2. Introduction and Chapter Objectives


 Evelyn Quinn
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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
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