Resistive Network Analysis. The Node Voltage Method - 1

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Resistive Network Analysis. The Node Voltage Method - 1"

Transcription

1 esste Network Anlyss he nlyss of n electrcl network conssts of determnng ech of the unknown rnch currents nd node oltges. A numer of methods for network nlyss he een deeloped, sed on Ohm s Lw nd Krchoff s Lw - we wll look t seerl of these. Generl pproch: Defne ll relent rles n systemtc wy. Identfy the known nd unknown rles. Construct set of equtons reltng these rles. Sole the equtons, usng the smllest set of equtons needed to sole for ll the unknown rles. he Node Voltge Method - hs s the most generl method for nlysng crcuts. Bss: defne oltge t ech node s n ndependent rle. One node s selected s reference node (often, ut not necessrly, ground). Ech of the other node oltges s referenced to ths node. Ohm s Lw s ppled etween ny two djcent nodes to determne the oltge flowng n ech rnch. Ech rnch current s expressed n terms of one or more node oltges, so currents do not pper n the equtons. rnch current : Krchoff s Current Lw s ppled t ech of the n nodes. hs ges n- ndependent lner equtons for the n- ndependent node oltges (n th reference node).

2 he Node Voltge Method - Methodology: () Select reference node (usully ground). All other node oltges re referenced to ths node. () Defne the remnng n- node oltges s the ndependent rles. () Apply Krchoff s Current Lw t ech of the n- nodes, expressng ech current n terms of the djcent node oltges. (4) Sole the lner system of n- equtons n n- unknowns. Exmple of pplyng KCL t one node (): d 0 c d 0 c he Node Voltge Method - Exmple: Drecton of current flow chosen ssumes s s poste current. eference node c. wo other nodes: nd. Sole for these. Applcton of KCL t node : S 0 Applcton of KCL t node : 0 Applcton of KCL t node : 0 (not ndependent!) S node node s s s node c c 0

3 4 Exmple contnued: Defne currents s f(,): Susttute these n the two nodl equtons: Sole for, n terms of s,,, : c c 0 0 S S c s s 0 0 he Node Voltge Method - 4 he Node Voltge Method - 5 he node oltge method s esy to pply when current sources re present ecuse they re drectly ccounted for y KCL. Howeer, the node oltge method cn lso e ppled when oltge sources re present, nd ths cse s ctully smpler. Exmple: One node oltge s known: s. Cn sole for nd c n terms of s, nd the four resstnces. c s s 4 0 c : 0 : 4 c S c c S

4 he Mesh Current Method - hs s n effcent nd systemtc method for nlysng crcuts ecuse meshes re esly dentfed n crcut. Bss: defne current n ech mesh s n ndependent rle. he current flowng through resstor n specfed drecton defnes the polrty of the oltge cross the resstor. Current, defned s flowng from left to rght, determnes the polrty of the oltge cross. o od confuson, defne poste mesh currents s clockwse. he sum of the oltges round closed crcut must equl zero y Krchoff s Voltge Lw. Apply KVL to ech mesh to otn set of n equtons, one for ech mesh. Brnch currents nd oltges cn e dered from mesh currents. he Mesh Current Method - Methodology: () Defne ech mesh current consstently. Generlly defne mesh currents clockwse, for conenence. () Apply KVL round ech mesh, expressng ech oltge n terms of one or more mesh currents. () Sole the resultng lner system of equtons, wth mesh currents s the ndependent rles. Once the drecton of current flow s selected, KVL requres: 0 Note: n rtrry drecton cn e ssumed for ny current n crcut s long s sgns re ppled consstently. If the nswer for the current s negte then the chosen reference drecton s just opposte to the drecton of ctul current flow. mesh 5

5 he Mesh Current Method - Exmple: wo-mesh crcut wth two unknowns: nd. Consder ech mesh ndependently. Mesh Voltges nd round the mesh he een ssgned ccordng the clockwse drecton of mesh current. Note: mesh current s flowng through ( rnch current for ) ut s not the rnch current for. hs s -. So ( - ). Apply KVL for mesh : ( ) s 0 s 4 s 4 Consder mesh he Mesh Current Method - 4 Exmple contnued: Mesh Voltges,, nd 4 round the mesh he een ssgned ccordng the clockwse drecton of mesh current. Note: mesh current s the rnch current for nd 4, ut not for. hs s -. So ( - ). hs s the opposte to mesh ecuse the mesh currents flow through n opposng drectons. Apply KVL for mesh : Comne the equtons for the two meshes to get: ( ) ( Sole for mesh currents nd. Dere other currents nd oltges. ) 4 S ( ) 4 0 s 0 4 Consder mesh 4 6

6 he Mesh Current Method - 5 he mesh current method s ery effecte when ppled to crcuts tht contn only oltge sources. Howeer, t my lso e ppled to crcuts contnng oth oltge nd current sources - just e creful to dentfy the correct current n ech mesh. Exmple: Sole for unknown oltge x cross the current source. Presence of current source requres: S A KVL for mesh : 0 5 x 0 5 Ω Ω KVL for mesh : x 4 0 Sole to get: A s A x s 0 A 0 V 6 0 V x 4Ω Dependent or controlled sources Dependent Sources re sources whose current or oltge output s functon of some other oltge or current n crcut (unlke del sources whch re ndependent of ny other element n crcut) exmple: trnsstor mplfers s Source type oltge-controlled oltge source (VCVS) current-controlled oltge source (CCVS) oltge-controlled current source (VCCS) current-controlled current source (CCCS) eltonshp s A x s A x s A x I s A x s 7

7 Crcut Anlyss wth Dependent Sources - he node oltge nd mesh current methods cn lso e ppled to dependent sources, wth mnor modfcton. When dependent source s present n crcut, t cn e treted ntlly s n del source, wth the node or mesh equtons then wrtten down s preously descred. An ddtonl constrnt equton wll lso e needed, reltng the dependent source to one of the crcut oltges or currents. hs full set of equtons cn then soled. Note: once the constrnt equton hs een susttuted nto the ntl set of equtons, the numer of unknowns remns the sme. Crcut Anlyss wth Dependent Sources - Exmple: smplfed model of polr trnsstor mplfer wo oous nodes - pply node oltge nlyss. KCL t node : KCL t node : S β 0 S C Current cn e determned y consderng current dder: S S S Insert ths nto eqn to get two equtons tht cn e soled for nd. S s S node s node β c o 8

8 Prncple of Superposton - In lner crcut contnng N sources, ech rnch oltge nd current s the sum of N oltges nd currents, ech of whch my e computed y settng ll ut one source equl to zero nd solng the crcut contnng the sngle source. hs s conceptul d rther thn precse nlyss technque lke the mesh current nd node oltge methods. Useful n sulzng the ehour of crcut contnng multple sources. Apples to ny lner system. Whle t cn esly nd sometmes effectely e ppled to crcuts wth multple sources, other methods re often more effcent. Prncple of Superposton - Consder crcut wth two oltge sources connected n seres. B B B _ B B _ B (from zzon Fgure.6) Net current through sum of nddul source currents: B B B B B More formlly: B B hs crcut s equlent to the comnton of two crcuts, ech contnng sngle source. A short crcut s susttuted for the mssng source n ech sucrcut. A short crcut sees zero oltge cross tself, so ths s equlent to zerong the output of one of the oltge sources. B 9

9 Zerong Voltge nd Current Sources When pplyng the Prncple of Superposton oltge sources re zeroed y susttutng short crcuts current sources re zeroed y susttutng open crcuts (no current cn flow through n open crcut, so ths s equlent to zerong the output of the current source). In order to set oltge source equl to zero, we replce t wth short crcut. S _ S S A crcut he sme crcut wth S 0. In order to set current source equl to zero, we replce t wth n open crcut. S _ S S _ (from zzon Fgure.7) A crcut he sme crcut wth S 0 Equlent Crcuts Becuse Ohm s Lw nd Krchoff s Lws re lner, ny DC crcut cn e replced y smplfed equlent crcut. Applyng Ohm s Lw to comnton of resstors cn ge n equlent resstor. Applyng Krchoff s Lws to comnton of crcut elements cn ge n equlent crcut. It s useful to ew ech source or lod s two-termnl dece descred y n - cure. hs confgurton s clled one-port network. 0

10 Exmple of n Equlent Crcut S _ Source Lod (from zzon Fgure.9) EQ nd s EQ heenn s Equlent Crcut he heenn heorem As fr s lod s concerned, ny network composed of del oltge nd current sources, nd of lner resstors, my e represented y n equlent crcut consstng of n del oltge source,, n seres wth n equlent resstor,. Source Lod _ Lod (from zzon Fgure.)

11 he Norton heorem Norton s Equlent Crcut As fr s lod s concerned, ny network composed of del oltge nd current sources, nd of lner resstors, my e represented y n equlent crcut consstng of n del current source, N, n prllel wth n equlent resstor, N. Source Lod N N Lod (from zzon Fgure.) Determnng heenn & Norton Equlent esstnce he frst step n computng heenn or Norton equlent crcut s to fnd the equlent resstnce presented y the crcut t ts termnls. hs s done y settng ll sources n the crcut equl to zero nd clcultng the effecte resstnce etween the termnls. Voltge nd current sources n the crcut re set to zero usng the sme pproch s wth the Prncple of Superposton oltge sources re replced y short crcuts current sources re replced y open crcuts

12 Exmple of heenn esstnce - Exmple: Wht s the equlent resstnce tht lod L sees etween termnls nd? emoe lod resstnce from the crcut nd replce oltge source s y short crcut. Exmple of heenn esstnce - Equlent resstnce seen y the lod: nd re n prllel (connected etween sme two nodes) Let e totl resstnce etween termnls nd. hen (from zzon Fgure.4)

13 Exmple of heenn esstnce - Alternte method of determnng the heenn resstnce: Hypothetcl -A current source s connected etween nd. Voltge x cross - wll then equl (ecuse s A). he source current encounters s resstor n seres wth the prllel comnton of &. Wht s the totl resstnce the current S wll encounter n flowng round the crcut? x S S S (from zzon Fgure.5) Clcultng Equlent esstnce Methodology for clcultng equlent resstnce of one-port network (heenn or Norton): () emoe the lod. () Zero ll ndependent oltge nd current sources. () Compute the totl resstnce etween lod termnls, wth the lod remoed. hs resstnce s equlent to tht whch would e encountered y current source connected to the crcut n plce of the lod. Note tht ths procedure ges result tht s ndependent of the lod. hs s wht we wnt, ecuse once the equlent resstnce hs een clculted for source crcut, the equlent crcut s unchnged f dfferent lod s connected. 4

14 Determnng the heenn Voltge - he equlent heenn source oltge,, s equl to the open crcut oltge present t the lod termnls wth lod remoed..e., n order to clcute, t s suffcent to remoe the lod nd to compute the open crcut oltge t the one-port termnls he open crcut oltge, OC, nd the heenn oltge,, must e the sme f the heenn heorem s true (see elow). hs s ecuse n the crcut contnng nd, the oltge OC must equl ecuse no current flows through nd so the oltge cross s zero. From KVL: (0) OC OC One-port network OC Determnng the heenn Voltge - Methodology: () emoe the lod, leng the lod termnls open-crcuted. () Defne the open-crcut oltge OC cross the open lod termnls. () Apply ny preferred method (e.g., nodl nlyss) to sole for OC. (4) he heenn oltge s OC. 5

15 A Crcut nd Its heenn Equlent L L S _ L S _ L (from zzon Fgure.49) A crcut Its héenn equlent hs s the crcut we consdered erler, long wth ts heenn equlent. he two crcuts re equlent n tht the current drwn y the lod, L, s the sme n oth: L S ( ) L L Determnng the Norton Current - he Norton equlent current, N, s equl to the short crcut current tht would flow f the lod were replced y short crcut. Consder the one-port network nd ts Norton equlent crcut: Current SC flowng through the short crcut replcng the lod s the sme s the Norton current N ecuse ll of the source current n ths crcut must flow through the short crcut. One-port network SC N N SC (from zzon Fgure.57) 6

16 Determnng the Norton Current - Let s fnd current SC n ths crcut. S _ Short crcut SC (from zzon Fgure.58) replcng the lod Mesh Current Method: Let nd SC e the mesh currents n the crcut. wo mesh equtons (sole for SC ): Node Voltge Method: Nodl equton (sole for ): hus: N S ( ) ( S SC ) SC S 0 Determnng the Norton Current - Methodology: () eplce the lod wth short crcut. () Defne the short crcut current SC to e the Norton equlent current. () Apply ny preferred method (e.g., nodl nlyss) to sole for SC. (4) he Norton current s N SC. 7

17 Source rnsformtons - Source trnsformtons cn e useful for determnng equlent crcuts sometmes llow replcement of current sources wth oltge sources nd ce ers. he heenn nd Norton heorems stte tht ny one-port network cn e represented y oltge source n seres wth resstor, or y current source n prllel wth resstor, nd tht ether of these representtons s equlent to the orgnl crcut. One-port heenn Norton network _ N equlent equlent (from zzon Fgure.6) Source rnsformtons - Implcton: ny heenn equlent crcut cn e replced y Norton equlent crcut, f we use the reltonshp: N S _ SC S SC he sucrcut on the left of the dshed lne cn e replced y ts Norton equlent, s shown. Current SC cn e esly found ecuse the three resstors re n prllel wth the current source - use smple current dder. SC flows through, so: SC N S (from zzon Fgure.64) S 8

18 Source rnsformtons - Sucrcuts menle to source trnsformton: Node or _ S S or S S _ Node heenn é sucrcuts Norton sucrcuts (from zzon Fgure.65) Fndng heenn & Norton Equlents Expermentlly heenn nd Norton equlent crcuts cn e eluted expermentlly usng smple technques. Bsc de: heenn oltge s n open-crcut oltge Norton current s short-crcut current herefore possle to mke mesurements to determne these qunttes. Once nd N re known, the heenn resstnce of the crcut cn e found usng Need to mesure nd N. N 9

19 Fndng heenn & Norton Equlents Expermentlly Mesurement of open-crcut oltge nd short-crcut current for n rtrry network connected to ny lod: Unknown network An unknown network connected to lod Unknown network Network connected for mesurement of short-crcut current Unknown network OC Network connected for mesurement of open-crcut oltge SC A V rm Lod Note: Do not short crcut network y nsertng n mmeter n seres - ths could dmge the crcut or the mmeter! rm (from zzon Fgure.7) Fndng heenn & Norton Equlents Expermentlly hese mesurements requre cre ecuse the mesurng nstruments re nondel. In the presence of fnte meter resstnce r m, ths quntty must e tken nto ccount when determnng the open-crcut oltge nd the short-crcut current. Qunttes OC nd SC he quotton mrks to ndcte tht the mesured lues re ffected y r m nd re not the true lues. he true lues cn e clculted usng (proe ths to yourself!): N r m "SC" " OC" rm where N del Norton current, del heenn oltge, nd true heenn resstnce. 0

20 Fndng heenn & Norton Equlents Expermentlly 4 r m N "SC" " OC" rm ecll For n del mmeter, r m should pproch zero (short crcut). For n del oltmeter, r m should pproch nfnty (open crcut). So these two equtons cn e used to fnd the true heenn nd Norton equlent sources from n mperfect mesurement of the open-crcut oltge nd the short-crcut current, proded tht the nternl meter resstnce r m s known. In prctce, the nternl resstnce of oltmeters s hgh enough to e consdered nfnte relte to the equlent resstnce of most crcuts. Howeer, t s mpossle to uld n mmeter wth zero nternl resstnce: need to know r m to determne the short crcut current. Mxmum Power rnsfer - he heenn nd Norton models mply tht some of the power generted y source wll e dsspted y the nternl crcuts wthn the source. Gen ths unodle power loss, how much power cn e trnsferred to the lod from the source under the most del condtons? Or, wht s the lod resstnce tht wll sor mxmum power from the source? Mxmum Power rnsfer heorem

21 Mxmum Power rnsfer - Power trnsfer etween source nd lod: Prctcl source s represented y ts heenn equlent crcut Gen nd, wht lue of L wll llow for mxmum power trnsfer? (from zzon Fgure.7) Prctcl source L L _ L Power sored y the lod: Lod current: L Comne to get lod power: L P P L L L Lod L ( ) L L Source equlent Mxmum Power rnsfer - Dfferentte P L w.r.t. L to fnd fnd the lue of L tht mxmzes the lod power (ssumng constnt nd ). ( ) ( ) dpl L L L 0 4 dl ( L ) Hence: ( L ) L ( L ) 0 For whch the soluton s: L o trnsfer the mxmum power to lod, the equlent source nd lod resstnces must e mtched (.e., equl). hus, n order to trnsfer mxmum power to lod, gen fxed equlent source resstnce, the lod resstnce must mtch ths equlent source resstnce.

22 Voltge Source Lodng Effects Voltge source lodng: When prctcl oltge source s connected to lod, the current tht flows from the source to the lod wll cuse oltge drop cross the nternl source resstnce, nt. As result, the oltge seen y the lod wll e lower thn the open-crcut (heenn) oltge of the source. Lod oltge s then: L nt hus wnt smll nternl resstnce n prctcl oltge source. nt _ L Source Lod (from zzon Fgure.74) Current Source Lodng Effects Current source lodng: When prctcl current source s connected to lod, the nternl source resstnce wll drw some current, nt, wy from the lod. As result, the lod wll recee only prt of the short-crcut (Norton) current lle from the source. Lod current s then: L N hus wnt lrge nternl resstnce n prctcl current source. nt N L (from zzon Fgure.74) Source Lod

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

(6)(2) (-6)(-4) (-4)(6) + (-2)(-3) + (4)(3) + (2)(-3) = -12-24 + 24 + 6 + 12 6 = 0

(6)(2) (-6)(-4) (-4)(6) + (-2)(-3) + (4)(3) + (2)(-3) = -12-24 + 24 + 6 + 12 6 = 0 Chapter 3 Homework Soluton P3.-, 4, 6, 0, 3, 7, P3.3-, 4, 6, P3.4-, 3, 6, 9, P3.5- P3.6-, 4, 9, 4,, 3, 40 ---------------------------------------------------- P 3.- Determne the alues of, 4,, 3, and 6

More information

Vector Geometry for Computer Graphics

Vector Geometry for Computer Graphics Vector Geometry for Computer Grphcs Bo Getz Jnury, 7 Contents Prt I: Bsc Defntons Coordnte Systems... Ponts nd Vectors Mtrces nd Determnnts.. 4 Prt II: Opertons Vector ddton nd sclr multplcton... 5 The

More information

Basics of Counting. A note on combinations. Recap. 22C:19, Chapter 6.5, 6.7 Hantao Zhang

Basics of Counting. A note on combinations. Recap. 22C:19, Chapter 6.5, 6.7 Hantao Zhang Bscs of Countng 22C:9, Chpter 6.5, 6.7 Hnto Zhng A note on comntons An lterntve (nd more common) wy to denote n r-comnton: n n C ( n, r) r I ll use C(n,r) whenever possle, s t s eser to wrte n PowerPont

More information

LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit

LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE

More information

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS? WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they

More information

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Version 001 CIRCUITS holland (1290) 1

Version 001 CIRCUITS holland (1290) 1 Version CRCUTS hollnd (9) This print-out should hve questions Multiple-choice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire

More information

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are: polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Chapter 3 - Vectors. Arithmetic operations involving vectors. A) Addition and subtraction. - Graphical method - Analytical method Vector components

Chapter 3 - Vectors. Arithmetic operations involving vectors. A) Addition and subtraction. - Graphical method - Analytical method Vector components Chpter 3 - Vectors I. Defnton II. Arthmetc opertons nvolvng vectors A) Addton nd sutrcton - Grphcl method - Anltcl method Vector components B) Multplcton Revew of ngle reference sstem 90º

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

Chapter Solution of Cubic Equations

Chapter Solution of Cubic Equations Chpter. Soluton of Cuc Equtons After redng ths chpter, ou should e le to:. fnd the ect soluton of generl cuc equton. Ho to Fnd the Ect Soluton of Generl Cuc Equton In ths chpter, e re gong to fnd the ect

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Answer, Key Homework 10 David McIntyre 1

Answer, Key Homework 10 David McIntyre 1 Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your

More information

Systematic Circuit Analysis (T&R Chap 3)

Systematic Circuit Analysis (T&R Chap 3) Systematc Crcut Analyss TR Chap ) Nodeoltage analyss Usng the oltages of the each node relate to a ground node, wrte down a set of consstent lnear equatons for these oltages Sole ths set of equatons usng,

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

Boolean Algebra. ECE 152A Winter 2012

Boolean Algebra. ECE 152A Winter 2012 Boolen Algebr ECE 52A Wnter 22 Redng Assgnent Brown nd Vrnesc 2 Introducton to Logc Crcuts 2.5 Boolen Algebr 2.5. The Venn Dgr 2.5.2 Notton nd Ternology 2.5.3 Precedence of Opertons 2.6 Synthess Usng AND,

More information

Irregular Repeat Accumulate Codes 1

Irregular Repeat Accumulate Codes 1 Irregulr epet Accumulte Codes 1 Hu Jn, Amod Khndekr, nd obert McElece Deprtment of Electrcl Engneerng, Clforn Insttute of Technology Psden, CA 9115 USA E-ml: {hu, mod, rjm}@systems.cltech.edu Abstrct:

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

CIRCUIT ELEMENTS AND CIRCUIT ANALYSIS

CIRCUIT ELEMENTS AND CIRCUIT ANALYSIS EECS 4 SPING 00 Lecture 9 Copyrght egents of Unversty of Calforna CICUIT ELEMENTS AND CICUIT ANALYSIS Lecture 5 revew: Termnology: Nodes and branches Introduce the mplct reference (common) node defnes

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

The Full-Wave Rectifier

The Full-Wave Rectifier 9/3/2005 The Full Wae ectfer.doc /0 The Full-Wae ectfer Consder the followng juncton dode crcut: s (t) Power Lne s (t) 2 Note that we are usng a transformer n ths crcut. The job of ths transformer s to

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Bipolar Junction Transistor (BJT)

Bipolar Junction Transistor (BJT) polar Juncton Transstor (JT) Lecture notes: Sec. 3 Sedra & Smth (6 th Ed): Sec. 6.1-6.4* Sedra & Smth (5 th Ed): Sec. 5.1-5.4* * Includes detals of JT dece operaton whch s not coered n ths course F. Najmabad,

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Fuzzy Clustering for TV Program Classification

Fuzzy Clustering for TV Program Classification Fuzzy Clusterng for TV rogrm Clssfcton Yu Zhwen Northwestern olytechncl Unversty X n,.r.chn, 7007 yuzhwen77@yhoo.com.cn Gu Jnhu Northwestern olytechncl Unversty X n,.r.chn, 7007 guh@nwpu.edu.cn Zhou Xngshe

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw

Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw Newton-Rphson Method o Solvng Nonlner Equton Autr Kw Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility

More information

Circuit Analysis. Lesson #2. BME 372 Electronics I J.Schesser

Circuit Analysis. Lesson #2. BME 372 Electronics I J.Schesser Ciruit Anlysis Lesson # BME 37 Eletronis J.Shesser 67 oltge Division The voltge ross impednes in series divides in proportion to the impednes. b n b b b b ( ; KL Ohm's Lw BME 37 Eletronis J.Shesser i i

More information

3. Bipolar Junction Transistor (BJT)

3. Bipolar Junction Transistor (BJT) 3. polar Juncton Transstor (JT) Lecture notes: Sec. 3 Sedra & Smth (6 th Ed): Sec. 6.1-6.4* Sedra & Smth (5 th Ed): Sec. 5.1-5.4* * Includes detals of JT dece operaton whch s not coered n ths course EE

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Lesson 28 Psychrometric Processes

Lesson 28 Psychrometric Processes 1 Lesson 28 Psychrometrc Processes Verson 1 ME, IIT Khrgpur 1 2 The specfc objectves of ths lecture re to: 1. Introducton to psychrometrc processes nd ther representton (Secton 28.1) 2. Importnt psychrometrc

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

Optimal Pricing Scheme for Information Services

Optimal Pricing Scheme for Information Services Optml rcng Scheme for Informton Servces Shn-y Wu Opertons nd Informton Mngement The Whrton School Unversty of ennsylvn E-ml: shnwu@whrton.upenn.edu e-yu (Shron) Chen Grdute School of Industrl Admnstrton

More information

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

More information

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information

Incorporating Negative Values in AHP Using Rule- Based Scoring Methodology for Ranking of Sustainable Chemical Process Design Options

Incorporating Negative Values in AHP Using Rule- Based Scoring Methodology for Ranking of Sustainable Chemical Process Design Options 20 th Europen ymposum on Computer Aded Process Engneerng ECAPE20. Perucc nd G. Buzz Ferrrs (Edtors) 2010 Elsever B.V. All rghts reserved. Incorportng Negtve Vlues n AHP Usng Rule- Bsed corng Methodology

More information

EN3: Introduction to Engineering. Teach Yourself Vectors. 1. Definition. Problems

EN3: Introduction to Engineering. Teach Yourself Vectors. 1. Definition. Problems EN3: Introducton to Engneerng Tech Yourself Vectors Dvson of Engneerng Brown Unversty. Defnton vector s mthemtcl obect tht hs mgntude nd drecton, nd stsfes the lws of vector ddton. Vectors re used to represent

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

Three-Phase Induction Generator Feeding a Single-Phase Electrical Distribution System - Time Domain Mathematical Model

Three-Phase Induction Generator Feeding a Single-Phase Electrical Distribution System - Time Domain Mathematical Model Three-Phse Induton Genertor Feedng Sngle-Phse Eletrl Dstruton System - Tme Domn Mthemtl Model R.G. de Mendonç, MS. CEFET- GO Jtí Deentrlzed Unty Eletrotehnl Coordnton Jtí GO Brzl 763 L. Mrtns Neto, Dr.

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

Applications of the Laplace Transform

Applications of the Laplace Transform EE 4G Noe: hper 6 nrucor: heung Applcon of he plce Trnform Applcon n rcu Anly. evew of eve Nework Elemen Superpoon Pge 6 PDF reed wh dekpdf PDF Wrer Trl :: hp://www.docudek.com EE 4G Noe: hper 6 nrucor:

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones. Physics 3P41 Chris Wiebe Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

More information

Formal Languages and Automata Exam

Formal Languages and Automata Exam Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

More information

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value 8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at

More information

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Positive Integral Operators With Analytic Kernels

Positive Integral Operators With Analytic Kernels Çnky Ünverte Fen-Edeyt Fkülte, Journl of Art nd Scence Sy : 6 / Arl k 006 Potve ntegrl Opertor Wth Anlytc Kernel Cn Murt D KMEN Atrct n th pper we contruct exmple of potve defnte ntegrl kernel whch re

More information

ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING

ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING Unversty of Wsconsn Mlwukee Pper No. 09-1 Ntonl Center for Freght & Infrstructure Reserch &

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions MSSCHUSES INSIUE OF ECHNOLOGY Deprtment of hysics 8.0 W02D3_0 Group roblem: ulleys nd Ropes Constrint Conditions Consider the rrngement of pulleys nd blocks shown in the figure. he pulleys re ssumed mssless

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

CANKAYA UNIVERSITY FACULTY OF ENGINEERING MECHANICAL ENGINEERING DEPARTMENT ME 212 THERMODYNAMICS II HW# 11 SOLUTIONS

CANKAYA UNIVERSITY FACULTY OF ENGINEERING MECHANICAL ENGINEERING DEPARTMENT ME 212 THERMODYNAMICS II HW# 11 SOLUTIONS CNKY UNIVESIY FCULY OF ENGINEEING MECHNICL ENGINEEING DEMEN ME HEMODYNMICS II HW# SOLUIONS Deterne te ulton reure of wter or t -60 0 C ung dt lle n te te tle. Soluton Ste tle do not ge turton reure for

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

Multiple stage amplifiers

Multiple stage amplifiers Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

Chapter 12 Inductors and AC Circuits

Chapter 12 Inductors and AC Circuits hapter Inductors and A rcuts awrence B. ees 6. You may make a sngle copy of ths document for personal use wthout wrtten permsson. Hstory oncepts from prevous physcs and math courses that you wll need for

More information

Lecture 2: Single Layer Perceptrons Kevin Swingler

Lecture 2: Single Layer Perceptrons Kevin Swingler Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Mechanics Cycle 1 Chapter 5. Chapter 5

Mechanics Cycle 1 Chapter 5. Chapter 5 Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

WiMAX DBA Algorithm Using a 2-Tier Max-Min Fair Sharing Policy

WiMAX DBA Algorithm Using a 2-Tier Max-Min Fair Sharing Policy WMAX DBA Algorthm Usng 2-Ter Mx-Mn Fr Shrng Polcy Pe-Chen Tseng 1, J-Yn Ts 2, nd Wen-Shyng Hwng 2,* 1 Deprtment of Informton Engneerng nd Informtcs, Tzu Ch College of Technology, Hulen, Twn pechen@tccn.edu.tw

More information

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.

More information