1300 Henley C. Pullman, WA 99163 509.334.6306 www.sore.digilen.com 6 Inroducion and Chaper Objecives So far, we have considered circuis ha have been governed by algebraic relaions. These circuis have, in general, conained only power sources and resisive elemens. All elemens in hese circuis, herefore, have eiher supplied power from exernal sources or dissipaed power. For hese resisive circuis, we can apply eiher imevarying or consan signals o he circui wihou really affecing our analysis approach. Ohm s law, for example, is equally applicable o imevarying or consan volages and currens: V = I R v() = i() R Since he governing equaion is algebraic, i is applicable a every poin in ime volages and currens a a poin in ime are affeced only by volages and currens a he same poin in ime. We will now begin o consider circui elemens, which are governed by differenial equaions. These circui elemens are called dynamic circui elemens or energy sorage elemens. Physically, hese circui elemens sore energy, which hey can laer release back o he circui. The response, a a given ime, of circuis ha conain hese elemens is no only relaed o oher circui parameers a he same ime; i may also depend upon he parameers a oher imes. This chaper begins wih an overview of he basic conceps associaed wih energy sorage. This discussion focuses no on elecrical sysems, bu insead inroduces he opic qualiaively in he conex of sysems wih which he reader is already familiar. The goal is o provide a basis for he mahemaics, which will be inroduced subsequenly. Since we will now be concerned wih imevarying signals, secion 6.2 inroduces he basic signals ha we will be dealing wih in he immediae fuure. This chaper concludes wih presenaions of he wo elecrical energy sorage elemens ha we will be concerned wih: capaciors and inducors. The mehod by which energy is sored in hese elemens is presened in secions 6.3 and 6.4, along wih he governing equaions relaing volage and curren for hese elemens. Afer compleing his chaper, you should be able o: Qualiaively sae he effec of energy sorage on he ype of mahemaics governing a sysem Define ransien response Define seadysae response Wrie he mahemaical expression for a uni sep funcion Skech he uni sep funcion Skech shifed and scaled versions of he uni sep funcion Wrie he mahemaical expression for a decaying exponenial funcion Define he ime consan of an exponenial funcion Skech a decaying exponenial funcion, given he funcion s iniial value and ime consan Use a uni sep funcion o resric an exponenial funcion o imes greaer han zero Chaper 6 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 1 of 58
Wrie he circui symbol for a capacior Sae he mechanism by which a capacior sores energy Sae he volagecurren relaionship for a capacior in boh differenial and inegral form Sae he response of a capacior o consan volages and insananeous volage changes Wrie he mahemaical expression describing energy sorage in a capacior Deermine he equivalen capaciance of series and parallel combinaions of capaciors Skech a circui describing a nonideal capacior Wrie he circui symbol for an inducor Sae he mechanism by which an inducor sores energy Sae he volagecurren relaionship for an inducor in boh differenial and inegral form Sae he response of an inducor o consan volages and insananeous curren changes Wrie he mahemaical expression describing energy sorage in an inducor Deermine he equivalen inducance of series and parallel combinaions of inducors Skech a circui describing a nonideal inducor 6.2 Fundamenal Conceps This secion provides a brief overview of wha i mean by energy sorage in erms of a sysemlevel descripion of some physical process. Several examples of energy sorage elemens are presened, for which he reader should have an inuiive undersanding. These examples are inended o inroduce he basic conceps in a qualiaive manner; he mahemaical analysis of dynamic sysems will be provided in laer chapers. We have previously inroduced he concep of represening a physical process as a sysem. In his viewpoin, he physical process has an inpu and an oupu. The inpu o he sysem is generaed from sources exernal o he sysem we will consider he inpu o he sysem o be a known funcion of ime. The oupu of he sysem is he sysem s response o he inpu. The inpuoupu equaion governing he sysem provides he relaionship beween he sysem s inpu and oupu. A general inpuoupu equaion has he form: y() = f{u()} Eq. 6.1 The process is shown in block diagram form in Fig. 6.1. Figure 6.1. Block diagram represenaion of a sysem. The sysem of Fig. 6.1 ransfers he energy in he sysem inpu o he sysem oupu. This process ransforms he inpu signal u() ino he oupu signal y(). In order o perform his energy ransfer, he sysem will, in general, conain elemens ha boh sore and dissipae energy. To dae, we have analyzed sysems which conain only energy dissipaion elemens. We review hese sysems briefly below in a sysems conex. Subsequenly, we inroduce sysems ha sore energy; our discussion of energy sorage elemens is mainly qualiaive in his chaper and presens sysems for which he reader should have an inuiive undersanding. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 2 of 58
6.2.1 Sysems wih no Energy Sorage In previous chapers, we considered cases in which he inpuoupu equaion is algebraic. This implies ha he processes being performed by he sysem involve only sources and componens which dissipae energy. For example, oupu volage of he invering volage amplifier of Fig. 6.2 is: V OUT = ( R f R in V in ) Eq. 6.2 This circui conains only resisors (in he form of Rf and Rin) and sources (in he form of Vin and he opamp power supplies) and he equaion relaing he inpu and oupu is algebraic. Noe ha he opamp power supplies do no appear in equaion (6.2), since linear operaion of he circui of Fig. 6.2 implies ha he oupu volage is independen of he opamp power supplies. R f R in V in Figure 6.2. Invering volage amplifier. One side effec of an algebraic inpuoupu equaion is ha he oupu responds insananeously o any changes in he inpu. For example, consider he circui shown in Fig. 6.3. The inpu volage is based on he posiion of a swich; when he swich closes, he inpu volage applied o he circui increases insananeously from 0V o 2V. Fig. 6.3 indicaes ha he swich closes a ime = 5 seconds; hus, he inpu volage as a funcion of ime is as shown in Fig. 6.4(a). For he values of Rf and Rin shown in Fig. 6.3, he inpuoupu equaion becomes: V OUT () = 5V in () Eq. 6.3 And he oupu volage as a funcion of ime is as shown in Fig. 6.4(b). The oupu volage responds immediaely o he change in he inpu volage. V OUT = 5 sec 1 kw 2V V in 5 kw V OUT Figure 6.3. Swiched volage amplifier. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 3 of 58
V in V ou 2V 0V 5 Time, sec 0V 5 Time, sec 10V (a) Inpu volage (b) Oupu volage Figure 6.4. Inpu and oupu signals for circui of Figure 3. 6.2.2 Sysems wih Energy Sorage We now consider sysems, which conain energy sorage elemens. The inclusion of energy sorage elemens resuls in he inpuoupu equaion for he sysem, which is a differenial equaion. We presen he conceps in erms of wo examples for which he reader mos likely has some expecaions based on experience and inuiion. Example 6.1: Massdamper sysem As an example of a sysem, which includes energy sorage elemens, consider he massdamper sysem shown in Fig. 6.5. The applied force F() pushes he mass o he righ. The mass s velociy is v(). The mass slides on a surface wih sliding coefficien of fricion b, which induces a force, which opposes he mass s moion. We will consider he applied force o be he inpu o our sysem and he mass s velociy o be he oupu, as shown by he block diagram of Fig. 6.6. This sysem models, for example, pushing a salled auomobile. The sysem of Fig. 6.5 conains boh energy sorage and energy dissipaion elemens. Kineic energy is sored in he form of he velociy of he mass. The sliding coefficien of fricion dissipaes energy. Thus, he sysem has a single energy sorage elemen (he mass) and a single energy dissipaion elemen (he sliding fricion). In secion 4.1, we deermined ha he governing equaion for he sysem was he firs order differenial equaion: m dv() bv() = F() Eq. 6.4 d The presence of he energy sorage elemen causes he inpuoupu equaion o be a differenial equaion. Mass s velociy, v() Exernally applied force, F() Mass, m Surface wih sliding coefficien of fricion, b Figure 6.5. Sliding mass on surface wih fricion coefficien, b. Inpu, F() Sysem Oupu, v() Figure 6.6. Massdamper sysem represened as a block diagram. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 4 of 58
We will examine he effec ha he energy sorage elemen has upon he sysem response in qualiaive erms, raher han explicily solving equaion (6.4). If we increase he force applied o he mass, he mass will accelerae and he velociy of he mass increases. The sysem, herefore, is convering he energy in he inpu force o a kineic energy of he mass. This energy ransfer resuls in a change in he oupu variable, velociy. The energy sorage elemens of he sysem of Fig. 6.5 do no, however, allow an insananeous change in velociy o an insananeous change in force. For example, say ha before ime = 0 no force is applied o he mass and he mass is a res. A ime = 0 we suddenly apply a force o he mass, as shown in Fig. 6.7(a) below. A ime = 0 he mass begins o accelerae bu i akes ime for he mass o approach is final velociy, as shown in Fig. 6.7(b). This ransiory sage, when he sysem is in ransiion from one consan operaing condiion o anoher is called he ransien response. Afer a ime, he energy inpu from he exernal force is balanced by he energy dissipaed by he sliding fricion, and he velociy of he mass remains consan. When he operaing condiions are consan, he energy inpu is exacly balanced by he energy dissipaion, and he sysem s response is said o be in seadysae. We will discuss hese erms in more deph in laer chapers when we perform he mahemaical analysis of dynamic sysems. F() =0 Figure 6.7(a). Force applied o mass. v() Final Velociy =0 Transien Response Seadysae Response Figure 6.7(b). Velociy of mass. Example 6.2 Our second example of a sysem, which includes energy sorage elemens, is a body ha is subjeced o some hea inpu. The overall sysem is shown in Figure 6.8. The body being heaed has some mass m, specific hea Cp, and emperaure TB. Some hea inpu qin is applied o he body from an exernal source, and he body ransfers hea qou o is surroundings. The surroundings are a some ambien emperaure T0. We will consider he inpu o Oher produc and company names menioned may be rademarks of heir respecive owners. Page 5 of 58
our sysem o be he applied hea inpu qin and he oupu o be he emperaure of he body TB, as shown in he block diagram of Fig. 6.9. This sysem is a model, for example, of he process of heaing a frying pan on a sove. Hea inpu is applied by he sove burner and he pan dissipaes hea by ransferring i o he surroundings. Body wih: mass m, specific hea c P, emperaure T B Hea Dissipaion, q ou Ambien Temperaure, T 0 Hea Inpu, q in Figure 6.8. Body subjeced o heaing. Inpu, q in () Sysem Oupu, T B () Figure 6.9. Sysem block diagram. The sysem of Fig. 6.8 conains boh energy sorage and energy dissipaion elemens. Energy is sored in he form of he emperaure of he mass. Energy is dissipaed in he form of hea ransferred o he surroundings. Thus, he sysem has a single energy sorage elemen (he mass) and a single energy dissipaion elemen (he hea dissipaion). The governing equaion for he sysem is he firs order differenial equaion: d(t mc B T 0 ) p q d OUT = q in Eq. 6.5 The presence of he energy sorage elemen causes he inpuoupu equaion o be a differenial equaion. We again examine he response of his sysem o some inpu in qualiaive raher han quaniaive erms in order o provide some insigh ino he overall process before immersing ourselves in he mahemaics associaed wih analyzing he sysem quaniaively. If he hea inpu o he sysem is increased insananeously (for example, if we suddenly urn up he hea seing on our sove burner) he mass s emperaure will increase. As he mass s emperaure increases, he hea ransferred o he ambien surroundings will increase. When he hea inpu o he mass is exacly balanced by he hea ransfer o he surroundings, he mass s emperaure will no longer change and he sysem will be a a seadysae operaing condiion. Since he mass provides energy sorage, he emperaure of he mass will no respond insananeously o a sudden change in hea inpu he emperaure will rise relaively slowly o is seadysae operaing condiion. (We know from experience ha changing he burner seing on he sove does no immediaely change he emperaure of our pan, paricularly if he pan is heavy.) The process of changing he body s emperaure from one seady sae operaing condiion o anoher is he sysem s ransien response. The process of changing he body s emperaure by insananeously increasing he hea inpu o he body is illusraed in Fig. 6.10. The signal corresponding o he hea inpu is shown in Fig. 6.10(a), while he resuling emperaure response of he body is shown in Fig. 6.10(b). Oher produc and company names menioned may be rademarks of heir respecive owners. Page 6 of 58
q in () T B () Final Temperaure Iniial Temperaure =0 =0 (a) Hea inpu (b) Temperaure response Figure 6.10. Temperaure response o insananeous hea inpu. Secion Summary Sysems wih energy sorage elemens are governed by differenial equaions. Sysems ha conain only energy dissipaion elemens (such as resisors) are governed by algebraic equaions. The responses of sysems governed by algebraic equaions will ypically have he same shape as he inpu. The oupu a a given ime is simply dependen upon he inpu a ha same ime he sysem does no remember any previous condiions. The responses of sysems governed by differenial equaions will no, in general, have he same shape as he forcing funcion applied o he sysem. The sysem remembers previous condiions his is why he soluion o a differenial equaion requires knowledge of iniial condiions. The response of a sysem ha sores energy is generally considered o consis of wo pars: he ransien response and he seadysae response. These are described as follows: o The ransien response ypically is shaped differenly from he forcing funcion. I is due o iniial energy levels sored in he sysem. o The seadysae response is he response of he sysem as. I is he same shape as he forcing funcion applied o he sysem. In differenial equaions courses, he ransien response corresponds (approximaely) o he homogeneous soluion of he governing differenial equaion, while he seadysae response corresponds o he paricular soluion of he governing differenial equaion. 6.1 Exercises 1. A mass is sliding on a surface wih an iniial velociy of 5 meers/seconds. All exernal forces (excep for he fricion force on he surface) are removed from he mass a ime = 0 seconds. The velociy of he mass as a funcion of ime is shown below. Wha is he seadysae velociy of he mass? Oher produc and company names menioned may be rademarks of heir respecive owners. Page 7 of 58
v() 5 m/sec 0 ime 6.2 Basic Timevarying Signals Since he analysis of dynamic sysems relies upon imevarying phenomenon, his chaper secion presens some common imevarying signals ha will be used in our analyses. Specific signals ha will be presened are sep funcions and exponenial funcions. 6.2.1 Sep Funcion We will use a sep funcion o model a signal, which changes suddenly from one consan value o anoher. These ypes of signals can be very imporan. Examples include digial logic circuis (which swich beween low and high volage levels) and conrol sysems (whose design specificaions are ofen based on he sysem s response o a sudden change in inpu). We define a uni sep funcion, uo() as follows: 0, < 0 u 0 () = { 1, > 0 Eq. 6.6 The uni sep funcion is illusraed in Fig. 6.11 below. For now, i will no be necessary o define a value for he sep funcion a ime =0. u 0 () 1 0 Figure 6.11. Uni sep funcion. Physically, he sep funcion models a swiching process. For example, he oupu volage Vou of he circui shown in Figure 6.12, in which a consan 1V source supplies volage hrough a swich which closes a ime =0, is a uni sep funcion. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 8 of 58
= 0 sec 1V V OUT Figure 6.12. Circui o realize a uni sep funcion. The uni sep funcion can be scaled o provide differen ampliudes. Muliplicaion of he uni sep funcion by a consan K resuls in a signal which is zero for imes less han zero and K for imes greaer han zero, as shown in Fig. 6.13. Ku 0 () K 0 Figure 6.13. Scaled sep funcion Ku0(); K>0. The sep funcion can also be shifed o model processes which swich a imes oher han =0. A sep funcion wih ampliude K which occurs a ime =a can be wrien as Ku0( a): 0, < a Ku 0 ( a) = { K, > a Eq. 6.7 The funcion is zero when he argumen a is less han zero and K when he argumen a is greaer han zero, as shown in Fig. 6.14. If a>0, he funcion is shifed o he righ of he origin; if a<0, he funcion is shifed o he le of he origin. Ku 0 (a) K 0 a Figure 6.14. Shifed and scaled sep funcion Ku0(a); K>0 and a>0. Swiching he sign of he above argumen in equaion (6.7) resuls in: K, < a Ku 0 ( a) = Ku a (a ) = { 0, > a Eq. 6.8 And he value of he funcion is K for <a and zero for >a, as shown in Fig. 6.15. As above, he ransiion from K o zero is o he righ of he origin if a>0 and o he lef of he origin if a<0. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 9 of 58
Ku 0 (a) K 0 a Figure 6.15. The funcion sep funcion Ku0(a); K>0 and a>0. Sep funcions can also be used o describe finieduraion signals. For example, he funcion: 0, < 0 f() = { 1,0 < < 2 0, > 2 Illusraed in Fig. 6.16, can be wrien in erms of sums or producs of uni sep funcions as follows: f() = u 0 () u 0 ( 1) Or f() = u 0 () u 0 (2 ) f() 1 0 2 Figure 6.16. Finieduraion signal. The sep funcion can also be used o creae oher finieduraion funcions. For example, he finieduraion ramp funcion: 0, < 0 f() = {, 0 < < 1 0, > 1 Shown in Fig. 6.17, can be wrien as a single funcion over he enire range < < by using uni sep funcions, as follows: f() = [u 0 () u 0 ( 1)] f() 1 0 1 Figure 6.17. Finieduraion ramp signal. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 10 of 58
6.2.2 Exponenial Funcions A funcion ha appears commonly in he analysis of linear sysems is he decaying exponenial: f() = Ae a Where a>0. The funcion f() is illusraed in Fig. 6.18. The value of he funcion is A a =0 and decreases o zero as. As he funcion increases wihou bound. The consan a dicaes he rae a which he funcion decreases as ime increases. f() A 0 Figure 6.18. Decaying exponenial funcion. We will usually be ineresed in his funcion only for posiive values of ime. We will also commonly wrie our exponenial funcion in erms of a ime consan, τ, raher han he consan a. Thus, he decaying exponenial funcion we will generally use is: f() = { 0, < 0 Ae 1 τ, > 0 Eq. 6.10 Or, using he uni sep funcion o limi he funcion o posiive values of ime: f() = Ae τ u 0 () Eq. 6.11 The funcion of equaions (6.10) and (6.11) is illusraed in Fig. 6.19. The ime consan, τ, is a posiive number which dicaes he rae a which he funcion will decay wih ime. When he ime = τ, f()ae 1 = 0.368A and he funcion has decayed o 36.8% of is original value. In fac, he funcion decreases by 36.8% every τ seconds. Therefore, a signal wih a small ime consan decays more rapidly han a signal wih a large ime consan, as illusraed in Fig. 6.20. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 11 of 58
f() A 0.368A 0 Figure 6.19. Exponenial funcion f() = Ae τ u 0 (). f() increasing 0 Figure 6.20. Exponenial funcion variaion wih ime cons. Secion Summary Sep funcions are useful for represening condiions (generally inpus), which change from one value o anoher insananeously. In elecrical engineering, hey are commonly used o model he opening or closing of a swich ha connecs a circui o a source, which provides a consan volage or curren. Mahemaically, an arbirary sep funcion can be represened by: K, < a Ku 0 ( a) = Ku 0 (a ) = { 0, > a So ha he sep funcion urns on a ime =a, and has an ampliude K. An exponenial funcion, defined for >0, is mahemaically defined as: f() = Ae τ u 0 () Oher produc and company names menioned may be rademarks of heir respecive owners. Page 12 of 58
The funcion has an iniial value, A, and a ime consan, τ. The ime consan indicaes how quickly he funcion decays; he value of he funcion decreases by 63.2% every τ seconds. Exponenial funcions are imporan o use because he soluions of linear, consan coefficien, ordinary differenial equaions ypically ake he form of exponenials. 6.2 Exercises 1. Express he signal below in erms of sep funcions. f() 1 1 1 2 3 4 Time, sec 1 2 2. The funcion shown below is a decaying exponenial. Esimae he funcion from he given graph. f() 6 4 2 0 0 10 20 30 Time, milliseconds 6.3 Capaciors We begin our sudy of energy sorage elemens wih a discussion of capaciors. Capaciors, like resisors, are passive woerminal circui elemens. Tha is, no exernal power supply is necessary o make hem funcion. Capaciors consis of a nonconducive maerial (or dielecric) which separaes wo elecrical conducors; capaciors sore energy in he form of an elecric field se up in he dielecric maerial. In his secion, we describe physical properies of capaciors and provide a mahemaical model for an ideal capacior. Using his ideal capacior model, we will develop mahemaical relaionships for he energy sored in a capacior and governing relaions for series and parallel connecions of capaciors. The secion concludes wih a brief discussion of pracical (nonideal) capaciors. 6.3.1 Capaciors Two elecrically conducive bodies, when separaed by a nonconducive (or insulaing) maerial, will form a capacior. Figure 6.21 illusraes he special case of a parallel plae capacior. The nonconducive maerial Oher produc and company names menioned may be rademarks of heir respecive owners. Page 13 of 58
beween he plaes is called a dielecric; he maerial propery of he dielecric, which is currenly imporan o us, is is permiiviy, ε. When a volage poenial difference is applied across he wo plaes, as shown in Fig. 6.21, charge accumulaes on he plaes he plae wih he higher volage poenial will accumulae posiive charge q, while he plae wih he lower volage poenial will accumulae negaive charge, q. The charge difference beween he plaes induces an elecric field in he dielecric maerial; he capacior sores energy in his elecric field. The capaciance of he capacior is a quaniy ha ells us, essenially, how much energy can be sored by he capacior. Higher capaciance means ha more energy can be sored by he capacior. Capaciance has unis of Farads, abbreviaed F. The amoun of capaciance a capacior has is governed by he geomery of he capacior (he shape of he conducors and heir orienaion relaive o one anoher) and he permiiviy of he dielecric beween he conducors. These effecs can be complex and difficul o quanify mahemaically; raher han aemp a comprehensive discussion of hese effecs, we will simply claim ha, in general, capaciance is dependen upon he following parameers: The spacing beween he conducive bodies (he disance d in Fig. 6.21). As he separaion beween he bodies increases, he capaciance decreases. The surface area of he conducive bodies. As he surface area of he conducors increases, he capaciance increases. The surface area referred o here is he area over which boh he conducors and he dielecric overlap. The permiiviy of he dielecric. As he permiiviy increases, he capaciance increases. The parallelplae capacior shown in Fig. 6.21, for example, has capaciance: C = ε A d Upper conducive plae wih area, A, and charge q v() Elecric Field, E d Lower conducive plae wih area, A, and charge q Dielecric maerial wih permiiviy, e Figure 6.21. Parallel plae capacior wih applied volage across conducors. Mahemaically, he capaciance of he device relaes he volage difference beween he plaes and he charge accumulaion associaed wih his volage: q() = CV() Eq. 6.12 Capaciors ha obey he relaionship of equaion (6.12) are linear capaciors, since he poenial difference beween he conducive surfaces is linearly relaed o he charge on he surfaces. Please noe ha he charges on he upper and lower plae of he capacior in Fig. 6.21 are equal and opposie hus, if we increase he charge on one plae, he charge on he oher plae mus decrease by he same amoun. This is consisen wih our previous assumpion elecrical circui elemens canno accumulae charge, and curren enering one erminal of a capacior mus leave he oher erminal of he capacior. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 14 of 58
Since curren is defined as he ime rae of change of charge, i() = dq(), equaion (6.12) can be rewrien in erms of he curren hrough he capacior: d i() = d [Dv()] Eq. 6.13 d Since he capaciance of a given capacior is consan, equaion (6.13) can be wrien as: i() = C dv() d Eq. 6.14 The circui symbol for a capacior is shown in Fig. 6.22, along wih he sign convenions for he volagecurren relaionship of equaion (6.14). We use our passive sign convenion for he volagecurren relaionship posiive curren is assumed o ener he erminal wih posiive volage polariy. i() v() C Figure 6.22. Capacior circui symbol and volagecurren sign convenion. Inegraing boh sides of equaion (6.14) resuls in he following form for he capacior s volagecurren relaionship: v() = 1 i(ξ)dξ v( C 0 0) Eq. 6.15 Where v(0) is a known volage a some iniial ime, 0. We use a dummy variable of inegraion, ξ, o emphasize ha he only which survives he inegraion process is he upper limi of he inegral. Imporan resul: The volagecurren relaionship for an ideal capacior can be saed in eiher differenial or inegral form, as follows: i() = C dv() v() = 1 C i(ξ)dξ v( 0) Example 6.3 d 0 If he volage as a funcion of ime across a capacior wih capaciance C=1μF is as shown below, deermine he curren as a funcion of ime hrough he capacior. 10 Volage, V 5 0 1 2 3 4 Time, sec 5 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 15 of 58
0 < < 1: The volage rae of change is 10 V/s. Thus, C dv() = (1 10 6 F) (10 V ) = 10μA. s 1 < < 2: The volage is consan. Thus, C dv() = 0A. d 2 < < 3: The volage rae of change is 15V/s. Thus, C dv() = (1 10 6 F) ( 15 V ) = 15μA. s 3 < < 4: The volage is consan. Thus, C dv() = 0A A plo of he curren hrough he capacior as a funcion of ime is shown below. d d d 10 Curren, ma 0 1 2 3 4 Time, sec 15 Example 6.4 If he curren as a funcion of ime hrough a capacior wih capaciance C=10mF is as shown below, deermine he volage as a funcion of ime across he capacior. Assume ha he volage across he capacior is 0V a ime =0. 10 Curren, ma 0 10 1 2 3 4 Time, sec A ime =0, he volage is given o be 0V. In he ime period 0<<1 second, he curren increases linearly and he volage will increase quadraically. The oal volage change during his ime period is he inegral of he curren, which is simply he area (10 10 3 A)(1s) under he curren curve divided by he capaciance: 1 = 0.5V. 2 0.01F In he ime period 1<<2 seconds, he curren is consan a 10 ma. The volage change is he area under he curren curve divided by he capaciance: (10 10 3 A) (1s) is, hen, 0.5V 1V = 1.5V. 0.01F = 1V. The oal volage a =2 seconds In he ime period 2<<3 seconds, he curren is consan a 10 ma. The volage change is he negaive of he volage change from 1<<2 sec. The oal volage a =3 seconds is, hen, 1.5V 1V = 0.5V. In he ime period 3<<4 seconds, he curren is zero. The inegral of zero over any ime period is zero, so here is no change in volage during his ime range and he volage remains consan a 0.5V. A plo of he volage across he capacior as a funcion of ime is shown below. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 16 of 58
1.5 Volage, V 1 0.5 0 1 2 3 4 Time, sec I is ofen useful, when analyzing circuis conaining capaciors, o examine he circui s response o consan operaing condiions and o insananeous changes in operaing condiion. We examine he capacior s response o each of hese operaing condiions below: Capacior response o consan volage: o If he volage across he capacior is consan, equaion (6.14) indicaes ha he curren hrough he capacior is zero. Thus, if he volage across he capacior is consan, he capacior is equivalen o a open circui. o This propery can be exremely useful in deermining a circui s seadysae response o consan inpus. If he inpus o a circui change from one consan value o anoher, he ransien componens of he response will evenually die ou and all circui parameers will become consan. Under hese condiions, capaciors can be replaced wih open circuis and he circui analyzed relaively easily. As we will see laer, his operaing condiion can be useful in deermining he response of circuis conaining capaciors and in doublechecking resuls obained using oher mehods. Capacior response o insananeous volage changes: o If he volage across he capacior changes insananeously, he rae of change of volage is infinie. Thus, by equaion (6.14), if we wish o change he volage across a capacior insananeously, we mus supply infinie curren o he capacior. This implies ha infinie power is available, which is no physically possible. Thus, in any pracical circui, he volage across a capacior canno change insananeously. o Any circui ha allows an insananeous change in he volage across an ideal capacior is no physically realizable. We may someimes assume, for mahemaical convenience, ha an ideal capacior s volage changes suddenly; however, i mus be emphasized ha his assumpion requires an underlying assumpion ha infinie power is available and is hus no an allowable operaing condiion in any physical circui. Imporan Capacior Properies: Capaciors can be replaced by opencircuis, under circumsances when all operaing condiions are consan. Volages across capaciors canno change insananeously. No such requiremen is placed on currens. 6.3.2 Energy Sorage The power dissipaed by a capacior is: p() = v() i() Eq. 6.16 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 17 of 58
Since boh volage and curren are funcions of ime, he power dissipaion will also be a funcion of ime. The power as a funcion of ime is called he insananeous power, since i provides he power dissipaion a any insan in ime. Subsiuing equaion (6.14) ino equaion (6.16) resuls in: p() = C v() dv() d Eq. 6.17 Since power is, by definiion, he rae of change of energy, he energy is he ime inegral of power. Inegraing equaion (6.17) wih respec o ime gives he following expression for he energy sored in a capacior: dv( ) 1 W C ( ) = Cv( ) d = Cv( ) dv( ) = Cv 2 ( ) d 2 Where we have se our lower limis of inegraion a = o avoid issues relaive o iniial condiions. We assume ha no energy is sored in he capacior a ime = so ha: W C () = 1 2 Cv2 () Eq. 6.18 From equaion (6.18) we see ha he energy sored in a capacior is always a nonnegaive quaniy, so WC() 0. Ideal capaciors do no dissipae energy, as resisors do. Capaciors sore energy when i is provided o hem from he circui; his energy can laer be recovered and reurned o he circui. Example 6.5 Consider he circui shown below. The volage applied o he capacior by he source is as shown. Plo he power absorbed by he capacior and he energy sored in he capacior as funcions of ime. v() C=1mF v(), V 10 5 0 0 1 2 3 4 Time, ms Power is mos readily compued by aking he produc of volage and curren. The curren can be deermined from equaion (6.14). The curren as a funcion of ime is ploed below. 10 Curren, A 5 0 0 1 2 3 4 Time, ms 5 The power absorbed by he capacior is deermined by aking a poinbypoin produc beween he volage and curren. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 18 of 58
100 p(), W 0 50 1 2 3 4 Time, ms Recall ha power is absorbed or generaed based on he passive sign convenion. If he relaive signs beween volage and curren agree wih he passive sign convenion, he circui elemen is absorbing power. If he relaive signs beween volage and curren are opposie o he passive sign convenion, he elemen is generaing power. Thus, he capacior in his example is absorbing power for he firs microsecond. I generaes power (reurns power o he volage source) during he second microsecond). Afer he second microsecond, he curren is zero and he capacior neiher absorbs nor generaes power. The energy sored in he capacior can be deermined eiher from inegraing he power or from applicaion of equaion (6.18) o he volage curve provided in he problem saemen. The energy in he capacior as a funcion of ime is shown below: WC(), mj 50 0 0 1 2 3 4 Time, ms During he firs microsecond, while he capacior is absorbing power, he energy in he capacior is increasing. The maximum energy in he capacior is 50 mj, a 1ms. During he second microsecond, he capacior is releasing power back o he circui and he energy in he capacior is decreasing. A 2ms, he capacior sill has 12.5 mj of sored energy. Afer 2ms, he capacior neiher absorbs nor generaes energy and he energy sored in he capacior remains a 12.5mJ. 6.3.3 Capaciors in Series Consider he series connecion of N capaciors shown in Fig. 6.23. v 1 () v 2 () v() C 1 C 2 C N v N () Figure 6.23. Series connecion of N capaciors. Applying Kirchhoff s volage law around he loop resuls in: v() = v 1 () v 2 () v N () Eq. 6.19 Using equaion (6.15) o wrie he capacior volage drops in erms of he curren hrough he loop gives: Oher produc and company names menioned may be rademarks of heir respecive owners. Page 19 of 58
v() = [ 1 1 1 i(ξ)dξ v C 1 ( 0 )] [ i(ξ)dξ v 1 C 2 ( 0 )] [ i(ξ)dξ v 2 C N ( 0 ) N 0 0 0 ] = [ 1 i(ξ)dξ 1 i(ξ)dξ 1 i(ξ)dξ] [v C 1 C 2 C 1 ( 0 ) v 2 ( 0 ) v N ( 0 )] N 0 This can be rewrien using summaion noaion as: 1 0 0 = ( 1 C 1 1 C 2 1 C N ) i(ξ)dξ v( 0 ) 0 N v() = ( k=1 ) Eq. 6.21 C k Thus, he circuis of Fig. 6.23 and Fig. 6.24 are equivalen circuis, if he equivalen capaciance is chosen according o equaion (6.21). v() C eq Figure 6.24. Equivalen circui o Figure 3. For he special case of wo capaciors C1 and C2 in series, equaion (6.21) simplifies o: C eq = C 1C 2 C 1 C 2 Eq. 6.22 Equaions (6.21) and (6.22) are analogous o he equaions, which provide he equivalen resisance of parallel combinaions of resisors. 6.3.4 Capaciors in Parallel Consider he parallel combinaion of N capaciors, as shown in Fig. 6.25. i 1 () i 2 () i N () i() C 1 C 2 C N v() Figure 6.25. Series connecion of N capaciors. Applying Kirchhoff s curren law a he upper node resuls in: i() = i 1 () i 2 () i N () Eq. 6.23 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 20 of 58
Using equaion (6.14) o wrie he capacior currens in erms of heir volage drop gives: Using summaion noaion resuls in: i() = C 1 dv() d dv() dv() C 2 C d N d = (C 1 C 2 C N ) dv() d i() = ( N k=1 C k ) dv() d Eq. 6.24 This is he same equaion ha governs he circui of Fig. 6.26, if: C eq = N k=1 C k Eq. 6.25 Thus, he equivalen capaciance of a parallel combinaion of capaciors is simply he sum of he individual capaciances. This resul is analogous o he equaions, which provide he equivalen resisance of a series combinaion of resisors. v() C eq Summary: Series and Parallel Capaciors Figure 6.26. Equivalen circui o Figure 5. The equivalen capaciance of a series combinaion of capaciors C1, C2,, CN is governed by a relaion which is analogous o ha providing he equivalen resisance of a parallel combinaion of resisors: 1 N C eq = 1 C k The equivalen of a parallel combinaion of capaciors C1, C2,, CN is governed by a relaion which is analogous o ha providing he equivalen resisance of a series combinaion of resisors: k=1 6.3.5 Pracical Capaciors N C eq = C k Commercially available capaciors are manufacured in a wide range of boh conducor and dielecric maerials and are available in a wide range of capaciances and volage raings. The volage raing of he device is he maximum volage, which can be safely applied o he capacior; using volages higher han he raed value will damage he capacior. The capaciance of commercially available capaciors is commonly measured in microfarads (μf; one microfarad is 10 6 of a Farad) or picofarads (pf; one picofarad is 10 12 of a Farad). Large capaciors are available, bu are relaively infrequenly used. These are generally called supercapaciors or ulracapaciors k=1 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 21 of 58
and are available in capaciances up o ens of Farads. For mos applicaions, however, using one would be comparable o buying a car wih a 1000 gallon gas ank. Several approaches are used for labeling a capacior wih is capaciance value. Large capaciors ofen have heir value prined plainly on hem, such as 10μF (for 10 microfarads). Smaller capaciors, appearing as small disks or wafers, ofen have heir values prined on hem in an encoded manner. For hese capaciors, a hreedigi number indicaes he capacior value in picofarads. The firs wo digis provides he base number, and he hird digi provides an exponen of 10 (so, for example, 104 prined on a capacior indicaes a capaciance value of 10 x 104 or 100000 pf). Occasionally, a capacior will only show a wodigi number, in which case ha number is simply he capacior value in pf. (For compleeness, if a capacior shows a hreedigi number and he hird digi is 8 or 9, hen he firs wo digis are muliplied by.01 and.1 respecively). Capaciors are generally classified according o he dielecric maerial used. Common capacior ypes include mica, ceramic, Mylar, paper, Teflon and polysyrene. An imporan class of capaciors which require special menion are elecrolyic capaciors. Elecrolyic capaciors have relaively large capaciances relaive o oher ypes of capaciors of similar size. However, some care mus be exercised when using elecrolyic capaciors hey are polarized and mus be conneced o a circui wih he correc polariy. The posiive lead of he capacior mus be conneced o he posiive lead of he circui. Connecing he posiive lead of he capacior o he negaive lead of a circui can resul in unwaned curren leakage hrough he capacior or, in exreme cases, desroy he capacior. Polarized capaciors eiher have a dark sripe near he pin ha mus be kep a he higher volage, or a near he pin ha mus be kep a a lower volage. Pracical capaciors, unlike ideal capaciors, will dissipae some power. This power loss is primarily due o leakage currens. These currens are due o he fac ha real dielecric maerials are no perfec insulaors some small curren will end o flow hrough hem. The overall effec is comparable o placing a high resisance in parallel wih an ideal capacior, as shown in Fig. 6.27. Differen ypes of capaciors have differen leakage currens. Mica capaciors end o have low leakage currens, he leakage currens of ceramic capaciors vary according o he ype of capacior, and elecrolyic capaciors have high leakage currens. R C Figure 6.27. Model of pracical capacior including leakage curren pah. Secion Summary Capaciors sore elecrical energy. This energy is sored in an elecric field beween wo conducive elemens, separaed by an insulaing maerial. Capacior energy sorage is dependen upon he volage across he capacior, if he capacior volage is known, he energy in he capacior is known. The volagecurren relaionship for a capacior is: i() = C dv() d Oher produc and company names menioned may be rademarks of heir respecive owners. Page 22 of 58
Where C is he capaciance of he capacior. Unis of capaciance are Farads (abbreviaed F). The capaciance of a capacior, very roughly speaking, gives an indicaion of how much energy i can sore The above volagecurren relaion resuls in he following imporan properies of capaciors: o If he capacior volage is consan, he curren hrough he capacior is zero. Thus, if he capacior volage is consan, he capacior can be modeled as an open circui. o Changing he capacior volage insananeously requires infinie power. Thus (for now, anyway) we will assume ha capaciors canno insananeously change heir volage. Capaciors placed in series or parallel wih one anoher can be modeled as a single equivalen capaciance. Thus, capaciors in series or in parallel are no independen energy sorage elemens. 6.3 Exercises 1. Deermine he maximum and minimum capaciances ha can be obained from four 1μF capaciors. Skech he circui schemaics ha provide hese capaciances. 2. Deermine volage divider relaionships o provide v1 and v2 for he wo uncharged series capaciors shown below. Use your resul o deermine v2 if C1=C2=10μF. 6.4 Inducors We coninue our sudy of energy sorage elemens wih a discussion of inducors. Inducors, like resisors and capaciors, are passive woerminal circui elemens. Tha is, no exernal power supply is necessary o make hem funcion. Inducors commonly consis of a conducive wire wrapped around a core maerial; inducors sore energy in he form of a magneic field se up around he currencarrying wire. In his secion, we describe physical properies of inducors and provide a mahemaical model for an ideal inducor. Using his ideal inducor model, we will develop mahemaical relaionships for he energy sored in an inducor and governing relaions for series and parallel connecions of inducors. The secion concludes wih a brief discussion of pracical (nonideal) inducors. 6.4.1 Inducors Passing a curren hrough a conducive wire will creae a magneic field around he wire. This magneic field is generally hough of in erms of as forming closed loops of magneic flux around he currencarrying elemen. This physical process is used o creae inducors. Figure 6.28 illusraes a common ype of inducor, consising of a coiled wire wrapped around a core maerial. Passing a curren hrough he conducing wire ses up lines of magneic flux, as shown in Fig. 6.28; he inducor sores energy in his magneic field. The inducance of he inducor is a quaniy, which ells us how much energy can be sored by he inducor. Higher inducance means ha he inducor can sore more energy. Inducance has unis of Henrys, abbreviaed H. The amoun of inducance an inducor has is governed by he geomery of he inducor and he properies of he core maerial. These effecs can be complex; raher han aemp a comprehensive discussion of hese effecs, we will simply claim ha, in general, inducance is dependen upon he following parameers: The number of imes he wire is wrapped around he core. More coils of wire resuls in a higher inducance. The core maerial s ype and shape. Core maerials are commonly ferromagneic maerials, since hey resul in higher magneic flux and correspondingly higher energy sorage. Air, however, is a fairly commonly used core maerial presumably because of is ready availabiliy. The spacing beween urns of he wire around he core. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 23 of 58
Magneic flux, y i() v() Coiled conducive wire Core Figure 6.28. Wirewrapped inducor wih applied curren hrough conducive wire. We will denoe he oal magneic flux creaed by he inducor by ψ, as shown in Fig. 6.28. For a linear inducor, he flux is proporional o he curren passing hrough he wound wires. The consan of proporionaliy is he inducance, L: ψ() = Li() Eq. 6.26 Volage is he ime rae of change of magneic flux, so: v() = dψ() d Eq. 6.27 Combining equaions (6.26) and (6.27) resuls in he volagecurren relaionship for an ideal inducor: v() = L di() d Eq. 6.28 The circui symbol for an inducor is shown in Fig. 6.29, along wih he sign convenions for he volage curren relaionship of equaion (6.28). The passive sign convenion is used in he volagecurren relaionship, so posiive curren is assumed o ener he erminal wih posiive volage polariy. i() v() L Figure 6.29. Inducor circui symbol and volagecurren sign convenion. Inegraing boh sides of equaion (6.28) resuls in he following form for he inducor s volagecurren relaionship: Oher produc and company names menioned may be rademarks of heir respecive owners. Page 24 of 58
i() = 1 v(ξ)dξ i( L 0 0) Eq. 6.29 In equaion (6.29), i(0) is a known curren a some iniial ime 0 and ξ is used as a dummy variable of inegraion o emphasize ha he only which survives he inegraion process is he upper limi of he inegral. Imporan Resul The volagecurren relaionship for an ideal inducor can be saed in eiher differenial or inegral form, as follows: v() = L di() i() = 1 L v(ξ)dξ i( 0) Example 6.6 d 0 A circui conains a 100mH inducor. The curren as a funcion of ime hrough he inducor is measured and shown below. Plo he volage across he inducor as a funcion of ime. 10 Curren, ma 5 0 1 2 3 4 5 Time, msec 5 In he ime range 0<<1ms, he rae of change of curren is 10 A/sec. Thus, from equaion (3), he volage is v() = (0.1H)(10A s) = 1V. In he ime range 1ms < < 2ms, he rae of change of curren is 5A/sec. The volage is 0.5V. In he ime range 2ms < < 3ms, he curren is consan and here is no volage across he inducor. In he ime range 3ms < < 5ms, he rae of change of curren is 5A/sec. The volage is 0.5V The plo of volage vs. ime is shown below: 1 Volage, V 0 1 1 2 3 4 5 Time, msec Power is he produc of volage and curren. If he signs of volage and curren are he same according o he passive sign convenion, he circui elemen absorbs power. If he signs of volage and curren are no he same, he circui elemen generaes power. From he above volage and curren curves, he inducor is absorbing power from he circui during he imes 0<<1ms and 4ms<<5ms. The inducor reurns power o he circui during he imes 1ms<<2ms and 3ms<<4ms. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 25 of 58
10 Power, mw 5 0 1 2 3 4 5 Time, msec 5 Example 2: If he volage as a funcion of ime across an inducor wih inducance L = 10 mh is as shown below, deermine he curren as a funcion of ime hrough he capacior. Assume ha he curren hrough he capacior is 0A a ime =0. 10 Volage, V 0 10 1 2 3 4 Time, msec A ime =0, he curren is given o be 0A. In he ime period 0<<1 msec, he volage is consan and posiive so he curren will increase linearly. The oal curren change during his ime period is he area under he volage curve curve, divided by he 1 inducance: (10V)(1 0.01 10 3 s) = 1A In he ime period 1<<2 msec, he volage is decreasing linearly. The curren during his ime period is a quadraic curve, concave downward. The maximum value of curren is 1.25A, a =1.5 msec. The curren a he end of his ime period is 1A. In he ime period 2<<3 seconds, he volage is consan a 10V. The curren change during his ime period is he area under he volage curve, divided by he inducance: ( 10V)(1 0.01 10 3 s) = 1A. The oal curren a =3 seconds is, hen, 1A 1A = 0A. In he ime period 3<<4 seconds, he volage is zero. The inegral of zero over any ime period is zero, so here is no change in curren during his ime range and he curren remains consan a 0A. A plo of he curren hrough he inducor as a funcion of ime is shown below. 1 Curren, A 1 0 0 1 2 3 4 Time, msec I is ofen useful, when analyzing circuis conaining inducors, o examine he circui s response o consan operaing condiions and o insananeous changes in operaing condiion. We examine he inducor s response o each of hese operaing condiions below: Oher produc and company names menioned may be rademarks of heir respecive owners. Page 26 of 58
Inducor response o consan curren: o If he curren hrough he inducor is consan, equaion (6.28) indicaes ha he volage across he inducor is zero. Thus, if he curren hrough he inducor is consan, he inducor is equivalen o a shor circui. Inducor response o insananeous curren changes: o If he curren hrough he inducor changes insananeously, he rae of change of curren is infinie. Thus, by equaion (6.28), if we wish o change he curren hrough an inducor insananeously, we mus supply infinie volage o he inducor. This implies ha infinie power is available, which is no physically possible. Thus, in any pracical circui, he curren hrough an inducor canno change insananeously. o Any circui ha allows an insananeous change in he curren hrough an ideal inducor is no physically realizable. We may someimes assume, for mahemaical convenience, ha an ideal inducor s curren changes suddenly; however, i mus be emphasized ha his assumpion requires an underlying assumpion ha infinie power is available and is hus no an allowable operaing condiion in any physical circui. Imporan Inducor Properies Inducors can be replaced by shorcircuis, under circumsances when all operaing condiions are consan. Currens hrough inducors canno change insananeously. No such requiremen is placed on volages. 6.4.2 Energy Sorage The insananeous power dissipaed by an elecrical circui elemen is he produc of he volage and curren: p() = v() i() Eq. 6.30 Using equaion (6.28) o wrie he volage in equaion (6.30) in erms of he inducor s curren: p() = L i() di() d Eq. 6.31 As was previously done for capaciors, we inegrae he power wih respec o ime o ge he energy sored in he inducor: W L () = Li(ξ) di(ξ) d d Which, afer some manipulaion (comparable o he approach aken when we calculaed energy sorage in capaciors), resuls in he following expression for he energy sored in an inducor: W L () = 1 2 Li2 () Eq. 6.32 6.4.3 Inducors in Series Consider he series connecion of N inducors shown in Fig. 6.30. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 27 of 58
v 1 () v 2 () L 1 L 2 v() L N v N () Figure 6.30. Series connecion of N inducors. Applying Kirchhoff s volage law around he loop resuls in: v() = v 1 () v 2 () v N () Eq. 6.33 Using equaion (6.28) o wrie he inducor volage drops in erms of he curren hrough he loop gives: Using summaion noaion resuls in: v() = L 1 di() d di() di() L 2 L d N d = (L 1 L 2 L N ) di() d v() = ( N k=1 L_k ) di() d Eq. 6.34 This is he same equaion ha governs he circui of Fig. 6.31, if: L eq = L k Thus, he equivalen inducance of a series combinaion of inducors is simply he sum of he individual inducances. This resul is analogous o he equaions which provide he equivalen resisance of a series combinaion of resisors. N k=1 v() L eq Figure 6.31. Equivalen circui o Figure 3. 6.4.4 Inducors in Parallel Consider he parallel combinaion of N inducors, as shown in Fig. 6.32. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 28 of 58
i 1 () i 2 () i N () i() L 1 L 2 L N v() Figure 6.32. Parallel combinaion of I inducors. Applying Kirchhoff s curren law a he upper node resuls in: i() = i 1 () i 2 () i N () Eq. 6.36 Using equaion (6.29) o wrie he inducor currens in erms of heir volage drops gives: i() = [ 1 1 1 v(ξ)dξ i L 1 ( 0 )] [ v(ξ)dξ i 1 L 2 ( 0 )] [ v(ξ)dξ i 2 L N ( 0 ) N 0 0 0 ] = [ 1 v(ξ)dξ 1 v(ξ)dξ 1 v(ξ)dξ] [i L 1 L 2 L 1 ( 0 ) i 2 ( 0 ) i N ( 0 )] N 0 This can be rewrien using summaion noaion as: 1 0 0 = ( 1 1 1 ) v(ξ)dξ i( L 1 L 2 L 0 ) N 0 N i() = ( k=1 ) v(ξ)dξ i( 0 ) Eq. 6.37 0 L k This is he same equaion ha governs he circui of Fig. 6.31, if: 1 L eq = L 1L 2 L 1 L 2 Eq. 6.39 Equaions (6.38) and (6.39) are analogous o he equaions which provide he equivalen resisance of parallel combinaions of resisors. Summary: Series and Parallel Inducors The equivalen inducance of a series combinaion of inducors L1, L2,, LN is governed by a relaion which is analogous o ha providing he equivalen resisance of a series combinaion of resisors: L eq = L k The equivalen inducance of a parallel combinaion of inducors L1, L2,, LN is governed by a relaion which is analogous o ha providing he equivalen resisance of a parallel combinaion of resisors: N k=1 1 N L eq = 1 L k k=1 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 29 of 58
6.5 Pracical Inducors Mos commercially available inducors are manufacured by winding wire in various coil configuraions around a core. Cores can be a variey of shapes; Fig. 6.28 in his chaper shows a core, which is basically a cylindrical bar. Toroidal cores are also fairly common a closely wound oroidal core has he advanage ha he magneic field is confined nearly enirely o he space inside he winding. Inducors are available wih values from less han 1 microhenry (1μH = 10 6 Henries) up o ens of Henries. A 1H inducor is very large; inducances of mos commercially available inducors are measured in millihenries (1mH = 10 3 Henries) or microhenries. Larger inducors are generally used for lowfrequency applicaions (in which he signals vary slowly wih ime). Aemps a creaing inducors in inegraedcircui form have been largely unsuccessful; herefore many circuis ha are implemened as inegraed circuis do no include inducors. Inclusion of inducance in he analysis sage of hese circuis may however, be imporan. Since any currencarrying conducor will creae a magneic field, he sray inducance of supposedly noninducive circui elemens can become an imporan consideraion in he analysis and design of a circui. Pracical inducors, unlike he ideal inducors discussed in his chaper, dissipae power. An equivalen circui model for a pracical inducor is generally creaed by placing a resisance in series wih an ideal inducor, as shown in Fig. 6.33. R L Secion Summary Figure 6.33. Equivalen circui model for a pracical inducor. Inducors sore magneic energy. This energy is sored in a magneic field (ypically) generaed by a coiled wire wrapped around a core maerial. Inducor energy sorage is dependen upon he curren hrough he inducor, if he inducor curren is known, he energy in he inducor is known. The volagecurren relaionship for an inducor is: v() = L di() d Where L is he inducance of he inducor. Unis of inducance are Henries (abbreviaed H). The inducance of an inducor, very roughly speaking, gives an indicaion of how much energy i can sore. The above volagecurren relaion resuls in he following imporan properies of inducors: o If he inducor curren is consan, he volage across he inducor is zero. Thus, if he inducor curren is consan, he inducor can be modeled as a shor circui. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 30 of 58
o Changing he inducor curren insananeously requires infinie power. Thus (for now, anyway) we will assume ha inducors canno insananeously change heir curren. Inducors placed in series or parallel wih one anoher can be modeled as a single equivalen inducance. Thus, inducors in series or in parallel are no independen energy sorage elemens. 6.4 Exercises 1. Deermine he equivalen inducance of he nework below: 4H 3H 2H 6H L eq Oher produc and company names menioned may be rademarks of heir respecive owners. Page 31 of 58
Real Analog Chaper 6: Lab Projecs 6.2.1: Timevarying Signals This assignmen will focus on using an arbirary waveform generaor o generae imevarying signals and using an oscilloscope o measure ime varying signals. In chaper 6 of he ex book, we deal analyically only wih sep funcions and exponenial funcions. This lab will, however inroduce us o a larger class of imevarying waveforms. The abiliy o apply and measure ime varying signals will be crucial hroughou he remainder of your career. I is srongly recommended ha you no only complee he specific seps oulined in his assignmen, bu ha you spend some addiional ime playing wih he ools we inroduce in his assignmen i is guaraneed o be ime well spen! Before beginning his lab, you should be able o: Define a sep funcion. Sae Ohm s law for imevarying signals Afer compleing his lab, you should be able o: Use a swich o creae a sep funcion Use he Analog Discovery waveform generaor o apply square, riangular, and sinusoidal waveforms Use he Analog Discovery oscilloscope o measure and display imevarying waveforms This lab exercise requires: Analog Discovery module Digilen Analog Pars Ki Symbol Key: Demonsrae circui operaion o eaching assisan; eaching assisan should iniial lab noebook and grade shee, indicaing ha circui operaion is accepable. Analysis; include principle resuls of analysis in laboraory repor. General Discussion: Numerical simulaion (using PSPICE or MATLAB as indicaed); include resuls of MATLAB numerical analysis and/or simulaion in laboraory repor. Record daa in your lab noebook. Once we begin o deal in earnes wih sysems which include energy sorage elemens, i will be crucial apply imevarying power o our elecrical circuis and measure he circuis responses as funcions of ime. This lab inroduces he conceps necessary for applicaion, measuremen, and inerpreaion of imevarying signals. Since we have no ye been inroduced o dynamic sysems, he elecrical circui of ineres in his assignmen will be he volage divider shown in Figure 1. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 32 of 58
v IN () R 1 R 2 v OUT () Figure 1. Volage divider circui. In Figure 1, he oupu volage, vout() is relaed o he inpu volage vin() via he volage divider relaion: R 2 v OUT () = v IN () Eq. 1 R 1 R 2 Noice ha he relaionship beween vin() and vout() is algebraic he value of vout a a paricular ime depends only upon he value of vin a ha same ime. In order o familiarize ourselves wih he fundamenals of applying and measuring imevarying signals, we will resric ourselves o some of he mos common signals encounered in engineering applicaions: sinusoidal waves, square waves, and riangular waves. The basic shapes of hese signals are shown in Fig. 2. The signals of Fig. 2 are all periodic signals ha is, hey repea hemselves a regular inervals. This inerval is called he period (commonly denoed mahemaically as T). The period of each of he signals of ineres o us is indicaed on Fig. 2. The oher primary aribue of he signals we will be dealing wih is heir ampliude (which we will denoe as A). The ampliude of he signal is essenially he maximum (and minimum) value ha he signal achieves 1. v() T A (a) Sinusoidal wave. 1 For now, our signals will be symmeric wih respec o he ime axis. Tha is, heir average value (also called he offse) will be zero. For he signals of immediae ineres o us, his means ha heir minimum value will be he negaive of heir maximum value. Laer labs will explore he effecs of a nonzero offse o he signal, and signals which are no symmeric wih respec o he ime axis. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 33 of 58
v() A T (b) Triangular wave v() A T (c.) Square wave Figure 2. Basic signal shapes. Alhough we have used he signals period as a fundamenal parameer defining he signal, i is more common for elecrical insrumens o use he frequency of he signal as a defining characerisic. The frequency provides essenially he same informaion as he period; he frequency is jus he inverse of he period: f = 1 T Eq. 2 As defined in equaion (2), he unis of frequency are in Herz (abbreviaed Hz) or cycles per second. Sinusoidal signals, however, are more accuraely defined mahemaically in erms of heir radian frequency, denoed as. Since here are 2 radians in one cycle, he conversion beween frequency and radian frequency is: ω = 2πf = 2π T Eq. 3 Mahemaically, he sinusoidal wave of Fig. 2(a) can be represened as: v() = Acos(ω θ) = Acos(2πf θ) Eq. 4 Where is he phase angle of he signal; i ranslaes he sinusoid in ime. We will concern ourselves wih phase laer in he course. Prelab: In he circui of Fig. 1, if R1 = R2, overlay skeches using he inpu and oupu volages (vin() and vout()) for he following cases: (a) vin() is a sinusoidal wave wih ampliude A and period T. (b) vin() is a riangular wave wih ampliude A and period T. (c) vin() is a square wave wih ampliude A and period T. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 34 of 58
Lab Procedures: Label he ampliude and period of boh he inpu and oupu waveforms on your skech. These values may be funcions of A, T, R1 and R2. (a) Tes he response of he circui o a sinusoidal inpu volage wih 2kHz frequency and 2V ampliude. Deails are below: i. Se vin() in he circui of Fig. 1 o be a sinusoidal volage wih ampliude 2V and frequency 1kHz across he volage divider. The average value of he sinusoid should be zero vols. To do his, open he WaveGen insrumen in he waveforms file. Click on he Basic ab (if i is no already seleced) and hen click on he Sandard opion. There should be a series of icons in a column below his opion, indicaing he shape of he associaed waveform. Click on he icon o selec a sinusoidal waveform. Choose 1kHz as he frequency (you can choose he desired frequency by selecing i from he dropdown menu, yping he desired value in he ex box, or using he slider bar) and 2V as he ampliude 2. The plo window on he waveform generaor insrumen will display one period he waveform you have se. Use his plo window o double check ha your signal has he correc frequency and ampliude. Noe on selecing parameers: When choosing parameers describing signals(e.g. frequency, ampliude, offse, and symmery) he allowable values are limied o he range specified by he values above and below he slider bar, as indicaed on he figure o he righ for he frequency parameer. When selecing a value, he desired value mus be beween he maximum and minimum values shown. If you wan a value ouside he displayed range, simply rese he range using he appropriae dropdown menus. If he waveform generaor will no le you se a desired value, be sure o check ha he desired value is wihin he allowable range. Desired value Maximum allowable value Minimum allowable value ii. Use he oscilloscope o display he volages vin() and vout() of Figure 1. To do his, open he Scope insrumen. Se he horizonal scale (or he ime axis scale) o be 1msec/div. Horizonal axis seings are se in he ime axis seings box on he oscilloscope window; his box and he desired seings for his lab are shown below: Trigger ime Time base Horizonal (ime axis) seings Se he verical axis seings on boh channel 1 and channel 2 (C1 and C2) o 500mv/div. Verical axis seings are se in he channel axis seings boxes on he oscilloscope window; 2 The offse should be zero, he symmery 50%, and he phase 0 degrees. These are he defaul values, and should no need o be rese. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 35 of 58
he seings box for channel 1 and is desired seings are shown below. Use he same seings for channel 2. C1 Offse C1 scale Verical axis (volage) seings Click on o acquire and display he daa. Record an image of he oscilloscope main ime window o a file for laer documenaion. iii. From he ime plos displayed in he oscilloscope window, deermine he period and ampliude of vin() and vout(). From your measured period, calculae he signal s frequency in Herz. Creae a able, showing he expeced ampliude and frequency of vin() and vout() and your measured ampliude and frequency of vin() and vout(). iv. Click on he buon on he oscilloscope window o open a measuremens window. Use he measuremen window o measure he ampliude, period, and frequency of vin() and vout(). Record he image of he oscilloscope window, showing he waveforms and heir measured ampliudes, periods, and frequencies 3. Commen on he agreemen beween he oscilloscope s measuremens and he measuremens you made in par iii above. v. Demonsrae operaion of your circui o he Teaching Assisan. Have he TA iniial he appropriae page(s) of your lab noebook and he lab checklis. vi. Vary he ampliude and frequency of he sinusoidal waveform using he waveform generaor. Change he horizonal and verical axis scales in he oscilloscope. Verify ha he changes resul in daa ha agree wih your expecaions. Familiarizing yourself wih hese insrumens now will be rewarded in laer experimens you can only inerpre he resuls of fuure experimens if you are comforable wih measuring he daa upon which he resuls depend! (b) Tes he response of he circui o a riangular inpu volage wih 1kHz frequency and 3V ampliude. i. Perform all he seps you did above for he sinusoidal inpu. ii. Demonsrae operaion of your circui o he Teaching Assisan. Have he TA iniial he appropriae page(s) of your lab noebook and he lab checklis. (c) Tes he response of he circui o a square wave inpu volage wih 500Hz frequency and 2.5V ampliude. i. Perform all he seps you did above for he sinusoidal inpu. ii. Demonsrae operaion of your circui o he Teaching Assisan. Have he TA iniial he appropriae page(s) of your lab noebook and he lab checklis. 3 Holding down he Al key and pressing Prin Screen (commonly labeled as PrScn or PrSc on compuer keyboards) will copy he currenly acive window o he clipboard. You can hen pase his image o a documen. The buon on he oscilloscope insrumen allows you o copy an image of he main ime window o he clipboard or save i o a file in a variey of formas. This opion will no, however, display he measuremen window if you use his approach, you will wan o record he measured values elsewhere. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 36 of 58
Real Analog Chaper 6: Lab Workshees 6.2.1: Timevarying Signals (40 poins oal) 1. Aach o his workshee he inpu and oupu volage skeches you creaed in he prelab for sinusoidal, riangular, and square waves. (7 ps) 2. Aach o his workshee an image of he oscilloscope window, showing he sinusoidal waveforms and heir measured ampliudes, periods, and frequencies. In he space below, provide he ampliudes, periods, and frequencies deermined direcly from he ime plo in he oscilloscope window. Commen on he agreemen beween he wo ses of daa. (8 ps) 3. DEMO: Have a eaching assisan iniial his shee, indicaing ha hey have observed your circui s operaion for sinusoidal inpus. (3 ps) TA Iniials: 4. Aach o his workshee an image of he oscilloscope window, showing he riangular waveforms and heir measured ampliudes, periods, and frequencies. In he space below, provide he ampliudes, periods, and frequencies deermined direcly from he ime plo in he oscilloscope window. Commen on he agreemen beween he wo ses of daa. (8 ps) 5. DEMO: Have a eaching assisan iniial his shee, indicaing ha hey have observed your circui s operaion for riangular inpus. (3 ps) TA Iniials: 6. Aach o his workshee an image of he oscilloscope window, showing he square waveforms and heir measured ampliudes, periods, and frequencies. In he space below, provide he ampliudes, periods, and frequencies deermined direcly from he ime plo in he oscilloscope window. Commen on he agreemen beween he wo ses of daa. (8 ps) 7. DEMO: Have a eaching assisan iniial his shee, indicaing ha hey have observed your circui s operaion for square wave inpus. (3 ps) TA Iniials: Oher produc and company names menioned may be rademarks of heir respecive owners. Page 37 of 58
Real Analog Chaper 6: Lab Projecs 6.3.1: Capacior Volagecurren Relaions In his assignmen, we will measure he relaionship beween he volage difference across a capacior and he curren passing hrough i. We will apply several ypes of imevarying signals o a series combinaion of a resisor and a capacior. The volage difference across he resisor, in conjuncion wih Ohm s law, will provide an esimae of he curren hrough he capacior. This curren can be relaed o he volage difference across he capacior. The resuls of our volagecurren measuremens will be compared o analyical expecaions. Before beginning his lab, you should be able o: Sae volagecurren relaionships for capaciors in boh differenial and inegral form Apply he capacior volagecurren relaions o calculae a capacior s volage from is curren and viceversa Use he Analog Discovery s arbirary waveform generaor and oscilloscope o apply and measure imevarying waveforms (Lab 6.2.1) Afer compleing his lab, you should be able o: Use he Analog Discovery oscilloscope s mah funcion o calculae he curren hrough a known resisor from he measured volage difference. Verify a capacior s volagecurren relaions using measured daa This lab exercise requires: Analog Discovery module Digilen Analog Pars Ki Symbol Key: Demonsrae circui operaion o eaching assisan; eaching assisan should iniial lab noebook and grade shee, indicaing ha circui operaion is accepable. Analysis; include principle resuls of analysis in laboraory repor. General Discussion: Numerical simulaion (using PSPICE or MATLAB as indicaed); include resuls of MATLAB numerical analysis and/or simulaion in laboraory repor. Record daa in your lab noebook. We will use he circui of Figure 1 in his lab assignmen. Boh he volage difference across he capacior and he resisor (vc() and vr()) will be measured. From his daa, we can readily compare he volage across he capacior wih he curren hrough he capacior. Since he volage across he resisor is measured, we can readily infer he curren hrough he resisor via Ohm s law: i R () = v R() R Eq. 1 The resisor and capacior are in series, so he curren hrough he capacior is he same as he curren hrough he resisor, so: i C () = v R() R Eq. 2 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 38 of 58
Since we are also measuring he volage difference across he capacior, vc(), we can readily compare hese parameers wih our expecaions based on our mahemaical models of he capacior volagecurren relaionships. i R () v R () v IN () R C i C () v C () Figure 1. Series RC circui. Prelab: For he circui of Figure 1, use he inducor volagecurren relaions o overlay skeches of he capacior volage and he capacior curren (vc() and ic()) if he capacior volage is: (a) A sinusoidal wave, v(), wih frequency (f) and ampliude (A) as shown in Figure 2 (a) (b) A riangular wave, v(), wih frequency (f) and ampliude (A) as shown in Figure 2 (b). Label your skech o show he ampliude and period of he capacior curren for boh of he above cases. Your resuls may be dependen up on he parameers A, f, R, and C. Be sure ha your skeches of volage and curren share he same ime axis! v() v() A A 1 f (a) Sinusoidal waveform 1 f (b) Triangular waveform Lab Procedures: Consruc he circui of Figure 1, using R = 100W and C = 1mF. Use channel 1 of your oscilloscope o measure he resisor volage difference, and channel 2 of your oscilloscope o measure he capacior volage difference. Use channel 1 of your waveform generaor (W1) o apply he volage vin() in Figure 1. Se up a mah channel o calculae he curren hrough he capacior per equaion (2) in he Oher produc and company names menioned may be rademarks of heir respecive owners. Page 39 of 58
prelab 4. Se he oscilloscope measuremens o provide a leas he ampliude of each of he hree displayed waveforms. 1. Apply a sinusoidal inpu volage wih frequency = 1kHz, ampliude = 2V, and offse = 0V o he circui of Figure 1. Use your oscilloscope o display he daa lised above (waveforms corresponding o C1, C2, and M1; measuremen window displaying ampliudes of C1, C2, and M1). Record he image of he oscilloscope window, showing he waveforms and heir measured ampliudes. 2. Apply a sinusoidal inpu volage wih frequency = 2 khz, ampliude = 2V, and offse = 0V o he circui of Figure 1. Use your oscilloscope o display he daa lised above (waveforms corresponding o C1, C2, and M1; measuremen window displaying ampliudes of C1, C2, and M1). Record he image of he oscilloscope window, showing he waveforms and heir measured ampliudes. 3. Apply a riangular inpu volage wih frequency = 100 Hz, ampliude = 4V, and offse = 0V o he circui of Figure 1. Use your oscilloscope o display he daa lised above (waveforms corresponding o C1, C2, and M1; measuremen window displaying ampliudes of C1, C2, and M1). Record he image of he oscilloscope window, showing he waveforms and heir measured ampliudes. 4. Demonsrae operaion of your circui o he Teaching Assisan. Have he TA iniial he appropriae page(s) of your lab noebook and he lab checklis. Poslab Exercises: For he hree cases in he lab procedures (1kHz sinusoid, 2kHz sinusoid, 100Hz riangular wave), use your prelab resuls o skech he expeced capacior curren waveforms corresponding o he capacior volage waveforms you measured in he lab procedures. Commen briefly on he agreemen beween he measured and expeced capacior currens for each of hese cases. In your commens, be sure o include a quaniaive comparison (including percen difference) beween he expeced and measured ampliudes of he capacior 4 Deailed insrucions for doing his are provided in Appendix A. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 40 of 58
Real Analog Chaper 6: Lab Workshees 6.3.1: Capacior Volagecurren Relaions (35 poins oal) 1. Aach o his workshee he skeches of he capacior volage and curren for boh sinusoidal and riangular inpus. (6 ps) 2. Aach o his workshee an image of he oscilloscope window, showing he capacior volage and curren waveforms and he measured ampliudes of he waveforms for a 1kHz sinusoidal inpu. (8 ps) 3. Aach o his workshee an image of he oscilloscope window, showing he capacior volage and curren waveforms and he measured ampliudes of he waveforms for a 2kHz sinusoidal inpu. (8 ps) 4. Aach o his workshee an image of he oscilloscope window, showing he capacior volage and curren waveforms and he measured ampliudes of he waveforms for a 100Hz riangular inpu. (8 ps) 5. DEMO: Have a eaching assisan iniial his shee, indicaing ha hey have observed your circui s operaion for he riangular inpu. (5 ps) TA Iniials: Oher produc and company names menioned may be rademarks of heir respecive owners. Page 41 of 58
Real Analog Chaper 6: Lab Projecs 6.3.2: Leakage Currens and Elecrolyic Capaciors Volagecurren relaionships for ideal capaciors do no always adequaely explain measured capacior behavior. In his assignmen, we will focus on he effecs of leakage currens on capacior behavior. As we saw in our discussion of nonideal capaciors in secion 6.3 of he ex, models for realisic capaciors ofen include a resisor in parallel wih an ideal capacior; his resisor allows us o model leakage currens, which explain among oher effecs he inabiliy of a capacior o hold a charge indefiniely, even if he capacior erminals are opencircuied. An imporan effec of leakage currens is in he case of elecrolyic capaciors. These capaciors are aracive in many cases, since a relaively large capaciance can be provided in a small package. However, one mus be aware, when using elecrolyic capaciors, ha heir leakage currens can be significan and ha hey are no symmeric relaive o he capacior s polariy. Thus, reversing he polariy of he capacior in a circui can aler he behavior of he capacior his makes elecrolyic capaciors undesirable in some applicaions (such as filering) in which he behavior of he capacior should be independen of he polariy of he volage applied o he capacior. Before beginning his lab, you should be able o: Sae volagecurren relaionships for capaciors in boh differenial and inegral form Define he ime consan of an exponenial waveform Use he Analog Discovery o apply and measure imevarying waveforms (Lab 6.2.1) Afer compleing his lab, you should be able o: Model a nonideal capacior as an ideal capacior in parallel wih a resisor Idenify some effecs of nonideal capaciors Describe how he polariy of an elecrolyic capacior affecs he capacior s leakage curren. This lab exercise requires: Analog Discovery module Digilen Analog Pars Ki Symbol Key: Demonsrae circui operaion o eaching assisan; eaching assisan should iniial lab noebook and grade shee, indicaing ha circui operaion is accepable. Analysis; include principle resuls of analysis in laboraory repor. General Discussion: Numerical simulaion (using PSPICE or MATLAB as indicaed); include resuls of MATLAB numerical analysis and/or simulaion in laboraory repor. Record daa in your lab noebook. In large par, his lab will be concerned wih elecrolyic capaciors. Elecrolyic capaciors are polarized ha is, one of heir erminals is inended o always be a a higher volage han he oher. The erminal which is inended o be a he higher volage is called he anode, while he erminal which is o be a he lower polariy is he cahode. A symbol for an elecrolyic capacior is shown in Figure 1 he cahode side is indicaed as a curved line. Physically, elecrolyic capaciors are readily idenifiable: he lead conneced o he anode is a longer wire han Oher produc and company names menioned may be rademarks of heir respecive owners. Page 42 of 58
ha of he cahode, and (if he capacior is physically large enough) a bar is prined on he cahode side of he capacior 5. i() v() Anode C Cahode Figure 1. Elecrolyic capacior circui symbol. Elecrolyic capaciors are desirable in ha heir capaciance can be large relaive o heir volume. However, hey also have some undesirable qualiies. Chief among hese is ha hey can fail raher specacularly if he cahode volage is significanly higher han he anode volage for an exended period of ime. More suble drawbacks include he fac ha leakage currens can be large if he polariy of he capacior is reversed 6. I is his laer characerisic ha we will explore in his assignmen. Cauion: In order o explore leakage effecs, we will be applying volages wih he opposie polariy as required by he elecrolyic capacior. Due o he volage levels we will use, i is unlikely ha we will cause a failure of he capacior. However, i is recommended ha you wear eye proecion while doing his lab assignmen. In his lab assignmen, we will measure he volage across an elecrolyic capacior for boh of he cases shown in Fig. 2. In Fig. 2(a), he polariy of he capacior is correc; he anode is always a he higher volage. In Fig. 2(b), he polariy of he capacior is reversed he cahode is now a he higher volage. We will use a swich o change he volage applied o he capacior our swich will be implemened simply by unplugging he posiive volage erminal of our power supply from he res of he circui. The resisor R in he circui of Fig. 2 simply limis he amoun of curren he capacior demands when i is being iniially charged. Capaciors require a large amoun of curren o charge rapidly; wihou he resisor, he capacior will aemp o draw more curren from he power supply han is available. 5 In addiion, he bar may have a negaive sign prined on i, which furher indicaes ha he cahode is o be a he lower (or negaive) volage. 6 The nonsymmery of he leakage currens relaive o capacior volage makes elecrolyic capaciors poor choices for filer circuis. Filer circuis ypically require heir operaion o be idenical for posiive and negaive volage inpus, unless special biasing schemes are inroduced o ensure ha he volage polariy does no change. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 43 of 58
=0 R =0 R 5V C v C () 5V C v C () (a) Correc polariy (b) Reversed polariy. Figure 2. Capacior configuraions used in his lab. Prelab: None Lab Procedures: Using R = 100Ω and C = 10µF, implemen he circui of Figure 2(a). (Recall ha, in Figure 2(a), he anode is a he higher volage. Thus, he capacior erminal wih he longer lead is conneced o he resisor R and he shorer lead is conneced o ground.) Use V o apply he 5V supply. i. Use channel 1 of your oscilloscope o measure he volage across he capacior, vc(). We will be monioring he amoun of ime required for he capacior o discharge once we open he swich in Figure 2(a); his will ake a relaively long ime, so se he ime scale on your oscilloscope o 5 s/div. Se he verical scale of your scope o 1 V/div, wih a 2V offse. ii. Turn on he power supply and click o sar acquisiion of daa. The oscilloscope should indicae a 5V volage across he capacior. iii. Open he swich in Figure 2(a) by unplugging he power supply erminal from he circui. (Simply pull he V connecor ou of he breadboard.) The capacior volage displayed on he oscilloscope screen should decay exponenially. Measure he ime consan of he waveform 7. Record he image of he oscilloscope window, showing he waveform. iv. Demonsrae operaion of your circui o he Teaching Assisan. Have he TA iniial he appropriae page(s) of your lab noebook and he lab checklis. Sill using R = 100Ω and C = 10µF, implemen he circui of Figure 2(b). (In Figure 2(b), he cahode is a he higher volage. Thus, he capacior erminal wih he shorer lead is conneced o he resisor R and he longer lead is conneced o ground.) Use V o apply he 5V supply. Noe ha his circui can be easily creaed from he circui of Figure 2(a) by removing he capacior, reversing he leads, and replacing i again. i. Measure he volage across he capacior, vc(), as in par 1 Turn on he power supply and click across he capacior. o sar acquisiion of daa. The oscilloscope should indicae a 5V volage ii. Open he swich in Figure 2(b) by unplugging he power supply erminal from he circui. (Simply pull he V connecor ou of he breadboard.) The capacior volage displayed on he oscilloscope screen should decay approximaely exponenially. Measure he ime consan of he waveform. Record he image of he oscilloscope window, showing he waveform. 7 Recall ha he ime consan is he amoun of ime required for an exponenial waveform o decay o 36.8% of is iniial value. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 44 of 58
iii. Demonsrae operaion of your circui o he Teaching Assisan. Have he TA iniial he appropriae page(s) of your lab noebook and he lab checklis. In he circuis of Figure 2, here is no way for he capacior s volage o decay he charge difference on he plaes canno leak away, since here is (in our model, anyway) no pah for he charge o ge from one plae o he oher. The circuis of Figure 2 are no realisic, based on he capacior behavior we observed in pars 1 and 2 above. We mus modify how we hink abou capaciors behavior in order o explain our daa! In secion 6.3 of he exbook, we modeled nonideal capaciors as a resisance in parallel wih an ideal capacior. Using his model, he circuis of Figure 2 can be modified o become Figure 3 below. Now here is a pah he resisor RC which allows he capacior volage o decay afer he swich opens, allowing us o explain our previous daa! =0 R Nonideal capacior 5V R C C v C () Figure 3. Circui of Figures 2, wih nonideal capacior. As we observed in pars 1 and 2 above, he leakage rae in elecrolyic capaciors changes, based on he polariy of he capacior volage. In essence, his means ha he capacior resisance, RC, in he model of Figure 3 depends on he polariy of he capacior volage 8! To obain an idea as o he variaion in he capacior resisance when he capacior polariy is reversed, le s normalize he ime consans we measured in pars 1 and 2 above. To do his, simply divide he ime consan by he capaciance value: R C τ C Eq. 1 Poslab Exercises: As we will see laer, he unis of equaion (1) are consisen. i. Tabulae he resuls you obained in pars 1, 2, and 3 above for each polariy, lis he measured ime consan, and he esimaed capacior resisance as deermined by equaion (1). Briefly commen on he magniude of hese resisances and he differences beween he capacior resisances for he wo cases. (Include a percen change in resisance induced by changing he capacior polariy.) Suppose ha we modify he circui of Figure 2(a) so ha here is a pah for he capacior o dissipae is volage afer he swich opens, as shown in Figure 4. Based on he capacior resisances you esimaed in par (c) of he lab procedures, how large would he resisor R need o be in order for he capacior resisance RC o change he rae a which he capacior volage dissipaes by abou 10%? 8 This makes modeling elecrolyic capaciors whose volage changes polariy edious, o say he leas. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 45 of 58
(E.g. how large would R need o be before he measured ime consan of he capacior volage decay changes by 10% when he resisance RC is included?) Hin: he resisances R and RC are in parallel. They can be combined o a single equivalen resisance seen by he capacior. =0 Nonideal capacior V S R R C C v C () Figure 4. Circui wih exernal resisor conneced across nonideal capacior. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 46 of 58
Real Analog Chaper 6: Lab Workshees 6.3.2: Elecrolyic Capacior Leakage Currens (35 poins oal) 1. Aach o his workshee an image of he oscilloscope window, showing he capacior volage, resuling from opening he swich in Fig. 2(a). (5 ps) 2. DEMO: Have a eaching assisan iniial his shee, indicaing ha hey have observed your circui s operaion when he elecrolyic capacior is conneced wih he correc polariy. (5 ps) TA Iniials: 3. Aach o his workshee an image of he oscilloscope window, showing he capacior volage, resuling from opening he swich in Figure 2(b). In he space below, provide your esimae of he ime consan of he circui. (5 ps) 4. DEMO: Have a eaching assisan iniial his shee, indicaing ha hey have observed your circui s operaion when he elecrolyic capacior is conneced in reversed polariy. (5 ps) TA Iniials: 5. In he space below, provide a able giving your esimaed ime consans and he calculaed capacior resisance for boh capacior polariies. Commen on he differences beween he wo cases, including a percen change in resisance. (8 ps) 6. In he space below, provide your esimae of he resisance required in he circui of Figure 4 which changes he ime consan of he capacior by 10%. (7 ps) Oher produc and company names menioned may be rademarks of heir respecive owners. Page 47 of 58
Real Analog Chaper 6: Lab Projecs 6.4.1: Inducor Volagecurren Relaions In his assignmen, we will measure he relaionship beween he volage difference across a capacior and he curren passing hrough i. We will apply several ypes of imevarying signals o a series combinaion of a resisor and a capacior. The volage difference across he resisor, in conjuncion wih Ohm s law, will provide an esimae of he curren hrough he capacior. This curren can be relaed o he volage difference across he capacior. The resuls of our volagecurren measuremens will be compared o analyical expecaions. Before beginning his lab, you should be able o: Sae volagecurren relaionships for inducors in boh differenial and inegral form Apply he inducor volagecurren relaions o calculae a inducor s volage from is curren and viceversa Use he Analog Discovery s arbirary waveform generaor and oscilloscope o apply and measure imevarying waveforms (Lab 6.2.1) Afer compleing his lab, you should be able o: Use he Analog Discovery oscilloscope s mah funcion o calculae he curren hrough a known resisor from he measured volage difference. Expor daa acquired by he Analog Discovery o files for posprocessing by oher programs Verify a inducor s volagecurren relaions using measured daa This lab exercise requires: Analog Discovery module Digilen Analog Pars Ki Symbol Key: Demonsrae circui operaion o eaching assisan; eaching assisan should iniial lab noebook and grade shee, indicaing ha circui operaion is accepable. Analysis; include principle resuls of analysis in laboraory repor. General Discussion: Numerical simulaion (using PSPICE or MATLAB as indicaed); include resuls of MATLAB numerical analysis and/or simulaion in laboraory repor. Record daa in your lab noebook. We will use he circui of Figure 1 in his lab assignmen. Boh he volage difference across he inducor and he resisor (vl() and vr()) will be measured. From his daa, we can readily compare he volage across he inducor wih he curren hrough he inducor. Since he volage across he resisor is measured, we can readily infer he curren hrough he resisor via Ohm s law: i R () = v R() R Eq. 1 The resisor and inducor are in series, so he curren hrough he inducor is he same as he curren hrough he resisor, so: i L () = v R () R Eq. 2 Oher produc and company names menioned may be rademarks of heir respecive owners. Page 48 of 58
Since we are also measuring he volage difference across he inducor, vl(), we can readily compare hese parameers wih our expecaions based on our mahemaical models of he capacior volagecurren relaionships. i R () v R () R i L () v IN () L v L () Figure 1. Series RL circui. Prelab: In his lab, we will apply sinusoidal signals o he inducor of Fig. 1. Mahemaically, he form of he inducor curren will be: i L () = Acos(2πf) Eq. 3 Where A is he ampliude of he sinusoid (in vols) and f is he frequency (in Hz). The waveform is shown graphically in Fig. 2. For he circui of Fig. 1, use he inducor volagecurren relaions o calculae he inducor volage resuling from applicaion of he volage of equaion (3). Your resuls may be dependen up on he parameers A, f, and L. i() 1 f A Lab Procedures: Figure 2. Basic waveform used in his lab. Consruc he circui of Fig. 1 wih L = 1mH and R = 100Ω. Use channel 1 of your oscilloscope o measure he resisor volage difference, and channel 2 of your oscilloscope o measure he inducor volage difference. Use channel 1 of your waveform generaor (W1) o apply he volage vin() in Fig. 1. Se up a mah channel o calculae he curren hrough he inducor per equaion (2) in he pre Oher produc and company names menioned may be rademarks of heir respecive owners. Page 49 of 58
lab 9. Se he oscilloscope measuremens o provide a leas he ampliude of each of he hree displayed waveforms. 1. Apply a sinusoidal inpu volage wih frequency = 1kHz, ampliude = 2V, and offse = 0V o he circui of Fig. 1. Use your oscilloscope o display he daa lised above (waveforms corresponding o C1, C2, and M1; measuremen window displaying ampliudes of C1, C2, and M1). Expor he daa in he oscilloscope ime window o a.csv file for laer processing. 2. Apply a sinusoidal inpu volage wih frequency = 2 khz, ampliude = 2V, and offse = 0V o he circui of Fig. 1. Use your oscilloscope o display he daa lised above (waveforms corresponding o C1, C2, and M1; measuremen window displaying ampliudes of C1, C2, and M1). Expor he daa in he oscilloscope ime window o a.csv file for laer processing. 3. Demonsrae operaion of your circui o he Teaching Assisan. Have he TA iniial he appropriae page(s) of your lab noebook and he lab checklis. Poslab Exercises: Impor he daa acquired in he lab procedures ino your favorie numerical analysis sofware (e.g. Excel, Malab, Ocave, ec.). Use he sofware and he resuls of your prelab analysis o calculae he expeced inducor volage waveforms corresponding o he inducor curren waveforms you measured in he lab procedures. Use he sofware o overlay plos of he expeced and measured inducor volages for each of he cases esed in he lab procedures. Commen briefly on he agreemen beween he measured and expeced inducor volages for each of he cases. In your commens, be sure o include a quaniaive comparison (including percen difference) beween he expeced and measured ampliudes of he inducor volages. Appendix A: Mah channel o calculae curren from resisor s volage The analog discovery provides capabiliies for performing mahemaical operaions on he displayed waveforms and displaying he resul. Essenially, here are wo basic ypes of mahemaical operaions which can be performed: Simple and Cusom. The simple mah operaions consis of addiion, subracion, or muliplicaion of he wo channels. The cusom operaions are much more wideranging. In order o deermine he resisor curren, we wan o divide he resisor volage by a consan (he resisance value), so we will creae a cusom mah channel. To do his, follow he seps below: 1. Click on Add Channel. Selec Add Mahemaic Channel from he resuling dropdown menu and choose Cusom. 2. A cusom mah funcion window will open, as shown below. Type he desired mah funcion (ypically a funcion of he scope channels, C1 and C2) in he ex box in his window or use he buons in he window o creae he funcion. We are using channel 1 (C1) o measure he resisor volage. The curren hrough he resisor is simply he resisor volage divided by he resisance value (100Ω), so our funcion is: C1/100, also shown in he Figure below. Click OK o display he funcion in he main window. 9 Deailed insrucions for doing his are provided in Appendix A. Oher produc and company names menioned may be rademarks of heir respecive owners. Page 50 of 58
3. The properies of he mah channel display can be adjused using he channel s conrol box, jus as any wih any oher channel displayed by he scope. A ypical conrol box is shown below: 4. The unis of our mah channel are amperes. I is nice o have he displayed unis agree wih he acual unis of he measuremen. To change he unis, click on he Unis icon on he conrol box and selec Unis from he resuling dropdown menu. Vols, will ypically be he defaul uni; if you wan he verical axis in amps, click he dropdown arrow icon and selec A from he resuling menu 10. 10 Choices of unis are vols (V), amps (A), and was (W). Oher produc and company names menioned may be rademarks of heir respecive owners. Page 51 of 58