# Chapter 3: Capacitors, Inductors, and Complex Impedance

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 haptr 3: apacitors, Inductors, and omplx Impdanc In this chaptr w introduc th concpt of complx rsistanc, or impdanc, by studying two ractiv circuit lmnts, th capacitor and th inductor. W will study capacitors and inductors using diffrntial quations and Fourir analysis and from ths driv thir impdanc. apacitors and inductors ar usd primarily in circuits involving tim-dpndnt voltags and currnts, such as A circuits. I. A oltags and circuits Most lctronic circuits involv tim-dpndnt voltags and currnts. An important class of tim-dpndnt signal is th sinusoidal voltag (or currnt, also known as an A signal (Altrnating urrnt. Kirchhoff s laws and Ohm s law still apply (thy always apply, but on must b carful to diffrntiat btwn tim-avragd and instantanous quantitis. An A voltag (or signal is of th form: (t p cos(ωt (3. whr ω is th angular frquncy, p is th amplitud of th wavform or th pak voltag and t is th tim. Th angular frquncy is rlatd to th frguncy (f by ω πf and th priod (T is rlatd to th frquncy by T/f. Othr usful voltags ar also commonly dfind. Thy includ th pak-to-pak voltag ( pp which is twic th amplitud and th MS voltag ( MS which is /. Avrag powr in a rsistiv A dvic is computd using MS quantitis: MS p PI MS MS I p p /. (3. This is important nough that voltmtrs and ammtrs in A mod actually rturn th MS valus for currnt and voltag. Whil most ral world signals ar not sinusoidal, A signals ar still usd xtnsivly to charactriz circuits through th tchniqu of Fourir analysis. Fourir Analysis On convnint way to charactriz th rat of chang of a function is to writ th tru function as a linar combination of a st of functions that hav particularly asy charactristics to dal with analytically. In this cas w can considr th trigonomtric functions. It turns out that w can writ any function as an intgral of th form ~ ( t cos( ωt + φ dω (3.3 whr ~ and φ ar functions of th frquncy ω. This procss is calld Fourir analysis, and it mans that any function can b writtn as an intgral of simpl sinusoidal - 7 -

2 functions. In th cas of a priodic wavform this intgral bcoms a sum ovr all th harmonics of th priod (i.. all th intgr multiplicativ frquncis of th priod. A cos + n ( nωt ( t (3.4 n φ n An implication of this mathmatical fact is that if w can figur out what happns whn w put pur sinusoidal voltags into a linar circuit, thn w will know vrything about its opration vn for arbitrary input voltags. omplx Notation In complx notation w rplac our sinusoidal functions by xponntials to mak th calculus and bookkping asir still. Thn w can includ both phas and magnitud information. W ll dfin whr i. iφ cos φ + isinφ, (3.5 Th gnral procdur for using this notation is:. hang your problm into complx algbra (i.. rplac cos ωt with. Solv th problm. 3. Tak th ral part of th solution as your answr at th nd. iωt II. apacitors On of th most basic ruls of lctronics is that circuits must b complt for currnts to flow. This wk, w will introduc an xcption to that rul. Th capacitor is actually a small brak in a circuit. Try masuring th rsistanc of a capacitor, you will find that it is an opn circuit. Howvr, at th insid nds of th capacitor s lad, it has littl plats that act as charg rsrvoirs whr it can stor charg. For short tims, you do not notic that th brak is thr. Ngativ charg initially flows in to on sid and out from out th othr sid just as if th two lads wr connctd. For fast signals, th capacitor looks lik a short-circuit. But aftr a whil th capacitor s rsrvoirs fill, th currnt stops, and w notic that thr rally is a brak in th circuit. For slow signals, a capacitor looks lik an opn circuit. What is fast, and what is slow? It dpnds on th capacitor and th rst of th circuit. This wk, you will larn how to dtrmin fast and slow for yourslvs. apacitors srv thr major rols in lctrical circuits (although all thr ar just variations of on basic ida: harg intgrators; High or low frquncy filtrs; D isolators

3 In ordr to prform ths functions analytically, w will nd to introduc a numbr of nw concpts and som significant mathmatical formalism. In this procss w will also dvlop a numbr of nw concpts in analyzing lctronic circuits. apacitanc A capacitor is a dvic for storing charg and lctrical nrgy. It consists of two paralll conducting plats and som non-conducting matrial btwn th plats, as shown in figur 3. on th right. Whn voltag is applid positiv charg collcts on on plat and ngativ charg collcts on th othr plan. Sinc thy ar attractd to ach othr this is a stabl stat until th voltag is changd again. A capacitor s charg capacity or capacitanc ( is dfind as: Q (3.6 which rlats th charg stord in th capacitor (Q to th voltag across its lads (. apacitanc is masurd in Farads (F. A Farad is a vry larg unit and most applications us µf, nf, or pf sizd dvics. Many lctronics componnts hav small parasitic capacitancs du to thir lads and dsign. Th capacitor also stors nrgy in th lctric fild gnratd by th chargs on its two plats. Th potntial nrgy stord in a capacitor, with voltag on it, is E (3.7 W usually spak in trms of currnt whn w analyz a circuit. By noting that th currnt is th rat of chang of charg, w can rwrit th dfinition of capacitanc in trms of th currnt as: or Q Idt Figur 3.: A capacitor consist of two paralll plats which stor qual and opposit amounts of charg (3.8 d I & (3.9 dt This shows that w can intgrat a function I(t just by monitoring th voltag as th currnt chargs up a capacitor, or w can diffrntiat a function (t by putting it across a capacitor, and monitoring th currnt flow whn th voltag changs

4 A Simpl ircuit W will start by looking in dtail at th simplst capacitiv circuit, which is shown in figur 3. on th right. An circuit is mad by simply putting a rsistor and a capacitor togthr as a voltag dividr. W will put th rsistor in first, so w can connct th capacitor to ground. By applying Kirchhoff s aws to this circuit, w can s that:. Th sam currnt flows through both th rsistor and th capacitor, and. Th sum of th voltag drops across th two lmnts qual th input voltag. This can b put into a formula in th following quation: IN I + Idt. (3. which can also b writtn as IN I + Idt. (3. W can also put this into th form of a diffrntial quation in th following way: or d di I IN + (3.a dt dt & IN I& + I. (3.b Ths quations show that tims ar masurd in units of, and that what you s dpnds on how quickly things chang during on tim intrval. If th currnt changs quickly, thn most of th voltag will show up across th rsistor, whil th voltag across th capacitor slowly chargs up as it intgrats th currnt. If th voltag changs slowly, thn most of th voltag shows up across th capacitor as it chargs. Sinc this usually rquirs a small currnt, th voltag across th rsistor stays small. But, what happns at intrmdiat tims? To dtrmin this quantitativly w will hav to dvlop som mor sophisticatd mathmatical tchniqus. IN OUT Figur 3.: A simpl circuit which intgrats currnt. Solutions to ircuit athr than produc th gnral solution, w will concntrat on two spcial cass that ar particularly usful. Th first will b for a constant voltag and th scond will b a sinusoidal input. - -

5 To study a constant supply voltag on an circuit, w st th lft sid of quation 3. qual to a constant voltag. Thn w hav a simpl homognous diffrntial quation with th simpl solution for th currnt of a dcaying xponntial, I ( t / I, (3.3 which will account for any initial conditions. Aftr a tim of a fw tim priods, this solution will hav dcayd away to th supply voltag. And now lt us considr th othr solution. In th prior sction, w argud that if w can undrstand th circuit s bhavior for sinusoidal input w can dal with any arbitrary input. Thrfor, this is th important on. t s look at our simpl circuit and suppos that w apply (or driv a simpl sin wav into th input: IN ( ωt cos. (3.4 In complx notation, this mans that w will st th driv voltag to IN ( iωt xp, (3.5 and w just hav to rmmbr to tak th al part at th nd of our calculation. If w put this driv voltag into th diffrntial quation (quation 3., thn it bcoms a rlativly simpl inhomognous diffrntial quation: din di iω xp( iωt + I. (3.6 dt dt This is rlativly simpl bcaus it shows up so oftn in physics that you might as wll mmoriz th solution or at last th way to gt th solution. Not that mathmatically it looks just lik a drivn harmonic oscillator. W can obtain th solution by using th standard rcip for first ordr linar diffrntial quations. W start by rwriting quation as di dt iω + I xp( iωt, (3.7 which w thn multiply by ( t / xp to obtain di xp( t / iω xp( t / + I xp[( iω + t]. (3.8 dt Th lft hand-sid of this quality can b rwrittn undr th form of a total drivativ (multiplication rul so that w now hav d dt I( txp( t iω This quation is asily intgrabl and can b rwrittn as xp[( iω + t]. (3.9 t iω I( txp( i t dt xp[( ω + ]. (3. - -

6 Th intgral is straightforward and yilds th following xprssion: iω I( t xp( iωt + st xp( + iω t (3. Th first trm rprsnt th stady stat oscillatory bhavior of th drivn circuit, whil th scond trm dscribs th transint bhavior of th currnt aftr switching on th driving voltag. Sinc w ar only intrstd in th long-trm bhavior of th circuit, w nglct th scond trm and concntrat on th first. Aftr a littl bit of algbra, w can rwrit th stady-stat currnt as iω ω ω + i I( t xp( iωt xp( iωt (3. i + ω + ( ω + ( ω Th scond fraction can b intrprtd as a phas trm with xprssion for th currnt bcoms with tan φ, so that th ω I ( t I xp( iω t + φ (3.3 ω I cos( φ (3.4 + ( ω Th ral solution of this simpl circuit can b obtaind by taking th ral part of quation 3, and is lft as an xrcis to th radr. Th solution of th simpl circuit appars to b rathr complicatd and involvd, howvr it simplifis considrably whn w plug quation 3 back in to th original intgral quation from Kirchhoff s loop law (quation. Aftr intgrating th xponntial and a littl bit of algbra, w obtain in ( t; ω I( t + I( t (3.5 i ω This rmarkably simpl xprssion looks a lot lik th standard Kirchhoff s loop law for rsistors, xcpt that th capacitor trm bhavs with a frquncy dpndnt imaginary rsistanc. Impdanc W will obtain th sam solution as th on w obtaind for th original voltag dividr, as long as w assign an imaginary, frquncy dpndnt, rsistanc to th capacitor. Th imaginary part just mans that it will produc a π phas shift btwn th voltag and th currnt for a sinusoidal input. W will call this impdanc Z. (3.6 iω - -

7 Now, th solution for an dividr bcoms somwhat simplifid. W can comput th total currnt flowing through th circuit as I Z in tot in + Z + / iωt iω iωt iω iωt ( iω + iω ( + iω cos( φ ( ωt+ φ i (3.7 Th voltag across an lmnt is just this currnt tims th lmnt s impdanc. For th voltag drop across th rsistor it is largly th sam as bfor: ( ωt+ φ i I cos( φ. (3.8 For th capacitor, w gt th following voltag drop: IZ I iω cos( φ iω cos( φ i ω i( ωt+ φ i sin( φ ( ωt+ φ ( ωt+ φ π / i( ωt+ φ π / i sin( φ (3.9 If vrything is corrctly calculatd thn th sum of th voltag drops across th two lmnts should b qual to th input voltag. t s try it: + ( i( ω t + φ i φ i( ω i t + φ i ω cos( sin( φ t φ (3.3 mmbr, you gt th actual wavforms by taking th ral parts of ths complx solutions. Thrfor cosφ cos(ωt+φ and (3.3 sinφ cos(ωt+φ-π/ - sinφ sin(ωt+φ (3.3 This looks complicatd, but th limits of high frquncy and low frquncy ar asy to rmmbr. At high frquncis ( φ, th capacitor is lik a short, and all th voltag shows up across th rsistor. At low frquncis ( φ π /, th capacitor is lik an opn circuit, and all th voltag shows up across th capacitor. If you considr th lading trms for th lmnts with th small voltags, you find that ( iω + ( ω ω ( i + ω + ( ω ( as ω iω as ω i ω (3.33 Thus, at high frquncy, th voltag across th capacitor is th intgral of th input voltag, whil at low frquncy th voltag across th rsistor is th drivativ of th input voltag. This says that as long as all th important frquncis ar high, th capacitor will intgrat th input voltag. If all th important frquncis ar small, th rsistor will diffrntiat th voltag. If thr ar intrmdiat frquncis, or a mixtur of som high - 3 -

8 and som low frquncis, th rsult will not b so simpl but it can b dtrmind from th voltag dividr algbra using complx notation. W finish by noting that th voltag on th capacitor is always -π/ out of phas with th voltag on th rsistor. III. Inductors An inductors is a coil of wir, or solnoid, which can b usd to stor nrgy in th magntic fild that it gnrats (s figur 3.3 on th right. It is mathmatically similar to a capacitor, but has xactly th opposit bhavior: it bhavs as a short circuit for low frquncis and as an opn circuit for high frquncis (i.. it passs low frquncy signals and blocks high frquncy signals. Th nrgy stord in th fild of an inductor with inductanc is givn by th following formula: B I E (3.34 Th SI unit of inductanc is th Hnry (H. ommrcially availabl inductors hav inductancs that rang from nh to mh. Small millimtr-siz and cntimtr siz solnoids typically hav inductancs in th rang of µh, whil magntic fild coils can hav a inductancs in th mh rang, and can somtims hav inductancs of up to svral H. Most lctronics componnts hav small parasitic inductancs du to thir lads and dsign (for xampl, wir-wound powr rsistors. Figur 3.3: An inductor consists of a coild wir, also calld a solnoid. Th dashd arrow B, rprsnt th magntic fild gnratd by th currnt in th inductor. In an lctric circuit, a voltag, or lctromotiv potntial, is gnratd across th trminals of th inductor whn th currnt changs du to Faraday s law. Th voltag drop is givn by th following simpl xprssion: di (3.35 dt From this quation, w s that th inductor oprats xactly opposit to a capacitor: an inductor diffrntiats th currnt and intgrats th voltag. Th circuit W can analyz th circuit in much th sam way that w drivd th opration of th circuit. W start by applying Kirchhoff s loop law to th circuit in figur 3.4 blow, and w find that - 4 -

9 di IN I +. (3.36 dt If w apply a constant voltag th solution can b calculatd using th tchniqus dvlopd for th circuit and w calculat that I( t I xp( t. (3.37 Th circuit approachs th stady stat currnt I IN / with a tim constant of /. impdanc Instad of solving th diffrntial quation for th circuit with a sinusoidal applid input voltag such as that givn by quations 4 and 5, as w did with th circuit, w will just assum that th currnt has th form I ( t I xp( iω t + φ (3.38 W plug this ansatz solution back into th diffrntial quation of quation 3 and find that IN I( t + iωi, (3.39 from which w dduc that th inductor bhavs as a rsistor with frquncy dpndnt imaginary rsistanc. Th impdanc of an inductor is thrfor Z iω (3.4 Just as with th circuit, w can apply Ohm s law to th circuit to calculat th total currnt. Sinc and ar in sris, w obtain iω iωt iωt in iωt i( ωt φ I ( t cos( φ (3.4 Ztotal + Z + iω + ω whr th phas is givn by tan( φ ω. W calculat th voltag drop across th rsistor using th xprssion for th currnt and find that i( ωt φ I( t cos( φ (3.4 Th voltag drop across th inductor is calculatd th sam way, and w find i( ωt φ i( ωt φ i( ωt φ + π / iωi ( t iω cos( φ i sin( φ cos( φ (3.43 IN OUT Figur 3.4: A simpl circuit

10 If vrything is corrctly calculatd thn th sum of th voltag drops across th two lmnts should b qual to th input voltag. t s try it: + ( i( ω t φ i φ i( ω i t φ i ω cos( + sin( φ t φ (3.44 You gt th actual wavforms by taking th ral parts of ths complx solutions. Thrfor cosφ cos(ωt-φ and (3.45 sinφ cos(ωt-φ+π/ sinφ sin(ωt-φ (3.46 This looks complicatd, but th limits of high frquncy and low frquncy ar asy to rmmbr. At high frquncis ( φ π, th inductor is lik an opn circuit, and all th voltag shows up across th inductor. At low frquncis ( φ, th inductor is lik a short circuit or just a plain wir, and all th voltag shows up across th rsistor. It should also b pointd out that th voltag on th inductor is always +π/ out of phas with th voltag on th rsistor. I. Transformrs Transformrs ar an ingnious combination of two inductors. Thy ar usd to transfr powr btwn two circuits by magntic coupling. Th transformr changs an input voltag, without affcting th signal shap, similar to th voltag dividr of last wk. Howvr it has svral important diffrncs: a It can incras as wll as dcras a signal s amplitud (i.. A voltag. b It rquirs a tim-varying (A input to work. c It is much hardr to fabricat. d It usually dos not work wll for vry fast signals (sinc inductors block high frquncis. Transformrs ar commonly usd as a major componnt in a D powr supplis sinc thy can convrt a A wall voltag into a smallr voltag that is closr to th dsird D voltag (.g. 5 or ±5. Th schmatic symbol for a transformr is shown in figur 3.5, abov. Transformrs ar passiv dvics that simultanously chang th voltag and currnt of a circuit. Thy hav (at last four trminals: two inputs (calld th primary and two outputs (calld th scondary. Thr is no ral diffrnc btwn th input and output for a transformr, you could simply flip it around and us th scondary as th input and th primary as th output. Howvr, for th sak of clarity, w will always assum that you us th primary for input and th scondary for output. IN OUT Figur 3.5: Th schmatic symbol for a transformr

11 Th coupling btwn th input and output is don magntically. This allows transformrs to hav a numbr of intrsting bnfits including: Thr is no D connction btwn input and output, so transformrs ar oftn usd to isolat on circuit from anothr. f Transformrs only work for tim varying signals, whn th inductiv coupling btwn th coils is gratr than th rsistiv losss. Sinc thy hav no xtrnal powr th output powr can not b gratr than th input powr P P IP S IS. (3.47 Usually, w will assum quality but thr ar small rsistancs (and hnc rsistiv losss in th coils and a poorly or chaply dsignd transformr many not hav th input and output sufficintly strongly coupld to ach othr. Dpnding on th dvic and th signal th output powr may wll b lss than th input powr. Transformrs ar most commonly usd to chang lin voltag ( MS at 6 Hz into a mor convnint voltag. High powr transmission lins us transformrs to incras th voltag and dcras th currnt. This rducs I powr losss in th transmission wirs. For our circuits w will us a transformr that rducs th voltag and incrass th currnt. Transformrs ar charactrizd by th ratio of th numbr of turns on th input and output windings. Th magntic coupling in an idal transformr will insur that th numbr of turns tims th currnt flowing is th sam for th input and output: N I I N S P P P NSI S I P N (3.48 S Sinc th voltag must chang in th opposit mannr to kp th input and output powr, th ratio of th voltags is th sam as th ratio of th turns: S P N N S (3.49 P Transformrs ar usually calld stp-up or stp-down according to whthr th output voltag incrass or dcrass. A transformr also transforms th impdanc of a circuit, sinc it changs th ratio of /I. Using our ruls abov, th ratio of output impdanc to input impdanc is th squar of th ratio of turns: ZS S I P N S ZP IS P NP (3.5 So, if you us a transformr as a stp-up transformr, it incrass th voltag and th impdanc at its output rlativ to its input. If you us a transformr as a stp-down transformr, it dcrass th voltag and th impdanc at its output

12 Dsign Exrciss Dsign Exrcis 3-: Using Kirchhoff s laws, driv a formula for th total capacitanc of two capacitors in paralll and a formula for th total capacitanc of two capacitors in sris. (Hint: prtnd that you ar working with an A signal of frquncy ω. Dsign Exrcis 3-: Using Kirchhoff s laws, driv a formula for th total inductanc of two inductors in paralll and a formula for th total inductanc of two inductors in sris. (Hint: prtnd that you ar working with an A signal of frquncy ω. Dsign Exrcis 3-3: alculat out as a function of in in th circuit of figur 3.6 on th right, using th formulas for Z, Z, and Z (do not us Mapl / Mathmatica / MATAB / Mathad for ths calculations and show all stps. in is a prfct A voltag signal with a frquncy of ω. IN OUT Plot th magnitud and phas of out as a function of ω for kω, µf, and µh. What happns to th magnitud and th phas of out at ω? (Mapl / Mathmatica / MATAB / Mathad ar prmittd for th plots. Figur 3.6: An filtr circuit. Dsign Exrcis 3-4: alculat out as a function of in in th circuit dpictd on th right, using th formulas for Z, Z, and Z (do not us Mapl / Mathmatica / MATAB / Mathad for ths calculations and show all stps. in is a prfct A voltag signal with a frquncy of ω. IN OUT Plot th magnitud and phas of out as a function of ω for kω, µf, and µh. What happns to th magnitud and th phas of out at ω? (Mapl / Mathmatica / MATAB / Mathad ar prmittd for th plots. Figur 3.7: Anothr filtr circuit

13 ab 3: A signals, omplx Impdanc, and Phas Sction : Introduction to transformrs In this sction, w us a transformr to chang th impdanc of an A signal. a. Masur th input impdanc of a spakr. Masur th output impdanc of a signal gnrator with a.5 amplitud sinusoid output of khz. mmbr you ar using A signals. How do you masur currnt with an oscilloscop? What dos an A currnt rading from a DM man in trms of th wavform? hck this with th oscilloscop. If masurmnts do not match call instructor for discussion bfor you procd any furthr. b. Masur th signal gnrator output without any load. Thn connct th signal gnrator to a spakr and masur th signal amplitud. Th voltag drops so much bcaus of th impdanc mismatch. Masur th powr into th spakr. c. Us a transformr to dcras th output voltag, whil incrasing th output currnt into th spakr. Masur out and I out of th signal gnrator, and in, and I in for th spakr. How wll dos th transformr transmit powr? Dos out / in I in /I out? Estimat th ratio of primary turns to scondary turns? d. Masur th output impdanc of th signal gnrator plus transformr circuit. Dos th masurd valu agr with what you xpct thortically? What should b th ratio of th transformr for th idal impdanc matching of th signal gnrator to th spakr? Sction : Th circuit In this sction, w tak a first look at th classic circuit and th concpt of phas.. Gt two capacitors and masur thir individual capacitancs. Masur th total capacitanc with a capacitanc mtr whn thy ar in sris, and whn thy ar in paralll. Do you gt good IN agrmnt with what you xpct? f. onstruct th circuit to th right, with componnt rangs - kω and.-. µf. St th function OUT gnrator at approximatly ω./ with a squar wav and dscrib what you s. Masur th tim constant of th xponntial and us it to dtrmin th capacitanc of ( should b dtrmind with a multimtr. Figur 3.8: An filtr g. (Sam st-up St th function gnrator to sinusoidal circuit. output at ω/ and masur th magnitud of in and out. Do you gt what you xpct? Masur th phas of out with rspct to in and mak a issajou plot of out and in

14 - 3 -

### Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

### New Basis Functions. Section 8. Complex Fourier Series

Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

### Lecture 3: Diffusion: Fick s first law

Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

### The Matrix Exponential

Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

### LABORATORY 1 IDENTIFICATION OF CIRCUIT IN A BLACK-BOX

LABOATOY IDENTIFICATION OF CICUIT IN A BLACK-BOX OBJECTIES. To idntify th configuration of an lctrical circuit nclosd in a two-trminal black box.. To dtrmin th valus of ach componnt in th black box circuit.

### 3. Yes. You can put 20 of the 6-V lights in series, or you can put several of the 6-V lights in series with a large resistance.

CHAPTE 6: DC Circuits sponss to Qustions. Evn though th bird s ft ar at high potntial with rspct to th ground, thr is vry littl potntial diffrnc btwn thm, bcaus thy ar clos togthr on th wir. Th rsistanc

### Question 3: How do you find the relative extrema of a function?

ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

### 14.3 Area Between Curves

14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and

### Exponential Growth and Decay; Modeling Data

Exponntial Growth and Dcay; Modling Data In this sction, w will study som of th applications of xponntial and logarithmic functions. Logarithms wr invntd by John Napir. Originally, thy wr usd to liminat

### The Normal Distribution: A derivation from basic principles

Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

### Section 7.4: Exponential Growth and Decay

1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

### Lecture 20: Emitter Follower and Differential Amplifiers

Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.

### Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc

### e = C / electron Q = Ne

Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and

### II. Equipment. Magnetic compass, magnetic dip compass, Helmholtz coils, HP 6212 A power supply, Keithley model 169 multimeter

Magntic fild of th arth I. Objctiv: Masur th magntic fild of th arth II. Equipmnt. Magntic compass, magntic dip compass, Hlmholtz s, HP 6212 A powr supply, Kithly modl 169 multimtr III Introduction. IIIa.

### QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

### by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

### Simulated Radioactive Decay Using Dice Nuclei

Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd,

### CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:

### Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

### 5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

### [ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)

MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris

### Principles of Humidity Dalton s law

Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

### Factorials! Stirling s formula

Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

### Genetic Drift and Gene Flow Illustration

Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.

### Ground Fault Current Distribution on Overhead Transmission Lines

FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 19, April 2006, 71-84 Ground Fault Currnt Distribution on Ovrhad Transmission Lins Maria Vintan and Adrian Buta Abstract: Whn a ground fault occurs on

### 14.02 Principles of Macroeconomics Problem Set 4 Solutions Fall 2004

art I. Tru/Fals/Uncrtain Justify your answr with a short argumnt. 4.02 rincipls of Macroconomics roblm St 4 Solutions Fall 2004. High unmploymnt implis that th labor markt is sclrotic. Uncrtain. Th unmploymnt

### Foreign Exchange Markets and Exchange Rates

Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

### ME 612 Metal Forming and Theory of Plasticity. 6. Strain

Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

### Noble gas configuration. Atoms of other elements seek to attain a noble gas electron configuration. Electron configuration of ions

Valnc lctron configuration dtrmins th charactristics of lmnts in a group Nobl gas configuration Th nobl gass (last column in th priodic tabl) ar charactrizd by compltly filld s and p orbitals this is a

### Gas Radiation. MEL 725 Power-Plant Steam Generators (3-0-0) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi

Gas Radiation ME 725 Powr-Plant Stam Gnrators (3-0-0) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbing-mitting mdia Whn a mdium is transparnt to radiation,

### Current and Resistance

Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...6-6.1.1 Currnt Dnsity...6-6. Ohm s Law...6-4 6.3 Elctrical Enrgy and Powr...6-7 6.4 Summary...6-8 6.5 Solvd Problms...6-9 6.5.1 Rsistivity of a Cabl...6-9

### Chi-Square. Hypothesis: There is an equal chance of flipping heads or tails on a coin. Coin A. Expected (e) (o e) (o e) 2 (o e) 2 e

Why? Chi-Squar How do you know if your data is th rsult of random chanc or nvironmntal factors? Biologists and othr scintists us rlationships thy hav discovrd in th lab to prdict vnts that might happn

### Solutions to Homework 8 chem 344 Sp 2014

1. Solutions to Homwork 8 chm 44 Sp 14 .. 4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt

### http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

Chaptr 9 Nuclar Radiation 9. Natural Radioactivity Radioactiv Isotops A radioactiv isotop has an unstabl nuclus. mits radiation to bcom mor stabl. can b on or mor of th isotops of an lmnt 2 Nuclar Radiation

### TN Calculating Radiated Power and Field Strength for Conducted Power Measurements

Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts TN100.04 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts Copyright Smtch 007 1 of 9 www.smtch.com Calculating adiatd

### 10/06/08 1. Aside: The following is an on-line analytical system that portrays the thermodynamic properties of water vapor and many other gases.

10/06/08 1 5. Th watr-air htrognous systm Asid: Th following is an on-lin analytical systm that portrays th thrmodynamic proprtis of watr vapor and many othr gass. http://wbbook.nist.gov/chmistry/fluid/

### Renewable Energy Sources. Solar Cells SJSU-E10 S John Athanasiou

Rnwabl Enrgy Sourcs. Solar Clls SJSU-E10 S-2008 John Athanasiou 1 Rnwabl Enrgy Sourcs Rnwabl: Thy can last indfinitly 1. Wind Turbin: Convrting th wind nrgy into lctricity Wind, Propllr, Elctric Gnrator,

### (Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

### AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

### Modelling and Solving Two-Step Equations: ax + b = c

Modlling and Solving To-Stp Equations: a + b c Focus on Aftr this lsson, you ill b abl to φ modl problms φ ith to-stp linar quations solv to-stp linar quations and sho ho you ord out th ansr Cali borrod

### L13: Spectrum estimation nonparametric and parametric

L13: Spctrum stimation nonparamtric and paramtric Lnnart Svnsson Dpartmnt of Signals and Systms Chalmrs Univrsity of Tchnology Problm formulation Larning objctivs Aftr today s lctur you should b abl to

### Traffic Flow Analysis (2)

Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

### Statistical Machine Translation

Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm

### The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

### A Note on Approximating. the Normal Distribution Function

Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

### Poisson Distribution. Poisson Distribution Example

Poisson Distribution Can b usd to valuat th probability of an isolatd vnt occurring a spcific numbr of tims in a givn tim intrval,.g. # of faults, # of lightning stroks tim intrval Rquirmnts: -Evnts must

### Settlement of a Soil Layer. One-dimensional Consolidation and Oedometer Test. What is Consolidation?

On-dimnsional Consolidation and Odomtr Tst Lctur No. 12 Octobr 24, 22 Sttlmnt of a Soil Layr Th sttlmnt is dfind as th comprssion of a soil layr du to th loading applid at or nar its top surfac. Th total

### THE FUNDAMENTALS OF CURRENT SENSE TRANSFORMER DESIGN. Patrick A. Cattermole, Senior Applications Engineer MMG 10 Vansco Road, Toronto Ontario Canada

, Snior Alications nginr MMG 10 Vansco Road, Toronto Ontario Canada Abstract Th following ar will first rviw th basic rincils of oration of a Currnt Sns Transformr and thn follow a simlifid dsign rocdur.

### A Derivation of Bill James Pythagorean Won-Loss Formula

A Drivation of Bill Jams Pythagoran Won-Loss Formula Ths nots wr compild by John Paul Cook from a papr by Dr. Stphn J. Millr, an Assistant Profssor of Mathmatics at Williams Collg, for a talk givn to th

### 7 Timetable test 1 The Combing Chart

7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing

### Sigmoid Functions and Their Usage in Artificial Neural Networks

Sigmoid Functions and Thir Usag in Artificial Nural Ntworks Taskin Kocak School of Elctrical Enginring and Computr Scinc Applications of Calculus II: Invrs Functions Eampl problm Calculus Topic: Invrs

### 5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

### Introduction to Finite Element Modeling

Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation

### Basis risk. When speaking about forward or futures contracts, basis risk is the market

Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also

### Chapter 5 Capacitance and Dielectrics

5 5 5- Chaptr 5 Capacitanc and Dilctrics 5.1 Introduction... 5-3 5. Calculation of Capacitanc... 5-4 Exampl 5.1: Paralll-Plat Capacitor... 5-4 Exampl 5.: Cylindrical Capacitor... 5-6 Exampl 5.3: Sphrical

### I. INTRODUCTION. Figure 1, The Input Display II. DESIGN PROCEDURE

Ballast Dsign Softwar Ptr Grn, Snior ighting Systms Enginr, Intrnational Rctifir, ighting Group, 101S Spulvda Boulvard, El Sgundo, CA, 9045-438 as prsntd at PCIM Europ 0 Abstract: W hav dvlopd a Windows

### Physics 106 Lecture 12. Oscillations II. Recap: SHM using phasors (uniform circular motion) music structural and mechanical engineering waves

Physics 6 Lctur Oscillations II SJ 7 th Ed.: Chap 5.4, Rad only 5.6 & 5.7 Rcap: SHM using phasors (unifor circular otion) Physical pndulu xapl apd haronic oscillations Forcd oscillations and rsonanc. Rsonanc

### Incomplete 2-Port Vector Network Analyzer Calibration Methods

Incomplt -Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar

### SAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII

SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 6-7 Tim allowd : 3 hours Maimum Marks : Gnral Instructions: (i) All qustions ar compulsor. (ii) This qustion papr contains 9 qustions. (iii) Qustion - 4

### Deer: Predation or Starvation

: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,

### Free ACA SOLUTION (IRS 1094&1095 Reporting)

Fr ACA SOLUTION (IRS 1094&1095 Rporting) Th Insuranc Exchang (301) 279-1062 ACA Srvics Transmit IRS Form 1094 -C for mployrs Print & mail IRS Form 1095-C to mploys HR Assist 360 will gnrat th 1095 s for

### Financial Mathematics

Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution

### CHAPTER EIGHT. Making use of the limit formula developed in Chapter 1, it can be shown that

CONTINUOUS COMPOUNDING CHAPTER EIGHT In prvious stions, th as of m ompoundings pr yar was disussd In th as of ontinuous ompounding, m is allowd to tnd to infinity Th mathmatial rlationships ar fairly asy

### Parallel and Distributed Programming. Performance Metrics

Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

### CIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method)

6. CIRCUITS AND ELECTRONICS Basic Circuit Analysis Mthod (KVL and KCL mthod) Cit as: Anant Agarwal and Jffry Lang, cours matrials for 6. Circuits and Elctronics, Spring 7. MIT 6. Fall Lctur Rviw Lumpd

### Optical Modulation Amplitude (OMA) and Extinction Ratio

Application Not: HFAN-.. Rv; 4/8 Optical Modulation Amplitud (OMA) and Extinction Ratio AVAILABLE Optical Modulation Amplitud (OMA) and Extinction Ratio Introduction Th optical modulation amplitud (OMA)

### Bipolar Junction Transistor

Bipolar Juntion Transistor Bipolar Juntion Transistor: A bipolar juntion transistor (BJT) is widly usd in disrt iruits as wll as in IC dsign, both analog and digital. Its main appliations ar in amplifiation

### System earthing. Industrial Electrical Engineering and Automation. Anna Guldbrand. Dept. of Industrial Electrical Engineering and Automation

CODN:UTDX/(T-76)/-/(6) ndustrial lctrical nginring and Automation Systm arthing Anna Guldbrand Dpt. o ndustrial lctrical nginring and Automation und Univrsity ntroduction... Solidly arthd systms... solatd

### Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl

### Long run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange

Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity

### Differential Equations (MTH401) Lecture That a non-homogeneous linear differential equation of order n is an equation of the form n

Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind Coffiints-Surosition Aroah Rall. That a non-homognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n +

### Far Field Estimations and Simulation Model Creation from Cable Bundle Scans

Far Fild Estimations and Simulation Modl Cration from Cabl Bundl Scans D. Rinas, S. Nidzwidz, S. Fri Dortmund Univrsity of Tchnology Dortmund, Grmany dnis.rinas@tu-dortmund.d stphan.fri@tu-dortmund.d Abstract

### The Fourier Transform

Th Fourir Transfor Larning outcos Us th Discrt Fourir Transfor to prfor frquncy analysis on a discrt (digital) signal Eplain th significanc of th Fast Fourir Transfor algorith; Eplain why windowing is

### Improving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost

Economy Transdisciplinarity Cognition www.ugb.ro/tc Vol. 16, Issu 1/2013 50-54 Improving Managrial Accounting and Calculation of Labor Costs in th Contxt of Using Standard Cost Lucian OCNEANU, Constantin

### AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com

AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos

### C H A P T E R 1 Writing Reports with SAS

C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd

### Adverse Selection and Moral Hazard in a Model With 2 States of the World

Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

### 5. Design of FIR Filters. We want to address in this Chapter the issue of approximating a digital filter with a desired frequency response

5. Dsign of FIR Filtrs Rfrnc: Sctions 7.2-7.4 of Txt W want to addrss in this Chaptr th issu of approximating a digital filtr with a dsird frquncy rspons Hd ( ω ) using filtrs with finit duration. 5.1

### Module 7: Discrete State Space Models Lecture Note 3

Modul 7: Discrt Stat Spac Modls Lctur Not 3 1 Charactristic Equation, ignvalus and ign vctors For a discrt stat spac modl, th charactristic quation is dfind as zi A 0 Th roots of th charactristic quation

### Notes for Geometry Conic Sections

Nots for Gomtry Conic Sctions Th nots is takn from Gomtry, by David A. Brannan, Matthw F. Espln and Jrmy J. Gray, 2nd dition 1 Conic Sctions A conic sction is dfind as th curv of intrsction of a doubl

### Inference by Variable Elimination

Chaptr 5 Infrnc by Variabl Elimination Our purpos in this chaptr is to prsnt on of th simplst mthods for gnral infrnc in Baysian ntworks, known as th mthod of Variabl Elimination. 5.1 Introduction Considr

### SPECIAL VOWEL SOUNDS

SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)

### SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* Rostov-on-Don. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs

### The first test of a transistor

Bipolar Transistors I Pag 1 Bipolar Transistors I Th first tst of a transistor This la uss an NPN transistor, th. Lik any NPN transistor, it onsists of two PN juntions, as shown in Fig. 1 () a) ) ) Figur

### Theoretical approach to algorithm for metrological comparison of two photothermal methods for measuring of the properties of materials

Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN 187 8196 Thortical approach to algorithm for mtrological comparison of two photothrmal mthods for masuring of th proprtis

### EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut

### II.6. THE DETERMINATION OF THE RYDBERG CONSTANT

II.6. TE DETERMINATION OF TE RYDBERG CONSTANT. Work purpos Th dtrination of th constant involvd in th spctral sris of th hydrognoid atos / ions.. Thory Th atos of ach lnt it, whn thy ar xcitd (for instanc

### Category 7: Employee Commuting

7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

### Chapter 19: Permanent Magnet DC Motor Characteristics

Chaptr 19: Prmannt Magnt DC Motor Charactristics 19.1: ntroduction Dirct currnt (DC) motors compris on of th most common typs of actuator dsignd into lctromchanical systms. hy ar a vry straightforward

### Vibrational Spectroscopy

Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) + + + + 6 3

### SPECIFIC HEAT AND HEAT OF FUSION

PURPOSE This laboratory consists of to sparat xprimnts. Th purpos of th first xprimnt is to masur th spcific hat of to solids, coppr and rock, ith a tchniqu knon as th mthod of mixturs. Th purpos of th

### intro Imagine that someone asked you to describe church using only the bible. What would you say to them?

intro Imagin that somon askd you to dscrib church using only th bibl. What would you say to thm? So many of th things w'v mad church to b arn't ssntial in scriptur. W'r on a journy of r-imagining what

### ! derived in Part (a): HW12 Solutions (due Tues, Apr 28)

HW1 Solutions (u Tus, Apr 8) 1. T&M 8.P.4 A unior 1.-T agntic il is in th +z irction. A conucting ro o lngth 15 c lis paralll to th y axis an oscillats in th x irction with isplacnt givn by x (. c) cos

### CHAPTER 88 THE BINOMIAL AND POISSON DISTRIBUTIONS

CHAPTER 88 THE BINOMIAL AND POISSON DISTRIBUTIONS EXERCISE Pag 97 1. Concrt blocks ar tstd and it is found that on avrag 7% fail to mt th rquird spcification. For a batch of nin blocks dtrmin th probabilitis

### Flow Switch Diaphragm Style Flow Switch IFW5 10 N Diaphragm style flow switch. Thread type. Model Body size Set flow rate

Flo Sitch Diaphragm Styl Flo Sitch Sris IFW5 [Option] Th flo sitch, sris IFW is usd for dtction and confirmation of th flo as a rlaying dvic for th gnral atr applications in som various uipmnt such as

### Discrete Time Signals and Fourier series

Discrt Tim Signals and Fourir sris In prvious two chaptrs w discussd th Fourir sris for continuous-tim signals. W showd that th sris is in fact an altrnat rprsntation of th signal. This rprsntation can