Chapter 6 Inductance, Capacitance, and Mutual Inductance

Transcription

1 Chapter 6 Inductance Capactance and Mutual Inductance 6. The nductor 6. The capactor 6.3 Seres-parallel combnatons of nductance and capactance 6.4 Mutual nductance 6.5 Closer look at mutual nductance

2 Oerew In adon to oltage sources current sources resstors here we wll dscuss the remanng types of basc elements: nductors capactors. Inductors and capactors cannot generate nor dsspate but store energy. Ther current-oltage (-) relatons nole wth ntegral and derate of tme thus more complcated than resstors.

3 Key ponts Why the - relaton of an nductor s Why the - relaton of a capactor s C d d?? Why the energes stored n an nductor and a capactor are: w C respectely? 3

4 Secton 6. The Inductor. Physcs. - relaton and behaors 3. Power and energy 4

5 Fundamentals An nductor of nductance s symbolzed by a solenodal col. Typcal nductance ranges from 0 H to 0 mh. The - relaton of an nductor (under the passe sgn conenton) s: d 5

6 Physcs of self-nductance () Consder an N -turn col C carryng current I. The resultng magnetc feld (Bot- Saart law) wll pass through C tself causng a B ( ) r N B S B ( r ) ds ( N I r ) flux lnkage where P N P s the permeance. P N I. I 6

7 Physcs of self-nductance () The rato of flux lnkage to the drng current s defned as the self nductance of the loop: N P I whch descrbes how easy a col current can ntroduce magnetc flux oer the col tself. 7

8 Examples Solenodal & torodal cols: ~ cm = 70 H = 36 H RG59/U coaxal cable: = 35 nh/m. 8

9 The - relaton Faraday s law states that the electromote force (emf n unts of olt) nduced on a loop equals the tme derate of the magnetc flux lnkage : d d d. Note: The emf of a loop s a non-conserate force that can dre current flowng along the loop. In contrast the current-drng force due to electrc charges s conserate. 9

10 Behaors of nductors d DC-current: nductor behaes as a short crcut. Current cannot change nstantaneously n an nductor otherwse nfnte oltage wll arse. Change of nductor current s the ntegral of oltage durng the same tme nteral: ( t) ( t t 0) t 0 ( ) d. 0

11 Inducte effect s eerywhere! Nearly all electrc crcuts hae currents flowng through conductng wres. Snce t s dffcult to sheld magnetc felds nducte effect occurs een we do not purposely add an nductor nto the crcut.

12 Example 6.: Inductor dren by a current pulse 0 t 0 t) t 0te ( 5 t 0 The nductor oltage s: d 0 t 0 ( t) 5t e ( 5t) t 0 Inductor oltage can ump! Memory-less n steady state.

13 Power & energy () Consder an nductor of nductance. The nstantaneous power n the nductor s: p Assume there s no ntal current (.e. no ntal energy) (t =0)=0 w(t =0)=0. We are nterested n the energy W when the current ncreases from zero to I wth arbtrary (t). d. 3

14 Power & energy () p dw d dw d W dw 0 0 I d W I 0 I.e. w How the current changes wth tme doesn t matter. It s the fnal current I determnng the fnal energy. Inductor stores magnetc energy when there s nonzero current. 4

15 Example 6.3: Inductor dren by a current pulse t < 0. p>0 w chargng. t > 0. p<0 w dschargng. In steady state (t ) 0 0 p0 w0 (no energy). 5

16 Example 6.3: Inductor dren by a oltage pulse ( t) t ( ) d (0) 0 p>0 w always chargng. In steady state (t) A 0 p0 w00 mj (sustaned current and constant energy). Wth memory n steady state. 6

17 Secton 6. The Capactor. Physcs. - relaton and behaors 3. Power and energy 7

18 Fundamentals A capactor of capactance C s symbolzed by a parallel-plate. Typcal capactance C ranges from 0 pf to 470 F. The - relaton of an capactor (under the passe sgn conenton) s: C d. 8

19 Physcs of capactance () If we apply a oltage V between two solated conductors charge Q wll be properly dstrbuted oer the conductng surfaces such that the resultng electrc feld E(r ) B E(r ) E( r) dl satsfes: V E whch s ald for any ntegral path lnkng the two conductng surfaces. 9

20 Physcs of capactance () If V Q Q whle the spatal dstrbuton of charge remans such that E( r) V E( r) V E( r ) dl V The rato of the deposted charge to the bas oltage s defned as the capactance of the conductng par: Q C V descrbng how easy a bas oltage can depost charge on the conductng par.. 0

21 Examples Ceramc dsc & electrolytc: RG59/U coaxal cable: C = 53 pf/m.

22 The - relaton From the defnton of capactance: Q d d d C q( t) C( t) q C C. V Note: Charge cannot flow through the delectrc between the conductors. Howeer a tme-aryng oltage causes a tme-aryng electrc feld that can slghtly dsplace the delectrc bound charge. It s the tme-aryng bound charge contrbutng to the dsplacement current.

23 Polarzaton charge 3

24 Behaors of capactors DC-oltage: capactor behaes as an open crcut. C d Voltage cannot change nstantaneously n an capactor otherwse nfnte current wll arse. Change of capactor oltage s the ntegral of current durng the same tme nteral: t ( t) ( t0) ( ) d. C t0 4

25 Capacte effect s eerywhere! A Metal-Oxde-Semconductor (MOS) transstor has three conductng termnals (Gate Source Dran) separated by a delectrc layer wth one another. Capacte effect occurs een we do not purposely add a capactor nto the crcut. (nfo.tuwen.ac.at) charges 5

26 Power & energy () Consder a capactor of capactance C. The nstantaneous power n the capactor s: d p C. Assume there s no ntal oltage (.e. no ntal energy) (t =0)=0 w(t =0)=0. We are nterested n the energy W when the oltage ncreases from zero to V wth arbtrary (t). 6

27 Power & energy () p dw C d dw C d W dw C 0 0 V d W C V 0 CV.e. w C How the oltage ncreases wth tme doesn t matter. It s the fnal oltage V determnng the fnal energy. Capactor stores electrc energy when there s nonzero oltage. 7

28 Example 6.4: Capactor dren by a oltage pulse Capactor current can ump! t < p>0 w chargng. t > p<0 w dschargng. In steady state (t) 0 0 p0 w0 (no energy). Memory-less n steady state. 8

29 Secton 6.3 Seres-Parallel Combnatons. Inductors n seres-parallel. Capactors n seres-parallel 9

30 30 Inductors n seres n eq eq d d d d d 3 3 3

31 3 Inductors n parallel n eq t t eq t t t t t t t d t d t d t d ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

32 3 Capactors n seres n eq C C = ) ( ) ( t d C t d C t t eq t t

33 33 Capactors n parallel n C eq C = 3 3 d C d C eq

34 Secton Mutual Inductance. Physcs. - relaton and dot conenton 3. Energy 34

35 Fundamentals Mutual nductance M s a crcut parameter between two magnetcally coupled cols. The alue of M satsfes M k where 0 k s the magnetc couplng coeffcent. The emf nduced n Col due to tme-aryng current n Col s proportonal to M d. 35

36 The - relaton () Col of N turns s dren by a tme-aryng current whle Col of N turns s open. The flux components lnkng () only Col () both cols and (3) total flux lnkng Col are: B r ) ds P N I B ( r ) ds P N ( I S S P NI P P NI P P P.. 36

37 The - relaton () Faraday s law states that the emf nduced on Col (when remans constant) s: d d d d ( PN ) NNP M. N N One can show that the emf nduced on Col (when remans constant) s: d NNP M d. For nonmagnetc meda (e.g. ar slcon plastc) P =P M =M =M= N N P. 37

38 38 Mutual nductance n terms of self-nductance. et. f k M k M P P P P k P P P P P P P N N P P P P N N PP N N P N P N The two self nductances and ther product are: The couplng coeffcent () k=0 f P =0 (.e. no mutual flux) () k= f P =P =0 (.e. = =0 = = no flux leakage).

39 Dot conenton () leaes the dot of the + polarty of s referred the termnal of wthout a dot. The total oltage across s: d d M. M ( t) M ( t ) 39

40 Dot conenton () M ( t) enters the dot of the + polarty of s referred the termnal of wth a dot. The total oltage across s: d d M. M ( t ) 40

41 Example 6.6: Wrte a mesh current equaton d d ( 4 H) (0 )( ) (5 )( g ) (8 H) ( g ) 0 Self-nductance passe sgn conenton Mutual-nductance g enters the dot of 6-H nductor 4

42 Example 6.6: Steady-state analyss In steady state (t ) nductors are short the 3 resstors are n parallel (R eq =3.75 ). et =0. () =(6A)(3.75 )=60 V. () = (60V)/(5) =A =(6-)=4 A (not zero!). (3) ' =(60V)/(0)=3 A ' =(+3)=5 A. 4 A 60 V ' A 3 A ' 4

43 Energy of mutual nductance () Assume =0 ntally. Fx =0 whle ncreasng from 0 to some constant I. The energy stored n becomes: W I 0 d I. 43

44 Energy of mutual nductance () Now fx =I whle ncreasng from 0 to I. Durng ths perod emf s wll be nduced n loops and due to the tme-aryng. The total power of the two nductors s: d d p( t) I M ( t). An extra energy of W +W s stored n the par: W W I M I 0 d I 0 d MI I I. 44

45 Energy of mutual nductance (3) The entre process contrbutes to a total energy W tot I MI I I for the two-nductor system. W tot only depends on the fnal currents I I [ndependent of the tme eoluton of (t) (t)]. 45

46 Key ponts Why the - relaton of an nductor s Why the - relaton of a capactor s C d d?? Why the energes stored n an nductor and a capactor are: w C respectely? 46