MAE140 Linear Circuits (for non-electrical engs)

Transcription

1 MAE4 Lnear Crcuts (for non-electrcal engs) Topcs coered Crcut analyss technques Krchoff s Laws KVL, KCL Nodal and Mesh Analyss Théenn and Norton Equalent Crcuts Resste crcuts, RLC crcuts Steady-state and dynamc responses Impulse and step responses Laplace transforms Snusodal steady-state response Crcut desgn technques Acte crcut elements dependent sources and operatonal amplfers Feedback bascs Sgnal flterng f we get tme MAE4 Lnear Crcuts

2 What do I expect you to know? Prerequstes Mathematcs 2D or 2D (some are dong ths concurrently) Soluton of sets of lnear equatons Ax=b equatons for ector x gen matrx A and ector b Soluton of constant coeffcent lnear ordnary dfferental equatons Laplace transform ntroducton Intal condtons and forced response Complex analyss Numbers, arthmetc, poles, zeros Physcs 2B Behaor of crcut elements: R, L and C Underlyng physcal prncples MAE4 Lnear Crcuts 2

3 Why should you be excted about MAE4? Lnear crcuts are exceptonally well descrbed by smple and powerful mathematcs The qualty of concordance between measured physcal behaor and mathematcal descrpton s amazng Ths s an analog computer from around 963 It was used to sole nonlnear ODEs a analog crcutry It contaned: about 2 ntegrators based on ales or transstors, some nonlnear functon blocks usng dodes, comparators Ths was the state of the art for much physcal system smulaton It reles on deas underpnnng MAE4 The tools of MAE4 are mmedately useful n desgn MAE4 Lnear Crcuts 3

4 Why should you stll be excted about MAE4? The front and back ends of your dgtal cell phone are comprsed of analog crcuts Ths s pretty much true of all dgtal technology Why does the stagecoach wheel appear to rotate backwards? Alasng A hgh frequency masqueradng as a low frequency Ant-alasng flters MUST be used n all sampled data systems After MAE4 you wll be able to start desgnng such ant-alasng flters The real thng MAE4 Lnear Crcuts 4

5 Why should you stll be excted about MAE4? You wll learn how these knobs and plugs work: We wll not hae tme to coer ths one: MAE4 Lnear Crcuts 5

6 Crcut arables (T&R Chap ) Charge and Energy - coulomb (C) = electrons charge q dq ampere (A) = coulomb/second = dt Current s a measure of charge passng Drecton of flow s of poste charge Ths s mportant for understandng some deces Voltage/potental dfference/electromote force/tenson Voltage measures the energy ganed by a charge joule(j)/coulomb = olt (V) Voltage s measured between two ponts (potental dfference) = Ths s akn to gratatonal potental and flud flow or temperature and heat flow dw dq MAE4 Lnear Crcuts 6

7 Crcut arables (contd) dw Power s the rate of energy change per unt tme p = watts (W) dt & dw# dq Note the chan rule p = & # = % dq "% dt " The nstantaneous electrcal power assocated wth a oltage and a current flow s gen by the product Passe sgn conenton Power s poste when a dece absorbs power Sgns are measured as llustrated Voltage s appled (across) and current results (nto) or ce ersa Note that (t) or (t) can be negate Ths just defnes the sense rest of crcut In the llustraton the current nto the dece s the negate of that nto the rest of the crcut MAE4 Lnear Crcuts 7 (t) - (t) dece

8 An example The fgure at rght depcts the crcut nsde my flashlght (t) The deces are the battery and the lamp _ A oltmeter reads (t) s 2V An ammeter reads (t) s.5a A (t) B Whch dece s the battery and whch s the lamp? How much power s beng consumed by the lamp? Usng the passe sgn conenton The power absorbed by dece A s W = -8 W A s the battery and B s the lamp, whch consumes 8W MAE4 Lnear Crcuts 8

9 Crcut arables (contd) Ground Because all oltages are measured between two ponts, there s no absolute oltage - t s a potental In crcuts we often refer all oltages of ponts as beng relate to a fxed pont or ground oltage Ths s depcted a the symbol A (t) B (t) C (t) A B C The terms A etc mean the oltage at A relate to G Ths s lke referrng heghts relate to sea leel Graty s also a potental Check out the eleator buttons n the Math & App Phys Buldng _ G MAE4 Lnear Crcuts 9

10 Crcut arables (contd) A crcut s a collecton of nterconnected electrcal deces For us: All electromagnetc nteractons n the crcut take place wthn the deces (Ths s a lumped-parameter crcut) The crcut deces are connected together by wres whch are deal They hae the same oltage at both ends nstantaneously They propagate current wthout loss nstantaneously They may be stretched arbtrarly wthout changng propertes All crcut deces hae at least two termnals and are assumed not to accumulate charge current n equals current out A node s the juncton of termnals of two or more deces A loop s a closed path formed by tracng thorough an ordered sequence of nodes wthout passng through any node more than once MAE4 Lnear Crcuts

11 Crcut arables The mportant physcal crcut arables are currents nto and oltages across crcut elements and ther assocated power consumpton or producton. Crcut analyss conssts of solng for the crcut arables n a gen crcut Crcut desgn or synthess conssts of constructng a crcut whose crcut arables exhbt a specfed behaor The topology of a crcut s the key structure It can be stretched and wrapped around wthout changng crcut arables Nodes can be stretched and shrunk to coer many termnals MAE4 Lnear Crcuts

12 Crcut Topology Ths s the crcut dagram of an audo amplfer It shows the relatonshp between parts but not ther actual physcal locaton The ground/earth s lkely the chasss or case The output transstors BD92 and BD9 wll need to be attached to an external heat snk The op-amp OPA64 wll be board mounted The olume control needs to be accessble MAE4 Lnear Crcuts 2

13 Crcut Topology hgh power amplfer wth complementary HEXFETS Crcut Dagram Crcut Board Layout HEXFET power MOSFET from Internatonal Rectfer MAE4 Lnear Crcuts 3

14 Basc Crcut Analyss (T&R Chap 2) Krchoff s Current Law (KCL) The algebrac sum of currents enterng a node s zero at eery nstant Ths s a restatement of the prncple of conseraton of charge Alternately, but not so nce, The sum of currents enterng a node s equal to the sum of currents leang a node at eery nstant KCL prodes lnear constrants between the currents n a crcut In a crcut contanng a total of N nodes there are only N- ndependent KCL connecton equatons MAE4 Lnear Crcuts 4

15 MAE4 Lnear Crcuts 5 T&R Exercse 2- p 22 Wrte KCL at nodes A, B, C, D Node A: Node B: Node C: Node D: If = ma, 3 =.5mA, 6 =.2mA, fnd the rest 2 =ma 4 =.5mA 5 =.3mA Note that we hae 4 nodes But only 3 ndependent currents D C A B = = = = " # % & = " # % & " # % & ' ' ' ' ' '

16 MAE4 Lnear Crcuts 6 Krchoff s Voltage Law (KVL) The algebrac sum of oltages around a loop s zero at eery nstant Ths s really a restatement of oltage as a potental, effectely a statement of the prncple of conseraton of energy Example: _ Loop Loop 2 Loop 3 Loop3: Loop2 : Loop: = = = " # % & = " # % & " # % & ' ' '

17 Crcut Analyss One way to consder the analyss of a crcut (.e. the computaton of the currents n and oltage across the crcut elements) s through the specfcaton of constrants on these arables KCL and KVL prode one set of constrants on crcut arables reflectng connecton topology The - propertes of the deces themseles prode further constrants N nodes and E crcut elements N- ndependent KCL relatons E-N ndependent KVL relatons (number of ndependent loops) E ndependent - characterstcs 2E crcut arables If we keep to lnear deces then we hae lnear crcut analyss 2E Lnear algebrac equatons for 2E arables MAE4 Lnear Crcuts 7

18 Lnear Crcut Elements Lnearty n - relatons Doublng doubles and ce ersa =α Straght lnes n - plane Note that =αβ descrbes an affne relaton Resstor =R or =G Ohm s Law R resstance n ohms (Ω), G conductance n semens (S) Power P= 2 R= 2 /R= 2 G A resstor always absorbs power - lne has poste slope Ths s an nstantaneous lnear element We wll look at crcut elements wth memory lke C and L later MAE4 Lnear Crcuts 8

19 Tme out puzzle We hae three lamps red, green and blue connected to three swtches as n the dagram 32V S S 2 S 3 The lamps are outsde the swtch room and we are alone What sngle test can we perform by playng wth the swtches to determne whch swtch s connected to whch lght? I know of two solutons, one of physcsts one of engneers MAE4 Lnear Crcuts 9

20 Lnear Crcut Elements (contd) Independent Voltage and Current Source (IVS and ICS) IVS = s for any s s ICS = s for any These are affne elements rather than lnear. But we admt them here. Open crcut and short crcut arants of ICS and IVS resp. Open crcut = for any Same as R= Short crcut = for any Same as R= These are truly lnear snce the cures pass through (,) Swtches ether o.c. or s.c. MAE4 Lnear Crcuts 2

21 Example 2- (T&R p. 3) Wrte a complete set of equatons for each element A =3V; = ; 2 =2 2 ; 3 =3 3 ; 3V A A 3 Ω Ω - 2 2Ω Wrte a complete set of connecton equatons - A =; - 2 =; -3 3 =; 2-3 =; Sole these equatons for all currents and oltages A =3V; =V; 2 =2V; 3 =3V; A =-2mA; = 2 = 3 =ma; MAE4 Lnear Crcuts 2

22 All you eer wanted to know about resstors Resstors are color coded to ndcate ther alue Black Brown Red Orange Yellow Green Blue Volet Grey Whte and tolerance 5% % 2% Gold Sler Empty Ths s a KΩ resstor wth 5% tolerance Resstors come at standard alues (e.g at % tolerance) Nomnal Lower Upper MAE4 Lnear Crcuts 22

23 Een more than you eer wanted to know about resstors The dstrbuton of resstances Why should ths be so? How can we get a 258.3Ω resstor? Keep testng the ones n the box A really fnely tunable potentometer -% % (arable resstor) a trmpot Nomnal alue A coarsely tunable potentometer n parallel wth a lower R Make one from hgh precson components Another mportant characterstc of R s the power ratng Why? MAE4 Lnear Crcuts 23