Energy Storage Elements: Capacitors and Inductors

Transcription

1 CHAPTER 6 Energy Storage Elements: Capactors and Inductors To ths pont n our study of electronc crcuts, tme has not been mportant. The analyss and desgns we hae performed so far hae been statc, and all crcut responses at a gen tme hae depended only on the crcut nputs at that tme. In ths chapter, we shall ntroduce two mportant passe crcut elements: the capactor and the nductor Introducton and A Mathematcal Fact Capactors and nductors, whch are the electrc and magnetc duals of each other, dffer from resstors n seeral sgnfcant ways. Unlke resstors, whch dsspate energy, capactors and nductors do not dsspate but store energy, whch can be retreed at a later tme. They are called storage elements. Furthermore, ther branch arables do not depend algebracally upon each other. Rather, ther relatons nole temporal derates and ntegrals. Thus, the analyss of crcuts contanng capactors and nductors nole dfferental equatons n tme An mportant mathematcal fact: Gen d f(t) = g(t), dt 77

2 78 6. ENERGY STORAGE ELEMENTS: CAPACITORS AND INDUCTORS 6.2. Capactors A capactor s a passe element desgned to store energy n ts electrc feld. The word capactor s dered from ths element s capacty to store energy When a oltage source (t) s connected across the capactor, the amount of charge stored, represented by q, s drectly proportonal to (t),.e., q(t) = C(t) where C, the constant of proportonalty, s known as the capactance of the capactor. The unt of capactance s the farad (F) n honor of Mchael Faraday. 1 farad = 1 coulomb/olt Crcut symbol for capactor of C farads: C C (a) (b) Snce = dq dt, then the current-oltage relatonshp of the capactor s (6.2) = C d dt. Note that n (6.2), the capactance alue C s constant (tme-narant) and that the current and oltage are both functons of tme (tme-aryng). So, n fact, the full form of (6.2) s (t) = C d dt (t). Hence, the oltage-current relaton s (t) = 1 C t t o (τ)dτ (t o )

3 6.2. CAPACITORS 79 Slope = C 0 d/dt where (t o ) s the oltage across the capactor at tme t o. Note that capactor oltage depends on the past hstory of the capactor current. Hence, the capactor has memory The nstantaneous power delered to the capactor s p(t) = (t) (t) = (C ddt ) (t) (t). The energy stored n the capactor s w(t) = t p(τ)dτ = 1 2 C2 (t). In the aboe calculaton, we assume ( ) = 0, because the capactor was uncharged at t = Typcal alues (a) Capactors are commercally aalable n dfferent alues and types. (b) Typcally, capactors hae alues n the pcofarad (pf) to mcrofarad (µf) range. (c) For comparson, two peces of nsulated wre about an nch long, when twsted together, wll hae a capactance of about 1 pf Two mportant mplcatons of (6.2): (a) A capactor s an open crcut to dc. When the oltage across a capactor s not changng wth tme (.e., dc oltage), ts derate wrt. tme s d dt = 0 and hence the current through the capactor s (t) = C d dt = C 0 = 0.

4 80 6. ENERGY STORAGE ELEMENTS: CAPACITORS AND INDUCTORS (b) The oltage across a capactor cannot jump (change abruptly) Because = C d dt, a dscontnuous change n oltage requres an nfnte current, whch s physcally mpossble. t t Remark: An deal capactor does not dsspate energy. It takes power from the crcut when storng energy n ts feld and returns preously stored energy when delerng power to the crcut. Example If a 10 µf s connected to a oltage source wth (t) = 50 sn 2000t determne the current through the capactor. V Example Determne the oltage across a 2-µF capactor f the current through t s (t) = 6e 3000t ma Assume that the ntal capactor oltage (at tme t = 0) s zero.

5 6.2. CAPACITORS 81 Example Obtan the energy stored n each capactor n the fgure below under dc condtons. 2 mf 2 kω 5 kω 6 ma 3 kω 4 kω 4 mf

6 82 6. ENERGY STORAGE ELEMENTS: CAPACITORS AND INDUCTORS 6.3. Seres and Parallel Capactors We know from resste crcuts that seres-parallel combnaton s a powerful tool for smplfyng crcuts. Ths technque can be extended to seres-parallel connectons of capactors, whch are sometmes encountered. We desre to replace these capactors by a sngle equalent capactor C eq The equalent capactance of N parallel-connected capactors s the sum of the nddual capactance. C eq = C 1 C 2 C N N C 1 C2 C3 CN The equalent capactance of N seres-connected capactors s the the recprocal of the sum of the recprocals of the nddual capactances. 1 = C eq C 1 C 2 C N C 1 C 2 C 3 C N N Example Fnd the C eq. 5 µf 60 µf a 20 µf 6 µf 20 µf C eq b

7 6.4. INDUCTORS Inductors An nductor s a passe element desgned to store energy n ts magnetc feld Inductors fnd numerous applcatons n electronc and power systems. They are used n power supples, transformers, rados, TVs, radars, and electrc motors Crcut symbol of nductor: L L L If a current s allowed to pass through an nductor, the oltage across the nductor s drectly proportonal to the tme rate of change of the current,.e., (6.3) (t) = L d dt (t), where L s the constant of proportonalty called the nductance of the nductor. The unt of nductance s henry (H), named n honor of Joseph Henry. 1 henry equals 1 olt-second per ampere By ntegraton, the current-oltage relaton s (t) = 1 L t where (t o ) s the current at tme t o. t o (τ) dτ (t o ), The nstantaneous power delered to the nductor s p(t) = (t) (t) = (L ddt ) (t) (t)

8 84 6. ENERGY STORAGE ELEMENTS: CAPACITORS AND INDUCTORS Slope = L The energy stored n the nductor s 0 d/dt w(t) = t p(τ) dτ = 1 2 L2 (t) Lke capactors, commercally aalable nductors come n dfferent alues and types. Typcal practcal nductors hae nductance alues rangng from a few mcrohenrys (µh), as n communcaton systems, to tens of henrys (H) as n power systems Two mportant mplcatons of (6.3): (a) An nductor acts lke a short crcut to dc. When the current through an nductor s not changng wth tme (.e., dc current), ts derate wrt. tme s d dt = 0 and hence the oltage across the nductor s (t) = L d dt = L 0 = 0. (b) The current through an nductor cannot change nstantaneously. Ths opposton to the change n current s an mportant property of the nductor. A dscontnuous change n the current through an nductor requres an nfnte oltage, whch s not physcally possble. (a) t Remark: The deal nductor does not dsspate energy. The energy stored n t can be retreed at a later tme. The nductor takes (b) t

9 6.4. INDUCTORS 85 power from the crcut when storng energy and delers power to the crcut when returnng preously stored energy. Example If the current through a 1-mH nductor s (t) = 20 cos 100t ma, fnd the termnal oltage and the energy stored. Example Fnd the current through a 5-H nductor f the oltage across t s { 30t 2, t > 0 (t) = 0, t < 0. In addton, fnd the energy stored wthn 0 < t < 5 s.

10 86 6. ENERGY STORAGE ELEMENTS: CAPACITORS AND INDUCTORS Example The termnal oltage of a 2-H nductor s (t) = 10(1 t) V. Fnd the current flowng through t at t = 4 s and the energy stored n t wthn 0 < t < 4 s. Assume (0) = 2 A. Example Determne C, L and the energy stored n the capactor and nductor n the followng crcut under dc condtons. 1 Ω 5 Ω L 4 Ω 12 V 2 H C 1 F

11 6.5. SERIES AND PARALLEL INDUCTORS 87 Example Determne C, L and the energy stored n the capactor and nductor n the followng crcut under dc condtons. L 6 H 4 A 6 Ω 2 Ω C 4 F 6.5. Seres and Parallel Inductors The equalent nductance of N seres-connected nductors s the sum of the nddual nductances,.e., L eq = L 1 L 2 L N L 2 L 3 L N L N

12 88 6. ENERGY STORAGE ELEMENTS: CAPACITORS AND INDUCTORS The equalent nductance of N parallel nductors s the recprocal of the sum of the recprocals of the nddual nductances,.e., 1 = L eq L 1 L 2 L N :59 AM Page N L 1 L 2 L 3 L N Remark: Note that (a) nductors n seres are combned n exactly the same way as resstors n seres and (b) nductors Chapter 6 Capactors n parallel and Inductors are combned n the same way as resstors n parallel. TABLE 6.1 Important characterstcs of the basc elements. Relaton Resstor (R) Capactor (C) Inductor (L) -: -: p or w: R R p 2 R 2 R 1 C t dt (t 0 ) t 0 d C dt w 1 2 C2 Seres: C eq C 1C 2 R eq R 1 R 2 L eq L 1 L 2 C 1 C 2 Parallel: L eq L 1L 2 R eq R 1R 2 C eq C 1 C 2 R 1 R 2 L 1 L 2 At dc: Same Open crcut Short crcut Crcut arable that cannot change abruptly: Not applcable d L dt 1 L t w 1 2 L2 t 0 dt (t 0 ) Passe sgn conenton s assumed. It s approprate at ths pont to summarze the most mportant characterstcs of the three basc crcut elements we hae studed. The summary s gen n Table 6.1. The wye-delta transformaton dscussed n Secton 2.7 for resstors can be extended to capactors and nductors.

13 6.6. APPLICATIONS: INTEGRATORS AND DIFFERENTIATORS 89 Example Fnd the equalent nductance L eq of the crcut shown below. 4 H 20 H a L eq 7 H 12 H 8 H 10 H b 6.6. Applcatons: Integrators and Dfferentators Capactors and nductors possess the followng three specal propertes that make them ery useful n electrc crcuts: (a) The capacty to store energy makes them useful as temporary oltage or current sources. Thus, they can be used for generatng a large amount of current or oltage for a short perod of tme. (b) Capactors oppose any abrupt change n oltage, whle nductors oppose any abrupt change n current. Ths property makes nductors useful for spark or arc suppresson and for conertng pulsatng dc oltage nto relately smooth dc oltage. (c) Capactors and nductors are frequency senste. Ths property makes them useful for frequency dscrmnaton. The frst two propertes are put to use n dc crcuts, whle the thrd one s taken adantage of n ac crcuts. In ths fnal part of the chapter, we wll consder two applcatons nolng capactors and op amps: ntegrator and dfferentator.

14 90 6. ENERGY STORAGE ELEMENTS: CAPACITORS AND INDUCTORS An ntegrator s an op amp crcut whose output s proportonal to the ntegral of the nput sgnal. We obtan an ntegrator by replacng the feedback resstor R f n the nertng amplfer by a capactor. C C R R a 0 Ths ges whch mples d dt o(t) = 1 RC (t), t o (t) = 1 (τ)dτ o (0). RC 0 To ensure that o (0) = 0, t s always necessary to dscharge the ntegrators capactor pror to the applcaton of a sgnal. In practce, the op amp ntegrator requres a feedback resstor to reduce dc gan and preent saturaton. Care must be taken that the op amp operates wthn the lnear range so that t does not saturate.

15 6.6. APPLICATIONS: INTEGRATORS AND DIFFERENTIATORS A dfferentator s an op amp crcut whose output s proportonal to the dfferentaton of the nput sgnal. We obtan a dfferentator by replacng the nput resstor n the nertng amplfer by a capactor. Ths ges R R C C a 0 o (t) = RC d dt (t). Dfferentator crcuts are electroncally unstable because any electrcal nose wthn the crcut s exaggerated by the dfferentator. For ths reason, the dfferentator crcut aboe s not as useful and popular as the ntegrator. It s seldom used n practce.