--3LabManualLab5.doc LAB 5: ompuer imulaion of RL ircui Response using Ppice PURPOE To use a compuer simulaion program (Ppice) o invesigae he response of an RL series circui o: (a) a sinusoidal exciaion. (b) a sep exciaion under condiions of underdamping, criical damping and overdamping. EQUIPMENT (i) (ii) uden compuer nework saion. Orad apure Ppice circui simulaion sofware. This lab is o be done individually unless equipmen shorages require oherwise. BAKGROUND Par. Frequency Domain Response of RL ircui o inusoidal Exciaion Figure shows a series RL circui driven by an ideal volage source,. Figure. eries RL circui. The loop curren I is given by: I = R + jω L + = R + jx jω () where: X ω L ω = () The magniude of he curren phasor is: ELE 5 Lab 5 Page of 8
--3LabManualLab5.doc I R = (3) + X The curren has a maximum value when X =, which occurs when: ω L = or ω = ω = ω (4) This phenomenon is called resonance. A he resonan frequency ω, he magniude of curren akes he value: I L = I = (5) MAX R and he volage across he reacance X, i.e. he volage across he combined impedance of L and in series, is zero. For frequencies higher or lower han ω, he curren is reduced below is peak value. A he resonan frequency, he volage across he capacior is: = I MAX (6) jω From (4) and (5): We define he raio: ω L = = (7) ω R R L = = Q (8) where Q is he qualiy facor of a series RL circui, and is given by: Q ω L = ω R R = (9) If R is small, Q can be greaer han uniy, i.e. volages appearing across and L are larger han he applied volage,. This represens a volage magnificaion equal o Q. Par. Time Domain Response of RL ircui o ep Exciaion The heory of he RL circui behavior when excied by a sep volage is beyond he scope of his course. However, we presen here a brief descripion required o complee he laboraory assignmen. The applicaion of KL around he loop of he RL circui shown in Figure gives: di ir + id + L = v ( ) d () for zero iniial condiions. ELE 5 Lab 5 Page of 8
--3LabManualLab5.doc When differeniaed wih respec o ime, we ge: d i L + d di R d + i = dv d () This is a second-order linear differenial equaion. The soluion of his equaion describes he response of he circui. Recall ha he response of he R and RL series circuis o a volage sep yielded a firs-order linear differenial equaion, whose soluion was of he form: v( ) = K e where K and K are consans. Using inuiion, he response of he RL circui mus be of he form: s s v( ) K e e τ () = (3) where K, K and K are consans, and s and s are called he naural frequencies of he circui. The acual response of he circui depends on he relaion beween and he naure of s and s. There are hree possible cases ha mus be considered: ase : s and s are boh real and unequal. In his case, he soluion is given by: v( ) = K e τ e τ (4) This indicaes ha he response of he RL circui is he sum of wo decaying exponenials. The circui is said o be overdamped. ase : s and s are complex conjugaes of each oher. In his case, he soluion is given by: v( ) ( σ jω ) ( σ + jω ) = K e e (5) = K + e σ ( K e jω e jω The response is an exponenially damped sinusoid. The circui is said o be underdamped, and is behavior is described as ringing. ase 3: s and s are boh real and equal. In his case, he soluion is given by: 3 v( ) = K 3e ( A + A ) ) (6) τ (7) The response is similar o ase, bu i reaches is peak earlier and also decays faser. The circui is said o be criically damped. The hree possible responses are shown in Figure. The RL circui is excied by a sep inpu volage, and he oupu volage is measured across he capacior. ELE 5 Lab 5 Page 3 of 8
--3LabManualLab5.doc Figure. omparison of overdamped, criically damped and underdamped responses. PRELAB. (_/.5) alculae he values of L (from Par, sep.3), R (from Par, sep.), and R 3 (from Par, sep.) o five significan digis. Recall xxx represens he las hree digis of your suden number.. (_/.5) alculae he resonan frequency ω for he circui given in Figure using he componen values as given in Par, sep.3. Also conver he resonan frequency ino Herz for f. 3. (_/.5) alculae he value of he qualiy facor Q for he circui given in Figure using he componen values as given in Par, sep.3. 4. (_/.5) Draw a phasor diagram of he circui in Figure a resonance using he componen values as given in Par, sep.3. PROEDURE This lab is mean o be an inroducion o circui simulaion using Ppice. The Ppice graphical user inerface is inuiive, and follows he sandard menus ha are common o many Windows applicaions. The lab manual does no give you every single sep required o complee he exercises. However, you are given enough ime o explore, learn and experimen wih he differen opions and seings of he Ppice simulaor. Use he following resources o help you wih Ppice since he version available for your use changes from ime o ime: hp://www.doe.carleon.ca/~nagui/web5/ppiceprimer.hml. Ppice help menu and web uorials. Par. Frequency Domain Response of RL ircui o inusoidal Exciaion. For your firs ime on a paricular worksaion: Open adence Release 6.5 onfigure. lick Nex, selec accep erms, click Nex, click Insall, click Finish. ELE 5 Lab 5 Page 4 of 8
--3LabManualLab5.doc Open adence Release 6.5 OrAD apure I Demo. The firs ime i is opened selec Allegro PB Librarian XL and click OK. reae a new projec wih he following parameers: Include your name in he projec name (his will enable you o disinguish your prinous from ohers ). hoose Analog or Mixed A/D opion in he projec dialogue box. elec he H: drive o sore your design hoose reae a blank projec in he reae Ppice Projec dialogue box.. In he chemaic window ha opens up, noe he icons locaed on he righ: Place par add a circui componen ( capacior, R resisor, L inducor, ec.). When choosing his icon for he firs ime, press Add Library icon (or al-a) and add he ANALOG and OURE libraries. To change he value of a componen, double click on he displayed value. The values of he elemens can be specified using scaling facors (n 9, u 6, m 3, k 3, meg 6 ec.). Add wire add an inerconnecing wire beween circui elemens. Place ne alias assign names o nes or nodes. GND add a ground o he circui..3 Draw he circui shown in Figure using he following values: R = 4 Ω L = 8 e xxx 8 8 mh =. µf = For he A volage source use he A componen from he ource library..4 Place he ground on an appropriae node in your circui..5 Label each node in your circui wih an appropriae label, e.g. NODE, NODE, ec. using he Place ne alias icon. This will help you in ploing in he following seps..6 In his par of he lab, we invesigae he frequency response of he circui o sinusoidal exciaion. Thus we mus specify he desired frequency range, which can be done by going o Ppice / New simulaion profile. hoose logarihmic A weep/noise for he analysis ype. Firs, we are ineresed in looking a a very wide range of frequencies o see he rough shape of he response. Therefore, se he frequency range o Hz khz, wih inervals of frequency poins per decade..7 imulae your circui (Ppice / Run) and make sure here are no errors..8 (_/.5) Once he circui is simulaed wih no errors, he Ppice A/D Demo window will come up. Graph he phase difference beween and by going o Trace / Add race and choosing he appropriae variables and funcions. Make sure he horizonal axis is se o log scale..9 (_/.5) Add a second plo o he exising one (Plo / Add plo o Window). Graph / in decibels on he op plo. Prin he resulan plos and include hem in your lab repor.. From your plo in sep.9, you should be able o idenify a narrower frequency range of ineres (approximaely f -.5kHz o f +.5kHz). e he frequency sweep o his narrower range wih abou 5 poins per decade. This will allow you o see a more precise shape of he magniude and phase plos around he resonance poin of he circui. ELE 5 Lab 5 Page 5 of 8
--3LabManualLab5.doc. imulae he circui and obain he wo graphs as in seps.8-.9. e he horizonal axis range o f -.5kHz o f +.5kHz by double-clicking on he axis, if needed. Do no prin your plos a his ime.. (_/.5) If Q is large, hen he values of ω and 3 db bandwidh can be obained from he magniude plo as shown in Figure 3. Figure 3. Definiion of 3 db bandwidh. Explore he opions in he graphing uiliy program (Ppice A/D Demo), and figure ou how o use cursors (Trace / ursors) o obain measuremens of he ploed daa. Using cursors, find he value of ω, he resonan frequency, and mark he relevan poin on he graph. ompare his value o he one calculaed in quesion of he prelab..3 (_/) Using cursors, find he 3 db bandwidh, as defined in Figure 3. Mark he relevan poins and heir values on he graphs. Prin he resulan plos and include hem wih your lab repor..4 (_/.5) alculae Q based on he linear relaionship: ω Q 3 db bandwidh ompare his value o he one calculaed in quesion 3 of he prelab. = () Par. Time Domain Response of RL ircui o ep Exciaion. reae a new projec as in Par, and include your name in he projec name.. Figure 4 shows hree independen circuis, conneced o a common inpu signal. This configuraion will enable you o observe he under damped, criically damped and over damped cases simulaneously. In a new file, draw he circui shown in Figure 4 using he following values: L = L = L 3 = mh R = xxx + 4 Ω R 3 = R +4 Ω ELE 5 Lab 5 Page 6 of 8
--3LabManualLab5.doc = = 3 = 54 nf R = 7 Ω IN = sep For he sep volage source use he PULE componen from he ource library. pecify appropriae values for he, and TD parameers of PULE ( and correspond o he minimum and maximum volages of he pulse, respecively, and TD corresponds o he delay from ime zero o he firs rising edge). Label he nodes of he circui as shown in Figure 4..3 To specify he ime range for he simulaion, go o Ppice / New imulaion Profile and choose Time domain (ransien) analysis ype. Make sure you specify an appropriae ime range o display he oupus clearly. You may need o ierae hrough seps.3 and.4 a number of imes o obain he required oupu plo (in ha case, go o Ppice / Edi simulaion profile o edi he seings)..4 imulae he circui and, on he same plo, graph he inpu signal v IN, as well as he hree oupu signals, v 3, v 5 and v 7, as specified in Figure 4..5 (_/) Idenify which one of he circuis is under damped, which one is over damped, and which one is criically damped..6 (_/) In he case of he under damped circui: (a) Wha is he frequency of he oscillaory response? (b) Does he frequency approximaely agree wih he resonance condiion in equaion (4)?.7 (_/) In he case of he over damped and he criically damped circuis, measure he rise-ime, ha is, he ime each oupu akes o reach 9% of is maximum volage. Mark he measured poins and heir values on he graph. Prin he plo and include i wih your lab repor..8 (_/) ommen on your resuls. ELE 5 Lab 5 Page 7 of 8
--3LabManualLab5.doc 6 Figure 4. ircui configuraion for Par. ELE 5 Lab 5 Page 8 of 8