecture 9: Diffusio, Electrostatics review, ad Capacitors EECS 5 Sprig 4, Lecture 9 Cotext I the last lecture, we looked at the carriers i a eutral semicoductor, ad drift currets
More informationChem 253A. Crystal Structure. Chem 253B. Electronic Structure
Chem 53, UC, Bereley Chem 53A Crystl Structure Chem 53B Electroic Structure Chem 53, UC, Bereley 1 Chem 53, UC, Bereley Electroic Structures of Solid Refereces Ashcroft/Mermi: Chpter 1-3, 8-10 Kittel:
More informationCrushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
More informationNonequilibrium Excess Carriers in Semiconductors
Lecture 8 Semicoductor Physics VI Noequilibrium Excess Carriers i Semicoductors Noequilibrium coditios. Excess electros i the coductio bad ad excess holes i the valece bad Ambiolar trasort : Excess electros
More informationConvergence rates of approximate sums of Riemann integrals
Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki
More information( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationThe Reimann Integral is a formal limit definition of a definite integral
MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More informationThe aim of the course is to give an introduction to semiconductor device physics. The syllabus for the course is:
Semicoductor evices Prof. Rb Robert tat A. Taylor The aim of the course is to give a itroductio to semicoductor device physics. The syllabus for the course is: Simple treatmet of p- juctio, p- ad p-i-
More informationProblems for HW X. C. Gwinn. November 30, 2009
Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationOptions: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.
O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More informationSection 2.2. Matrix Multiplication
Mtri Alger Mtri Multiplitio Setio.. Mtri Multiplitio Mtri multiplitio is little more omplite th mtri itio or slr multiplitio. If A is the prout A of A is the ompute s follow: m mtri, the is k mtri, 9 m
More information1. Do the following sequences converge or diverge? If convergent, give the limit. Explicitly show your reasoning. 2n + 1 n ( 1) n+1.
Solutio: APPM 36 Review #3 Summer 4. Do the followig sequeces coverge or iverge? If coverget, give the limit. Eplicitly show your reasoig. a a = si b a = { } + + + 6 c a = e Solutio: a Note si a so, si
More informationTrapezoidal Rule of Integration
Trpezoidl Rule o Itegrtio Civil Egieerig Mjors Authors: Autr Kw, Chrlie Brker http://umericlmethods.eg.us.edu Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/00 http://umericlmethods.eg.us.edu
More information