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Modern Physics and Advanced Electrical Laboratory

by: Marjorie Hahn

Modern Physics and Advanced Electrical Laboratory PHYSICS 111

Marketplace > University of California - Berkeley > Physics 2 > PHYSICS 111 > Modern Physics and Advanced Electrical Laboratory
Marjorie Hahn

GPA 3.95

J. Siegrist

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J. Siegrist
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This 51 page Class Notes was uploaded by Marjorie Hahn on Thursday October 22, 2015. The Class Notes belongs to PHYSICS 111 at University of California - Berkeley taught by J. Siegrist in Fall. Since its upload, it has received 36 views. For similar materials see /class/226691/physics-111-university-of-california-berkeley in Physics 2 at University of California - Berkeley.

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Date Created: 10/22/15
Physics 111 BSC Lecture 5 Jim Siegrist Phone 4864397 Email JLSiegristlblgov Room at LBL 504055 Advice Today JFETI lec 6 TH Feb 8 lec 7 TH Feb 15 lec 8 TH Feb 22 solutions posted hand in late labs Instruments amp Devices 0 Now move to 3terminal semiconductor devices Why 7 Enormously useful because the use of a control signal voltage or current depending on the device between 2 terminals is used to control current in the third terminal 0 Two major device varieties bipolar junction transistor BJT emitter symbol base collector E p n p C Pup emitter base Collector also npn works B Page 1 of 8 Physics 111 BSC Lecture 5 0 Second is a eld effect transistor FET a number of varieties of these simplest is junction FET or JFET drain D l o nchannel IF ET gate G o l HmZZgtEO gate pinches off current ow in channel see Sedra More important MOSFET CMOS is basis of much circuitry eg Your laptop P N forward bias MetalOXideSemiconductor Field Effect Transistor section 92 MOSFET insulating layer gate vgsgtv VG lt z39D K VDS ms ND ptypebulk to source reverse bias PN junctions diodes one is back biased for any VDS nothing happens Convert region near surface from p to n via field electrons drawn from source region Gate insulated almost no input current Voltage operated device Enhancement mode device 7 charge carriers added to VIg 0 3 no conduction Page 2 of 8 Physics 111 BSC Lecture 5 depletion mode 7 opposite source source of charge carriers iD mA IV Characteristics 10 6 V dev1ce model VgsltVT3iD0 triode region 0 curves for different Vgs ohm1c reglon Kw I 2 4 T 10V Vns depends on device saturation region 2 3 induced channel pinchoff region vanishes someplace VS gt VG Power limits max voltages FET Facts 1 operation depends only on majority carrier ow 3 unipolar not bipolar like transistor 2 more compact to fabricate 3 high packing density on Si 3 can function as a resistor 3 no other components needed for digital circuits 4 hi input impedance Disadvantages noisy at low frequencies lower speed than transistor generally lower transconductance depletion enhancement JFET MOS MOS D D D 11 channel G GOJ B GOJ B S S S D p channel G GOJ B GOJ B S U2 U2 Page 3 of 8 Physics 111 BSC Lecture 5 J FET Large Signal Characteristics D T Note this device technically inferior to n MOSFET 7 the pn junction is reverse biased for operation 3 leakage current GeD is 10 A but is 0 for MOSFET G p p since gate is insulated MOSFETs easy to destroy by static discharge 3 you ll use JFET in lab insulating layer gets punched through i Also pn junction 3 strong temperature dependence S On JFET If a voltage vdS is applied between drain amp source current iD flows for vigs 0 If we lower vigs 3 electrons repelled from channel 3 less current ows Negative vigs said to deplete the channel of charge carriers vigs negative enough 3 channel depleted amp you pinch off the channel see Sedra 511 for drawings amp discussion For positive vgs gatechannel pn junction forward biased amp gate can t control the channel 3 0V is maximum vigs for JFET nchannel IV characteristics similar to MOSFET depletion devices Parameters of device speci ed in terms of pinchoff voltage Vp amp drainsource current with gate shorted to the source IDss Operating regions n channel l cutoff vgs SVP iD 0 2 v 2 triode region VP S vgs S 0 VDS S vgs VP z39D IDSS21Vi5vJ S P P P VP 3 Vgs s 0 V s 2 v 3 saturatlon pmchoff reglon I39D DSS 1 g 1 L VDS Z Vgs VP VP A VA E Early Voltage N 100V 3 ignore eg IDSS 50 mA Vp 7 4V Page 4 of 8 Physics 111 BSC Lecture 5 80 A 60 lt E Vgs 0 V Hg 40 l0 V 20 Vgs 25 V 2 4 6 8 10 VDs V JFET Small Signal Model G D Vgs nggs iD S Voltagecontrolled current source z39D gmng 1 gm E transconductance gm id ng 3 units of conductivity JFET is a voltageoperated deVice 2 639 V 5 gm 61D from 1 so 1n saturatlon reglon iD m IDSS1 Vg J amp vgs BiD 6ng v gm 21DSS l g5 Vgs N 1V 750mNV for gm P VP Independent of VDs linear in Vgsl NOTE gm Vp etc depend on physical deVice characteristics Look at data sheets 7 also capacitances associated with junction etc In triode region DeVice Variation FET characteristics parameters show much greater spread with FAB process than e g BJT eg spread in Vp may be 75 to 73V on sample deVices from same batch Page 5 of 8 Physics 111 BSC Lecture 5 3 Most design circuits to get around device variation to make circuits with predictable characteristics Various methods can be used in device layout to match characteristics interleaving channels so thermal amp FAB effects cancel problem in obtaining uniform channel doping over large areas amp sensitive thermal dependence This is why people use bipolar for precision circuits Concept Feedback 0 The best way to get around device variation is through the use of feedback feedback E compare output of the system with the desired output to generate a control signal to input e g you get bad HW grades you do better next time Negative feedback 3 couple some output back so as to cancel input At first sounds stupid but negative feedback improves circuit characteristics e g very high gain amplifier X 106 with strong negative feedback will have characteristics that depend on the property of the feedback elements only we ll see this later Note also negative 3 180 phase shift between input amp output Positive feedback also possible 7 leads to oscillations unwanted positive feedback 7 parasitic oscillations n dependent due to stray caps in circuit Circuit Analysis JFET Current Sources 7 Simplest example V From IV characteristics iD reasonably constant for vigs 0 for VDS gt 139 ID So JFET looks like a current source However IDSS varies from device to device 3 value of i1 not D predictable G S So 1t 1s a pretty good current source but we don t know the current Page 6 of 8 Physics 111 BSC Lecture 5 o A variation gives an adjustable current source R is a self biasing resistor 7 back bias gate by IDR amp brings JFET closer to pinchoff lowers lam A form of currentsensing feedback Still some variation of output current with voltage N 2 I vs V variation 3 not ideal Vary R with each device to set idmin lt IDSS Source Follower Take advantage of high input impedance of JFET 7 ig m 0 V We ll have S0 V0142 RLgm Vm V0ut salve for V0142 3 R or v0 Lg39quot v lRLgm m For RL gtgt yg vm m VW hence follower with very very high input impedance Use to isolate sub circuit components Page 7 of 8 Physics 111 BSC Lecture 5 NB define rs g 3 R Lvm amp equlvalent elrcult looks 11ke rs RL amp device behaves to keep vm m vow within rs 2 IIS V0142 Page 8 of 8 Physics 111 BSC Lecture 1 Jim Siegrist Phone 4864397 Email JLSiegristlblgov Room at LBL 504055 Room Campus 309 Old Leconte Administrative Issues Lectures Reserve T amp Th 56 PM location Only about 12 lectures for the semester mostly they will be on Thursdays Must watch for lecture dates Current Plan next 3 lectures Today lec l Thur Jan 18 lec 2 Tues Jan 23 lec 3 Thurs Jan 25 lec 4 Tues Jan 30 Course Grades 1 Must hand in all labs to pass including PF labs 2 See yellow sheet for grade breakdown Grading Full WriteUps Introduction 10 points Conclusion 10 points 3 or 4 random questions X 15 points Prelab Questions 20 points 100 points Will not grade all the problems all the time up to TAs Supplemental Problems 2 points each 2 points per question with no answer 73 points per day late Work will not be accepted after the solutions have been posted PF Labs 50 points if you attempt all the questions 72 points per question with no answer 73 points per day late Final Projects MUST be in on time Send me an email if you don t want your paper posted Course Content 0 Goal of this course is to teach the basics of modern electronics You ll need this for the next semester of Advanced Lab We assume you know electricity and magnetism at the level of Page 1 of 9 Physics 111 BSC Lecture 1 Physics 7 and take you to the point that you can design your own circuits by the end of the semester 0 Main elements that we ll learn about in lecture Concepts Instruments amp Devices Measuring Techniques Circuit Analysis Techniques Tools 0 Topics in lectures tend to jump around among these 7 I ll try to note which area I am talking about as I change topics 0 I ll also give advice and information at the start of each lecture 7 the most important stuff will be in the front of the notes so I hope you don t miss it Notes will be posted to the web after lecture Advice for the course Reading material for each lab at front of each handout Labs are quite long so come in early This course is harder than you think 7 give it plenty of time Find a partner sign up for 2 aftemoonsweek yellow form Course grade strongly correlated with lecture attendance will work some problems in lecture Some labs PF 7 see handout No introduction or conclusion writeup needed for PF labs Office Hours by appointment Schedule of writeup due dates in handout All writeups must be NEAT Example writeup in handout is minimum acceptance for PF labs 7 really need diagrams and descriptions Label axes on plots Indicate units Note hand symbol for TA signoff 7 get before proceeding gtllt gtllt gtllt gtllt 9696969696 Advice this week Scope manuals in lab if needed 7 triggering xy mode DMM description in notes 7 read it Read this amp next week s lab 7 I ll talk about next week on Thursday Why is the course goal important Physics is Experimental Science 3 What we know about nature determined by what we can measure Experiment crucial to guide theoretical development 7 witness development of Quantum Mechanics in 20 s Standard Model in 70 s What limits experiment 7 Instrumentation brains Why is electronics important in this Page 2 of 9 Physics 111 BSC Lecture 1 Huge technology base industry around electronics 3 opportunity to exploit for m science N All modern instruments depend crucially on electronics for read out and data handling Many electronic devices operate at or near the theoretical limit allowed by quantum mechanics or thermodynamics e g position amp momentum noise in ampli ers Beating limits of measurement therefore is a major industry for physical science 7 other limits cost size sensitivity etc exist and also have to be beat Therefore you need to know this to be a physicist theory or experimental once I t I Circuits De nitions 7 Units Charge electron charge is 716x10 19 Coulombs Current 11 9 ows in arbitrary reference direction 11 is positive if positive charges move in the reference direction else negative Charge dQ moves past a point in time dt 3 current 61 in Ampere E M t sec ands W Voltage Vx 7 Unit E Volts Zero is arbitrary Zero E reference point 3 common Only Differences Matter Power Charge Q moves from V1 to V2 3 a charge in energy of QVz 7 V1 Joules For a steady state current I P E 1V17 V2 E IV E Joulessec E Watt E Amp 0 Volt Circuits imply a closed loop In the lab be sure you have made complete circuits 39 Circuits Kirchofl s Laws Page 3 of 9 Physics 111 BSC Lecture 1 D B Fl l 4 De nitions Wire from C 7 B has no resistance ideal wire 3 Vc VB Node point at which 2 or more circuit elements are connected Branch twoterminal circuit element connected between nodes ABOVE 3 nodes 4 branches Nodes are NOT separated by wires Kirchoff s Current Law no charge accumulates on a wire 3 sum of all currents entering a node is zero charge conservation Kirchoff s Voltage Law sum of all voltage drops around a complete loop is zero potential at any instant is well de ned 3 physical size of circuit is much leg than 7 at frequency in question Circuit Elements amp Smbols W 0 S RM S Rab Ohms V IR R WI or variable version NOTE Stored energy U CV2 so be C VQC Ili Farads C dV careful 7 big capacitor can give you a 3 I C nasty shock Q CV L VL Henries dt Real signals come from transducers eg phototube photodiode radiation detector etc 0 Learn more about these mostly next semester Page 4 of 9 Physics 111 BSC Lecture 1 0 Here consider ideal sources DC Voltage V ideal 3 V independent of I S ourc e AC Voltage V v0 V0 sin cat or V0 cos oat Source 0 lowercase usually Current ideal 3 I independent of Source R load resistor RL L compliance 7 limit source can provide Useful to review microscopic de nition of resistance 7 Berkeley Series Vol II Sec 43 7 47 Component Specs 0 Res1stors A Watt carbon compos1tlon 1n lab Color Code Tolerances Resistance varies with T V time humidity etc Power 7 IV 7 12R Example 60V What s the power dissipated in the 1k 29k I V 2mA amp I is the same for both R 29k 1k resistors since they are in series P 12R 2mA 1k 2 4mW rt 0t MW be NEAT o Capacitors Electrolytic Ceramic Mica Tantalum NOTE Some capacitors have polarity amp must be connected in the right direction 0 List of other components amp symbols in the lab book 0 See section in Hayes amp Horowitz on capacitor values pp 51 7 53 Signals amp Waveforms as seen on scope Page 5 of 9 Physics 111 BSC Lecture 1 Sine Wave V already discussed t Voltage Ramp V x quotquotquot 39 39 limit Saw Tooth V repeating ramp Triangle V linear sine K 7 K t Noise V 4Afv417v4v47 v4 t Square Wave V t Step V V t idealised t V V reality t Pulse Logic levels 7 0 or 1 for digital circuits 7 we ll discuss later AC Circuits Amplitude amp Phase Page 6 of 9 Physics 111 BSC Lecture 1 27239 V VU sm cot VUsm 27z39ft VUsm T a 2727 15140 f 1 HZ 3 T0 2 Tt7 a phase angle rad V amplitude Vs V time Vs time t phase di erence 7427i amplitude Vt7 peak voltage V7 Vpip 2V0 1 7 Vm root mean square VS t dt over 1 period 0 0 V 7 do Integral over J5 Using Xy mode of the scope you can nd the phase A between 2 signals Chl A sin oat Page 7 of 9 Physics 111 BSC Lecture 1 Ch2 B sin oat 4 Ch2 ymt ymax At wt0 Asinwt03yint Bsin At ymx sinat l so ymzx B amp y sin or arcsinyi use in lab I Also look at these signals in the frequency domain 7 eg look at response of linear circuits to 96 96 96 pure sine waves Linear 3 output driven by sum of 2 signals equals output sum if driven by each signal independently e g OAB OA 0B Linear circuit driven by sin results in a sin but phase amp amplitude of the output changes 3 frequency response is a very use ll concept 7 how do amplitude amp phase change with n A useful hint to compare amplitudes of 2 signals or of 1 signal relative to a standard reference 7 decibel dB defined by 613 20 log10 1 Measuring Technig ues In working through labs this semester think carefully about errors on your measurements What limits the accuracy of the measurements you make Are component values amp their variation important Can such errors be reduced many circuits are designed to avoid such problems 7 feedback Precision vs Accuracy 7 You can make a very precise inaccurate measurement 7 systematic errors can mess you up time variation thermal variation etc eg Voltage across a resistor 7 Horowitz amp Hill pg 430 All resistors have a noise voltage Johnson noise across their terminals frequency spectrum is at same noise power in each Hz of frequency up to some limit Page 8 of 9 Physics 111 BSC Lecture 1 3 call white noise mm 4kTRB 2 k is Boltzmann constant T is temperature R is resistance B is frequency bandwidth to which you are sensitive 72on 4kT 16gtlt10 9 At room T V amp 4kTR 2 13x10quot R 2 42 So 10k R at room temperature looked at by lOkHz bandpass oscilloscope has Vm 13 uV pretty small Many other kinds of noise thermal noise discussed here related to fundamental constants others are not fundamental noise in circuits usually not at thermal limit Many folks don t even know such limits exist sec 711 HampH Instruments amp Tools ONE NOTE On ammeter must connect to circuit in series to measure I break amp insert meter to measure I DMM Real Life Use a shunt Lowresistance element permanently in circuit measure AV to get I 31A Page 9 of 9 Physics 111 BSC Lecture 4 Jim Siegrist Phone 4864397 Email JLSiegristlblgov Room at LBL 504055 Advice Get papers back from TAs Best labs posted across hall from 111 labs in glass case each week No labs accepted after solution posted Anyone objecting to having their labs posted 7 let me know via email Today Diodes lec 5 TH Feb 1 lec 6 TH Feb 8 lec 7 TH Feb 15 Utility of following discussion Complaints about not enough solid state detail 3 small signal models not well motivated I will give some detail but you promise to remember good design 3 circuit M sensitive to details of device parameters Instruments amp Devices Semiconductor Diode Pure semiconductor covalent bonds all 4 valence electrons used 11 eV band gap silicon 3 thermal excitations important T 0 K 6 free holes amp electrons m 1010 ca1rierscm3 for Si 300 K 1023 for metal 3 semiconductor Doping Add impurities to control the ehole balance 5 valence electrons 3 extra electron gets loose donor 3 valence electrons 3 extra hole gets loose acceptor o Donor 7 phosphorus arsenic antimony electrons majority carrier holes minority carrier E ntype Page 1 ofll Physics 111 BSC Lecture 4 o Acceptor 7 boron gallium aluminum indium holes majority carrier electrons minority carrier E ptype immobile ions Z 3 GB 69 69 lt mobilee NTYPE 69 T mobile hole net charge 0 immobile ions 1 9 Q 6 mobile e ee A I mobile hole PTYPE 1015 carriers 3 m 1015 donor atomscm3 Si has N 5 X 1022 atomscm3 3 l impurity50 X 106 Si 20 ug of phosphoruskg SI 3 extreme care in manufacture Application of a voltage causes a drift of charge carried by both holes amp electrons Total current is the sum of the 2 currents As in a metal but 2 carriers Diffusion Current Suppose concentration of holes varies with position P0 P00 0 O gtJ O 0 O o O o 0 O O o O gt X 39 dP 3 El concentratlon grad1ent Ax Page 2 of 11 Physics 111 BSC Lecture 4 Density of holes on one side of surface gt on the other Because of random thermal motion they want to drift toward lower concentration region This current is proportional to the concentration gradient J P qDP dx Dp diffusion constant for holes dp ix lt 0 JP gt 0 in x direction An electric eld will be set up within the Si block to hold the equilibrium current to zero Very different from metal NB Drift current density under in uence of E given by jdn qpup myquot 5 pn holeelectron density JPH holeelectron mobility PN Junction Electrons amp holes diffuse both ways 3 depletion region 05 pm thick is generated or space charge layer P N E X diffusion Ivu x contact potential Page 3 ofll Physics 111 BSC Lecture 4 Reverse Bias anode cathode N electrons carry current P holes carry current gt gt I direction of current V easy to remember This polarity causes holes amp electrons to move away from the junction m 0 current some current because thermal ehole generation Holes anywhere on n side drift to gap amp are then pulled by the eld reverse saturation current E lg n independent of V depends on T NB Addition of minority carriers T inferences this ism 7 important for transistor operation Forward Bias Now current ows sum of electron amp hole current l l I V V Contacts metal 7 semiconductors are ohmic voltage drop across crystal ideally 0 then applied V appears to add to junction potential 0 Time varying signals charge goes intoout of space charge layer 3 looks like a capacitance E transition capacitance CT function of reverse voltage 10 gt 200 pF Important for transistor operation Field is reduced in the spacecharge layer allows majority carriers to diffuse across the junction to the side where they are in the minority injection of minority carriers 0 Time varying signals diffusion or storage capacitance CD more later CD CT N 10 HF however rCD 139 N nsec so time constant is small Page 4 of 11 Physics 111 BSC Lecture 4 Diode behavior in both regions described by eXponential diode V I 14547 1 q electron charge k boltzmarm s constant 138 X103923J K T K kTq 26mV 300K V gtgt kT q 3 neglect l qV Forward b1as 3 I Ise 1 gtgt IS S Reverse bias 3 V ltlt kT q 3 139 1 Silicon Is N 10 pA qV nte 1 4 1 illgr k7TlnSj IS NIOpA V1 20mAN 26mV1nl 20ml HOP1 N 56V E VT on 3 N 06V drop off3open ln ln Real Diode Characteristics I Power Limitations IV lt max 3v good operating region VT m 6 V for silicon Page 5 ofll Physics 111 BSC Lecture 4 High forward currents ohmic contacts take over amp diode behaves as a resistor High reverse voltages minority carriers gain sufficient speed to knock loose additional electrons in deplection region where field is high 3 larger reverse current 7 avalanche multiplication Maximum reverse voltage VR E maximum reverse blocking voltage Zeners have a well controlled VR independent of 139 can be used as a voltage references 68 V IInstruments amp Devices Circuit Analyses Diodes as Network Elements I nonlinear theorems don t work I Graphical Method L gt V1 Thevenin Eqv linear VOC VL VD net Page 6 of 11 Physics 111 BSC Lecture 4 If we can nd 11 V1 we are done We know VD vs ID from characteristic curve manufacturer s or measured We know VL VOC 7 ILRTH Plot this on the i v characteristic line 2 points open circuit voltage amp shortcircuit current VL39IL characteristic load line diode characteristic see example page 157 SampS Source is time varying see page 164 SampS RTH remains constant VOC varies 3 Load line has constant slope but the intercept Voc moves as a function of TH time so the operating point moves Page 7 of 11 Physics 111 BSC Lecture 4 AC Amplitude M x Wocai t V 1me v0 5 gt slope lRTH smallsignal approx output Small Signal Approx Taylor expand about some operating point 2 use tangents to curve at that point to describe gt the behavior gt rescue linear circuit laws by linearizing 911111 II Linearize the behavior for small deviations about the oeratin oint 3 taylor expand keep lst term B doesn t work over whole ran e Approximate diode behavior with linear elements over limited i v range Ideal diode switch 1i K forward bias reverse bias If voltages not large compared to 6 V then Page 8 ofll Physics 111 BSC Lecture 4 l i I l l slope lR v OR V R i L T VT l VT V l VT Reverse current zero usually ok lt Use simplest model to meet your needs Circuit Anal ses Examples Recti cation VS 2gt Vs VT 6V 3 v A vvv short Vs gt 0 ltgt open Vs lt 0 j ltgt Page 9 of 11 Physics 111 BSC Lecture 4 A recti er Another example matched pairs pg 49 HampH Peak Sampler gt reverse bias t 0 Vs gives positive diode a short 0 Past Vs maX diode is reverse biased amp we are stuck More realistic V discharge time constant RC Amount that it drops depends on frequency of Vs amp RC time constant NB 7 RC comes from step response LEDs Same as ordinary diodes but forward voltage drop in range 15 7 25 V 5 7 20 mA of current causes light to be emitted electronhole recombination Inverse also works 7 reversebiased diode can be lightsensitive photodiode Important for electric eyes Also basis for light amp particle detection liberate eh pairs within depletion region Page 10 ofll Physics 111 BSC Lecture 4 Summa Reverse bias open Forward bias closed gt06V 6V drop good diode model for most uses Pagell ofll Physics 111 BSC Lecture 3 Jim Siegrist Phone 4864397 Email JLSiegristlblgov Room at LBL 504055 Advice Today Linear Circuits II lec 4 TUE Jan 30 lec 5 TH Feb 1 lec 6 TH Feb 8 Get your labs handed in Read labs 2 amp 3 Return to AC signals steady state sinusoids For sinusoidal signals V 1 7c0s wt in hase with volta e R R R P g if Cal V aCVL7 sinmt dt wCVt7 c0sat 90 3 current in capacitor leads the voltage by 90 but still cc V0 3 Rescue Ohm s Law by representing amplitude amp phase of signal in the complex plane 0 Voltage amp currents represented by the complex quantities X amp l amplitude amp phase V7 c0sat gt gt Vgew e 0 Active voltages amp I obtained by multiplying v lby em amp taking real part Vt Retem It ReQe 3 Vt ReVc0s mt ImVsin mt 1t Rec0s mt Imsin mt Page 1 of 9 Physics 111 BSC Lecture 3 d Forget dlfferentlal equatlons reduce to algebra1c 1n a systemat1c way pulls out 139 a2 1 NB 6 cos isin NO j from me Add amp subtract in rectangular form a 1b Multiply amp divide in polar form 2 lzle 2122 I21 llzzle lw l z Z1 2 Below 22 Izzl 3 complex Ohm s Law IVZ Zohms If I phase lead by 90 voltage 1c imCV And the ratio L E C E Z O is the reactance K R for a resistor reactance imC I resistance E complex resistance Consider last time R IVVI Vuut v C Page 2 of 9 Physics 111 BSC Lecture 3 voltage divider L V 0142 m m R 1 1za2RC imC z 1 z V 1mRCZ quot 1wRCZ V 1 transfer function H I39m a le 1wRCZ 1 3db point wRC 1 Amplitude VWt TV 2 m 6 db down Page 3 of 9 Physics 111 BSC Lecture 3 39 61139 For an 1nductor v L dt g imL inductor 3 short a low a V l capacztor 3 short at hzgh a I iwC V R resistor I V Impedance Z E 7 steady state s1nus01ds between 2 pomts 1n the net as before In ohms act like resistors reactance resistance E types of impedance Further de nition 1 l Admittan ce E Y E G 139B Admittan ce E G is conductance B is susceptance Network theorems still hold Power ReIV ReV1 NB Power E 0 through pure reactive components IV N sincos I over one cycle E 0 Apply to this problem lt 01 ltgt R lt V ltgt C2 lt R Vent lt Page 4 of 9 Physics 111 BSC Lecture 3 ng 2 Vm ZlZ2 Z1Z2 E ltZ C gtR qu C ZR ZRZC parallel combination 3 R L Z zi a zL 2g R l 11Ra2C iwC R2 VLWZVM Z2 2 1z39R2a2C2 Vm ZlZZ R1 R2 lia2R1C1 lz39aR2C2 divide by numerator KMVK 1 lza2R2C2 R2 lia2R1C1 Page 5 of 9 Physics 111 BSC Lecture 3 Vm C1 a oo 2 V 1le1R2CZ C1 C2 inlRZC1 lia2R2C2 lia2R1C1 or take derivative 0 a independence K solution mRzCz leCl ratio a constant if RzCz R1C1 by inspection Fourier Analysis Note that m repetitive signal can be decomposed into an in nite series of harmonic sinusoids eg Ft i114quot sinnatBn cosnat n0 A 2jFtsinnatdt n T 0 with B iFtcosnatdt n T 0 17 ampB F d Tgmr where T is the period a 2 T You have seen this in your courses already my reference 7 Arfken Mathematical Methods for Physicists Single Time Constant Circuits 3 Circuits that can be reduced to or composed of one capacitor amp one resistor 3 Important issue is guring out the time constant Page 6 of 9 Physics 111 BSC Lecture 3 Quick method to nd equivalent circuit 1 reduce excitation to zero a short voltage source b open current source if circuit has one capacitor or inductor grab 2 terminals of C nd Req eg in example RM R4 R3 R2 R1 139 RMC 3 invert for one R many C or L N Read about step amp pulse response of STC 7 HampH 114 SampS App F Circuit Analysis Inductors and Transformers 0 Inductors L d rate of change of I 1n L depends on app11ed voltage V L L is inductance in Henries Power is not dissipated as heat IV but stored in the magnetic eld of L Z L imL Constructed as a wire wound around magnetic material to make a transformer two coupled coils primary secondary AC applied to primary 3 constant power through transformer 3 step down or up voltage multiplication cx turns ratio current multiplication cx turns ratio NB Turning off circuits with inductors 3 boom they don t like it so ramp down slowly e g an accelerator Resonant Circuits Page 7 of 9 Physics 111 BSC Lecture 3 Combine C with L in a circuit to generate a sharp peak in frequency characteristics C J 0 Vin L C Vout Reactance of the LC combination at frequency 03 is 1 ZLC ZL ZC imL z39 z39a2C i a 139 orZ LC 1 aC mL At an then ZLC 00 open 3 peak in the response of the circuit there VonV gt lt Af 39 w wedOE Quality factor QE resonant frequency m0 Af E width at the 3db points DeVices Instruments and Tools 1 Spectrum Analyzer 2 Network Analyzer Page 8 of 9 Physics 111 BSC Lecture 3 Measuring Technigues Vm F11ter 03 response sweep to measure V 0 142 straightforward for linear circuits mm a only phase and amplitude change 0142 For a real voltage source we said we found is by varying RL to measure IV characteristic 3 output impedance What are the best values of RL to use for a good measurement Note V0 IWRL RL V Rs R How does Vom change with changes in RL when we make the measurement de RS RL V VRL W Rs R 2 RSV RS RL 2 for RL Rs alng N 0 no sensitivity V011 V dRL dV V 7 7 for RL ltlt Rs 0 N no sens1t1v1ty amp ratlo can t be separated short N 1m N dRL RS V so short open gives Rs RS We are not using an ammeter 3 use RL N Rs N RL near Rs gives best sensitivity Reading for next week note Tools New tool in lab 3 N PSPICE circuit simulation package We d be using that to model circuit performance Discussion in Appendix C of Sedra Smith 0 Many many other simulations modeling and design packages used by real people but SPICE is earliest version amp an early standard For complex circuits absolutely crucial We ll only get a avor Page 9 of 9 Physics 111 BSC Lecture 8 Jim Siegrist Phone 4864397 Email JLSiegristg glblgov Room at LBL 504055 Advice NB problems 812 7 813 gt supplementary Please tell everyone Today OP AMP 11 lec 9 TH Mar 8 lec 10 TH Mar 15 lec 11 TH Mar 22 Report solutions posted Circuit Anal si Example 4 Summer 7 is grounded 3 V1 RF V1 V2 V3 47 0 v R1 R1 R2 R3 R F 2 K v R1R2R3R v3 RF R3 3V011 V1i V2i V3 R 7 R 3 7 sum of voltages w1th welghts F R Example 5 CurrenttoVoltage Converter Inverting ampli er useful when we require an output voltage proportional to the current owing into the input No current ows into input 3 11 through R1 goes through R f 11 vim 11Rf since other end at virtual ground Page 1 of 4 Physics 111 BSC Lecture 8 Example 6 C Inte grator current through resistor 4 current through cap 0 C de V1 quot0 1C dt R dt J4 dvm dt J vm tdt vm vm I integrate wrt time 3 0 dt RC 1 V0140 Ejvxntdtvaut0 VERY IMPORTANT PHYSICS CIRCUIT 7 Integrate charge from a detector element Use switches to set initial conditions vm 0 0 R0 For baseline restoration add a resistor l C If RC gtgt t the integration time this has little effect but discharges the capacitor for long times OpAmp Circuit Design Considerations 1 voltage offset V V by shorting inputs output at 0 2 offset currents input bias current DC current owing through two input terminals T N deal 1 Difference in two bias currents can lead to a voltage offset eg 1131 has no effect it s current just goes to vS ground nut Short V5 to ground ask what is the DC output Page 2 of 4 Physics 111 BSC Lecture 8 3 No voltage across R1 is grounded 3 7 ground 3 both ends of R1 are grounded V0142 IDZRF 3 Current ID ows through RF 3 1m lLamp4 RF lk 3 v0 1mV 3 Small DC offset from input offset current distinct from offset voltage 3 This stuff normally trimmed internally by trim circuits Temperature variation of nulling 3 DRIFT that s why internal compensation works better Slew Rate El limitations on the speed at which the output voltage of the opamp can change Input changes suddenly 3 output takes a finite time to reach the new value This process is called slewing maX rate that output can move to new value maX slew rate e g for follower you might have Vm 39 i i t usec 10 20 5 V MSR 106 V VM 5 usec sec t Sinusoidal signal vm VD cosmt d d v0 A vm 3 V A V A aVU s1n at vom dt dt 560 max dvmquot A mVU dt dv ltltMSR maX 1 0 dt W N MSR maX M dt 391 5V tome V V Note that in the second case opamp slews all the time first trying to catch up with the input in the uphill direction then chasing it downhill Slope oflines i MSR 3 you can measure on scope Characteristics 1 onset of slewrate distortion depends on output amplitude and frequency Page 3 of 4 Physics 111 BSC Lecture 8 2 output amplitude is reduced amp waveform is distorted Frequencv Response of OpAmp Circuits For a typical opamp no feedback open loop lAl db 1 100 80 20 10 103 105 107fHZ De ne bandwidth E range of frequencies over which lAl is within 3db of max 3 openloop bandwidth N 10 Hz Negative feedback makes gain insensitive to A for inverting amp A39 R1 Choose A39 N 10 then plot vsf3 lA l db 1 100 80quot bandwidth extends till 3 lA l W 20 o 10 103 105 107fHZ 3 A tradeoff between gain amp bandwidth or gainbandwidth product constant Typical amp has gainbandwidth product of 106 Hz To get more gain fixed bandwidth 3 add more stages of amplification NB Frequency limitation above 105 Hz not same as slewing problem onset depends on frequency not amplitude no output waveform distortion for sinusoids Page 4 of 4 Physics 111 BSC Lecture 2 Jim Siegrist Phone 4864397 Email JLSiegristlblgov Room at LBL 504055 Advice Today Linear Circuits I 7 lab 12 lec 3 TH Jan 25 lec 4 TUE Jan 30 lec 5 TH Feb 1 Notes Do prelab questions before labs Attempt to answer all of them Don t hand in incomplete labs 73 per day for late reports Office Hours 7 by appointment 7 JLSiegristlblgov Introduction amp conclusion content guidelines outside TA office 2 points for experiment evaluation forms on time Get prelab questions signed off lt Friday Circuit Anal sis Examle 39 39 Circuits Input amp Output I 1 Divide circuit into subunits 7 interested in terminal characteristics I vs V of subunits 3 replace them with a simpler equivalent circuit In general for a linear system original contains voltage amp current sources resistances an equivalent circuit is one with the same IV characteristic as the original at all frequencies of interest Model for a real DC voltage source Rs 0 for an ideal source V independent of I The relation between V across RL and the opencircuit voltage V0 is the voltage transfer V R V 1RL WW 3 V1 voltage divider S L 0 S L Page 1 of 10 Physics 111 BSC Lecture 2 RL Rs 3 V V0 3 N ideal voltage source RL ltlt Rs 3 current through the load N 1 short circuit current of the input network S 3 N current source Find Rs by varying RL to measure IV characteristic N output impedance resistance of the source Likewise measure for RL or the 2terminal linear device and nd is input impedance resistance for now gt Im a Iin V modeled by Vary Ii measure V vs Ii 3 nd Rin as Rm allyd the slope of the line Circuit Anal sis RC Circuits To introduce RC circuits cons1der C R Current through the capacitor is Q CV 3 ail Q I Cj V can t change voltage across C instantly t 1 Solution to this differential equation is V Aeizw So a capacitor charged to V0 with R placed across Page 2 of 10 Physics 111 BSC Lecture 2 37 tRC discharge waveform RC has units seconds time constant of the circuit Circuit Anal sis Exam le Freguency Response 7 Low Pass Filter Low Pass C 9 short at high 03 HiPass C I 0 R v 0 Vin V cos oat dQ dv open c1rcu1ted 3 Z currents at node A 0 3 Q CV 3 E I C V t Civm w 3 iv LVW Lyman dt R dt RC RC Page 3 of 10 Physics 111 BSC Lecture 2 11near dlfferentlal equatlon w1th constant coef clent drlven by Ccos wt 3 Vom has the form v0 N V1 cosat gt to solve write V1cosat A sin at B cos at so V1 cos mt cos sin mt sin Asin at B cos mt equate coefficients 3 A V1 sin B V1 cos takeratio3 tan amp V1AzBz going back to differential equation put in A V1 cos v0 As1n mt B cos at V1 cosat gt w1th B V1 s1n dvm so mA cos mt wB s1n wt and our dlfferentlal equatlon d 1 1 VW vm V cos mt becomes the algebralc equatlon dt RC RC 1 1 mA cos mt wBs1n at A s1n mt Bcos wt V cos at RC RC equate coefficients 3 A B 0 S111 a cos mAiL RC RC or B A wRC and Aa 1 2 L aRC RC so 12 BZLZL41 mRC7 QRC QRC mRC7 wRC mRC ml ycz or AV R 1a22RCZ 1a2RC2 Page 4 of 10 Physics 111 BSC Lecture 2 so from above VI A2 1132 A 1 MIC V QRC2 1 wRC L WRCY mRC V mRC mRC21 2 mRC2 1 K mRC V 1 1wRCZ and tan391 tan391 mRC tan391 wRC Keep V xed look at Vom vs nRC 7 VlV 3 transmits low 03 well high 03 not at 10 all 3 Low Pass Filter 2 12 3 4 RC Different frequencies give different amplitudes and phases for Vom Plot instead on a loglog scale Bode Plot VlV 10 1f roll off 101 slope 20 dbdecade 6 dboctave 10392 10 110 100 wRC m set of straight line segments where segments meet are comer or break frequencies 1 RC Page 5 of 10 Physics 111 BSC Lecture 2 As I said last time useful unit Voltage transfer b E db 2010g10 Vim J eg signal reduced by a factor of 10 has a gain of 720 db Put hi amp low pass together to make lters HampH 501505 prob 213 hi amp 1 low 03 pass band 11 More Concepts 0n Networks Networks ideal voltage amp current sources independent gt i gt i v is v real model 39 gti AAA gtl gt v5 v 15 gtR v gt s 0 dependent or controlled source 7 important for modeling 0 0 g V l 2 R V VS o 39 in using KVL 7 loop equations 7 there is a positive drop in the direction of current through a resistor and there is a positive drop in the direction to 7 terminal of a battery independent of the current direction The effective resistance between any 2 Page 6 of 10 Physics 111 BSC Lecture 2 points in the net is given by measuring the current I drawn by an external voltage source V R V I ideal voltage sources are shorted ideal current sources are opened and dependent sources are left ON Network Theorems 1 Superposition The response of a linear network containing several independent sources is found by considering each generator separately and then adding the individual responses ie rest are shut off as above 2 Thevenin s Theorem Any linear network may with respect to a pair of terminals be replaced by a voltage generator Vth equal to the opencircuit voltage in series with the resistance Rm seen between these terminals As above to nd Rm all independent voltage sources are short circuited and all independent current sources are opened 3 Norton s Theorem Any linear network may with respect to a pair of terminals be replaced by a current generator equal to the shortcircuit current in parallel with the resistance seen between the 2 terminals two generators on page 5 are equal if V v is 139 B R Rn R351 VthR 4 Open Circuit Voltage 7 Short Circuit Current Theorems i 139 l 0 155 a vac open circuit voltage a ifC short circuit current 0 Rm resistance between terminals V0 net 0 Then v V0 lscRm Rm M l s Page 7 of 10 Physics 111 BSC Lecture 2 slope lR 2 points determine the line Using the Concepts Simple Example V2 14V What s the voltage across R3 Short node 2 t0 4 then using superposition ZK SE R1 R2 6mA gml 6 16mA 44mA Page 8 of 10 Physics 111 BSC Lecture 2 Turn off voltage sources 24 resistance corresponds to R113 all in parallel 3 1 1 1 1 1 16 16mA Rm 1 2 9 Q A 44mA V 3 V R 16mA 24 1 th 28V Lab Page 2 What is Thevenin s equivalent for o MI gt nd equation for box voltage across AB is for A B 139 11 I2 2V 1V 2 2 4Q Page 9 of 10 Physics 111 BSC Lecture 2 short voltage sources AB resistance is 2 2 4 in 11 3 1 1 1 1 rm 2 2 4 rm Q so VAR 139SE R2q 3 V 4 Q 49 5 3 V 5 so 1 45 35V so for whole box 3 V0 AV rm A Q so 0 95 35 V L o Page 10 of10


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