Chemical Instrumentation Design and Control Applications
Chemical Instrumentation Design and Control Applications CHEM 628
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This 85 page Class Notes was uploaded by Ms. Jerrell Lind on Thursday September 17, 2015. The Class Notes belongs to CHEM 628 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 39 views. For similar materials see /class/205350/chem-628-university-of-wisconsin-madison in Chemistry at University of Wisconsin - Madison.
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Do Not Use Pena v a c 39 Do Not Staple Phase Course Chem 63quot Lecturer Hana45 Day M Date yq W Notes Taken by P3 M Page I of Lt Total Pages Submit a COPY of these notes for posting please L03 h M C0Mbr wmj glwl 3amp7 manila 1 Ca 7 b5 A LAM J Gz Anal 0 Pub3 Do Not Use Pencil Do Not39 StapleP1ease 39 39 Course 69quot Lecturer HM Day M Date I 31quot Notes Taken by I PINW Page 7 of Lt Total Pages Selbmit a COPY of these notes for posting please 0 A V H 450 NAIOD a Do Not Use Bend 39 39 39 39 39 L Do Not Stspfc Please Course Lecturer Haw Day M V Date 3 V Notes Taken by P1quot Page 3 of 439 Total Pages Submit a COPY of these notes for posting please 06 Marsavb ILL 5M 76 3w l 3wquot QQLCOS I WWU AND 9 012 lays7L ill WWW SI OW LPWB jD w 2 j Mu1e lcxcr Do Not Use Pencil i r V Do Not Seaplc Please Course 67quot Lecturer HQ 5 Day M A V Date 31 I Notes Taken by 95 M39W I Page 1 of Lt Total Pages J Submit a COPYof these notes for posting please quot 5 6 0m I Om quotvth H1 Iupud HMquot w W B39G39u LN 5 8 a Q J 1091qu g quot Ou fo a s 284 02 V ILM dado 3 W 5m M14 P f a7 Mwmirce ADQ39NOt UseEeIzc r DoNot StapleP1case39 Course Lecturer Q I k V Day 39 Date 0 29 Notes Taken by r 14 4 WW Page of Total Pages V Submit 3 COPY of these notes for posting please W spew 6f W39m i b MW vag new Wanms mww W 3 fa cmh We 39 6 M4 1 FM W NWT Awu zy 399 A W VMWZ CW WWW PryW yhkf 20 par3014 1 meW W 174 6 MEW a W XS ammof 1 Wham CmwvIceif Do Not Use P611617 v t I 4 Do Not Staple PIc Course Lecturer Day I Date 02 RelO Notes Taken by AZ V A Page I of Aquot T oral Pages Submita COPY of these notes for posting please 7 49 1 Okay 1 m s quotinfum Wf t M WW6 anW W ow P a WWW 1 14 em W 0 lt6 39 P A anquot bofwtc 39 lezzi 8 M WWW 409Wquot QM M51 AW 3ij gcwea SHPSYHD5 39 ypa 391 fe4W Q mm W393 Do Not Sizple Please 7 Do Not Use Bend Course Lecturer p05 Day e5 I 39 Date 0 L I a V Notes Taken by 39Pg fLOW UL WW Page of 39 V Total Pages Submit a 15 of these notes foryposting pleasa SMWSMB v F Le wv e l J 10er mo Haah 3 H 04quot 39 MiG49 N514 045 MW 5 1 n a quot lP T 6 quot54m W v m 013 om 0153 cg quot 37 gm k a G 39 me v3 sz quot We WWquot 37 7536qu WIquot 39Q 4m 1 Jim F QQW rmme vazqm Gangf m r Pl b bii j 39 39Do Not UseRen iI 39 u 1 Course Lecturer QJQ WM Day k Date 0 r Notes Taken by Km M L Ma 39 Page 3 of V Total Pages Submit a COPY of these notes for posting please WWW T MM W Work 07145 MAVNLV semch Do No Staple Plea r a Do Not Use P6110 r r Do Not Staple Plea e Coursc 39 V Lecturer I Day W M 39 Date 0 o Notes Taken by 9 amp Page of LI 39Total Pages Submit a COPYof these39notes 03 posting please V 0 Maia Do Not Use P611617 A I v 39 Do Not Staple Pleasequot Course 6 8 Lecturer p114 Day WWM W lt0 3 LI 9 Notes Taken by VI 7W 07 WM Page f of I Fetal Pages Submit a 4019 of these notes for posting please 16ng gagawjm39m gt 5 9 L L y squot i l39 mi J5 m1 maMNA MLMW ampW a 39 Mwbww iplw 39 39 ATPILCUWW SLAM 39 P W 1 0 B 5 39 r I gw mf WWWW ML QM 39 aw M5 7 am g QED i l Do Not Use Penal Course Lecturer 7 gt Day L Date 47 0 Notes Taken by In 0 Page 2quot of 3 otal Pages Submit a COPY of these notes for posting please Do Not Staple Please 39v CUM Mr 39 Luq I 34 my jg l owpw E 13 I 3 V 01554 WW I M V A TTL Twalw m M Le 397 xk L I HCHWS H gL39 le CMa Wlum My ch GmW er ugh P0 V xfgg E grpokm V I H611 1 I M ltth or 22 23V MW 1 MW mu 0gi T 036 1 0 PAtmnj fo mwm Do Not Use Bend Do Not Staple Please Course 3 39 LOD Lecturer 5 v H mt ir Day 2 3 f9 39 Date 39 9 A zp q I 1 f V e U Notes Taken by anlelk WW Page i of LIL T otal Pages Submit a COPY of these notes for posting please 39i pf39U W 5651 due L2g OP WS vo v4 w 5 r 1 5 t2 39 V RF 2 qrez VW QCM r W IQF QL We 4 1 RF HZ 392 vs Ly 167H M 39 LHLP39Z ww Maw 39 1f 614 7 7 quot Jm M 9 A 39 V ivgwa y g Vm4gt1 Um 5quot 2 63 1 over14W 5 9f 0p My 67H qu 3m J 00 Not Staple Please Do Not Use Pena Course 6 4 I Lecturer Day M 0 fa 4 39 V r Dage 0 2439 L 0 Notes Taken by 4 1A ALMAQ Page 1 I of oral Pages Submit a CaOPY of these notes for po39sting please a f P F V A 23 39 L a g a 6qu WW W HgtgtL V Wu akaRF r LPWW 00 W gmrga IN ENVWWVL 6 Wear Haw V i Do Not Use Pencil 39 I 39 Do Not Staple Please Course 44 l 39 Lecturer Ukm Day gmz f Date 27 gt1 0 I Notes Taken by l UmL4 W Page of Total Pages ubmjt a CgPYof these notes for posting please I22 2 529quot gt3 1 v 3 VVM V4 7 12f m 1wi LY 39 1 ZV UM M9 m RF RF I V3Um3vw Y 3 L 7 1F 39 My v L Pr r 911 3 39 M26969 T 1 Wv o thM V W V quot Do Not Use Pencil L V Do Not Staple Please Course Lecturer Day I l a I Date 0W Notes Taken by gt471 VI 4 WW39 Page 4 39 of g Total Pages Submit a C Yof these notes for posting please WW1 EWA 4 1 w 1 t 3 I21 1 2 k gtwbwm syw u quot WWquot quot 1 v39 quotMi4L 24v 4 Vw Hf quot4 Kai X W f Mr 4sz m We W WWW 5 XJVMquotRT 73454 JL 3 law 5ch W193 WM gymf Chem 628 Fall 2005 Lecture Notes Analysis of RLC Circuits For even relatively silnple circuits such as low pass and high pass filters most problems will typically involve solving a differential equation However even relatively complicated circuits consisting of resistors capacitors inductors and voltage sources can be analyzed much more easily by using complex notation This method will work for all RLC circuits To do this we first define a complex voltage V Bold face letters will denote complex quantities The real measureable voltage is then VmeasuredReV Likewise we define a complex current I The real measureable current is then lmeasuredReI Then we define the complex ilnpedance as V ZE I Why does this method work Complex notation allows us to include the fact that for an AC input signal the output signal will generally be shifted in phase with respect to the input signal but will still be proportional to the input signal ie the response is still linear Note that currrent and voltage are directly measureable quantities and so must be REAL numbers By representing the ilnpedance as a complex number we allow devices like inductors and capacitors to change the phase of the output signal relative to the input signal in addition to changing the amplitude For silnplicity we usually assume that the input signal in complex notation is VV0eiOJt V0cosnt isinnt Then ReV V0cosnt This is essentially the same as what we assumed in our earlier analysis of the low pass filter in which we assumed that the input signal was of the form VinV0cosmt and then solved a first order differential equation to get the output signal Example 1 Let39s look at a simple example first Z 7 l VinVocosnt c imC imt C Re V06 IRe Here from the definition of capacitance we have QCV Differentiating we get E C If V V0810 then E inOeZWI SO that I iaCVOeZwl iwCV 1 Then Z E r of a canaritnr V 1 This is defined as the cnmnlex I imC Since I imVOeiwI Imeasured ReI ReiaV0 cos mt wVO sinat aV0 sinat This is identical to what we found earlier as the solution to the differential equation Inductors V V As we saw before the inductance is defined by L E v so that dydt I If we assume that the input voltage has the form VappliedV0coswt then V V dydt 4L1 cos mt Then 1t 1005 un t The solution to this V is 1t L 0s1nat C where C is any constant We can often ignore this a constant value which is essentially a result of the boundary conditions and write V V It Jsin mt 41 cos wt 900 wL wL If we plot the current through the inductor as a function of time it looks like O A 3901 A 3901 Voltage current arbitrary units 0 0quot o I 2 0 2 4 6 8 1O 12 14 Time Here the current reaches a maxilnum value 11 2 or 90 degees after the voltage does Thus the current through an inductor is said to lag the voltage by 90 degrees If we use complex notation then we can again write the applied voltage as 1 1 39I 1 ital 1 V v 1wtw th h 1 tht V mdt V v m 08 e an ave LI 0 L1 08 mm 06 mm 0 Using the definition of complex i1npedance gives V E T T 139 wL This then becomes the definition of the complex impedance E of an inductor So we have Zresistor R anpacz39tor 1 WC wC Zinductor iwL Combinations of ResistorsI InductorI and Capacitors The real power of the complex notation method comes in when we deal with even more complicated situations Let39s look at some general rules Complex impedances in series VinVocosmt ReV0ei Dt Kirchoff s Laws give V V V 12 12 22 3IandVV1V2V3 Z1 Z2 Z3 V Iz1 IZ2 128 1Z1 Z2 Z3 Ize ec ve where Ze ec ve z1 Z2 Z3 1 nmnlpx I in parallel VinVocosmt ReV0ei Dt 00 V V V l l l V I 1112 I3 V where Z1 Z2 Z3 Z1 Z2 Z3 Ze eclive l l l l Ze ec ve Z1 Z2 Z3 Thus the rules for adding complex impedances in series and parallel are identical to the rules for adding resistances Example 2 Lowpass filter VinVocosmt ReV Oei wt Perfect Voltmeter In this case we know from Kichoff s Laws that VVABVBC and I ABIBC Using ohm s law for the resistor and definition of complex impedance for a capacitor which we just determined above gives 1 VABIR and VBCIanpacitor 1K zwc 1 1 V V IR IK IKR or I Substituting this expression for I sz sz 1 into the above expression for VBC gives V IZ V K 1 j V V 17 iaiRC BC ca acitor p R 1 le 1 1wRC 1 QRC2 iwC Now we remember that VBCWWSWM ReVBC However since both V and f 1 szC2 are complex we need to be careful K1aRC 1 wRC V 2w Reltvgt ImltVgt Bcmeasured BC 1 wiec2 1a2RC2 But VV0ei Jt so ReVV0cosoat and ImVV0sinoat so that sat 1 mR VBC co sinat measured 1 wRC2 1 wRC2 This is exactly the same solution as we found earlier In this case the complex notation saved us from having to solve a differential equation A more cnmnlicated The RC Band ass Filter This is an example where if you didn39t have complex notation available you39d probably just throw up your hands and give up With complex notation it39s straightforward although possibly tedious I I1 3gt A gt B I Z3 D o VinVocosmt Z2 12 Loop Z45 14 ReV Oeimt Direction irectiOn 1 l O C E Kirchoff s Current laws give I1 Iz I30 and I3I4 Kirchoff s Voltage Laws give V VAB VBC 11Z112Z2 VBcVBD VDE 0 12Z2 I3Z3 I4Z4 We thus have four equations in four unknowns I1I4lf we want to find the output voltage we need to find I4I3 The four equations above can be easily siInplified to three equations in three unknowns 0 11 I2 13 V IIZI 12Z2 0 12z2 13z3 Z4 or in matrix notation 1 1 1 Vl 0 VI Z2 W W 0 Z2 Z3 Z4 13 0 The solution to this equation is VZ 1413 2 Z1Z2 Z3 l l Substitutin in R Z Z Z R ives g Z1 1 2 WCZ 3 WC3 4 4 g F l V l 1 l 14 21m C I 1 1 1 I WCZ sz3 szlCZ V 4 39 R C 1 39R 39 M71 2 imR1C22 4 4R4 le2 Kg R1C2C3 WRICZ wC3 V 14 R1 1 R4 in1C2R4 wC3 C3 I V0cos mtisin mt 4 R1 R4 R1 i 1 wR1C2R4 C3 wC3 V0 cosalisin wt R1R4R1 amp 7139 i wR1C2R4 C3 wC3 I4 2 2 R1R4R1 l ia lC2R4 C3 wC3 Then the final voltage drop measured across R4 is given by C 71 V0R4cos 500 isin wt R1 R4 R1 J 7139 wR1C2R4 C3 wC3 VDE MM 7 2 2 I C2 71 R1R4 R1 aR1C2R4 C3 wC3 and finally C2 71 V0R4 R1R4R1 cosagtl V0R4 wR1C2R4 smagtl C3 wC3 2 2 2 2 C2 1 C2 1 R1R4R1 C CaR1C2R4 R1R4R1 C a2R1C2R4 3 a 3 3 Vout Re VDE wC3 A cos an B cos an Note that this result has the general form V0 V0R44A ZBM2 2 where C A R1 R4 R1 z C3 be and B 7 wR1C2R4 The amplitude of the output will then 60 3 Vmax V0R4 and the transfer function T will then be given by rms Peak Vout V t R4 R4 T k Virnms Vile A2B 2 C2 2 1 2 R1R4R1 wR1C2R47 C3 wC3 Do Not Use Pencil 7 Do Not Staple Please Course 39 7 Lecturer Haw f 39 Day m Date 1 I Notes Taken by P af Page I of 5 T eta Pages Eillbmit a COPY of these notes for stting piease 0 Amp 510541 I RF V 12 cm 771 V 1 lt V 39 4v Mquot I 1 oquot v3 b h 39 67 H 439 6 a V quotprewuiics W W quotquot mpg 14 w ML M a J pwmg I 139 PpMP wcz PA a we 091M G agitava 62H czar H L L v 3 671701 37 Ca fOOO 345 pm 39 9 3 1 Fr ap 77 4 p6 7 I 5 67w 40m Do Not Use Bend Do Not Staple Please 7 A f Course e Lecturer Ha quot5 Day L M Date 9M Notes Taken by pm 3 Page 2 6f 7 otal Pages 39 Submit 2 COPY of these notes fer posting please 3 an vow uan Lad 3ka Crngioum 3W5 g e f Aprl cr ew mam 7 Mmawgs SAWquot 0 OP Anfj Ham 67H 300l TL Mtgc Gy v can I at man n 4 Me 59603 Do Not Us fend 2 a s 39 39 Do Not Staple Please Course Lecturer H wawb Day gt V N Date Notes Taken by Mu3 Pag 3 of 1390th Pages Submita COPY of these notes for posting please mug 39 I m 0P Amp Has a PMS Sk V 539 v x r R Z V V A huh Pas3 We kag q Phase Slu39 Va rquot A I l W B WowK3 6 Pmiar op QMF Hm W Flatww w C kJ k clog 0 04 VI Do Not Use Pencil 39 r r 39 Do Not Staple Please 39 Course 7 61393quot Lecturer 39 Hi HI fquot Day M Date 2 Notes Taken by Po 3 L Page 1 of 4 Total Pages Submit a COPYof these notes for pesting please bk 2 30 74 06 acute f a F f ILH we 6x105 H I 639 651031104 gl b 3 L xO 390 353439 3 7 J xm 00 Not Use Pencil v 1 v 39 L 39 Do Not Staple Please Course I 1 9 V Lecturer Ham K Day M Date gK Notes Taken by s V3 Page r of 9 Total Pages a Submit a COPY of these notes for posting please alga O IIquot 39 d 4571 L Natsc Envirmi l l 3 39 39 Vol39h gc 39 I 1 7 7 MT 7 b t t C g finwa D0 th Use Frinail V 119 Not StapIe Please Course 39 L cturer H M Day L M Date 28 Notes Taken by 39 P 4 3 r Page 5 of 5 T otal Pages 39 Submit a COPY of these notes for posting please V 1 No 2N5 n 7 f I I MS A Eelwt c In x 339 3 52 ValL 39 1M 2 Ha For MnCDfrol c39lal foamtag 4 98 A Vw MW a v54 Avgquot 5 8 36 2 Cuuud39 Wadr6 Johan N036 39 7 W 7 Vow yawn Anf sz L 39 im 2 V q k l Niquot V Avmr ZRKW M attumu 9 VI39MZ aclug Do Not Use Pencil Course Day 39 Wt Notes Taken by 36 9 av 1 Do Not Staple Please Lectur x Ham 5 39 Date NzAa39ov Page I of 7 Total Pages Submit a COPYof these notes for posting please Dug F54 I M Problem 66 quot I I I X5 M 3 Xialy Wan 40 b5 7 105 4D 9 1055 I gt 75 100 XNQO Do Not Staple Please Do Not Use Pena n V39 Course Lecturer V fwquot 5 Day V39V Date 2 39 Notes Taken by R39v D Page 2 of 1 Total Pages Submit a COPYof these notes for posting please 39 121 5 2650 at 39 Q col quot L 6239 a QMLb F34 gnawed A 4 Resonate I LL03 J Do Not Use Pencil r 5 7 39 39 Do Not Staple Please Course 7 39 V Lecturer y 5 D357 W Vv Date 2 7 1 Notes Taken by P AMquot 3 39 Page 3 of v Fetal Pages Submit a COPY of these notes for posting please CMva s L7 IMMemut EM 39 Sm lt1 i C a I 1 kw cm ruJWu a 74 L J mi Gravy Wri I 3 E7 w Zhumc I s C 7 chamul im g Qt W ra c a FL EluJal 77 N0 R0440M390M I4 f x 7 3 9 v I 3 0 gwaL a 4 ak 3900 6M4 5 l 7Z5 For Hfak F0 6 a g iwoos1 b amp j 39 w use Do Net Use If ncz l Do NotStsze PIC33 Course Lecturer Haka Day Date 27 Notes Taken by 39 PS us Page I39 of V Total Pages Submit 3 COPY of these notes for posting please 33911 New E w 45 Noise in Analog Circuits As we discussed earlier the noise properties are often of paramount importance in analog circuit design Noise is typically described in terms of its frequency spectrum which describes how prevalent the noise is at different frequencies Noise is typically described in terms of the noise power However in this respect the jargon makes somewhat sloppy use of terminology and what is called noise power is often in units of V2 There are several types of noise a White noise White noise has a constant frequency spectrum b lf oneoverf noise sometimes called pink noise The noise power varies like lf c Environmental noise This refers to any noise source that arises from wellde ned sources in the environment the most common would be pickup at 60 Hz for example from the 110 VAC power lines Environmental noise f 2 White noise f 2 pink noise f 2 Vn l Vn Vn l l f1 f2 f1 f2 f1 f2 freq uenc frequency frequency If a source that has a voltage noise spectrum given by vnf is measured over some range of frequencies extending from frequency H to frequency f2 then the meansquare noise is given by f 2 f2 V V f 2 df and the RMS noise voltage is V V f 2 df f1 f1 From this relationship you can see that the noise voltage vnf has units of VoltssqrtHz commonly referred to as volts per root Hertz The meansquare noise is then given by the area under the curves that are shaded in the gure above Fundamental origins of noise Some sources of noise are fundamental and arise directly from thermodynamics and the quantized nature of electrical charge 1 Johnson Noise Johnson noise arises from the fact that in a resistive material the spatial distribution of electrons is uctuating in time At any given point in time there may be more electrons on the left than one the right Since the electrostatic potential depends on the spatial distribution of charges there is an accompanying uctuation in the potential voltage measured across a resistor So if you put a perfect voltmeter across a resistor you would see that the average potential would be zero but it would be uctuating above and below zero 8 More electrons on left than right More electrons on right than left Johnson Noise arises straight out of thermodynamics The uctuations in potential have a frequency dependence such that that the squared voltage noise density is given by v 2 4kTR units of voltssqrtHz f2 AVW I4kTRdf 1 4kTRAf where k is Boltzmann s constant T is absolute f1 temperature R is the resistance and Af is the bandwidth of the measurement Shot noise Similarly quot Shot Noise arises from the fact that current is not a quot uidquot ow but rather the motion of discrete electrons Current refers to the average rate of ow of electrons Shot noise is the statistical uctuation in the rate at which electrons move Like Johnson Noise shot noise is quotwhitequot in 2iq in units of ampszHz where I is current in amps and q is the charge on the electron 1 602x103919 Coulombs Then Ms J21qAf Noise in circuits The noise characteristics of electronic components and circuits are Virtually always eXpressed in terms of noise densities Because electronic circuits always have some input voltage and some input current we actually need two quantities to eXpress the noise properties These are the voltage noise density in units of VoltsVH2 and the current noise density in units of AmpsVH2 You should further note that we can eXpress the noise in terms of its effect on the output of a circuit or opamp or we could eXpress the noise as if it were superimposed on the input signal In the former case we speak of noise quotreferred to outputquot or quotRTOquot while in the latter case we speak of noise quotreferred to inpu quot or quotRTIquot In understanding the overall behavior of an electronic circuit we will usually be more interested in noise referred to the output or simply the total output noise However individual circuit elements usually have their noise properties speci ed referred to the input In any real circuit there will almost always be several sources of noise In order to determine the total noise we must know whether the individual sources of noise are correlated or whether they are uncorrelated That s equivalent to saying that we must know whether the circuit elements generating the noise behave independently If they behave independently then the noise generated by the components will be uncorrelated For uncorrelated sources of noise the total noise is determined as 2 2 2 Vnaise wml VN1VN2VN3 Let s look at how to predict the output noise for an electronic circuit We ll start with the non inverting ampli er The overall noise performance can be determined by considering the noise sources one by one assuming that they are uncorrelated and nally performing a sumofsquares as above In put voltage Noise Let s consider rst the effect of input voltage noise at the inverting input 0 V R R am Because of the input voltage noise the voltage at the inverting input will 1 F be uctuating by the amount vn If V50 and the opamp is functioning properly then the op amp will adjust Vout in order to keep VV0 This requires that the output of the opamp uctuate We have 0 u viz V2 R1 RF This can be easily solved for V0 to give R Vow vn Kl R F More correctly we should remember that vn is a uctuating voltage and l R write this asAV0mvoltage V vn l R F This number represents the uctuations in l the output voltage arising from the noise voltage at the inverting input What about noise at the 0 v v noninverting input Again we have W which again F R R gives Vow vn l EE or AVaulvoltage VJr vn l EE This number represents 1 l the uctuations in the output voltage arising from the noise voltage at the noninverting input Input Current Noise Input current noise can be interpreted as a uctuating current going into the inverting and non inverting inputs At the inverting input we have 7 in 7 0m where in represents the uctuating current into the inverting input R R1 But VV0 soAV0m inRF What is the effect of current noise at the noninverting input It depends on the output impedance of whatever is driving the opamp If the noninverting input is held at ground then noise current at V cannot create any voltage at V and the output uctuation will be zero However if we write the output impedance of the signal source as Z then through a similar equation we would getAV0m inZ Now let s look at the total noise arising from both voltage and current noise at both the inverting and noninverting inputs we have a r 12 F 72 LanHfTTJJ lvnl1ijl RFf Zf 2 2vl f i 622 There are several important points here First of all look at what happens if the source impedance is high In the limit of large Z the output noise becomes AVWI E inZ From this we can see a general rule At high source impedances the noise in electronic circuits is almost always dominated by the input current noise At low source impedances noise is usually dominated by input voltage noise Let s put in some typical values We ll consider the Burrbrown OP627 Operational ampli er It has an input current noise density of 17 fAHz and an input voltage noise density of about 5 nVHz Let s assume that we made an ampli er with a gain of 10000 using a 100 ohm and a lMegohm resistor The gainproduct bandwidth is 100 MHZ so a gain of 100 ampli er will operate out to frequencies of approximately 10 kHz Thus the quotbandwidthquot Af will be 104 Hz The input voltage noise will than be 5X10396 volts and the input current noise will be 17 X 103913 amp Let s assume rst that we simply ground the input so that the noninverting input is connected directly to ground The output noise will then be 2 7 106 713 6 2 25x10 1 l7x10 10 7lmi111volts Looking at the magnitudes of the numbers in the equation above you will see that the total noise is almost completely dominated by voltage noise because the source has zero impedance In this case you will see that the voltage noise continues to dominate the performance as long as Zsome is less than about 1010 ohms total Vaut We have thus far forgotten about the Johnson noise of the resistors How does this contribute Let s consider the Johnson noise of R1 We have O V V V0m R R but now the voltage across the res1stor 0V has some uncertainty in 1t 1 F Let s for the moment assume that Vin 0 Then 0V VJohnsonR1 VJahnsarlRl V0m R1 RF out F or AVJahnsarR1 R1 VJahnsadR l Now the effect of Johnson noise of the feedback resistor RF This appears as uctuating voltage between Vout and V and so appears as 0 V V7 Vaut VJahnsa RF out T RF Us1ng V0 g1ves AVJOhnSOKR 2 vJ0hnsmR2 Note that the Johnson noise of R1 gets quotampli edquot by the ratio of RFRl while the Johnson noise of R2 is not Through a similar analysis we d nd that the Johnson noise of the source resistance Z would appear at the output as AVESZMOKZ vJ0hnsaZ For a 100 ohm resistor at 300 Kelvin Av1msm12x10397 Volt For RF AVJohnsonl27X10395 Volt For AVSource we havel7xlO398SqrtZ Since again these noise sources are uncorrelated we can take them and using the sumofsquare approach calculate the total noise I 2 2 AVttbbial Alape amp AVJahnsan 1x1032 127x10 52 12x10 72 12x10 822murce Provided that Zsome lt 1010 ohms this again reduces to about 7 1 millivolts Noise in the current to Voltage converter In the currenttovoltage converter it is very common to use quite large resistor values in the feedback loop In that case the Johnson noise of the feedback network becomes limiting Since Kirchoffs law says that the current input and the current through the feedback resistor are the same this Johnson noise generates a current that looks just like a uctuating input current A V7 V0u2 VJ0hn50nRF m R F AVm VJohnsonRF V 4kTRAf One important point is that the noise in a currenttovoltage converter scales like R However the signal gets ampli ed by a factor of R So the signaltonoise ratio of a currenttovoltage converter scales like JR This has an important consequence It is very common in currenttovoltage converter to need a very high gain on the order of 109 voltsamp You could do this using a singlestage ampli er with a 10 ohm resistor or using a twostage ampli er using an IV converter with a 106 ohm resistor followed by a regular opamp voltage ampli er with a 1000 gain Both these systems would have the same overall ampli cation However they would give different signaltonoise ratios The single stage ampli er gives a noise level of AKn VJohnsonRF V 4kT109Af while the twostage ampli er has a noise level closer to AVquot ijgnmm 1000 V4kT106 Af ampli er con guration will be 30 times smaller than that of the twostage design You can see that the noise level in the single This leads to the general rule In any circuit the noise will be almost completely controlled by the transducer and the rst stage of ampli cation You will almost always achieve the best signaltonoise in any measurement by amplifying as much as possible in the very rst stage In a twostage currenttovoltage converter for example the signaltonoise is controlled almost entirely by the rst opamp because whatever noise is generated by this opamp is ampli ed by the second stage Chem 628 Grounding and Shielding In general extrinsic quotnoisequot in instruments comes from three sources 1 Poor grounding and or ground loops usually at high frequency 2 Magnetic pickup from transformers and power lines 3 Electrostatic pickup from various points in a circuit Grounding in electronic circuits and between instruments Grounding is a very important element in making electronic circuits work correctly Since all that we can every measure are differences in voltages between two points our quotreference pointsquot ie quotgroundsquot are just as important as our quotsignalquot wires Although we have a tendency in schematics to only include the signals and generally leave the ground as an quotunderstoodquot reference point the details of how ground wires are wired and connected can drastically alter the way a circuit performs Particularly in high gain circuits poor grounding techniques can lead to offset nonlinearity and oscillation problems With a few simple rules most grounding problems can be avoided However in some cases you can t optimize the grounding the way you d like For example even though you can do the grounding within any given circuit box correctly you might nd grounding problems that arise when you start connecting several instruments together Some times the very act of trying to observe a circuit using an oscilloscope can introduce new problems because of the grounds involved With an understanding of how grounding problems arise you will be able to solve most grounding problems scientifically The quotgolden rolequot for proper grounding is Circuits should be constructed such that the flow of current in the g39round lines does NOT produce IR drops at undesired places within a circuit This can be restated as quotAll grounds are created equal but some are more equal than othersquot To illustrate this we ll consider two circuits in detail We39ll consider the circuit with BAD grounding technique rst Bad grounding technique The following circuit is a twostage amplifier which delivers an output voltage to a load represented by the resistor RL The output voltage which will be measured is the voltage drop across this resistor The wires connecting the grounds have resistances RG1 and RG2 The input to the rst opamp is connected directly to ground we39ll assume that this wire has no resistance The quotbad techniquequot part of this circuit is that the quotgroundquot on the output connector is connected to the input of opamp 2 which is then connected to the ground of opamp 1 This kind of quotlinearquot chaining of grounds from the output connector to the opamp 2 to opamp 1 to ground is called quotdaisychainingquot We39ll see that it can easily give problems RG1 RG2 For opamp 1 there is no problem at all we have VoutR2R1Vin As always we write the equations for opamp 2 as Vout1V2 R3 Na Vout2 R4 Note however that V2 is connected to ground through a wire with resistance and the current ow through this wire gives rise to an IR drop at the noninverting input The voltage at V2 is determined by the quotvoltagedivider equationquot since RL RG2 and RGl form a voltage divider we39ll asssume that the opamps have zero input bias current The voltage at V2 is then V V 2 01422 RGIRGZ RL Substituting in the equation above then gives R R R 2 Vm V0142 GI Vautl GI R1 RGIRGZRL RG1RG2RL R R V 011 2 Finally we get V V V git m 01422 In R1123 1L1amp RGI RGZ RL Ri The output voltage measured by whatever we connect is given by the voltage drop R across the quotloadquot resistor RL Thus Vm ILRL VOW L R31 RGZ RL This can be written as RZR R V L m Vm RIRS RL R02 RGI R3 The rst two factors are what we would have if there was no load indeed if RL in nity the last factor goes to one and we have that Vout VjIlR2R4 R1R3 as we hoped for in the rst place To understand how the ground current affects the circuit we need to look at the last factor If RL in nity then this factor becomes 1 and there39s no problem To understand what happens when there is a nite load note that the denominator contains a term RG1R4 R3 This term re ects the effect of the IR drop across RG1 which is ampli ed by the opamp circuit If RG1 is small compared to RL then as RG1 is increased the current ow through the ground line remains approximately constant but the voltage at V increases We can also look at this as a circuit having two feedback loops the feedback from Voutz to V is quotnegativequot feedback which stabilizes the circuit The ground connection from Voutz to V represents positive feedback which tends to destabilize the circuit To see how serious the back grounding is let s assume that the rst stage has a gain of 10 and the second stage has a gain of 100 so R4IQ 100 Let39s also assume that the load resistor R1 is 1000 ohms and that the quotgroundquot resistances RG1 and RG2 are just 1 ohm each it39s easy to get 1 ohm of resistance in a solder connection According to our result above Vmeasured an 100010001000100 1111 Instead of the expected gain of 1000 we got a gain of 1111 or an quoterrorquot of more than 11 You can see that the output voltage is increased by our bad grounding because a fraction of the input voltage is returned to the NONinverting input where it tends to increase the output voltage To see a more drastic effect consider what happens if the load resistor is only 100 ohms Now Vm an 1000100110000 H The gain is 100 times larger than it should be if R1 is slightly smaller than 100 ohms the output gain goes to INFINITYL Again it s easy to understand what39s happening as the output current ows it increases the voltage at V further increasing the output voltage We39ve got POSITIVE feedback and the opamp will easy saturate at one of the power supply voltages or else oscillate between 15 and 15 Volts Either way you won t get what you want Now consider the quotcorrectquot way of grounding as shown below The main difference here is that instead of quotchainingquot the grounds together we can think of the grounds as coming together into a quotstarquot con guration This type of grounding in which all the individual grounds come together at a single point is called singlepoint grounding and is generally the best solution to grounding problems Let39s analyze the circuit Vou V2IQ V2 Vout2 R4 As before Vou VjnR2 R1 Now however V2 0 since it s connected directly to the quotsinglepointquot ground and because the op amp inputs draw no current at least for an ideal op amp We get Voutz R2R4 R1R3an The voltage measured at the output is still going to be the voltage drop across RL which is Vm Vout2 RLRLRG2 R2R4R1R3RLRLRG Vin For the gain of 1000 ampli er we considered earlier with a load resistance of 10000hms and a quotgroundquot resistance of 1 ohm w find that Vm Vin 100010001001999 compared with the quotidealquot value of 1000 or an error of only 01 compared with an error of more than 11 for the daisychained arrangement Even from a load resistance as small as 100 ohms the error is only 1 whereas in the daisychained arrangement the gain was off by a factor of 1000 Additionally you will note that this circuit is stable for all values of RL whereas the daisychained arrangement would oscillate for low values of RL From the above discussion you can see that the best grounding arrangement is one which minimizes the possibility of having current ow in one part of a circuit create IR drops which wind up at the inputs to other parts of the circuit particular in highgain ampli ers What happens when you connect two instruments together Unfortunately this is where things get a bit more dif cult and grounding sometimes takes on more an aspect of quotartquot than quotsciencequot The techniques which one might use to get close to quotsinglepointquot grounding are not always those that lead to the best rejection of interference from noise sources for example Exactly how the quotbestquot grounding is done between instruments requires an understanding of where the noise originates the frequencies of interest ie the bandwidth of the measurement and other considerations Ideal configuration The ideal con guration of grounding between instruments as far as grounding is concerned would involve having each instrument grounded internally to a single point and then having each instrumentjoined together at a second single point Signals would then be coupled from one instrument to another through singlewire cables since all instruments would use one universal ground as their reference Instrument 2 Instrument Instrument 1 Common Ground Common Ground Unfortunately this con guration has some problems with magnetic and electrostatic pickup Most of the interferences that arise whether from poor grounding or electronstatic and or magnetic pickup get worse as the distances between the instruments increases Therefore the best solution to solving grounding and or interference problems is usually to place quotsensitivequot instruments close to the signal source where the lowest level signals usually occur In many cases one can ianrove the grounding situtation by making a very low resistance ground connection from instrument to instrument such that resistances on ground lines become negligible Note here that you generally want a very low resistance and so you want to use thick wire and good solid mechanical contacts In my research group we routinely connect all our instruments together with 34quotwide quotgrounding strapquot to eliminate small ground loops which cannot be eliminated any other way Electrostatic Pickup Electrostatic pickup arises because signal and ground wires have capacitance to other wires in the circuit Voltages changing in one wire will therfore induce a current ow in adjacent wires according to the normal capacitance relationship ICdVdt Because the currents are proportional to the rate of change of voltage electrostatic pickup is generally most serious in highfrequency circuits but lowfrequency circuits can also be affected by electrostatic pickup If you imagine inserting a capacitor between a quotsignalquot source with finite output impedance and an amplifier assumed to have infinite input impedance as shown below it s easy to understand electrostatic pickup Changing the voltage on one plate of the capacitor which is actually an adjacent wire in the circuit allfluctuating dt induces a current I ucma ng C This current in turn induces a voltage dV uclualin g drop equal to AVin Zsaurcel uctuaz ing ZsaurceCT Where Zsource 15 the output impedance of the signal source Here we see one immediately important consequence of electrostatic pickup its importance in voltageamplification circuits scales with the impedance of the signal source Thevinin Model R1 R2 R1 for Signal Source Zsource Zsource Vsource i V R2 Vout I f I uctuating source V u ctuating V uctuating Stray Wires etc l Thus high impedance voltage sources such as pH electrodes and some kinds of optical detectors are particularly susceptible to electrostatic pickup In contrast electrostatic pickup between opamp stages for example is small because the output impedance of opamps is small In current amplification circuits such as an I V converter the capacitivelycoupled current looks identical to a quotrealquot signal current and will be amplified again this is easy to understand if you think of a current source as an ideal battery in series with a very large resistance infinite output impedance Electrostatic pickup can be minimized in several ways First one can put a quotshieldquot around signal wires with the shield connected to a good ground In that case the capacitive currents can ow through the shield without inducing an IR drops the signal wire sees that it is surrounded by a ground potential and does not see any uctuating electrostatic potentials This is basic idea behind quotcoaxquot cable and quotshiededquot twisted pair In both cases by surrounding the signal wires by a good conductor at ground potential the effects of electrostatic pickup can be reduced It s worth noting that the typical quotshieldquot of coax cable doesn39t actually cover the entire wire usually it covers 90 95 of the wire but you will still get some electrostatic coupling if you place another wire with a uctuating potential very close to your signal wires Thevinin Model R1 R2 R1 R2 for Signal Source V Vin out V1n Vom I ZSOUFce Zsource 1 R g 9 Load Vsource i Vsource 39 V uctuatin g Straywires etc I t f I lie uamg V u dilating Note that external uctuating voltages now induce currents in the ground line rather than the signal line Since the external quotloadquot usually connected to an opamp is on the order of kilohms or tens of kilohms rather than the Megohms often associated with devices such as pH electrodes photomultiplier tubes optical detectors etc the effects are reduced by many orders of magnitude If RloadltltZsoulce then it is easy to see that all this induced current will ow through Rload The opamp will keep Vout constant so that current induced on the ground line will NOT produce ANY change in the ouput voltage measured with respect to ground assuming of course that the ground line has zero resistancei One way of further reducing electrostatic pickup is to use quottriaxquot and or quotshielded twisted pairquot wire with a differential input on the voltage ampli er The idea here is that there are essentially two quotgroundsquot one serves as a reference point for the voltage measurement and should have only small amounts of current owing through it and the other quotshieldquot acts a s a capacitor plate to establish a constant potential and carry off the capacitivelyinduced currents to a good ground Any remaining electrostatic coupling which propagates into the two central conductors will usually affect both wires similarly so that a differential amplifier which responds to the difference in voltage between the two inputs can often get rid of the remaining quotcominonmodequot piclltup Magnetic Pickup Magnetic pickup is more insidious than electrostatic pickup and isn39t talked about too much in polite company For lownoise circuits operating at audio frequencies 1 Hz 100 kHz or so it s usually more problematic than the more commnlydiscussed electrostatic pickup Magnetic piclltup usually occurs at a few wellde ned frequencies 60 Hz and multiples thereof 120 Hz 180 Hz etc from 110Volt power lines 30 kHz 50 kHz or so from computer monitors and possibly higher frequencies hundreds of MHz from instruments lillte NMR39s From electrostatics you might remember that a loop of wire in a timevarying magnetic eld will have an induced voltage This is of course how transformers work and how electricity is generated at electric power plants If we assume that the magnetic eld B is constant in space but varies in time appropriate for a small loop of wire for example the voltage induced around the loop will be V AdBdt where V is the voltage induced in the loop B is the magnetic eld strength in Tesla where 1 Tesla 104 Gauss and A is the area of the loop in square meters The earth39s magnetic eld is 06 Gauss or 6x10395 Tesla First let s consider how big this effect is The magnetic field in the vicinity of a transformer can easily be 100 times the Earth39s magnetic eld or approximately 100 Gause 10392 Tesla If we assume that the magnetic field is varying at f 60 Hz then Bt001sinft and dBdt 0012pf cos2pftFor a loop wire 10 cm x 10 cm 001m2 the induced voltage is 6 millivolts Magnetic pickup is also quite serious from some other souces for example computer monitors use electrostatic coils to de ect an electron beam which strikes the phosphor screen the quotrasterscanquot of the electron beam is typically at 30 50kHz for a superVGA monitor which means that there39s a time varying magnetic eld at 30 50 kHz Although B might be small at this higher frequency dB dt can be quite large again producing voltages in your circuitry at 30 kHz If you don39t believe this can be signi cant just ask any of my students about 30 kHz monitor noise Magnetic pickup of this sort is quite problematic in circuits where one needs to measure timevarying signals in the range between 60 Hz and about 240 Hz While a lot of attention is typically given to grounding and shielding for highfrequency circuits the magneticallyinduced voltages can be a far more serious and dif cult to eliminate problem The stray magnetic elds from transformers and other coils basically looks like a magnetic dipole and decays rather slowly with distance approximately r0 r where r0 is the size of the coil generating the magnetic field Because 1 r isn39t a particularly strong function of distance it means that it s generally impractical to to eliminate magneticallyinduced voltage by simply moving the power supplies farther away although sometimes it solves the problem In general the primary solution to magneticallyinduced voltages is to make the circuit immune to pickup by making all circuit paths small effectively decreasing the area A Again when considering magnetic noise pickup in your circuit you must think about how your signal propagates through the circuit and returns through the ground line H You must think in terms of current loops not just voltages on quotsignalquot wires One way of minimizing the area A is to use quotcoaxquot cable in which the signal is applied to the central element and the return current ows through the quotshieldquot The insulator between the shield and the wire is about 1 mm thick 0001 meter making the effective quotareaquot of the loop about 0001cable length in meters Because the shield completely surrounds the central conductor you can think of this at least in two diJnensions as being like two loops one constituting the central conductor and the top part of the shield and the second loop comprised of the central conductor and the bottom part of the shield The current induced in each of these loops will be either clockwise or counterclockwise depending on whether the magnetic eld is incrreasing or decreasing which means the the currents induced in the top loop and the bottom loop will cancel one another Of course if the magnetic field is inhomogeneous such as a coax wire running right next to a transformer then the fields will not cancel and you will still have to contend with the magnetic pickup dBdT Coaxial cable radius r lenth 1 loop of area r1 1 Load L loop of area r1 1 dBdT ll dBdT A V Load U I i Another way of reducing the loop area A is to use quottwistedpairquot conductors in which the signal and ground wires are twisted around one another You can think of each quottwistquot as being two halftwists of 180 degrees each m Any current induced in the area of the first halftwist will be cancelled by the current induced in the next halftwist because the quotsignalquot and quotgroundquot wires are exchanged in spaceAs long as the magnetic eld B doesn39t change significantly on the distance scale of one twist the magnetic inducion will be zero Twistedpair usually does a better job than coax cable because the insulators are usually thinner making loop areas smaller Ways of reducing magneticallycoupled quotpickupquot in electronic circuits 1 Keep loop areas small by using coax or shielded twistedpair wire whenever possible and keeping wires short 2 Keep 110 V power lines away from your circuit Use twistedpair wire shielded twistedpair is even better for all 110 volt lines Assuming that you bring the 110V into an instrument through the back panel putting the 110volt quotonoffquot switch on the rear panel instead of the front panel will save you from having to snake 110V lines near your circuit and will almost always give you reduced 60Hz pickup 3 If you need really low noise at 60 Hz and multiples thereof consider getting rid of all AC voltages by removing the usually 110 VAC 15VDC power supplies from the box containing sensitive highgain electronics andputting them in a separate box This may create other problems due to pickup along the wire connecting the supplies and or IR drops along the wire connecting the power supply to your circuit Alternatively consider powering your circuit with batteries Small sealed rechargeable leadacid batteries are available in a range of voltages and can easily power a few opamps for 8 10 hours before being rechargedlt39s worth noting that commercial lownoise voltage and current amplifiers available from EGampGPrinceton Applied Research lthaco KrohnHite and others are all available with operation from 1215VDC sealed leadacid batteries Contrary to what you might have expected they use batteries not because the DC voltage produced is any quieter but simply because eliminating all 110VAC from the instrument reduces the 60 Hz magnetic pickup 4 If you want to keep the power supplies in the same box consider use power supplies with quottoroidalquot transformers instead of normal quotsquareframequot transformers The more symmetric design of the toroidal transformers reduces the fringing elds 5 Use speciallydesigned quotlownoisequot power supplies available from AAK and others these are usually quotpottedquot supplies which internally use toroidal transformers the designers have usually paid attention to magnetic shielding These cost about 23 times as much as regular supplies but have about 10x lower noise and reduced magnetic fringing elds might be worth it if you need really low noise 6 Magnetic shielding can help pieces of soft iron or quotmumetalquot strategically placed can reduce the fringe elds from power supplies This doesn39t always work as well as you might think it should dipolar magnetic elds can easily quotsquirtquot through holes in boxes and putting a magnetic shield in one place will often increase the eld somewhere else this generally requires some experimentation and can do more damage than good if you just do things blindly With some attention this can reduce magnetic elds by a factor of 3 or so without too much trouble 7 Pay attention to what else is in the vicinity of lownoise highgain electronics A computer monitor 3 feet away can be spewing out magnetic elds at 30 kHz oscilloscopes and other power supplies can also couple magnetically into your circuit over distances of a couple feet Because magnetic elds are usually dipolar elds turning a box on its side can often change the pickup dramatically this isn39t always a recommended solution but it sometimes helps in tracking things down Chem 628 Lecture Notes Passive Circuits Resistance Capacitance Inductance Our course in chemical instrumentation and electronics will begin with a quite overview of some things that you ve probably seen before but many of you will need a refresher For some it may be the first time you ve encountered the material The relationship between voltage and current in any material or device is described in terms of an impedance usually symbolized by Z The impedance refers to the extent to which a material or device impedes the ow of current In general impedance is thought of as arising from three different contributions Resistance Capacitance and Inductance The main goals here are first to give you a physical picture of what resistance capacitance and inductance arise Then we will want to analyze the relationship between voltage and current in perfect resistors capacitors and inductors We ll then look at combinations particlarly combinations of resistors and capacitors which are widely used as lters in circuits Ifwe have time we ll also look brie y at some properties of inductors which are of importance mainly in highfrequency circuits Along the way we ll develop some mathematical tools to make the analysis of these circuits simpler through the use of Kirchoff s Laws and to some extent through the use of complex notation Resistance Resistance is the tendency of a material to oppose the ow of steady current In any material the ow of electrical current requires partiallyoccupied electronic levels Let s look at the three primary types of solid materials insulators metals and semiconductors it t it it AIL IL IL IL Insulator Semiconductor Metal Egapgt40 eV Egap0025 40 eV Insulators are materials in which there is a large gap in energy Egap between the highest occupied electronic states and the lowestlying unoccupied electronic states In order for current to ow an electron must be able to move from one atom to the next however because electrons are fermions we know that no more than 2 electrons can occupy any given electronic energy level In an insulator the lowestlying energy level is completely filled so that it can t hold any more electrons In order for electrical current to ow in such a material an electron must be excited from the highestlying occupied energy state to the lowest unoccupied electronic energy level which requires a great deal of energy At room temperature the average thermal energy of an electron is approximately equal to kT 0026 eV Insulators have gaps which are many times larger than kT usually greater than about 4 eV Semiconductors are materials in which there is again a gap between the highest occupied set of energy levels and the lowest unoccupied energy levels but the gap is comparatively small As a result at room temperature some electrons will be thermally excited from the highestoccupied set of energy levels referred to as the valence band to the lowestlying set of unoccupied energy levels referred to as the conduction band As we will discuss later the electrical conductivity of 39 t is strongly r on r and can also be strongly affected by optical illumination making them useful for optical detectors Semiconductors are typically defined as those materials which have gaps which are between about 01 eV and about 4 eV Silicon has a bandgap of 11 eV Metals are materials in which there is no gap at all between the occupied and unoccupied energy levels The hybridization of s p and dorbitals leads to a continuous distribution of energy levels or an energy band which is only partially occupied As a result electrons can easily be excited into unoccupied energy levels leading to high electrical conductivity In all these materials once an electron is excited into an empty energy level it can propagate a relatively long distance In a perfect crystalline lattice of a metal at zero Kelvin the resistance would normally be quite small As the temperature increases several factors lead to increased resistance First the atomic nuclei begin to vibrate leading to natural vibrational modes for the atomic nuclei quotphononsquot In a metal at normal temperature resistance is primarily caused by electrons scattering from these phonons At lower temperatures the resistance is dominated by scattering from defects and other imperfections in the crystal lattice II e39 e 6 e e 0 39 e 0 Resistance These factors lead to two general conclusions for electrical conductivity in metals 1 Resistance increases as the temperature increases 2 Resistance increases as a material become more disordered For semiconductors the resistivity is a more complicated function of temperature At low temperatures such that kT is small compared with the bandgap Egap of the semiconductor an increase in the temperature increases the number of free electrons in the conduction band and decreases the resistance At higher temperatures scattering from phonons and defects again becomes important as in a metal and the resistance increases with increasing temperature An operational de nition of resistance is Ohm s quotLawquot RVI where V is the voltage across the material and I is the resulting current ow It is important to realize that Ohm s quotLawquot is not really a law at all but rather a definition of what we often call resistance It assumes that the current through a material or device is directly proportional ie linearly related to the applied voltage that is true for some materials but is not true for many important materials such as semiconductors More generally we will use the term quotimpedancequot as a way of relating current to voltage The resistance depends on a number of factors including the purity of the material its crystalline form the temperature and the applied voltage For metals the resistance is only weakly dependent on the applied voltage so we can usually de ne a resistance For semiconductors however the resistance depends very strongly on the applied voltage and there is no unique de nition of resistance The resistance ofa meterial can be related to the resistivity p For a material with crosssectional area A and length l the resistance and resistivity are related by the equationR l Note that resistivity is usually expressed in units of ohmcm or sometimes microohmcm Below are shown some values of resistivity for a few common metals Copper 17 x 10396 ohmcentimeters Silver 16 x 10 6 Tungsten 56 x 10 6 Stainless Steel 74 x 10396 Nichrome 960 x 10 6 often used as a resistive heater element in toasters etc Capacitance From a fundamental standpoint capacitance is the storage of charge It is expressed mathematically as CQV where Q is the stored charge in Coulombs V is the applied Voltage V and C is the capacitance Capacitance has units of CoulombsVolt often referred to as a Farad 1 Farad 1 CoulombVolt A Farad is a huge capacitance and more common values are 39 J J or r 39 J 39 ca mi r mi r arads or mmf particularly in older literature Capacitance arises anytime we have two conductors separated by an insulator Capacitance is easiest to understand in a metal To understand capacitance you must understand a basic law from electrostatics which states that an electric eld cannot exist within a conductor without a resulting ow of current For any static situtation ie no corrent ow this essentially means that no electric eld can exist within a conductor This is relatively easy to understand Suppose that we have a good conductor and somewhere outside of the conductor we bring a positive point charge e m 3 a a a o 5 g m f f fifth No electric ll n39 231w 9 e e e e To E 3 9 9 conductor 6996 eeee e 99 99 99 Electric field from charge Free electrons move in Redistributionofelectrons penetrates momentarily electric field toward surface cancels electrlc fleld In bulk by into bulk buildup of surface charge Q This point charge will tend to create an electric eld within the conductor However if an electric eld exists within a conductor the free electrons will be acted upon by a force FqE For a positive point charge outside the metal the electrons will tend to move toward the positive charge This redistribution of charge within the metal tends to oppose the electric eld induced by the positive charge by creating an excess of electrons at the surface of the metal The net result is that if we try to apply an electric eld to a conductor the free electrons within the conductor move to the surface in such a way as to cancel the electric eld Now consider the situation in which we have two metallic plates separated by a vacuum Ifwe apply a voltage between them we create an electric eld The electrons in one plate will tend to move toward the surface to cancel the electric eld and the electrons in the other will move away from the surface in order to cancel the electric field The buildup or deficiency of electrons near the surface of the conductor can be quantitatively described in terms of a buildup or depletion of charge Q The capacitance C is then de ned as CQV ie it is the ratio of stored charge to applied voltage It can easily be shown from electrostatics that the amount of stored charge is proportional to the applied voltage so that C is a constant nearly independent of voltage but dependent mainly on the physical arrangement of the metallic plates As you might expect the capacitance is proportional to the area of the plates and is inversely proportional to the separation between the plates Taking into account all the proportionality constants and units conversions the capacitance of a pair of parallel metal plates is given by C SoAD where 80 is a constant called the permittivity of free space A is the area of the plates and D is the distance between them 808854x103912 CZN391 m392 CCoulomb N Newton m meter Buildup of positive charge Q at surface of positive electrode Electric eld exists only in region between electrodes E Buildup ofnegative charge Q at surface of negative electrode If instead of leaving a vacuum in the region between the plates we instead put some kind of insulating material the electric field between the plates will induce a polarization in this material This increases the overall capacitance such that the new capacitance is given KDA D byC 0 where KD is the dielectric constant which is dimensionless Note that high dielectric constants lead to high capacitance and vice versa Below are shown some dielectric constants for important materials Material Dielectric Constant Vacuum 10000 by definition Te on 21 Polystyrene 32 Polyester 35 Mica 54 Aluminum Oxide 84 Tantalum Oxide 276 Water 78 Barium Titanate200 Practical Capacitors Capacitors used for electronics applications should have several properties 1 Capacitance should be indepedent of voltage temperature and humidity 2 There should be no inductance or resistance for an ideal capacitor 3 The capacitor should have a reasonable physical size In practice it is difficult to meet all of these conditions simultaneously As a result there are several kinds of capacitors available each with its own advantages and disadvantages One of the primary dif culties is that because of imperfections in the dielectrics used in real capacitors their resistance is not in nite ie when a constant voltage is applied to the capacitor some quotleakagequot current will ow Whether leakage is important or not depends on the application A good table of the advantages and disadvantages of each type of capacitor is given in Horowitz and Hill page 22 Inductance Inductance results from the fact that a ow of current produces a magnetic eld according to the quotrighthand rulequot However a changing magnetic field also tends to induce a voltageor current in a wire according to Faraday s Law VdltIgtdt where I is the magnetic ux This is the basic principle of magnetic induction The net result is that whenever we try to create a ow of current in a wire the resultant magnetic eld will tend to oppose the change in current that we re trying to make XXXXXX Magnetic eld produced by a current according tothe RightHand Rule The operational de nition of inductance is L L d dz The unit of inductance Voltsseconds Amp is called a quotHenryquot Typical inductances are in the microhenry to millihenry range Every wire in an electric circuit has some associated inductance inductance The electric device which we call an quotinductorquot is usually a coil of wire often wrapped around some ferromagnetic material such as iron Note that for a circuit with a large inductance it is difficult to change the current rapidly ie dIdt will be small Inductance is primarily something which we will deal with in high frequency circuits Important There is no such thing as a quotperfectquot resistor capacitor or inductor All electrical components will have some resistance capacitance and inductance The absence of quotperfectquot devices often limits the performance of circuits especially at high frequencies Thevinin39s Theorem Thevenin s theorem says that any circuit having only two terminals and consisting of any number of voltage sources and resistors can be replaced by a single equivalent voltage source in series with a single resistor These are sometimes called the quotThevenin equivalent voltagequot and the quotThevinin equivalent resistance II u quot39 1E gt E J o E if 1 I Vthevmln Rthevmln 1 i Kirchoff39s Laws Kirchoff s Laws are a general way of keeping track of voltages and currents in a circuits containing more than one resistor capacitor or inductor Kirchoff s laws are based upon thermodynamic laws of energy and mass conservation and so they can be applied to any electronic circuit no matter how complicated We will introduce them here using simple resistors and batteries but later they will be used for much more complicated circuits Kirchoff s Laws state that 1 At any point in a circuit 21 0 2 Around any closed loop 2 Voltagedropsz mem 0 Kirchoff s First Law is another way of expressing conservation of charge It simply says that at any point in a circuit we can t accumulate or deplete charge If we try to apply this to a point right on the surface of a capacitor plate we run into trouble but otherwise it s fine Kirchoff s Second Law is another way of expressing energy conservation It says that the total energy change experienced by an electron owing in a complete loop must be zero Or equivalently it says that the electrostatic potential is a thermodynamic state function and we can t violate the second law of thermodynamics For a circuit consisting only ofbatteries and resistors the quotvoltage drops are given by Ohm s Law VIR In using Kirchoff s Laws the biggest challenges are usually remembering that currents and voltages are signed quantities As a simple illustration of Kircho s Laws let s calculate the equivalent resistance of two resistors in parallel The rst step is to assign currents in each leg of the circuit with a direction as 11 12 etc An important point here is that in making these initial assignments we do not need to know the actual direction of current ow but we are simply making assignments for use in solving the mathematical equations If the nal currents which result from solution of the equations are positive it means that our initial assignment has the correct direction If the nal currents which result from solution of the equations are negative it means that our initial assignments have the wrong direction the magnitude of the quantities will still be correct Thus the choice of directions is completely arbitrary We will get the correct answer in any case Let s choose a really simple example rst We ll pick the point labeled a Here Kirchoff s rst law states that 13 IzI10 By convention we use positive signs for currents which we have assigned owing toward the point and negative for currents which are owing away For Kircho s Second Law we ll pick the loop which includes the battery and one resistor Note that we must specify an arbitrary but consistent direction for the loop In calculating voltage drops around a loop the drop is positive if the assigned current direction and the loop direction are the same39 otherwise the voltage drop is negative Likewise the sign of a voltage source battery is positive if the current loop passes from the positive battery pole through the circuit and returns to the negative pole of the battery In the case above we have Vbattery VaVb Vab Now we can solve the equation We know that I1VabR1 and IzVabR2 from Ohm s Law l l Then 13 11 12 VabT R j We can rewrite this in terms of an quotequivalentquot parallel resistance 1 Z V Vlmttzry 1 1 1 R R R as I b where or e u1valentl R 1 2 eq 3 RM R2 RM R1 R2 1 y 1 R1 R2 Thus we ve used Kirchoff s Laws to derive the wellknown expression for the Thevinin equivalent resistance of two parallel resistances R1 E Huh wmm O R2 Req R1 R2 Similarly we could nd the expression for the Thevinin equivalent resistance of two series resistances Req R1 R2 R1 R2 ojln NM MM o 0 Let s consider a harder case 2 At point a we have 1213I40 1 At point quotbquot we have 1113I40 2 For the loop labelled quotAquot we have V1V211R112R213R3 3 For the loop labelled quotBquot we have V213R314R4 4 We now have four equations in four unknowns and can solve this can be done in a variety of ways For more complicated circuits matrix algebra is the best For small systems we can solve by simple elimination Subtracting equations 1 and 2 gives I112 5 This gets us down to three equations 68 Substitution I112 into equation 1 gives I1I3I40 6 Substituting I112 into equation 3 gives V1 7 V2 11R1 R2 7 I3R3 7 and we still have equation 3 unchanged V213R3I4R4 8 Solving equation 8 for 14 gives I413R3V2R4 and substitution of this into equations 6 and 7 gives 1113 13R3 V2R4 0 9 V1V2I1R1R213R3 10 this is the same as equation 8 Now we re down to 2 equations 9 and 10 Solving equation 10 for 13 gives I3I1R1R2 V1V2R3 and substituting this into equation 9 gives 11 Rll RZ l1l2 11 Rll RZ l1l2 l2 7 R 11 0 R4 3 R4 Now it s just rearranging R R R R V 7 V V 7 V V 11 1MM LL 1 2 4 R3 R 4 R3 R 4 R 4 And nally assuming thatI haven t made any mistakes 1 1 V ER UT i1 3 41 31 1R1 R2 R3 R4 Now you can backsubstitute to find the other currents and voltages We won t do this here because I m getting tired of using the equation formatter and by now you should have the general idea Two more tips on using Kirchoff s Laws 1 Remember that you only need as many independent equations as unknowns In the above example we had 4 unknown currents and so we needed only four equations We could have generated more equations by choosing a different loop say but taking the large loop through V1R1R2 and R4 However those equations would not have been independent equations they would have been linear combinations of the four equations which we used So don t go overboard think about how many equations you need and use only that many Otherwise you ll run into trouble 2 Kirchoff s are in fact general and can include resistors capacitors inductors etc provided that you properly express the voltage across the capacitor or inductor Since for capacitors and inductors the current and voltage have different time dependences this will require a slightly different approach in which we use complex numbers to represent the voltages and currents Voltage Dividers Above we saw the basic formulas for the equivalent resistance of two series resistors and two parallel resistors These can be generalized to quotNquot resistors as follows Parallel Resistors l l qu R1 R2 R3 RN Series Resistors R2 R1 R2 R3RN Let s got back to the case of two series resistors and measure the voltage across R2 We ll call this VOL R2 Vout The current in both resistors is the same The Thevinin equivalent resistance is Req R1R2 so that I R ER and the voltage drop across resistor R2 is just 1R2 Then 1 2 Van2 R2 V 7 V m R1R2 R1R2 m This is known as the VOLTAGE DIVTDER EQUATION It s one of the equations that you should memorize The voltage divider is used in many applications By varying the values of the resistors the output voltage can be changed One common application of a voltage divider is as a potentiometer A potentiometer is a threeterminal resistor in which there is a movable slider contact The total resistance between two terminals is always constant but turning a knob or sliding a level the position of the slider can be changed thereby changing the ratio R2R1Rz quot 90 Vin 39 Vout Potentiometer as Petent39ometer Variable Voltage Divider The Voltage Divider is widely used as a way of conveniently adjusting an output voltage In other applications only two terminal of the potentiometer will be used in which case it is acting simply as a variable resistor In our study of electronics we will often try to think of complicated electronic circuits as being comprised of a number of smaller quotmodulesquot A voltage divider is a good example ofa simple module which has an input voltage and an output voltage 39 I o I v I m 5 Vout 0 i Potentiometer as An Instrument Module When we talk about instruments being composed of modules we will generally be interested in describing the operation of each module using some sort of common terminology The terms that we will use in describing the operation of any module are 1 Transfer Function The ratio of output to input including phase information 2 Input Impedance 3 Output Impedance Do Not Use Pencil r Do Not Staple Please r r I Course Law 61 V Lecturer HMWS Day W Date 34 Notes Taken by 9quot 7 Page I of 5 Total Pages Submit 2 COPY of these notes for posting please 39 1w L peufp Eef 39 JJ bl W Van n 81 v 4 3 1 clean 396 gd wc M 23m VAC W L499 akaHA 3295 Law Fag Do Not Staple Please Do Not Use Bend Course V 7 39 34 V Lecturer Haw Day W Date Notes Taken by w 51 Page 1 of g Total Pages Submit 3 COPY of these notes for posting pleas PM 00mm 0 39 vw w M lt2 wk 1 V I t V r t 39 I l g r I I 3 l f o 5 3140 1 L Do Not Use Rene1391 v Do Not Staple Please V COUISC I39 39 Lect r er HAM Day 39 W Date I 37 Notes Taken by PHI Page 3 of 5 Total Pages r Submit a COPYof these notes for posting please T8q1 1 c436 sml ue 9 Add amp 3 ergo Ow w n V Vat V50 3 j r ImLVM bk wad quot V IL L1quotquotLJf U VJ Do Not Use Pena 1 39 Do Not Staple Please Course 70quot I Lecturer 39 Han4396 7 Day 39 V 39 Date ZM Notes Taken by P Mwa Page q of b Total Pages Submit a COPY of these notes for posting please Peak oaolL anti Hwy SL542 Flue 5L3H PLQ D l J O f 17gt Filler a am PM 7 quotP4quot 60 7 400 rep Wa plwso CL cnv 4391 a Kimm NL N Luck Cam I ha JVVVW f Iv w I V0 th I wozza39 Ric 39 a4 rlovv L0 I V0 539 VlV r 61174 70 2 4L Lula w Vs Mam Ply shit 00 Do Not Use Pencil a I r Do NotStaplc I Iegse Course 7 1 7 Lecturer Have Day 39 V Date 31 Y Notes Takenrby 39 39 a 5 Page 539 of I b T Otal Pages Submit a COPYof these notes for posting please Bade Plo Qv cowl th b as J12 M914 7 w thrlkfa vuwa s UQL quot 1 Amp IC of Do Not Use Pena quot I Do Not Staple Pica36 39 Course I Iketurer Day M0 Date 2207 Notes Taken by 39 P NMf Page I of 3 oral Pages Egmit 3 COPY of thes notes for posting please 1 MW at meme gig5145 94m Dabs HanJMJ Lab H 1 D30 Cvmr 3W9 Raul Sm ownquot ampm egag 7t3 l i a 04LllosL P55 Pa sstvc 39 ChadS I Jou c 39 V Vd uquot5 239 omznhb A V 7 Cau lamb V V AMPeMS Awf gt 56 39 b w impeo cwa 39 v5 5 V I y M g 7 Law 7quot I quot R 7501 Aw Rasv stfS RmrgyLa39rs A R I M7 CaFchws V C 7 Iv ckttlW I L Do Not Use Pencil V gt Do Not Staple Please Course 6 2 Lecturer New 6 Day M V Date Notes Taken by Pm a 7 Page 2 39 of 3 Total Pages Submit a COPYof these notes for posting please Land ance39 bHPm 40 avy G our qu 39 V u MQ LGI r 56quot Couclud vr j n W W n quot quotI 11 1quotquot w H 7 Iquot H 1 H quot l I Qt I 1 M quot A 1 3quot B L if if 1 21 2 2 r 33 U Ea 4quot Em k 39 L a i gt k I V ala3901395 5aampeo3 quotL Elev 596 awn boan 39nj Do Not Use Pencil Do Not Staple Please Course 675 Lecturer H maJ Day M0 4 Date 12 Notes Taken by PM Page 3 of 3 Total Pages Submit 9 COPY of these notes for posting please Cupqcjhvs v 51love Cha ae onion Va Cr 2 FaraAS 37 V quot 6e m t 7 g 4 Q 4 62 3 5 e v I VX Mwy I 1 Qt quot IR 959 5010 19 y 4 ch 4quotquot veoA39 To L M3 f 1L39Pk C 6 3 D 0 LE Do Not Use Pena Do Not Staple PleaSel Course I anem Lecturer How5 m6 Day W Date 1 2 Notes Taken by PPMK Page I of 7 Total Pages Sgbmit a COPYof these notes for posting please 2 C Li CW LS nAwLaw be L Ath t mt TLIampQ3 ampW Marl39ads Tkwrom I Kgraho m Lm s gtlt x gt6 Tram F L 5 Of ce s a kauy Cw WWWL L Haw3934 I at ClaudeRalf 56M cuer33 5quot Cm 61645 Fight fwAanm 2 Singing Ma3 I39m 01 MN INN Ffle 26639quot39tr a ssaP J39e encraj mi VQWJ a V m if 5 5MPch ample anal5 CO b39amp Usainal Wang s TkaoreW 2 94 39 J Y Md i r k Do Not Staple PleaSe Do Not Use Pencil 39 quot Course I Lecturer Haw 5 Day W2 Date 1 Notes Taken by P N Page I Q of 5 otal Pages Submit a COPY of these notes for posting please Tch i v12 n gt ThomA Afblu a o MV LQ LCAoML 39 Simple Lh wJ 9 one uYwblh 439 6M 1 I I 3 WW 39 A vol39Hr ILquotG105 L W w 39 I A tatS Pdiw39l n ap39ro39lioi z 1 Const a bin 39 Aron l a3 39o39lasul loaf Zvaum on Zv 0 50am gt Z 6w 2 2 m Do Not Use Pencil 39 Do Not Staple Please Course Lecturer H quot quot 5 Day W Date 12 A Notes Taken by I P m r Page 3 of 7 Total Pages Submit 3 COPY of these notes for posting please LAVW SVS Alba tats 455 tango Do Not Use Pencil 39 Do Not Staple P149356 Course I zg Lecturer 39 H quot quotquot39 I 39 Day W Date 13 Notes Taken by PMNM 39 39 Page L of 6 I Total Pages J Submit a COPY of these notes for posting please 1 j m Draw 00 Cittalk Do39Not Use Pena 39 V 39 I Do Not Stzpf Please Course 4139 Lecturer Humws Day w Date lZ f39 39 Notes Taken by P 39 3 Page I g gt of 5 Total Pages Submit 2 COPY bf these notes for po39sting please I T 0M o VOW V U V n I n V Rug 7 Vmcf39fffo iquot gt y V 2 2 Z 39 1 o w j V 5 L I Vfas Vo Kw g ta d l 611 4L peca v 1 301 1 03 539 3 am quot31quot 39 2 v 39 04 V in an 39 r I 39Vz Law skw LCrgw 1 P 1 39 l 01355196 5quot 01 31 pk 0mm MAW Phuwg chemwaroedg u Do Not Use P612613 Do Not Staple Please Course 9 Lecturer 129 Day gt I 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M 4 4 Day Date 0 I 39 Notes Taken by WA LAJW Page 3 of Submit a COPY of these notes for posting please Total Pages frims 2 Vquot I D F L 2 T 9 MOW Cod2c WWW far WIZCltlt1 L quotLA BY MK 9 I 7 RC 1 QTUCRC yaw WKC f L T H C E Lia k QOJBoUW 39 ackme 5amp7 Home QXWQ fMthgjqxcgjl 39 L quot eas w bb Do Not Use Fencil I Do Not Staple Please Course 39 extM A Lectufer V Day Date 9 g 7 Notes Taken by 39 Page of I Total Pages Submit a CSj Yof these notes for posting please A i z 7 I VM V9 QUW V6 Mm71 mart 39 A AV A 7b w mammag we ym wwsw BMW 6101 I my Wm Rem74 1m22 1m2E 2 A 12090 2 j A C a WW CW1 V1Vo W i IA R 2 27 xv V L 6L 7 39 m2 We LL W C quot 1quot quot A w 39 2 l 2 x Kg Wig Way9351 we YEARWALK Do Not Use Pena u r Do Not Stapfc Pleascf Course thw gt Lecturer 39chnbts Day M i Date 07 Notes Taken by M3 413 Page I a 4 Total Pages Submit 1 COPY of these notes for posting please 3 VqPP VM Vo v v In LD 10 Va l R La 6 4 4 7 F 390 quot Va WY 4quot I3 V358 vquot i quotz a it Up V Y 4quot PVq o Vnzo r2 XWL amplt 0 3 VW V0 1 to 4 9 Q g sf Far Shqhtr o 5 v 9 v1 th5 var Ya Faf Amtquot L 3 Do Not Use Pencil r Do Not Staple Please v Course L Lecturer Hamcv Day M Datcr 6 Notes Taken by P1w39 3 Page 2 V of 6 Total Pages Submit 2 COPY of these notes for posting please 39 Hale 61quot quot k DI f jg quot9fquot thJC I V 39 Adv x Signal I 0J e 7 Low 39 39 R Ltzf r V 7397 26M 03 Lramaa g Phofwl mlc smwaj magic 2on LL59 V r M4 MW 7 palm1 5gbrbanM J quot a4 4 quotJuly E mh 91F Vowch69 paw Do Not Use Fahdquot r quot 39 Do Not Staple P162136 Course f V 39 g g Learner M quot 4 Day M Date b 33 thes Taken by P quot13 Page 3 of Total Paggs 39 Submit a CbPYof these notes for posting please 39 7 no 05V V H9514 wayg RaoL CI W Pmj 5w Aar v7 W7 11 p z 39 Y 6 air WC 39 Z Q 1 a I quotquot o lrquoox v quot f L v A Do Not Use Reneii V 39 7 7 V Do Not Staple Please 7 Course H gt 7 V Lecturer My 5 39 Day r M Date 55 Notes Tsken by P W Page A i of 6 39I39otalPages 39 Submit a COPY of these notes for posting 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