University of Cyprus Biomedical Imaging and Applied Optics. Appendix. DC Circuits Capacitors and Inductors AC Circuits Operational Amplifiers

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1 Universiy of Cyprus Biomedical Imaging and Applied Opics Appendix DC Circuis Capaciors and Inducors AC Circuis Operaional Amplifiers

2 Circui Elemens An elecrical circui consiss of circui elemens such as volage sources, resisances, inducances and capaciances ha are conneced in closed pahs by conducors 2

3 DC Circuis Curren: The rae of moion of charge in a circui Symbol: I (or someimes i). SI unis: C/s = ampere (A) Convenional curren Assumed o consis of he moion of posiive charges. Convenional curren flows from higher o lower poenial. AC/DC Direc curren (DC) flows in one direcion around he circui Alernaing curren (AC) sloshes back and forh (ime-varying ha changes is sign periodically) I q q dq d 3

4 DC Circuis Ohm s aw and esisance: The curren ha flows hrough an objec is direcly proporional o he volage applied across he objec The consan of proporionaliy,, is called he resisance of he objec. SI uni of resisance: ohm (Ω) esisance depends on he geomery of he objec, and a propery, resisiviy, of he maerial from which i is made esisiviy symbol: ρ SI unis of resisiviy: ohm m (Ω m) V I cross-secional area A A 4

5 DC Circuis Elecrical Power Power is he ime rae of doing work Volage is he work done per uni charge Curren is he ime rae a which charge goes by Combining Ohm s aw subsiuions allow us o wrie several equivalen expressions for power egardless of how specified, power always has SI unis of was (W) W P W V q q I P VI P W I 2 W q q V VI 2 5

6 DC Circuis Series Connecion A circui, or a se of circui elemens, are said o be conneced in series if here is only one elecrical pah hrough hem. The same curren flows hrough all series-conneced elemens. (Equaion of coninuiy) Parallel Connecion A circui, or a se of circui elemens, are said o be conneced in parallel if he circui curren is divided among hem. The same poenial difference exiss across all parallel-conneced elemens. eq eq 1 2 n n 6

(Energy conservaion) The Curren aw: A any poin in a circui, he oal of he currens

7 DC Circuis Kirchoff s aws The Volage aw: Around any closed loop in a circui, he sum of he poenial changes mus equal zero. (Energy conservaion) The Curren aw: A any poin in a circui, he oal of he currens flowing ino ha poin mus be equal o he oal of he currens flowing ou of ha poin. (Charge conservaion; equaion of coninuiy) 7

8 DC Circuis Node Volage Analysis Mesh Curren Analysis v1 v s From Kirchoff s Curren aw v v v v v 4 v v v 5 v 3 v From Kirchoff s Volage aw ( ) ( ) 0 i1 i3 2 i1 i2 3 v A ( i i ) v i B 2 4 ( ) 0 i1 i3 2 i31 v B 8

9 DC Circuis Superposiion Principle The superposiion principle saes ha he oal response is he sum of he responses o each of he independen sources acing individually 9

10 DC Circuis Thévenin Equivalen Circuis Circui wih is source zeroed V v oc v oc eq i sc 10

11 DC Circuis Noron Equivalen Circuis In i sc I n V 11

12 12 12 d dp V i P V I DC Circuis Maximum Power Transfer

13 DC Circuis Diode Semiconducor device Posiive and negaive polariies Semiconducor Diode ecifier A forward-biased diode circui A reverse-biased diode circui A bipolar square wave applied o a forward-biased diode circui 13

14 Summary Table of DC Conceps 14

15 Capaciors and Inducors Capaciance Abiliy o sore charge The SI uni of capaciance is he farad: 1 farad = 1 F = 1 Coulomb/Vol For a given charge, a capacior wih a larger capaciance will have a greaer poenial difference A 0 Q V CV C C d 0 A d k 0 A d (wih dielecric) 15

16 Capaciors and Inducors ike resisors, capaciors in circuis can be conneced in series, in parallel, or in more-complex neworks conaining boh series and parallel connecions C C C C eq 1 2 n Ceq C1 C2... C n 16

17 Capaciors and Inducors Inducance Abiliy o sore magneic energy The polariy of he volage is such as o oppose he change in curren (enz s law). Inducance, uni: henry [H] 17

18 Capaciors and Inducors ike resisors, inducors in circuis can be conneced in series, in parallel, or in more-complex neworks conaining boh series and parallel connecions. eq eq

19 C Circuis A capacior conneced in series wih a resisor is par of an C circui. esisance limis charging curren Capaciance deermines ulimae charge Unlike a baery, a capacior canno provide a consan source of poenial difference. This value is consanly changing as he charge leaves he plae. Curren due o a discharging capacior is finie and changes over ime. 19

20 C Circuis Charging he capacior Assume no iniial charge in he capacior A he insan he source is conneced, he capacior sars o charge. The capacior coninues o charge unil i reaches is maximum charge (Q = Cε) Once he capacior is fully charged, he curren in he circui is zero. The poenial difference across he capacior maches ha supplied by he baery The charge on he capacior increases exponenially wih ime τ is he ime consan τ = C C q( ) C 1 e Q0 1 e dq Ce C I() e Ioe d C 20

21 C Circuis Discharging he capacior Assume a fully charged capacior A he insan he swich closes, he capacior sars o charge. The capacior coninues o discharge unil i reaches 0 Once he capacior is fully discharged, he curren in he circui is zero. The charge on he capacior decreases exponenially wih ime τ is he ime consan τ = C C q() Ce Q e dq C I() e I0e d C 0 21

AC Circuis Insananeous volage Insananeous curren θ is he phase angle In phasor form Impedance In series In parallel v() = V max sin ω i() = I max sin (ω - θ) I max = V max / Z

22 AC Circuis Insananeous volage Insananeous curren θ is he phase angle In phasor form Impedance In series In parallel v() = V max sin ω i() = I max sin (ω - θ) I max = V max / Z θ =an -1 (X/) Z = V / I 2 Z X X V=V rms 0 I = I rms θ V rms =V max / 2, I rms =I max / 2 Z =+jx X=X -X c Z eq = ( )+j(x 1 +X 2 ) 1/Z eq =1/( 1 +jx 1 )+1/( 2 +jx 2 ) C 2 22

23 AC Circuis Volage and curren waveforms in a resisive circui V I Z Volage and curren waveforms in a capaciive circui V C Z I C C Z C 1 j C 1 jc 1 C 90 23

24 AC Circuis Volage and curren waveforms in an inducive circui V Z I V j I Z j 90 Volage and curren waveforms in an C series circui 24

25 AC Circuis Power can be expressed in recangle form P- real power Q reacive power S = P + jq P=V rms I rms cos(θ) =I 2 rms Q = V rms I rms sin(θ) = V 2 rms/x S 2 = P 2 + Q 2 Power facor Maximum Power Transfer PF = cos(θ) = cos(θ v -θ i ) cos(θ)=1 θ = 0 Z load = Z * 25

26 Summary Table of AC Conceps 26

27 Operaional Amplifier (Op-amp) The op-amp is a device for increasing he power of a signal. I does his by aking power from a power supply and conrolling he oupu o mach he inpu signal shape bu wih a larger ampliude (Amplificaion). The op-amp is used also o perform arihmeic operaions (addiion, subracion, muliplicaion) wih signals. The properies of he negaive feedback loop deermine he properies of he circui conaining an op-amp. I has wo inpus: he invering inpu (-) and he non-invering inpu (+), and one oupu. I has usually wo supplies (±Vss) bu i can work wih one. Invering inpu Non-invering inpu V s s +V s s Oupu Symbol of op-amplifier 27

28 Operaional Amplifier (Op-amp) Ideal Op-Amp Op-amp equivalen circui: The wo inpus are 1 and 2. A differenial volage beween hem causes curren flow hrough he differenial resisance d. The differenial volage is muliplied by A (he open-loop gain of he op amp) o generae he oupu-volage source Any curren flowing o he oupu erminal vo mus pass hrough he oupu resisance o. 28

29 Operaional Amplifier (Op-amp) Inside he Op-Amp (IC-chip) 20 ransisors 11 resisors 1 capacior 29

30 Operaional Amplifier (Op-amp) eal vs. Ideal Op-amp Parameer Ideal Op Amp Typical Op Amp Open-loop volage gain A Common mode volage gain Frequency response f 1-20 MHz Inpu impedance Z in 10 6 Ω (bipolar) Ω (FET) Oupu impedance Z ou Ω 30

31 Operaional Amplifier (Op-amp) Summing Poin Consrain In a negaive feedback sysem, he ideal op-amp oupu volage aains he value needed o force he differenial inpu volage and inpu curren o zero. Circui soluion 1. Verify ha negaive feedback is presen. 2. Assume ha he differenial inpu volage and he inpu curren of he op amp are forced o zero. (This is he summing-poin consrain.) 3. Apply sandard circui-analysis principles, such as Kirchhoff s laws and Ohm s law, o solve for he quaniies of ineres. 31

32 Operaional Amplifier (Op-amp) Applying he Summing Poin Consrain 32

33 Operaional Amplifier (Op-amp) Invering Amplifier Non-invering Amplifiers A v v v o in 2 1 A v v v o in

34 Operaional Amplifier (Op-amp) Volage Follower A v v v o in

35 Operaional Amplifier (Op-amp) Summing Amplifier 35

36 Operaional Amplifier (Op-amp) Differenial Amplifier In differenial mode you can signals common o boh inpu signals 1V 3V 2V f g 1 2 f Vou V2 V1 1 36

37 Operaional Amplifier (Op-amp) Insrumenaion Amplifier High gain and high-inpu impedance. Composed of 2 amplifiers in noninvering forma and a 3 rd amplifier as a differenial amplifier V ou V2 V 1 gain 2 37

38 Operaional Amplifier (Op-amp) Differeniaors Inegraors v o C dv d in v o 1 C 0 v in d 38

39 Operaional Amplifier (Op-amp) Acive Filers- ow-pass Filer A low-pass filer aenuaes high frequencies G i i + f o f / i f / i Gain G V V o i j f 1 j i 1 j f C f f c = 1/2 i C f freq 39

40 Operaional Amplifier (Op-amp) Acive Filers (High-Pass Filer) A high-pass filer aenuaes low frequencies and blocks dc. f G i C i i + o f / i f / i Gain G V V o i j f jici j i 1 jici f c = 1/2 i C f freq 40

41 Operaional Amplifier (Op-amp) Acive Filers (Band-Pass Filer) A bandpass filer aenuaes boh low and high frequencies. C f G i C i i + f o f / i f / i V V o i j j f Ci j 1 j C 1 j C f f i i f c = 1/2 i C i f ch = 1/2 f C f freq 41

Chapter 7 Response of First-order RL and RC Circuits

Chapter 7 Response of First-order RL and RC Circuits Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial