2. Introduction and Chapter Objectives


 Evelyn Quinn
 1 years ago
 Views:
Transcription
1 eal Analog Crcuts Chapter : Crcut educton. Introducton and Chapter Objectes In Chapter, we presented Krchoff s laws (whch goern the nteractons between crcut elements) and Ohm s law (whch goerns the oltagecurrent relatonshps for resstors). These analytcal tools prode us wth the ablty to analyze any crcut contanng only resstors and deal power supples. Howeer, we also saw n Chapter that a crcut analyss, whch reles strctly on a bruteforce applcaton of these tools can become complex rapdly we essentally must use as our unknowns the oltage dfferences across all resstors and the currents through all resstors. Ths generally results n a large number of unknowns and a correspondngly large number of equatons, whch must be wrtten and soled n order to analyze any but the smplest crcut. In the next few chapters, we wll stll apply Krchoff s laws and Ohm s law n our crcut analyss, but we wll focus on mprong the effcency of our analyses. Typcally ths mproement n effcency s acheed by reducng the number of unknowns n the crcut, whch reduces the number of equatons, whch must be wrtten to descrbe the crcut s operaton. In ths chapter, we ntroduce analyss methods based on crcut reducton. Crcut reducton conssts of combnng resstances n a crcut to a smaller number of resstors, whch are (n some sense) equalent to the orgnal resste network. educng the number of resstors, of course, reduces the number of unknowns n a crcut. We begn our dscusson of crcut reducton technques by presentng two specfc, but ery useful, concepts: seres and parallel resstors. These concepts wll lead us to oltage and current dder formulas. We then consder reducton of more general crcuts, whch typcally corresponds to dentfyng multple sets of seres and parallel resstances n a complex resste network. Ths chapter then concludes wth two mportant examples of the applcaton of crcut reducton technques: the analyss of nondeal power sources and nondeal measurement deces; wthout an understandng of these deces, t s mpossble to buld practcal crcuts or understand the consequences of a oltage or current measurement. After completng ths chapter, you should be able to: Identfy seres and parallel combnatons of crcut elements Determne the equalent resstance of seres resstor combnatons Determne the equalent resstance of parallel resstor combnatons State oltage and current dder relatonshps from memory Determne the equalent resstance of electrcal crcuts consstng of seres and parallel combnatons of resstors Sketch equalent crcuts for nondeal oltage and current meters Analyze crcuts contanng nondeal oltage or current sources Determne the effect of nondeal meters on the parameter beng measured 0 Analog Deces and Dglent, Inc.
2 eal Analog Crcuts Chapter.: Seres Crcut Elements and Voltage Dson. Seres Crcut Elements and Voltage Dson There are a number of common crcut element combnatons that are qute easly analyzed. These specal cases are worth notng snce many complcated crcuts contan these crcut combnatons as subcrcuts. ecognzng these subcrcuts and analyzng them approprately can sgnfcantly smplfy the analyss of a crcut. Ths chapter emphaszes two mportant crcut element combnatons: elements n seres and elements n parallel. Also dscussed s the use of these crcut element combnatons to reduce the complexty of a crcut s analyss. Seres Connectons: Crcut elements are sad to be connected n seres f all of the elements carry the same current. An example of two crcut elements connected n seres s shown n Fgure.. Applyng KCL at node a and takng currents out of the node as poste we see that: or = 0 = (.) Equaton (.) s a drect outcome of the fact that the (sngle) node a n Fgure. nterconnects only two elements there are no other elements connected to ths node though whch current can be derted. Ths obseraton s so apparent (n many cases ) that equaton () s generally wrtten by nspecton for seres elements such as those shown n Fgure. wthout explctly wrtng KCL. a Fgure.. Crcut elements connected n seres. When resstors are connected n seres, a smplfcaton of the crcut s possble. Consder the resste crcut shown n Fgure.(a). Snce the resstors are n seres, they both carry the same current. Ohm s law ges: (.) Applyng KVL around the loop: (.3) 0 If there s any doubt whether the elements are n seres, apply KCL! Assumng elements are n seres whch are not n seres can hae dsastrous consequences. 0 Analog Deces and Dglent, Inc.
3 eal Analog Crcuts Chapter.: Seres Crcut Elements and Voltage Dson Substtutng equatons (.) nto equaton (.3) and solng for the current results n (.4) Now consder the crcut of Fgure.(b). Applcaton of Ohm s law to ths crcut and soluton for the current I ges (.5) eq eq (a) Seres resstors (b) Equalent Crcut Fgure.. Seres resstors and equalent crcut. Comparng equaton (.4) wth equaton (.5), we can see that the crcuts of Fgures.(a) and.(b) are ndstngushable f we select eq (.6) Fgures.(a) and.(b) are called equalent crcuts f the equalent resstance of Fgure.(b) s chosen as shown n equaton (.6). eq of equaton (.6) s called the equalent resstance of the seres combnaton of resstors and. Ths result can be generalzed to a seres combnaton of N resstances as follows: A seres combnaton of N resstors,,, N can be replaced wth a sngle equalent resstance eq N. The equalent crcut can be analyzed to determne the current through the seres combnaton of resstors. Voltage Dson: Combnng equatons (.) wth equaton (.4) results n the followng expressons for and : (.7) 0 Analog Deces and Dglent, Inc. 3
4 eal Analog Crcuts Chapter.: Seres Crcut Elements and Voltage Dson (.8) These results are commonly called oltage dder relatonshps, because they state that the total oltage drop across a seres combnaton of resstors s dded among the nddual resstors n the combnaton. The rato of each nddual resstor s oltage drop to the oerall oltage drop s the same as the rato of the nddual resstance to the total resstance. The aboe results can be generalzed for a seres combnaton of N resstances as follows: The oltage drop across any resstor n a seres combnaton of N resstances s proportonal to the total oltage drop across the combnaton of resstors. The constant of proportonalty s the same as the rato of the nddual resstor alue to the total resstance of the seres combnaton. For example, the oltage drop of the k th resstance n a seres combnaton of resstors s gen by: k k N (.9) where s the total oltage drop across the seres combnaton of resstors. Example.: For the crcut below, determne the oltage across the 5 resstor,, the current suppled by the source,, and the power suppled by the source V 5 The oltage across the 5 resstor can be determned from our oltage dder relatonshp: 0 Analog Deces and Dglent, Inc. 4
5 eal Analog Crcuts Chapter.: Seres Crcut Elements and Voltage Dson 5 5 5V 5V. 5V The current suppled by the source can be determned by ddng the total oltage by the equalent resstance: 5V eq 5V 5V 0.5A The power suppled by the source s the product of the source oltage and the source current: P ( 0.5A)(5V ) 7. 5W We can doublecheck the consstency between the oltage and the current wth Ohm s law. Applyng Ohm s law to the 5 resstor, wth a 0.5 A current, results n ( 5)(0.5A). 5V, whch agrees wth the result obtaned usng the oltage dder relatonshp. Secton Summary: If only two elements connect at a sngle node, the two elements are n seres. A more general defnton, howeer, s that crcut elements n seres all share the same current ths defnton allows us to determne seres combnatons that contan more than two elements. Identfcaton of seres crcut elements allows us to smplfy our analyss, snce there s a reducton n the number of unknowns: there s only a sngle unknown current for all seres elements. A seres combnaton of resstors can be replaced by a sngle equalent resstance, f desred. The equalent resstance s smply the sum of the nddual resstances n the seres combnaton. Therefore, a seres combnaton of N resstors,,, N can be replaced wth a sngle equalent resstance eq N. If the total oltage dfference across a set of seres resstors s known, the oltage dfferences across any nddual resstor can be determned by the concept of oltage dson. The term oltage dson comes from the fact that the oltage drop across a seres combnaton of resstors s dded among the nddual resstors. The rato between the oltage dfference across a partcular resstor and the total oltage dfference s the same as the rato between the resstance of that resstor and the total resstance of the combnaton. If k s the oltage across the k th resstor, TOT s the total oltage across the seres combnaton, k s the resstance of the k th resstor, and TOT s the total resstance of the seres combnaton, the mathematcal statement of ths concept s: k TOT k TOT 0 Analog Deces and Dglent, Inc. 5
6 eal Analog Crcuts Chapter.: Seres Crcut Elements and Voltage Dson Exercses:. Determne the oltage V n the crcut below. 0k 4k V 6k V 4k 0 Analog Deces and Dglent, Inc. 6
7 eal Analog Crcuts Chapter.: Parallel Crcut Elements and Current Dson. Parallel Crcut Elements and Current Dson Crcut elements are sad to be connected n parallel f all of the elements share the same par of nodes. An example of two crcut elements connected n parallel s shown n Fgure.3. Applyng KVL around the loop of Fgure.3 results n: = (.0) Ths result s so common that equaton (.0) s generally wrtten by nspecton for parallel elements such as those shown n Fgure.3 wthout explctly wrtng KVL. a b Fgure.3. Parallel connecton of crcut elements. We can smplfy crcuts, whch consst of resstors connected n parallel. Consder the resste crcut shown n Fgure.4(a). The resstors are connected n parallel, so both resstors hae a oltage dfference of. Ohm s law, appled to each resstor results n: (.) Applyng KCL at node a: (.) Substtutng equatons (.) nto equaton (.): (.3) or (.4) 0 Analog Deces and Dglent, Inc. 7
8 eal Analog Crcuts Chapter.: Parallel Crcut Elements and Current Dson If we set eq, we can draw Fgure.4(b) as beng equalent to Fgure.4(b). We can generalze ths result for N parallel resstances: A parallel combnaton of N resstors,,, N can be replaced wth a sngle equalent resstance: eq N (.5) The equalent crcut can be analyzed to determne the oltage across the parallel combnaton of resstors. a eq (a) Parallel resstance combnaton (b) Equalent crcut Fgure.4. Parallel resstances and equalent crcut. For the specal case of two parallel resstances, and, the equalent resstance s commonly wrtten as: eq (.6) Current Dson: Substtutng equaton (.4) nto equatons (.) results n (.7) 0 Analog Deces and Dglent, Inc. 8
9 eal Analog Crcuts Chapter.: Parallel Crcut Elements and Current Dson smplfyng: (.8) Lkewse for the current, (.9) Equatons (.8) and (.9) are the oltage dder relatonshps for two parallel resstances, so called because the current nto the parallel resstance combnaton s dded between the two resstors. The rato of one resstor s current to the oerall current s the same as the rato of the other resstance to the total resstance. The aboe results can be generalzed for a seres combnaton of N resstances. By Ohm s law, eq. Substtutng our preous result for the equalent resstance for a parallel combnaton of N resstors results n:. (.0) N Snce the oltage dfference across all resstors s the same, the current through the k th resstor s, by Ohm s law, k (.) k where k s the resstance of the k th resstor. Combnng equatons (.0) and (.) ges: k k N (.) It s often more conenent to prode the generalzed result of equaton (.0) n terms of the conductances of the nddual resstors. ecall that the conductance s the recprocal of the resstance, G. Thus, equaton (.) can be reexpressed as follows: The current through any resstor n a parallel combnaton of N resstances s proportonal to the total current nto the combnaton of resstors. The constant of proportonalty s the same as the rato of the conductance of the nddual resstor alue to the total conductance of the parallel combnaton. For example, the current through the k th resstance n a parallel combnaton of resstors s gen by: Gk k (.3) G G GN where s the total current through the parallel combnaton of resstors. One fnal comment about notaton: two parallel bars are commonly used as shorthand notaton to ndcate that two crcut elements are n parallel. For example, the notaton ndcates that the resstors and are n 0 Analog Deces and Dglent, Inc. 9
10 eal Analog Crcuts Chapter.: Parallel Crcut Elements and Current Dson parallel. The notaton s often used as shorthand notaton for the equalent resstance of the parallel resstance combnaton, n leu of equaton (.6). Double checkng results for parallel resstances: The equalent resstance for a parallel combnaton of N resstors wll always be less than the smallest resstance n the combnaton. In fact, the equalent resstance wll always obey the followng nequaltes: mn eq mn N where mn s the smallest resstance alue n the parallel combnaton. In a parallel combnaton of resstances, the resstor wth the smallest resstance wll hae the largest current and the resstor wth the largest resstance wll hae the smallest current. Secton Summary: If seeral elements nterconnect the same two nodes, the two elements are n parallel. A more general defnton, howeer, s that crcut elements n paralell all share the same oltage dfference. As wth seres crcut elements, dentfcaton of parallel crcut elements allows us to smplfy our analyss, snce there s a reducton n the number of unknowns: there s only a sngle unknown oltage dfference for all of the parallel elements. A parallel combnaton of resstors can be replaced by a sngle equalent resstance, f desred. The conductance of the parallel combnaton s smply the sum of the nddual conductances of the parallel resstors. Therefore, a parallel combnaton of N resstors,,, N can be replaced wth a sngle eq equalent resstance:. N If the total current through a set of parallel resstors s known, the current through any nddual resstor can be determned by the concept of current dson. The term current dson comes from the fact that the current through a parallel combnaton of resstors s dded among the nddual resstors. The rato between the current through a partcular resstor and the total current s the same as the rato between the conductance of that resstor and the total conductance of the combnaton. If k s the oltage across the k th resstor, TOT s the total current through the parallel combnaton, G k s the conductance of the k th resstor, and G TOT s the total conductance of the parallel combnaton, the mathematcal statement of ths concept s: k TOT G G k TOT Exercses:. Determne the alue of I n the crcut below. 0 Analog Deces and Dglent, Inc. 0
11 eal Analog Crcuts Chapter.: Parallel Crcut Elements and Current Dson 5mA 3k 5k I. Determne the alue of n the crcut below whch makes I = ma. I 3mA k 0 Analog Deces and Dglent, Inc.
12 eal Analog Crcuts Chapter.3: Crcut educton and Analyss.3 Crcut educton and Analyss The preous results ge us an ablty to potentally smplfy the analyss of some crcuts. Ths smplfcaton results f we can use crcut reducton technques to conert a complcated crcut to a smpler, but equalent, crcut whch we can use to perform the necessary analyss. Crcut reducton s not always possble, but when t s applcable t can sgnfcantly smplfy the analyss of a crcut. Crcut reducton reles upon dentfcaton of parallel and seres combnatons of crcut elements. The parallel and seres elements are then combned nto equalent elements and the resultng reduced crcut s analyzed. The prncples of crcut reducton are llustrated below n a seres of examples. Example.: Determne the equalent resstance seen by the nput termnals of the resste network shown below eq The sequence of operatons performed s llustrated below. The 6 and 3 resstances are combned n parallel to obtan an equalent resstance. Ths resstance and the remanng 6 resstance are n seres, these are combned nto an equalent 8 resstance. Fnally, ths 8 resstor and the 4 resstor are combned n parallel to obtan an equalent 6 resstance. Thus, the equalent resstance of the oerall network s Analog Deces and Dglent, Inc.
13 eal Analog Crcuts Chapter.3: Crcut educton and Analyss Example.3: In the crcut below, determne the power delered by the source. 4 6V 3 6 In order to determne power delery, we need to determne the total current proded by the source to the rest of the crcut. We can determne current easly f we conert the resstor network to a sngle, equalent, resstance. A set of steps for dong ths are outlned below. Step : The four ohm and two ohm resstors, hghlghted on the fgure to the left below n grey, are n seres. Seres resstances add drectly, so these can be replaced wth a sngle sx ohm resstor, as shown on the fgure to the rght below. 4 6V 3 6 6V Step: The three ohm resstor and the two sx ohm resstors are now all n parallel, as ndcated on the fgure to 5 eq. the left below. These resstances can be combned nto a sngle equalent resstor The resultng equalent crcut s shown to the rght below. 6V V.5 The current out of the source can now be readly determned from the fgure to the rght aboe. The oltage drop 6V across the.5 resstor s 6V, so Ohm s law ges 4A. Thus, the power delered by the source s.5 P ( 4A)(6V ) 4W. Snce the sgn of the current relate to the current does not agree wth the passe sgn conenton, the power s generated by the source. 0 Analog Deces and Dglent, Inc. 3
14 eal Analog Crcuts Chapter.3: Crcut educton and Analyss Example.4: For the crcut shown below, determne the oltage, s, across the A source. A s 4 The two resstors and the two resstors are n seres wth one another, as ndcated on the fgure to the left below. These can be combned by smply addng the seres resstances, leadng to the equalent crcut shown to the rght below. A 4 A 4 4 The three remanng resstors are all n parallel (they all share the same nodes) so they can be combned usng the eq relaton. Note that t s not necessary to combne all three resstors smultaneously, the 4 4 same result s obtaned by successe combnatons of two resstances. For example, the two 4 resstors can be 4 4 combned usng equaton (4) to obtan: eq. The total equalent resstance can then be 4 4 determned by a parallel combnaton of eq and the resstor: eq. A 4 4 A s The oltage across the source can now be determned from Ohm s law: polarty of the source oltage s correct. s ( )( A) V. The assumed 0 Analog Deces and Dglent, Inc. 4
15 eal Analog Crcuts Chapter.3: Crcut educton and Analyss Example.5: Wheatstone brdge A Wheatstone brdge crcut s shown below. The brdge s generally presented as shown n the fgure to the left; we wll generally use the equalent crcut shown to the rght. A Wheatstone brdge s commonly used to conert a araton n resstance to a araton n oltage. A constant supply oltage Vs s appled to the crcut. The resstors n the crcut all hae a nomnal resstance of ; the arable resstor has a araton from ths nomnal alue. The output oltage ab ndcates the araton n the arable resstor. The arable resstor n the network s often a transducer whose resstance ares dependent upon some external arable such as temperature. Vs a ab b Vs a ab b By oltage dson, the oltages b and a (relate to ground) are ( ) b V S and a VS VS The oltage ab s then ab a b V S ( ) ( ) V ( ) S V ( ) S For the case n whch, ths smplfes to: ab Vs 4 and the output oltage s proportonal to the change n resstance of the arable resstor. Practcal Applcatons: A number of common sensors result n a resstance araton resultng from some external nfluence. Thermstors change resstance as a result of temperature changes; stran gages change resstance as a result of deformaton, generally due to applcaton of a load to the part to whch the gage s bonded; photoconducte transducers, or photoresstors, change resstance as a result of changes n lght ntensty. Wheatstone brdges are commonly used n conjuncton wth these types of sensors. Secton Summary: 0 Analog Deces and Dglent, Inc. 5
16 eal Analog Crcuts Chapter.3: Crcut educton and Analyss In a crcut, whch contans obous seres and/or parallel combnatons of resstors, analyss can be smplfed by combnng these resstances nto equalent resstances. The reducton n the oerall number of resstances reduces the number of unknowns n the crcut, wth a correspondng reducton n the number of goernng equatons. educng the number of equatons and unknowns typcally smplfes the analyss of the crcut. Not all crcuts are reducble. Exercses:. For the crcut shown, determne: a) eq (the equalent resstance seen by the source) b) The currents I and I I I 6V eq 0 Analog Deces and Dglent, Inc. 6
17 eal Analog Crcuts Chapter.4: NonIdeal Power Supples.4 NonIdeal Power Supples In secton., we dscussed deal power sources. In that secton, an deal oltage supply was characterzed as prodng a specfed oltage regardless of the current requrements made upon the dece. Lkewse, an deal current source was defned as prodng a specfed oltage regardless of the oltage potental dfference across the source. These models are not realstc snce an deal oltage source can prode nfnte current wth nonzero oltage dfference and an deal current source can prode nfnte oltage dfference wth nonzero current, ether dece s capable of delerng nfnte power. In many cases, the deal oltage and current source models wll be adequate, but n the cases where we need to more accurately replcate the operaton of realstc power supples, we wll need to modfy our models of these deces. In ths secton, we present smple models for oltage and current sources whch ncorporate more realstc assumptons as to the behaor of these deces. NonIdeal Voltage Sources: An deal oltage source was defned n secton. as prodng a specfed oltage, regardless of the current flow out of the dece. For example, an deal V battery wll prode V across ts termnals, regardless of the load connected to the termnals. A real V battery, howeer, prodes V across ts termnals only when ts termnals are open crcuted. As we draw current from the termnals, the battery wll prode less than V the oltage wll decrease as more and more current s drawn from the battery. The real battery thus appears to hae an nternal oltage drop whch ncreases wth ncreased current. We wll model a real or practcal oltage source as a seres connecton of an deal oltage source and an nternal resstance. Ths model s depcted schematcally n Fgure.5, n whch the nondeal oltage source contans an deal oltage source prodng oltage V s and an nternal resstance, s. The nondeal oltage source delers a oltage V and a current, where: V Vs s (.4) Equaton (.4) ndcates that the oltage delered by our nondeal oltage source model decreases as the current out of the oltage source ncreases, whch agrees wth expectatons. s V s V Nondeal oltage source Fgure.5. Nondeal oltage source model. 0 Analog Deces and Dglent, Inc. 7
18 eal Analog Crcuts Chapter.4: NonIdeal Power Supples Example.6: Consder the case n whch we connect a resste load to the nondeal oltage source. The fgure below prodes a schematc of the oerall system; L s the load resstance, V L s the oltage delered to the load, and L s the current delered to the load. s L V s V L L Vs L In the case aboe, the current delered to the load s and the load oltage s VL Vs. s L s L Thus, f the load resstance s nfnte (the load s an open crcut), V L = V s, but the power supply delers no current and hence no power to the load. If the load resstance s zero (the load s a short crcut), V L = 0 and the Vs power supply delers current L to the load; the power delered to the load, howeer, s stll zero. s 0 Analog Deces and Dglent, Inc. 8
19 eal Analog Crcuts Chapter.4: NonIdeal Power Supples Example.7: Chargng a battery We hae a dead car battery whch s prodng only 4V across ts termnals. We want to charge the battery usng a spare battery whch s prodng V across ts termnals. To do ths, we connect the two batteres as shown below: V 4V If we attempt to analyze ths crcut by applyng KVL around the loop, we obtan V = 4V. Ths s obously ncorrect and we cannot proceed wth our analyss our model dsagrees wth realty! To resole ths ssue, we wll nclude the nternal resstances of the batteres. Assumng a 3 nternal resstance n each battery, we obtan the followng model for the system: 3 3 V 4V Battery Battery Applyng KVL around the loop, and usng Ohm s law to wrte the oltages across the battery nternal resstances n terms of the current between the batteres results n: V (3) (3) 4V 0 whch can be soled for the current to obtan: V 4V. 33A 6 Notce that as the oltage of the dead battery ncreases durng the chargng process, the current delered to the dead battery decreases. NonIdeal Current Sources: 0 Analog Deces and Dglent, Inc. 9
20 eal Analog Crcuts Chapter.4: NonIdeal Power Supples An deal current source was defned n secton. as prodng a specfed current, regardless of the oltage dfference across the dece. Ths model suffers from the same basc drawback as our deal oltage source model the model can deler nfnte power, whch s nconsstent wth the capabltes of a real current source. We wll use the crcut shown schematcally n Fgure.6 to model a nondeal current source. The nondeal model conssts of an deal current source, s, placed n parallel wth an nternal resstance, s. The source delers a oltage V and current. The output current s gen by: V S (.5) S Equaton (.5) shows that the current delered by the source decreases as the delered oltage ncreases. s (t) s V Fgure.6. Nondeal current source model. 0 Analog Deces and Dglent, Inc. 0
21 eal Analog Crcuts Chapter.4: NonIdeal Power Supples Example.8: Consder the case n whch we connect a resste load to the nondeal current source. The fgure below prodes a schematc of the oerall system; L s the load resstance, V L s the oltage delered to the load, and L s the current delered to the load. L s (t) s V L L In the case aboe, the current delered to the load can be determned from a current dder relaton as s S L L s and the load oltage, by Ohm s law, s VL LL s. If the load resstance s zero s L s L (the load s a short crcut), L = s, but the power supply delers no oltage and hence no power to the load. In the case of nfnte load resstance (the load s an open crcut), L = 0. In ths case, we can neglect s n the S L denomnator of the load oltage equaton to obtan VL s so that VL s S. Snce the current s zero, L howeer, the power delered to the load s stll zero. S L s If we explctly calculate the power delered to the load, we obtan VL s. A plot of the s L s L power delered to the load as a functon of the load resstance s shown below; a logarthmc scale s used on the horzontal axs to make the plot more readable. As expected, the power s zero for hgh and low load resstances. The peak of the cure occurs when the load resstance s equal to the source resstance, L = s. PL L = S log 0 ( L ) Secton Summary: 0 Analog Deces and Dglent, Inc.
22 eal Analog Crcuts Chapter.4: NonIdeal Power Supples In many cases, power supples can be modeled as deal power supples, as presented n secton.. Howeer, n some cases representaton as a power supply as deal results n unacceptable errors. For example, deal power supples can deler nfnte power, whch s obously unrealstc. In ths crcut, we present a smple model for a nondeal power supply. a. Our nondeal oltage source conssts of an deal oltage source n seres wth a resstance whch s nternal to the power supply. b. Our nondeal current supply conssts of an deal current source n parallel wth a resstance whch s nternal to the power supply. Voltage and current dder formulas allow us to easly quantfy the effects of the nternal resstances of the nondeal power supples. Our analyss ndcates that the nondeal effects are neglgble, as long as the resstance of the load s large relate to the nternal resstance of the power supply. Exercses:. A oltage source wth an nternal resstance of Ω as shown below s used to apply power to a 3Ω resstor. What oltage would you measure across the 3Ω resstor? 5V Nondeal oltage source. The oltage source of exercse aboe s used to apply power to a kω resstor. What oltage would you measure across the kω resstor? 0 Analog Deces and Dglent, Inc.
23 eal Analog Crcuts Chapter.5: Practcal Voltage and Current Measurement.5 Practcal Voltage and Current Measurement The process of measurng a physcal parameter wll almost narably change the parameter beng measured. Ths effect s both undesrable and, n general, unaodable. One goal of any measurement s to affect the parameter beng measured as lttle as possble. The aboe statement s true of oltage and current measurements. An deal oltmeter, connected n parallel wth some crcut element, wll measure the oltage across the element wthout affectng the current flowng through the element. Unfortunately, any real or practcal oltmeter wll draw some current from the crcut t s connected to; ths loadng effect wll change the crcut s operatng condtons, causng some dfference between the measured oltage and the correspondng oltage wthout the oltmeter present n the crcut. Lkewse, an deal ammeter, connected n seres wth some crcut element, wll measure current wthout affectng the oltage n the crcut. A practcal ammeter, lke a practcal oltmeter, wll ntroduce loadng effects whch change the operaton of the crcut on whch the measurement s beng made. In ths secton, we ntroduce some effects of measurng oltages and currents wth practcal meters. Voltmeter and Ammeter Models: We wll model both oltmeters and ammeters as hang some nternal resstance and a method for dsplayng the measured oltage dfference or current. Fgure.7 shows schematc representatons of oltmeters and ammeters. The ammeter n Fgure.7(a) has an nternal resstance M ; the current through the ammeter s A and the oltage dfference across the ammeter s V M. The ammeter s oltage dfference should be as small as possble an ammeter, therefore, should hae an extremely small nternal resstance. The oltmeter n Fgure.7(b) s also represented as hang an nternal resstance M ; the current through the meter s V and the oltage dfference across the meter s V V. The current through the oltmeter should be as small as possble the oltmeter should hae an extremely hgh nternal resstance. The effects of nonzero ammeter oltages and nonzero oltmeter currents are explored n more detal n the followng subsectons. V A V V A M A V M V (a) Ammeter model (b) Voltmeter model Voltage Measurement: Fgure.7. Ammeter and oltmeter models. 0 Analog Deces and Dglent, Inc. 3
24 eal Analog Crcuts Chapter.5: Practcal Voltage and Current Measurement Consder the crcut shown n Fgure.8(a). A current source, s, prodes current to a crcut element wth resstance,. We want to measure the oltage drop, V, across the crcut element. We do ths by attachng a oltmeter across the crcut element as shown n Fgure.8(b). In Fgure.8(b) the oltmeter resstance s n parallel to the crcut element we wsh to measure the oltage across and the combnaton of the crcut element and the oltmeter becomes a current dder. The current through the resstor then becomes: M s (.6) M The oltage across the resstor s then, by Ohm s law, V M s (.7) M If M >>, ths expresson smplfes to V M s s (.8) M and neglgble error s ntroduced nto the measurement the measured oltage s approxmately the same as the oltage wthout the oltmeter. If, howeer, the oltmeter resstance s comparable to the resstance, the smplfcaton of equaton (.8) s not approprate and sgnfcant changes are made to the system by the presence of the oltmeter. V M s V s V V (a) orgnal crcut (b) crcut wth oltmeter Fgure.8. Voltage measurement 0 Analog Deces and Dglent, Inc. 4
25 eal Analog Crcuts Chapter.5: Practcal Voltage and Current Measurement Current Measurement: Consder the crcut shown n Fgure.9(a). A oltage source, V s, prodes power to a crcut element wth resstance,. We want to measure the current,, through the crcut element. We do ths by attachng an ammeter n seres wth the crcut element as shown n Fgure.9(b). In Fgure.9(b) the seres combnaton of the ammeter resstance and the crcut element whose current we wsh to measure creates a oltage dder. KVL around the sngle crcut loop prodes: V s ( M ) (.9) Solng for the current results n VS (.30) M If M <<, ths smplfes to VS (.3) and the measured current s a good approxmaton to current n the crcut of Fgure.9(a). Howeer, f the ammeter resstance s not small compared to the resstance, the approxmaton of equaton (.3) s not approprate and the measured current s no longer representate of the crcut s operaton wthout the ammeter. V A M A V s V V s V (a) Orgnal crcut (b) Crcut wth ammeter. Fgure.9. Current measurement. 0 Analog Deces and Dglent, Inc. 5
26 eal Analog Crcuts Chapter.5: Practcal Voltage and Current Measurement Cauton: Incorrect connectons of ammeters or oltmeters can cause damage to the meter. For example, consder the connecton of an ammeter n parallel wth a relately large resstance, as shown below. M V s A M VS In ths confguraton the ammeter current, M. Snce the ammeter resstance s typcally ery small, ths M can result n hgh currents beng proded to the ammeter. Ths, n turn, may result n excesse power beng proded to the ammeter and resultng damage to the dece. Ammeters are generally ntended to be connected n seres wth the crcut element(s) whose current s beng measured. Voltmeters are generally ntended to be connected n parallel wth the crcut element(s) whose oltage s beng measured. Alternate connectons can result n damage to the meter. Secton Summary: Measurement of oltage and/or current n a crcut wll always result n some effect on the crcut s behaor that s, our measurement wll always change the parameter beng measured. One goal when measurng a oltage or current s to ensure that the measurement effects are neglgble. In ths crcut, we present smple models for oltmeters and ammeters. (Voltage and current measurement deces, respectely.) a. Our nondeal oltmeter conssts of an deal oltmeter (whch has nfnte resstance, and thus draws no current from the crcut) n seres wth a resstance whch s nternal to the oltmeter. Ths model replcates the fnte current whch s necessarly drawn from the crcut by a real oltmeter. b. Our nondeal ammeter conssts of an deal ammeter (whch has zero resstance, and thus ntroduces no oltage drop n the crcut) n seres wth a resstance whch s nternal to the ammeter. Ths resstance allows us to model the fnte oltage drop whch s ntroduced nto the crcut by a real current measurement. Voltage and current dder formulas allow us to easly quantfy the effects of the nternal resstances of oltage and current meters. Our analyss ndcates that the nondeal effects are neglgble, as long as: a. The resstance of the oltmeter s large relate to the resstance across whch the oltage measurement s beng made. b. The resstance of the ammeter s small compared to the oerall crcut resstance. Exercses:. A oltmeter wth an nternal resstance of 0MΩ s used to measure the oltage ab n the crcut below. What s the measured oltage? What oltage measurement would you expect from an deal oltmeter? 0 Analog Deces and Dglent, Inc. 6
27 eal Analog Crcuts Chapter.5: Practcal Voltage and Current Measurement 5M a V 0M b 0 Analog Deces and Dglent, Inc. 7
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) = 1224 + 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+ + +   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 informationSystematic 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 informationThe 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 informationLOOP 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 informationOptical SignaltoNoise Ratio and the QFactor in FiberOptic Communication Systems
Applcaton ote: FA9.0. Re.; 04/08 Optcal Sgnaltoose Rato and the QFactor n FberOptc Communcaton Systems Functonal Dagrams Pn Confguratons appear at end of data sheet. Functonal Dagrams contnued at
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), 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 informationCIRCUIT 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 informationChapter 6 Inductance, Capacitance, and Mutual Inductance
Chapter 6 Inductance Capactance and Mutual Inductance 6. The nductor 6. The capactor 6.3 Seresparallel combnatons of nductance and capactance 6.4 Mutual nductance 6.5 Closer look at mutual nductance Oerew
More information3. Bipolar Junction Transistor (BJT)
3. polar Juncton Transstor (JT) Lecture notes: Sec. 3 Sedra & Smth (6 th Ed): Sec. 6.16.4* Sedra & Smth (5 th Ed): Sec. 5.15.4* * Includes detals of JT dece operaton whch s not coered n ths course EE
More informationBipolar Junction Transistor (BJT)
polar Juncton Transstor (JT) Lecture notes: Sec. 3 Sedra & Smth (6 th Ed): Sec. 6.16.4* Sedra & Smth (5 th Ed): Sec. 5.15.4* * Includes detals of JT dece operaton whch s not coered n ths course F. Najmabad,
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More informationThe 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 informationThe FullWave Rectifier
9/3/2005 The Full Wae ectfer.doc /0 The FullWae 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 informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationFORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER
FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 276957909 Introducton he convectve heat
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationThe Bridge Rectifier
9/4/004 The Brdge ectfer.doc 1/9 The Brdge ectfer Now consder ths juncton dode rectfer crcut: 1 Lne (t)  O (t) _ 4 3 We call ths crcut the brdge rectfer. Let s analyze t and see what t does! Frst, we
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationThe complex inverse trigonometric and hyperbolic functions
Physcs 116A Wnter 010 The complex nerse trgonometrc and hyperbolc functons In these notes, we examne the nerse trgonometrc and hyperbolc functons, where the arguments of these functons can be complex numbers
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More information1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits
E6 Electrcal Engneerng A rcut Analyss and Power ecture : Parallel esonant rcuts. Introducton There are equvalent crcuts to the seres combnatons examned whch exst n parallel confguratons. The ssues surroundng
More informationMultiple stage amplifiers
Multple stage amplfers Ams: Examne a few common 2transstor amplfers:  Dfferental amplfers  Cascode amplfers  Darlngton pars  current mrrors Introduce formal methods for exactly analysng multple
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationIT09  Identity Management Policy
IT09  Identty Management Polcy Introducton 1 The Unersty needs to manage dentty accounts for all users of the Unersty s electronc systems and ensure that users hae an approprate leel of access to these
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationFaraday's Law of Induction
Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy
More informationHALL EFFECT SENSORS AND COMMUTATION
OEM770 5 Hall Effect ensors H P T E R 5 Hall Effect ensors The OEM770 works wth threephase brushless motors equpped wth Hall effect sensors or equvalent feedback sgnals. In ths chapter we wll explan how
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages  n "Machnes, Logc and Quantum Physcs"
More informationPeak Inverse Voltage
9/13/2005 Peak Inerse Voltage.doc 1/6 Peak Inerse Voltage Q: I m so confused! The brdge rectfer and the fullwae rectfer both prode fullwae rectfcaton. Yet, the brdge rectfer use 4 juncton dodes, whereas
More informationSection 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 informationInterleaved Power Factor Correction (IPFC)
Interleaved Power Factor Correcton (IPFC) 2009 Mcrochp Technology Incorporated. All Rghts Reserved. Interleaved Power Factor Correcton Slde 1 Welcome to the Interleaved Power Factor Correcton Reference
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationIMPACT ANALYSIS OF A CELLULAR PHONE
4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationChapter 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 informationSmallSignal Analysis of BJT Differential Pairs
5/11/011 Dfferental Moe Sall Sgnal Analyss of BJT Dff Par 1/1 SallSgnal Analyss of BJT Dfferental Pars Now lets conser the case where each nput of the fferental par conssts of an entcal D bas ter B, an
More informationThe 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 informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationActivity Scheduling for CostTime Investment Optimization in Project Management
PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta San Sebastán, September 8 th 10 th 010 Actvty Schedulng
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCullochPtts 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 informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationDamage detection in composite laminates using cointap method
Damage detecton n composte lamnates usng contap method S.J. Km Korea Aerospace Research Insttute, 45 EoeunDong, YouseongGu, 35333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The contap test has the
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationCHAPTER 9 SECONDLAW ANALYSIS FOR A CONTROL VOLUME. blank
CHAPTER 9 SECONDLAW ANALYSIS FOR A CONTROL VOLUME blank SONNTAG/BORGNAKKE STUDY PROBLEM 91 9.1 An deal steam turbne A steam turbne receves 4 kg/s steam at 1 MPa 300 o C and there are two ext flows, 0.5
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationtotal A A reag total A A r eag
hapter 5 Standardzng nalytcal Methods hapter Overvew 5 nalytcal Standards 5B albratng the Sgnal (S total ) 5 Determnng the Senstvty (k ) 5D Lnear Regresson and albraton urves 5E ompensatng for the Reagent
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationChapter 31B  Transient Currents and Inductance
Chapter 31B  Transent Currents and Inductance A PowerPont Presentaton by Paul E. Tppens, Professor of Physcs Southern Polytechnc State Unversty 007 Objectves: After completng ths module, you should be
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationTime Value of Money Module
Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More informationMultiple discount and forward curves
Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of
More informationRELIABILITY, RISK AND AVAILABILITY ANLYSIS OF A CONTAINER GANTRY CRANE ABSTRACT
Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March RELIABILITY RIK AND AVAILABILITY ANLYI OF A CONTAINER GANTRY CRANE
More informationSection 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 informationColligative Properties
Chapter 5 Collgatve Propertes 5.1 Introducton Propertes of solutons that depend on the number of molecules present and not on the knd of molecules are called collgatve propertes. These propertes nclude
More informationUniversity Physics AI No. 11 Kinetic Theory
Unersty hyscs AI No. 11 Knetc heory Class Number Name I.Choose the Correct Answer 1. Whch type o deal gas wll hae the largest alue or C C? ( D (A Monatomc (B Datomc (C olyatomc (D he alue wll be the same
More informationRESEARCH ON DUALSHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.
ICSV4 Carns Australa 9 July, 007 RESEARCH ON DUALSHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract
More informationMultiplication Algorithms for Radix2 RNCodings and Two s Complement Numbers
Multplcaton Algorthms for Radx RNCodngs and Two s Complement Numbers JeanLuc Beuchat Projet Arénare, LIP, ENS Lyon 46, Allée d Itale F 69364 Lyon Cedex 07 jeanluc.beuchat@enslyon.fr JeanMchel Muller
More informationStudy on Model of Risks Assessment of Standard Operation in Rural Power Network
Study on Model of Rsks Assessment of Standard Operaton n Rural Power Network Qngj L 1, Tao Yang 2 1 Qngj L, College of Informaton and Electrcal Engneerng, Shenyang Agrculture Unversty, Shenyang 110866,
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationFuzzy Set Approach To Asymmetrical Load Balancing In Distribution Networks
Fuzzy Set Approach To Asymmetrcal Load Balancng n Dstrbuton Networks Goran Majstrovc Energy nsttute Hrvoje Por Zagreb, Croata goran.majstrovc@ehp.hr Slavko Krajcar Faculty of electrcal engneerng and computng
More informationAnalysis and Modeling of Magnetic Coupling
Analyss and Modelng of Magnetc Couplng Bryce Hesterman Adanced Energy Industres Tuesday, Aprl 7 Dscoery earnng Center Unersty Of Colorado, Boulder, Colorado Dener Chapter, IEEE Power Electroncs Socety
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More informationSafety instructions VEGAVIB VB6*.GI*******
Safety nstructons VEGAVIB VB6*.GI******* Kosha 14AV4BO0107 Ex td A20, A20/21, A21 IP66 T** 0044 Document ID: 48578 Contents 1 Area of applcablty... 3 2 General nformaton... 3 3 Techncal data... 3 4 Applcaton
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationComparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions
Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda
More informationTrafficlight a stress test for life insurance provisions
MEMORANDUM Date 006097 Authors Bengt von Bahr, Göran Ronge Traffclght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationAn Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems
STANCS73355 I SUSE73013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part
More informationVRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT09105, Phone: (3705) 2127472, Fax: (3705) 276 1380, Email: info@teltonika.
VRT012 User s gude V0.1 Thank you for purchasng our product. We hope ths userfrendly devce wll be helpful n realsng your deas and brngng comfort to your lfe. Please take few mnutes to read ths manual
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More information( ) B. Application of Phasors to Electrical Networks In an electrical network, let the instantaneous voltage and the instantaneous current be
World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 Analyss of Electrcal Networks Usng Phasors: A Bond Graph Approach
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mt.edu 5.74 Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 41 4.1. INTERACTION OF LIGHT
More informationLaws of Electromagnetism
There are four laws of electromagnetsm: Laws of Electromagnetsm The law of BotSavart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationLuby 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 informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationNOTE: The Flatpak version has the same pinouts (Connection Diagram) as the Dual InLine Package. *MR for LS160A and LS161A *SR for LS162A and LS163A
BCD DECADE COUNTERS/ 4BIT BINARY COUNTERS The LS160A/ 161A/ 162A/ 163A are hghspeed 4bt synchronous counters. They are edgetrggered, synchronously presettable, and cascadable MSI buldng blocks for
More informationSolutions to the exam in SF2862, June 2009
Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodcrevew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More information