# Capacitors and inductors

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear resisive circuis are applicable o circuis ha conain capaciors and inducors. Unlike he resisor which dissipaes energy, ideal capaciors and inducors sore energy raher han dissipaing i. Capacior: In boh digial and analog elecronic circuis a capacior is a fundamenal elemen. I enables he filering of signals and i provides a fundamenal memory elemen. The capacior is an elemen ha sores energy in an elecric field. The circui symbol and associaed elecrical variables for he capacior is shown on Figure. i C v Figure. Circui symbol for capacior The capacior may be modeled as wo conducing plaes separaed by a dielecric as shown on Figure 2. When a volage v is applied across he plaes, a charge q accumulaes on one plae and a charge q on he oher. insulaor plae of area A q and hickness s v E d q s Figure 2. Capacior model 6.07/22.07 Spring 2006, Chanioakis and Cory

2 If he plaes have an area A and are separaed by a disance d, he elecric field generaed across he plaes is and he volage across he capacior plaes is q E = (.) εα qd v= Ed = (.2) ε A The curren flowing ino he capacior is he rae of change of he charge across he dq capacior plaes i =. And hus we have, d dq d εa εa dv dv i = = v = = C (.3) d d d d d d The consan of proporionaliy C is referred o as he capaciance of he capacior. I is a funcion of he geomeric characerisics of he capacior plae separaion (d) and plae area (A) and by he permiiviy (ε) of he dielecric maerial beween he plaes. ε A C = (.4) d Capaciance represens he efficiency of charge sorage and i is measured in unis of Farads (F). The currenvolage relaionship of a capacior is dv i = C (.5) d The presence of ime in he characerisic equaion of he capacior inroduces new and exciing behavior of he circuis ha conain hem. Noe ha for DC (consan in ime) dv signals ( 0 d = ) he capacior acs as an open circui (i=0). Also noe he capacior does no like volage disconinuiies since ha would require ha he curren goes o infiniy which is no physically possible. If we inegrae Equaion (.5) over ime we have 6.07/22.07 Spring 2006, Chanioakis and Cory 2

3 dv id = C d (.6) d v= id C (.7) = id v(0) C 0 The consan of inegraion v(0) represens he volage of he capacior a ime =0. The presence of he consan of inegraion v(0) is he reason for he memory properies of he capacior. Le s now consider he circui shown on Figure 3 where a capacior of capaciance C is conneced o a ime varying volage source v(). i() v() C v Figure 3. Fundamenal capacior circui If he volage v() has he form Then he curren i() becomes v () = Acos( ω) (.8) dv i () = C d = CAωsin( ω) π = CωAcos ω 2 Therefore he curren going hrough a capacior and he volage across he capacior are 90 degrees ou of phase. I is said ha he curren leads he volage by 90 degrees. The general plo of he volage and curren of a capacior is shown on Figure 4. The curren leads he volage by 90 degrees. (.9) 6.07/22.07 Spring 2006, Chanioakis and Cory 3

4 Figure 4 If we ake he raio of he peak volage o he peak curren we obain he quaniy Xc = (.0) Cω Xc has he unis of Vols/Amperes or Ohms and hus i represens some ype of resisance. Noe ha as he frequency ω 0 he quaniy Xc goes o infiniy which implies ha he capacior resembles an open circui. Capaciors do like o pass curren a low frequencies As he frequency becomes very large ω he quaniy Xc goes o zero which implies ha he capacior resembles a shor circui. Capaciors like o pass curren a high frequencies Capaciors conneced in series and in parallel combine o an equivalen capaciance. Le s firs consider he parallel combinaion of capaciors as shown on Figure 5. Noe ha all capaciors have he same volage, v, across hem. i() i i2 i3 in v() v C C2 C3 Cn Figure 5. Parallel combinaion of capaciors. 6.07/22.07 Spring 2006, Chanioakis and Cory 4

5 By applying KCL we obain And since dv ik = Ck we have d i= i i2 i3 in (.) dv dv dv dv i = C C2 C3 Cn d d d d dv = C C2 C3 Cn Ceq d (.2) dv = Ceq d Capaciors conneced in parallel combine like resisors in series Nex le s look a he series combinaion of capaciors as shown on Figure 6. i() C C2 C3 Cn v v2 v3 vn v() Figure 6. Series combinaion of n capaciors. Now by applying KVL around he loop and using Equaion (.7) we have v= v v2 v3 vn = id () v(0) C C2 C3 Cn 0 Ceq = id () v(0) Ceq 0 (.3) Capaciors in series combine like resisors in parallel 6.07/22.07 Spring 2006, Chanioakis and Cory 5

6 By exension we can calculae he volage division rule for capaciors conneced in series. Here le s consider he case of only wo capaciors conneced in series as shown on Figure 7. i() v() v v2 C C2 Figure 7. Series combinaion of wo capaciors The same curren flows hrough boh capaciors and so he volages v and v2 across hem are given by: And KVL around he loop resuls in v = id C (.4) 0 v2 = id C2 (.5) 0 v () = id C C2 (.6) 0 Which in urn gives he volages v and v2 in erms of v and he capaciances: v C2 = v C C 2 v C 2 = v C C 2 (.7) (.8) Similarly in he parallel arrangemen of capaciors (Figure 8) he curren division rule is i C = i C C 2 i C2 2 = i C C 2 (.9) (.20) Assume here ha boh capaciors are iniially uncharged 6.07/22.07 Spring 2006, Chanioakis and Cory 6

7 i() i() v C i C2 i2 Figure 8. Parallel arrangemen of wo capaciors The insananeous power delivered o a capacior is P () = iv () () (.2) The energy sored in a capacior is he inegral of he insananeous power. Assuming ha = v( ) = 0 hen he energy sored he capacior had no charge across is plaes a [ ] in he capacior a ime is E () = P( τ) dτ = v( τ) i( τ) dτ dv( τ ) = v( τ ) C dτ dτ = Cv () 2 2 (.22) 6.07/22.07 Spring 2006, Chanioakis and Cory 7

8 Real Capaciors. If he dielecric maerial beween he plaes of a capacior has a finie resisiviy as compared o infinie resisiviy in he case of an ideal capacior hen here is going o be a small amoun of curren flowing beween he capacior plaes. In addiion here are lead resisance and plae effecs. In general he circui model of a nonideal capacior is shown on Figure 9 i C nonideal = i v C Rs Rp Figure 9. Circui of nonideal capacior The resisance Rp is ypically very large and i represens he resisance of he dielecric maerial. Resisance Rs is ypically small and i corresponds o he lead and plae resisance as well as resisance effecs due o he operaing condiions (for example signal frequency) In pracice we are concerned wih he in series resisance of a capacior called he Equivalen Series Resisance (ESR). ESR is a very imporan capacior characerisic and mus be aken ino consideraion in circui design. Therefore he nonideal capacior model of ineres o us is shown on i R(ESR) C Figure 0. Nonideal capacior wih series resisor. Typical values of ESR are in he mωω range. 6.07/22.07 Spring 2006, Chanioakis and Cory 8

9 A capacior sores energy in he form of an elecric field dv Currenvolage relaionship i = C, v id d = C In DC he capacior acs as an open circui The capaciance C represens he efficiency of soring charge. The uni of capaciance is he Farad (F). Farad=Coulomb/Vol 3 Typical capacior values are in he mf ( 0 2 F) o pf ( 0 F) The energy sored in a capacior is E = Cv 2 2 Large capaciors should always be sored wih shored leads. Example: A 47µF capacior is conneced o a volage which varies in ime as v ( ) = 20sin(200 π) vols. Calculae he curren i() hrough he capacior v() i() C v The curren is given by dv i = C d 6 d 6 = sin(200 π) = π cos(200 π) = 0.59cos(200 π) Amperes d 6.07/22.07 Spring 2006, Chanioakis and Cory 9

10 Example: Calculae he energy sored in he capacior of he circui o he righ under DC condiions. In order o calculae he energy sored in he capacior we mus deermine he volage across i and hen use Equaion (.22). k Ω 8 V uf 2k Ω We know ha under DC condiions he capacior appears as an open circui (no curren flowing hrough i). Therefore he corresponding circui is k Ω v 2k Ω 8 V And from he volage divider formed by he kω and he 2kΩ resisors he volage v is 2Vols. Therefore he energy sored in he capacior is Ec = Cv = = 72µJoule s /22.07 Spring 2006, Chanioakis and Cory 0

11 Example Calculae he energy sored in he capaciors of he following circui under DC condiions. C 50uF 0k Ω 25k Ω 0 V uf C2 C3 50k Ω 0uF Again DC condiions imply ha he capacior behaves like an open circui and he corresponding circui is C v 0 V 0k Ω C2 25k Ω C3 50k Ω v2 From his circui we see ha he volages v and v2 are boh equal o 0 Vols and hus he volage across capacior C is 0 Vols. Therefore he energy sored in he capaciors is: For capacior C: For capacior C2: For capacior C3: 0 Joules EC 2 = C2v = 0 0 = 50µJoules EC 3 = C3v = = 500µJoules /22.07 Spring 2006, Chanioakis and Cory

12 Inducors The inducor is a coil which sores energy in he magneic field Consider a wire of lengh l forming a loop of area A as shown on Figure. A curren i() is flowing hrough he wire as indicaed. This curren generaes a magneic field B which is equal o i () B () = µ (.23) l Where µ is he magneic permeabiliy of he maerial enclosed by he wire. i() B l Loop lengh A Area Figure. Curren loop for he calculaion of inducance The magneic flux, Φ, hrough he loop of area A is Φ = AB() Aµ = i () l = Li() Aµ Where we have defined L. l From Maxwell s equaions we know ha (.24) d v () d (.25) dli () = v () d (.26) And by aking L o be a consan we obain he currenvolage relaionship for his loop of wire also called an inducor. 6.07/22.07 Spring 2006, Chanioakis and Cory 2

13 di v = L (.27) d The parameer L is called he inducance of he inducor. I has he uni of Henry (H). The circui symbol and associaed elecrical variables for he inducor is shown on Figure 2 i L v Figure 2. Circui symbol of inducor. di For DC signals ( 0 d = ) he inducor acs as a shor circui (v=0). Also noe he inducor does no like curren disconinuiies since ha would require ha he volage across i goes o infiniy which is no physically possible. (We should keep his in mind when we design inducive devices. The curren hrough he inducor mus no be allowed o change insananeously.) If we inegrae Equaion (.27) over ime we have di vd = L d (.28) d i = vd L (.29) = vd i(0) L 0 The consan i(0) represens he curren hrough he inducor a ime =0. (Noe ha we have also assumed ha he curren a = was zero.) 6.07/22.07 Spring 2006, Chanioakis and Cory 3

14 Le s now consider he circui shown on Figure 3 where an inducor of inducance L is conneced o a ime varying curren source i(). i() i() v L Figure 3. Fundamenal inducor circui If we assume ha he curren i() has he form Then he volage v() becomes i () = I cos( ω) (.30) o di v () = L d = LI ωsin( ω) o π = LωIo cos ω 2 (.3) Therefore he curren going hrough an inducor and he volage across he inducor are 90 degrees ou of phase. Here he volage leads he curren by 90 degrees. The general plo of he volage and curren of an inducor is shown on Figure 4. Figure /22.07 Spring 2006, Chanioakis and Cory 4

15 Inducor conneced in series and in parallel combine o an equivalen inducance. Le s firs consider he parallel combinaion of inducors as shown on Figure 5. Noe ha all inducors have he same volage across hem. i() i i2 i3 in v() v L L2 L3 Ln By applying KCL we obain Figure 5. Parallel combinaion of inducors. And since i= i i2 i3 in (.32) ik = vd ik(0) Lk we have 0 i = vd i(0) vd i2(0) vd i3(0) vd in(0) L L2 L3 Ln = vd i(0) i2(0) i3(0) in(0) L L2 L3 Ln 0 i(0) Leq = vd i(0) Leq (.33) Inducors in parallel combine like resisors in parallel Nex le s look a he series combinaion of inducors as shown on Figure 6. i() L L2 L3 Ln v v2 v3 vn v() Figure 6. Series combinaion of inducors. 6.07/22.07 Spring 2006, Chanioakis and Cory 5

16 Now by applying KVL around he loop we have v= v v2 v3 vn di = L L2 L3 Ln (.34) Leq d di = Leq d Inducor in series combine like resisor in series The energy sored in an inducor is he inegral of he insananeous power delivered o he inducor. Assuming ha he inducor had no curren flowing hrough i a = i( ) = 0 hen he energy sored in he inducor a ime is [ ] E () = P( τ) dτ = v( τ) i( τ) dτ di( τ ) = L i ( τ ) dτ dτ = Li () 2 2 (.35) 6.07/22.07 Spring 2006, Chanioakis and Cory 6

17 Real Inducors. There are wo conribuions o he nonideal behavior of inducors.. The finie resisance of he wire used o wind he coil 2. The cross urn effecs which become imporan a high frequencies The nonideal inducor may hus be modeled as shown on Figure 7 i L v = v i L Rc resisance of coil (small value) Rf Frequency dependen urn o urn field effecs (imporan a high frequecnies) nonideal Figure 7. Circui momdel of nonideal inducor In addiion o he resisive nonidealiies of inducors here could also be capaciive effecs. These effecs usually become imporan a high frequencies. Unless saed oherwise, hese effecs will be negleced in ou analysis. A inducor sores energy in a magneic field di Currenvolage relaionship v = L, i vd d = L 2 The energy sored in an inducor is E = Li 2 In DC he inducor behaves like a shor circui The inducance L represens he efficiency of soring magneic flux. 6.07/22.07 Spring 2006, Chanioakis and Cory 7

18 Problems: Calculae he equivalen capaciance for he following arrangemens: Ceq F 00 F 5 F 2 F F Ceq C C C C 2C 2C 2C o infiniy Calculae he volage across each capacior and he energy sored in each capacior. 0 V 0uF 2uF 0uF 20uF 5uF In he circui below he curren source provides a curren of i = 0exp( 2 ) ma. Calculae he volage across each capacior and he energy sored in each capacior a ime =2 sec. v i() 0uF 20uF v2 5uF 6.07/22.07 Spring 2006, Chanioakis and Cory 8

19 6.07/22.07 Spring 2006, Chanioakis and Cory 9

### Chapter 7. Response of First-Order RL and RC Circuits

Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

### Inductance and Transient Circuits

Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

### RC (Resistor-Capacitor) Circuits. AP Physics C

(Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

### Circuit Types. () i( t) ( )

Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All

### Chapter 2: Principles of steady-state converter analysis

Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

### Physics 111 Fall 2007 Electric Currents and DC Circuits

Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels

### 9. Capacitor and Resistor Circuits

ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

### 4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F

efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos

### RC, RL and RLC circuits

Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

### Module 3. R-L & R-C Transients. Version 2 EE IIT, Kharagpur

Module 3 - & -C Transiens esson 0 Sudy of DC ransiens in - and -C circuis Objecives Definiion of inducance and coninuiy condiion for inducors. To undersand he rise or fall of curren in a simple series

### µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ

Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high

### Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

### Switching Regulator IC series Capacitor Calculation for Buck converter IC

Swiching Regulaor IC series Capacior Calculaion for Buck converer IC No.14027ECY02 This applicaion noe explains he calculaion of exernal capacior value for buck converer IC circui. Buck converer IIN IDD

### Using RCtime to Measure Resistance

Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a buil-in ADC (Analog o Digial Converer)

### Differential Equations and Linear Superposition

Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

### CHARGE AND DISCHARGE OF A CAPACITOR

REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

### Fourier Series Solution of the Heat Equation

Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS

EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67 - FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1 - RANIEN Uni conen 2 Undersand he ransien behaviour of resisor-capacior (R) and resisor-inducor (RL) D circuis

### RC Circuit and Time Constant

ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

### Chabot College Physics Lab RC Circuits Scott Hildreth

Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

### CAPACITANCE AND INDUCTANCE

CHAPTER 6 CAPACITANCE AND INDUCTANCE THE LEARNING GOALS FOR THIS CHAPTER ARE: Know how o use circui models for inducors and capaciors o calculae volage, curren, and power Be able o calculae sored energy

### A Mathematical Description of MOSFET Behavior

10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

### Basic Circuit Elements - Prof J R Lucas

Basic Circui Elemens - Prof J ucas An elecrical circui is an inerconnecion of elecrical circui elemens. These circui elemens can be caegorized ino wo ypes, namely acive elemens and passive elemens. Some

### Laboratory #3 Diode Basics and Applications (I)

Laboraory #3 iode asics and pplicaions (I) I. Objecives 1. Undersand he basic properies of diodes. 2. Undersand he basic properies and operaional principles of some dioderecifier circuis. II. omponens

### FE Review Basic Circuits. William Hageman

FE eview Basic Circuis William Hageman -8-04 FE opics General FE 4. Elecriciy, Power, and Magneism 7 A. Elecrical fundamenals (e.g., charge, curren, volage, resisance, power, energy) B. Curren and volage

### 11. Properties of alternating currents of LCR-electric circuits

WS. Properies of alernaing currens of L-elecric circuis. Inroducion So-called passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science

### Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

### Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

### 2 Electric Circuits Concepts Durham

Chaper 3 - Mehods Chaper 3 - Mehods... 3. nroducion... 2 3.2 Elecrical laws... 2 3.2. Definiions... 2 3.2.2 Kirchhoff... 2 3.2.3 Faraday... 3 3.2.4 Conservaion... 3 3.2.5 Power... 3 3.2.6 Complee... 4

### Transient Analysis of First Order RC and RL circuits

Transien Analysis of Firs Order and iruis The irui shown on Figure 1 wih he swih open is haraerized by a pariular operaing ondiion. Sine he swih is open, no urren flows in he irui (i=0) and v=0. The volage

### Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

### 8. TIME-VARYING ELECTRICAL ELEMENTS, CIRCUITS, & THE DYNAMICS

8. TIMEVARYING ELECTRICAL ELEMENTS, CIRCUITS, & THE DYNAMICS 8..1 Timevarying Resisors Le us firs consider he case of a resisor, a simple saic device. In addiion o he wo variables, volage and curren, which

### Understanding Sequential Circuit Timing

ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

### Voltage level shifting

rek Applicaion Noe Number 1 r. Maciej A. Noras Absrac A brief descripion of volage shifing circuis. 1 Inroducion In applicaions requiring a unipolar A volage signal, he signal may be delivered from a bi-polar

### ENE 104 Electric Circuit Theory

Elecric Circui heory Lecure 0: AC Power Circui Analysis (ENE) Mon, 9 Mar 0 / (EE) Wed, 8 Mar 0 : Dejwoo KHAWPARSUH hp://websaff.ku.ac.h/~dejwoo.kha/ Objecives : Ch Page he insananeous power he average

### 6.003 Homework #4 Solutions

6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d

### Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow.

Whies, EE 481 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih a ground

### LECTURE 9. C. Appendix

LECTURE 9 A. Buck-Boos Converer Design 1. Vol-Sec Balance: f(d), seadysae ransfer funcion 2. DC Operaing Poin via Charge Balance: I(D) in seady-sae 3. Ripple Volage / C Spec 4. Ripple Curren / L Spec 5.

### Brown University PHYS 0060 INDUCTANCE

Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide INTODUCTION INDUCTANCE Anyone who has ever grabbed an auomobile spark-plug wire a he wrong place, wih he engine running, has an appreciaion

### 1. The graph shows the variation with time t of the velocity v of an object.

1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

### Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

### Making Use of Gate Charge Information in MOSFET and IGBT Data Sheets

Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have

### DC-DC Boost Converter with Constant Output Voltage for Grid Connected Photovoltaic Application System

DC-DC Boos Converer wih Consan Oupu Volage for Grid Conneced Phoovolaic Applicaion Sysem Pui-Weng Chan, Syafrudin Masri Universii Sains Malaysia E-mail: edmond_chan85@homail.com, syaf@eng.usm.my Absrac

### Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

### The Transport Equation

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

### Lenz's Law. Definition from the book:

Lenz's Law Definiion from he book: The induced emf resuling from a changing magneic flux has a polariy ha leads o an induced curren whose direcion is such ha he induced magneic field opposes he original

### Chapter 2 Kinematics in One Dimension

Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

### PHYS245 Lab: RC circuits

PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of

### Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

### ORDER INFORMATION TO pin 300 ~ 360mV AMC7150DLF 300 ~ 320mV AMC7150ADLF 320 ~ 340mV AMC7150BDLF 340 ~ 360mV AMC7150CDLF

www.addmek.com DESCRIPTI is a PWM power ED driver IC. The driving curren from few milliamps up o 1.5A. I allows high brighness power ED operaing a high efficiency from 4Vdc o 40Vdc. Up o 200KHz exernal

### Signal Processing and Linear Systems I

Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

### Two Compartment Body Model and V d Terms by Jeff Stark

Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

### CHAPTER 5 CAPACITORS

CHAPTER 5 CAPACITORS 5. Inroducion A capacior consiss of wo meal plaes separaed by a nonconducing medium (known as he dielecric medium or simply he dielecric, or by a vacuum. I is represened by he elecrical

### Equation for a line. Synthetic Impulse Response 0.5 0.5. 0 5 10 15 20 25 Time (sec) x(t) m

Fundamenals of Signals Overview Definiion Examples Energy and power Signal ransformaions Periodic signals Symmery Exponenial & sinusoidal signals Basis funcions Equaion for a line x() m x() =m( ) You will

### 17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

### Thyristor Based Speed Control Techniques of DC Motor: A Comparative Analysis

Inernaional Journal of Scienific and Research Publicaions, Volume 2, Issue 6, June 2012 1 Thyrisor Based Speed Conrol Techniques of DC Moor: A Comparaive Analysis Rohi Gupa, Ruchika Lamba, Subhransu Padhee

### Signal Rectification

9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

### CHAPTER 21: Electromagnetic Induction and Faraday s Law

HAT : lecromagneic nducion and Faraday s aw Answers o Quesions. The advanage of using many urns (N = large number) in Faraday s experimens is ha he emf and induced curren are proporional o N, which makes

### State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

### Astable multivibrator using the 555 IC.(10)

Visi hp://elecronicsclub.cjb.ne for more resources THE 555 IC TIMER The 555 IC TIMER.(2) Monosable mulivibraor using he 555 IC imer...() Design Example 1 wih Mulisim 2001 ools and graphs..(8) Lile descripion

### MTH6121 Introduction to Mathematical Finance Lesson 5

26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

### AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

### Fourier series. Learning outcomes

Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

### Math 201 Lecture 12: Cauchy-Euler Equations

Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

### UMR EMC Laboratory UMR EMC Laboratory Technical Report: TR

UMR EMC Laboraory UMR EMC Laboraory Dep. of Elecrical & Compuer Engineering 870 Miner Circle Universiy of Missouri Rolla Rolla, MO 65409-0040 UMR EMC Laboraory Technical Repor: TR0-8-00 Effec of Delay

### cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

### s-domain Circuit Analysis

Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preered c decribed by ODE and heir Order equal number of plu number of Elemenbyelemen and ource ranformaion

### FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in

### The Torsion of Thin, Open Sections

EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such

### Part II Converter Dynamics and Control

Par II onverer Dynamics and onrol 7. A equivalen circui modeling 8. onverer ransfer funcions 9. onroller design 1. Inpu filer design 11. A and D equivalen circui modeling of he disconinuous conducion mode

### TLE 472x Family Stepper Motor Drivers. Current Control Method and Accuracy

Applicaion Noe, V 1.0, Augus 2001 ANPS063E TLE 472x Family Sepper Moor Drivers Curren Conrol Mehod and Accuracy by Frank Heinrichs Auomoive Power N e v e r s o p h i n k i n g. - 1 - TLE 472x sepper moor

### Stochastic Optimal Control Problem for Life Insurance

Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

Chaper 8 Copyrigh 1997-2004 Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon

### Graphing the Von Bertalanffy Growth Equation

file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

### Chapter 8: Regression with Lagged Explanatory Variables

Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

### Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

### PRESSURE BUILDUP. Figure 1: Schematic of an ideal buildup test

Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics PRESSURE BUILDUP I is difficul o kee he rae consan in a roducing well. This is no an issue in a buildu es since he well is closed.

### 4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

### 2. Dissipation mechanisms: Resistive Eddy currents Flux pinning Coupling currents

AC loss par I Fedor Gömöry Insiue of Elecrical Engineering Slovak Academy of Sciences Dubravska cesa 9, 84101 raislava, Slovakia elekgomo@savba.sk www.elu.sav.sk Ouline of Par I: 1. Wha is AC loss 2. Dissipaion

### Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

### THE PRESSURE DERIVATIVE

Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.

### AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

### Lecture-10 BJT Switching Characteristics, Small Signal Model

1 Lecure-1 BJT Swiching Characerisics, Small Signal Model BJT Swiching Characerisics: The circui in Fig.1(b) is a simple CE swich. The inpu volage waveform v s shown in he Fig.1(a) is used o conrol he

### 23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

### HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

### Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

### Diagnostic Examination

Diagnosic Examinaion TOPIC XV: ENGINEERING ECONOMICS TIME LIMIT: 45 MINUTES 1. Approximaely how many years will i ake o double an invesmen a a 6% effecive annual rae? (A) 10 yr (B) 12 yr (C) 15 yr (D)

### Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

### Switched Mode Converters (1 Quadrant)

(1 Quadran) Philippe Barrade Laboraoire d Elecronique Indusrielle, LEI STI ISE Ecole Polyechnique Fédérale de Lausanne, EPFL Ch-1015 Lausanne Tél: +41 21 693 2651 Fax: +41 21 693 2600 Philippe.barrade@epfl.ch

### Full-wave Bridge Rectifier Analysis

Full-wave Brige Recifier Analysis Jahan A. Feuch, Ocober, 00 his aer evelos aroximae equais for esigning or analyzing a full-wave brige recifier eak-eecor circui. his circui is commly use in A o D cverers,

### Brown University PHYS 0060 Physics Department LAB B Measuring the Earth s Magnetic Field

Measuring he Earh s Magneic Field As is well known he Earh has a small magneic field on he order of 0.5 Gauss. I is an ineresing and challenging lab exercise o ry and measure i. elow we lis 5 mehods wih

### 23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

### Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

### Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

### Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

### Converter Topologies

High Sepup Raio DCDC Converer Topologies Par I Speaker: G. Spiazzi P. Teni,, L. Rosseo,, G. Spiazzi,, S. Buso,, P. Maavelli, L. Corradini Dep. of Informaion Engineering DEI Universiy of Padova Seminar

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

### Present Value Methodology

Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer