# RC Circuit and Time Constant

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 capaciaive ime consan for he circui. esisor (100Ω), capacior (330 µf), wo volage sensors, power amplifier, pach-cords, compuer, PASO Inerface, and Daa Sudio Sofware. onsider he circui shown in Figure 1: FOWOD a S + _ b V V + _ + _ Figure 1. A resisor-capacior series circui. When he swich S is hrown o posiion a, charges sar o move and we have a curren flow. These charges move hrough he resisor and begin o charge he capacior. A any ime, he sum of he volages around he circui loop mus be zero (Kirchoff s ule), hence we have: (1) V V = 0 Where: = he applied volage (2) V = i = he volage drop across he resisor (3) (4) V = q he volage drop across he capacior When (2) is insered ino equaion (1) we obain: q = i + If we include he relaionship beween he curren and he charge, namely i = dq d 8M 1

2 (5) We obain: = q dq d = ( )I I 0 0( q) ( ) dq ( d ) quaion (5) conains wo variables, he charge on he capacior (q) and he ime (). Afer separaing hese variables we have: + q (6) 1 dq d = ( q) Seing equaion (6) up for inegraion and using he fac ha a he ime = 0 he charge q = 0 as he lower limi and he fac ha a some oher ime he charge is q as he upper limi we have: (7) 1 Le s now make a change of variable by leing u = q and hence du = dq. We will also need o change he limis of he q inegraion by noing ha when q = 0 he lower limi is u = and for he upper limi we replace q wih u = q. This allows us o rewrie (7) as follows: (8) 1 q du ( d = )I I ( u) 0 (9) (10) quaion (8) inegraes o: Insering he limis we have: Taking he anilog of equaion (10) we obain: 1 = lnu 1 q = ln( q) ln() = ln ( ) ( ) (11) e / = ( q) / Which may be solved for q o obain: ( ) (12) q = (1 e / ) Looking a equaion (12), we see ha afer ime >>, he capacior is fully charged and he charge on i is (13) q = = Q max If he charge on he capacior a any ime is given by equaion (12), hen he poenial difference across he capacior a any ime is (14) V = q / = (1 e / ) / = (1 e / ) A plo of V as a funcion of ime is shown in Figure 2. 0 q 8M 2

3 V V vs Figure 2. Plo of he poenial difference V across he capacior in a -circui as a funcion of ime while he capacior is charging The curren in he circui a any ime may be obained from equaion (12) as follows: (15) i = ( ) dq = d[(1 e / )] = e / / = I Max e d d where: (16) I max = / From equaion (15), noe ha a = 0 he curren has he maximum value of / and afer a long ime compared o he curren is zero. If he curren in he circui a any ime is given by equaion (15), he poenial difference across he resisor a any ime is (17) V = i = I max e / / = e A plo of V as a funcion of ime is shown in Figure 3. V V vs Figure 3. Plo of he poenial difference V across he resisor in a -circui as a funcion of ime while he capacior is charging. 8M 3

4 Figure 4 gives a summary of wha happens when he swich S is hrown o posiion a and he poenial difference is applied o he circui charging he capacior. vs (a) V = 0 V vs (b) = 0 V V vs (c) = 0 Figure 4. (a) Volage applied o he -circui as a funcion of ime. (b) Poenial difference V across he capacior as a funcion of ime. (c) Poenial difference V across he resisor as a funcion of ime. Noe ha a all imes V + V =. Now ha we have a good record of wha happens when he swich is hrown o posiion a and he capacior charges, le s hrow he swich o posiion b and le he capacior discharge. Once again, he sum of he volages around he loop mus be zero, hence (18) V + V = 0 where: (19) = O, here is no applied volage V = i = he volage drop across he resisor V = q / c = he volage drop across he capacior When (19) is insered ino equaion (18) we obain (20) q / c = i 8M 4

5 (21) Use he relaionship beween i and q [equaion (4)] o obain Separae he variables q and o obain q dq = d d (22) = dq q Seing equaion (22) up for inegraion and making use of he fac ha a = 0, q = Q max = as he lower limi and a some ime laer he charge on he capacior is q as he upper limi, obain (23) (1/) I d = I dq / q Inegrae equaion (23) o obain (24) (1/) = lnq Inser he upper and lower limis ino equaion (24) o obain (25) (1/) = lnq lnq max = ln (q / Q max ) Take he anilog of equaion (25) o obain (26) e / = (q / Q max ) 0 Solving equaion (26) for q, obain he following expression for he charge on he discharging capacior as a funcion of ime. / (27) q = Q max e If he charge of he capacior a any ime as i discharges is given by equaion (27) he poenial difference across he capacior a any ime as i discharges is (28) V = q / = (Q max / )e / / = e Looking a equaions (27) and (28) we see ha a = 0 he charge on he discharging capacior has is maximum value Q max and he poenial difference across he capacior has a maximum value of. Afer some ime >>, here is no charge on he capacior, he poenial difference across he capacior is zero, and he capacior is oally discharged. A plo of V as a funcion of ime is shown in Figure 5. 0 q Q max q Q max V V vs Figure 5. Plo of he poenial difference V across he capacior in a -circui as a funcion of ime while he capacior is discharging. As he capacior discharges, he curren a any ime obained from equaion (27) is as follows: (29) i = dq / d = d(q max e / ) / d = (Q max / )e / = ( / )e / / = I max e 8M 5

6 If he curren in he circui of he discharging capacior a any ime is given by (29). he poenial difference across he resisor a any ime is (30) V = i = I max e / / = e where: (31) = I max A plo of V as a funcion of ime is shown in Figure 6. V V vs Figure 6. Plo of he poenial difference V across he resisor in a -circui as a funcion of ime while he capacior is discharging. Figure 7 gives a summary of wha happens when he swich S is hrown o posiion b, he applied volage is disconneced, and he capacior is discharged. vs (a) V = 0 V vs (b) (c) = 0 = 0 V vs 8M 6 V Figure 7. (a) Volage applied o he -circui as a funcion of ime. (b) Poenial difference V across he capacior as a funcion of ime. (c) Poenial differencev across he resisor as a funcion of ime. Noe ha a all imes V +V =.

7 Now if he swich is flipped back and forh from a o b o a o b ec., he volage is alernaely applied and disconneced, and he capacior alernaely charges and discharges. All of he preceding informaion allows us o conclude ha a record of, V and V would look like Figure 8. vs (a) V = 0 V vs (b) = 0 V vs V (c) = 0 = 0 is applied Swich o a apacior charging is disconneced Swich o b apacior discharging is applied Swich o a apacior harging Figure 8. For a -circui, he quaniy is a characerisic ime called he capaciaive ime consan. The ime consan is he ime i akes he capacior o charge o 63% of is maximum charge or alernaely he ime for a discharging capacior o lose 63% of is charge. This is shown below by using he ime = in he expression for he charge on a charging capacior [equaion (12)] and in he expression for he charge on a discharging capacior [equaion (27)]. For a charging capacior For a discharging capacior (12) q = Q max (1 e / / ) (27) q = Q max e Insering he ime =, obain: q = Q max (1 e / ) Insering he ime =, obain: / Q max e = Q max (1 e 1 ) = Q max e 1 = (0.63)Q max = (0.37)Q max 8M 7

8 From he above we see ha afer a ime = he charge on a capacior is 63% of is maximum value and he discharging capacior has a charge of 37% of is maximum value (ha is i has los 63% of is maximum charge). As a maer of convenience, he ime consan is represened by he symbol τ, hence (31) τ = Since he poenial difference across he capacior a any ime is V = q /, we may also say ha he ime consan is he ime for he poenial difference across a charging capacior o reach 63% of he applied volage () and/or i is he ime for he poenial difference across a discharging capacior o fall o 37% of he applied volage. Anoher unique ime for a circui is he ime for he charge on he capacior or he poenial difference across a charging capacior o rise o one half is maximum value and/or o fall o one half is maximum value for a discharging capacior. We will refer o his ime as T 1/2. For a charging capacior For a discharging capacior (14) V = (1 e / / ) (28) V = e 1 2 If = T 1/2 hen V =. Also, τ =. If = T 1/2 hen V =. Also, τ =. 1 2 Insering hese values, we obain: 1 2 = (1 e T 1/2 / τ ) Insering hese values, we obain: 1 2 = e T 1/2 / τ 1 2 e T 1/2 / τ = e T 1/2 / τ = 1 2 T 1/2 = 1n2 τ τ T 1/2 = 1n2 (32) τ = T 1/2 (32) τ = T 1/2 1n2 1n2 This gives us an alernae mehod for finding he ime consan. We simply go o he V = / 2 posiion on he V vs plo of a charging or discharging capacior and he corresponding ime is T 1/2. We can hen deermine he ime consan using equaion (32). PODU In his aciviy, a power amplifier is used o produce a low frequency posiive only square wave. When his wave form is applied o a series circui, i has he same effec as connecing and hen disconnecing a D volage source. When he volage source is conneced and hen disconneced, he capacior charges and hen discharges. We will observe he oupu of he power amplifier and use volage sensors o measure he volage across he resisor and capacior as he capacior charges and discharges. We will use a graphical display of hese volages (i.e., V and V ) o invesigae he behavior of he circui while he charge is increasing, seady a is maximum value, and decreasing. We will also deermine he ime consan for he -circui direcly and indirecly. 8M 8

9 Par I. Iniial quipmen and Sofware Se-up 1. Sar Daa Sudio, and selec reae xperimen. 2. onnec he Power Amplifier o he compuer via an analog por on he PASO inerface. Plug he Power Amplifier ino an A oule, and urn he power amplifier on. onnec he power amplifier o an circui consising of a 100Ω esisor and a 330µF apacior. 3. Inform he sofware which analog por you plugged he Power Amplifier ino by selecing he Power Amplifier icon and dragging i o he appropriae analog por. 4. A signal generaor box should appear. hange he wave paern from he defaul Sine Wave o a Posiive Square Wave. Noe ha a Posiive Square Wave and a Square Wave are wo differen ypes of wave forms. Since we are using he Signal generaor as our swich, he wave form needs o be a Posiive Square Wave so ha he volage will alernae beween 0.00 and 4.00 vols. If a normal Square Wave is used, he volage will alernae beween and 4.00 vols. hange he frequency o 0.40Hz. hange he ampliude o 4.00V. The Auo buon should already be seleced. 5. onnec a volage sensor across he 100Ω resisor, and connec i o he compuer via an analog por on he PASO inerface. 6. Inform he sofware which analog por you plugged he volage sensor ino by selecing he volage sensor icon and dragging i o he appropriae analog por. 7. onnec a volage sensor across he 330µF capacior, and connec i o he compuer via an analog por on he PASO inerface. 8. Inform he sofware which analog por you plugged he volage sensor ino by selecing he volage sensor icon and dragging i o he appropriae analog por. 9. lick he Sar/Sop Opions buon, and selec he Auomaic Sop ab. Selec he ime opion and inpu 6 seconds. lick OK. 10. reae a graph of he Volage across he apacior vs. ime. 11. Double click somewhere wihin he body of he acual graph, or click on he Graph Seings buon locaed on he graph oolbar o open he Graph Seings window. Selec he layou ab, and under Group Measuremens selec he Do No Group opion. Nex selec he Tools ab. Under he Smar Tools, se he Daa Poin Graviy o 0. Selec OK. 8M 9

10 Par II. ollecion of Daa and Analysis of he Time onsan 1. lick he Sar buon. The sofware will collec daa for 6 seconds and auomaically sop. 2. lick he auoscale buon locaed on he graph ool bar. 3. Maximize he graph o fill he screen. 4. Magnify a secion of he graph where he capacior is charging. Srech he ime scale if necessary. 5. Deermine he amoun of ime i akes o reach one half of he maximum volage using he smar ool. lick on he Smar Tool buon. When he smar ool curser has he following appearance, i can be moved o any locaion on he graph. Place he smar curser a he locaion where he capacior begins o charge. By moving he mouse slighly o he 2 nd quadran of he smar ool curser, you should be able o change he curser o he dela curser. When he dela curser is presen, lef click and drag he curser o he poin where he volage is one half of he maximum. The difference in ime beween hese poins is he ime a half-max, or T 1/2. 6. lick he auoscale buon on he graph. 7. Magnify a secion of he graph where he capacior is discharging. 8. Deermine he amoun of ime i akes o reach one half of he maximum volage using he smar ool in a similar manner as sep lick on he smar ool buon so ha he smar ool is off. Par III. ompleion of Sofware Se-up and Daa Analysis 1. Add a graph of he Volage Across he esisor vs. Time o he exising graph. lick and drag he icon ha represens he Volage Across he esisor o he exising graph. When he enire graph is boxed in a doed line, drop he icon. 2. epea sep one for he Oupu Volage of he signal generaor. 3. Make sure ha he align x-axis lock buon has been seleced. 4. You should now have hree graphs ha look like Figure 8 in he Foreword of he experimen. However, he graphs will be in a differen order. 5. lick on each graph one a a ime and urn on he smar ool. 6. Make he Volage Across he apacior vs. ime graph acive by clicking on i. 8M 10

11 7. lick he auoscale buon on he graph. 8. Magnify a secion of he graph in an area where he volage of he apacior is increasing. 9. Pick a poin where he volage of he apacior is increasing and deermine he volage of he apacior using he smar ool. Also deermine he volage of he esisor and Oupu Volage for he same ime. Prin he graph wih he smar ool daa visible. ommen on he resul. 10. epea seps 6 and 9 for a poin where he volage of he apacior is remaining sable. Prin he graph wih he smar ool daa visible. Make sure you commen on he resuls. 11. epea seps 6 and 9 for a poin where he volage of he apacior is decreasing. Prin he graph wih he smar ool daa visible. Make sure you commen on he resuls. 8M 11

12 8M 12

13 NAM STION DAT DATA AND ALULATION SUMMAY harging apacior T sar = s Time when he volage across he charging capacior sars o increase. T half max = s Time when he volage across he charging capacior reaches he value V = / 2. T 1/2 = s Time for he volage across he charging capacior o rise o half is maximum value. (T 1/2 = T half max T sar ) τ = s Time consan for he circui. (τ = T 1/ 2 / In 2) = Ω esisance of he resisor. = F apaciance of he capacior. τ = s Time consan for he circui. (τ = ) Discharging apacior T sar = s Time when he volage across he discharging capaciy sars o decrease. T half max = s Time when he volage across he discharging capacior reaches he value V = / 2. T I/2 = s Time for he volage across he discharging capacior o fall o half is maximum value. (T 1/2 = T half max T sar ) τ = s Time consan for he circui. (τ = T 1/ 2 / In 2) 8M 13

14 Volages and commens for a ime when he volage across he capacior is increasing: = vols V vols V = vols Volages and commens for a ime when he volage across he capacior is sable: = vols V vols V = vols Volages and commens for a ime when he volage across he capacior is decreasing: = vols V vols V = vols Quesions 1. The ime o half-maximum volage is how long i akes he capacior o charge half-way. Based on your experimenal resuls, how long does i ake for he capacior o charge o 75% of is maximum? 2. Afer four "half-lives", o wha percenage of he maximum charge is he capacior charged? 3. Wha is he maximum charge for he capacior in his experimen? 4. Wha is he maximum curren flowing hrough he resisor in his experimen? 8M 14

### 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.

### 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

### 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

### 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

### 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:

### 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

### 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

### 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,

### 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

### 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

### Capacitors and inductors

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

### 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

### 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

### 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)

### 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

### 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

### 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

### 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

### 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

### 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

### and Decay Functions f (t) = C(1± r) t / K, for t 0, where

MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

### TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED

Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific half-life. Sudens will develop mahemaical

### 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

### 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

### Acceleration Lab Teacher s Guide

Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

### 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

### 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

### 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

### BD FACSuite Software Quick Reference Guide for the Experiment Workflow

BD FACSuie Sofware Quick Reference Guide for he Experimen Workflow This guide conains insrucions for using BD FACSuie sofware wih he BD FACSVerse flow cyomeer using he experimen workflow. Daa can be acquired

### 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

### 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

### 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

### LAB 6: SIMPLE HARMONIC MOTION

1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:

### 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

### Pulse-Width Modulation Inverters

SECTION 3.6 INVERTERS 189 Pulse-Widh Modulaion Inverers Pulse-widh modulaion is he process of modifying he widh of he pulses in a pulse rain in direc proporion o a small conrol signal; he greaer he conrol

### 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

### 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

### INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### NOTES ON OSCILLOSCOPES

NOTES ON OSCILLOSCOPES NOTES ON... OSCILLOSCOPES... Oscilloscope... Analog and Digial... Analog Oscilloscopes... Cahode Ray Oscilloscope Principles... 5 Elecron Gun... 5 The Deflecion Sysem... 6 Displaying

### 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

### 2. Waves in Elastic Media, Mechanical Waves

2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in

### 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,

### 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

### DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

### 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

### 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

### 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

### 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

### 11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure

### 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

### ( ) in the following way. ( ) < 2

Sraigh Line Moion - Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and man-made. Wrie down several of hem. Horizonal cars waer

### Permutations and Combinations

Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

### Rotational Inertia of a Point Mass

Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

### 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

### Week #9 - The Integral Section 5.1

Week #9 - The Inegral Secion 5.1 From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,

### A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

### 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 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

### 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

### Relative velocity in one dimension

Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

### MOTION ALONG A STRAIGHT LINE

Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,

### 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

### Chapter 15: Superposition and Interference of Waves

Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when

### The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,

### Fourier Series Approximation of a Square Wave

OpenSax-CNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier

### YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

### OPERATION MANUAL. Indoor unit for air to water heat pump system and options EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1

OPERAION MANUAL Indoor uni for air o waer hea pump sysem and opions EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1 EKHBRD011ABY1 EKHBRD014ABY1 EKHBRD016ABY1 EKHBRD011ACV1 EKHBRD014ACV1 EKHBRD016ACV1 EKHBRD011ACY1

### 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

### Trends in TCP/IP Retransmissions and Resets

Trends in TCP/IP Reransmissions and Reses Absrac Concordia Chen, Mrunal Mangrulkar, Naomi Ramos, and Mahaswea Sarkar {cychen, mkulkarn, msarkar,naramos}@cs.ucsd.edu As he Inerne grows larger, measuring

### 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

### Steps for D.C Analysis of MOSFET Circuits

10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.

### Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

### 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

### 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

### ECEN4618: Experiment #1 Timing circuits with the 555 timer

ECEN4618: Experimen #1 Timing circuis wih he 555 imer cæ 1998 Dragan Maksimović Deparmen of Elecrical and Compuer Engineering Universiy of Colorado, Boulder The purpose of his lab assignmen is o examine

### HT1668 LED Driver IC DESCRIPTION FEATURES APPLICATION BLOCK DIAGRAM

DESCRIPTION is an LED Conroller driven on a 1/7o 1/8 duy facor. Eleven segmen oupu lines, six grid oupu lines, 1 segmen/grid oupu lines, one display memory, conrol circui, key scan circui are all incorporaed

### 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

### Chapter 4: Exponential and Logarithmic Functions

Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion

### 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,

### 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

### Product Operation and Setup Instructions

A9 Please read and save hese insrucions. Read carefully before aemping o assemble, insall, operae, or mainain he produc described. Proec yourself and ohers by observing all safey informaion. Failure o

### Section 7.1 Angles and Their Measure

Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

### Section A: Forces and Motion

I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and

### 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

### Gate protection. Current limit. Overvoltage protection. Limit for unclamped ind. loads. Charge pump Level shifter. Rectifier. Open load detection

Smar ighside Power Swich for ndusrial Applicaions Feaures Overload proecion Curren limiaion Shor circui proecion Thermal shudown Overvolage proecion (including load dump) Fas demagneizaion of inducive

### AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

### Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINAL EXAMINATION. June 2009.

Name: Teacher: DO NOT OPEN THE EXMINTION PPER UNTIL YOU RE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINL EXMINTION June 2009 Value: 100% General Insrucions This examinaion consiss of wo pars. Boh pars

### 4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

### AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 1 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 1, he College

### 11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

### 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