Intro to ElectTest Instrum
Intro to ElectTest Instrum PHY 4744C
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This 12 page Class Notes was uploaded by Jairo Dooley on Wednesday September 23, 2015. The Class Notes belongs to PHY 4744C at University of South Florida taught by Myung Kim in Fall. Since its upload, it has received 63 views. For similar materials see /class/212683/phy-4744c-university-of-south-florida in Physics 2 at University of South Florida.
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Date Created: 09/23/15
BASIC THEORY OF AC CIRCUITS Sinusoidal signal Capacitance Inductance Impedance Power dissipated in an AC circuit RC circuit Resonance Signals ANATOMY OF SINUSOI DAL SIGNAL A A VVVV T 4 D vtVU Vl coswtp viti vt instantaneous voltage V n V1 dc bias offset amplitude 27 a 27rf T angular frequency f frequency s391 Hz hertz T period W 3 P 3amp1 ase angle Notational Convention fixed quantities are denoted in upper case timevarying andor small quantities are denoted in lower case 2006 M K Kim AC Basics Amplitude amplitude Vl eg 50 Vp peaktopeak voltage VP 2V eg 100 Vpp rms rootmeansquare voltage Vm Vl HE 0707Vl eg 35 Vrms COMPLEX NOTATION vt V coswt p Re Ke mw 5 vt Vl39e mw Review of Complex Algebra 2 XjY j E 471 Complex conjugate 2x Xin Absolute magnitude XZ YZ I Y 8tan 1 X Phase angle Real and imaginary parts X ReZZcosamp Y Imp l2lsin8 dim Z X1 mm 116 X1Xzjz n Multiplication ZZZ XljY1XXZ 1104ng Km XJZ Yle Division M z Xm39Yz X5 Euler Formula 2 2cosBjsin8 l le g Multiplication and division are especially simple using exponential notation AC Basics CAPACITANCE amp CAPACITORS Capacitance C F The charge stored on a capacitor is proportional to the voltage across it Q CV Capacitance C V parallel plate capacitor C Q Q A V d 40 cm 0 c Va gji d dt 2 energy storage U 1 E d capacitor combinations series amp parallel series C1 C2 C3 C parallel C C1 C2 03 Cn AC Response of a Capacitor Capacitive Impedance using real notation vtVncosat itCwcr sinwt W vt t it t using complex notation d vtgt View to 0 ijVneJ ne 2c 20 oc 7 27 voltage lags current by 90 Capacitors Usage of capacitors energy storage filter bypass amp blocking tuning timing Tolerance 50 typical Electrolytic capacitors oxide formed by electrochemical reaction do not reverse polarity big ones can explo e AC Basics aluminum tnatalum many p Nonelectrolytic capacitors plastics mica glass ceramic paper Variable capacitors tuning trimm39n Marking larger ones have values printed otherwise no consistent scheme Large capacitors in high power pulsed circuits can be lethal handle with caution INDUCTANCE L Inductance L H e magnetic ux stored in an inductor is proportional to the current through it CD LI CD solenoid inductor L MNA I gt LDQQ9QJ lt gt v 050 vtLg ink UH LIZ energy storage U T Inductive Reactance d 11 Ind vt L7 ijIUe w jVne m Z gt ij JrrZ ZC oc j e voltage leads current by 90 Inductors ferrite beads transformers AC Basics 5 IMPEDANCE Z Impedance a generalization of resistance is complex 1 ZRj7 QLRJX wC ZR R resistance XXCXL reactance XC 7L capacitive reactance wC XL wL inductive reactance Irn xLmLA Xc 1mC Ohm s law generalized vt itZ Z RJX ZeW it I e vt itZ I e Ze IZe1l lzl lquot l E In It Now all the basic circuit theorems can be applied to analysis of ac circuits as long as the resistors in the previous statements of theorems are understood to represent general complex impedances series and parallel connections of impedance voltage divider and current divider Kirchoff laws Thevenin s and Norton s theorems At dc a capacitor looks like an open circuit and and an inductor a short circuit a gt 0 X gt 00 X At high frequency a capacitor looks like a short and an inductor an open circuit a gt oo XC gt 039 AC Basics POWER DISSIPATED IN AN AC CIRCUIT in real notation it IO cosmt vt V cosat 11 instantaneous power dissipated Pt itvt Iolg cos2at 11 cos 11 average power P Iolg cos11 in complex notation it 106 vt neWW Pt JorgeMW 2 P 0 wrong P Rez39t Re Va Re emf 10V0 cos11 PIOV0 resistive load 11 0 3 Also note that P j 10 X t V0 mlm reactive load 11 7r 2 3 0 Capacitors and inductors do not dissipate any energy energy stored and extracted periodically in the electric or magnetic fields but there is no average dissipation of energy VC V1 139RV1 RCdVC dt chle VC dVR dt C VI VHR dt RC 0 rtRC 0 rtRC VR VRe 1 Vce VG V V V0ertRC 1 C AC Basics RC CIRCUIT Charging and Dicharging of Capacitor u o vt time constant 4 RC general solution for v switching from VD to V1 vctVC 4391equot Vl W V1 7 V5 quotC tm m W1N V VVV rltlt1f vct vRt t t rgtgt1f VCt VRt good to remember e l 37 e 2 14 e 3 50 e 4 18 e 5 07 4 1447 1m in AC Basics LowPass Filter 2 R ltgt Wt lr O c vct p o vt Ver output across capacitor v t vt CV i wCeJm g vt C R Z0 R 7 j 0C 0 Vcfwc g v RijwC lgcle amplitude gain 1 igcl we m inc 1l1wiec2 2 1 power gaIn G g c cl 1 M Cy Gain and Decibels I Amplitude gain gl or vm 1m P 2 2 2 Power gaIn G39 Ig forPocM or H Decibel Gin dB 1010gm G 201ogmg dB G 0 1 1 1 13 11 3 2 14 6 4 2 10 10 32 20 100 10 30 1000 32 40 10000 100 3 05 071 10 01 032 20 001 01 If there are a number of ampli cation stages then the total gain is the product of gains G G If the gain is expressed in dB then G q AC Basics 9 Bode Plot GC dB w 20dBdec n4 x n l l 102101 100 101 102 mm 102101 100 101 102 ch halfpower point or 3 dB frequency 50de lRC attenuation slope 20 dBdecade or 6 dBoctave HighPass Filter output across resistor R R t t V J t V14 Wm uRijwce EM R l a m gR v RijwC lgRle 1 11 tanquot1wRC El 11mCZ R 1 G Z R kkl 11aRCZ GRldE w 0 nZ 3 I n4 39 2ddBdec 39 o E 102101 100 lol 102 UJRC 102101 100 101 102 mm At low frequency a capacitor is like an open circuit There is very little current in the circuit and therefore very little voltage across the resistor and most of the signal appears across the capacitor At high frequency a capacitor is like a short circuit and most of the signal appears across the resistor Integrator and Differentiator q zm M C q 1 1 q 1 vctEE ttdtE vxn76JdINijlxntdt if ye ltlt vm d i N i vRt7zR7RdtqtiRCdtm zRRCdtvmt lvaltlt v current source input to capacitor would give true integrator AC Basics RESONANCE OJ 1 z LRC Fig A mo mo AC Basics SIGNALS oDC AC or sinusoidal mixture of DC amp AC Square wave Ramp sawtooth Triangle Pulse 0 Step 0 Analog signals 0 Digital or logic signals i as Comp dynalluc Sig offset bias dc Comp statlc 519 AC Basics 12 Noise Thermal noise Johnson generated in resistors by thermal electrons Interference stray or parasitic capacitance and inductance Ringing in transmission line coaxial cable due to repeated reflections of signal pulse corrected by impedance matching
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