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
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.
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
Secton 6. The Inductor. Physcs. - relaton and behaors 3. Power and energy 4
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
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
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
Examples Solenodal & torodal cols: ~ cm = 70 H = 36 H RG59/U coaxal cable: = 35 nh/m. 8
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
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
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.
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.
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
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
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
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
Secton 6. The Capactor. Physcs. - relaton and behaors 3. Power and energy 7
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
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
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
Examples Ceramc dsc & electrolytc: RG59/U coaxal cable: C = 53 pf/m.
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.
Polarzaton charge 3
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
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
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
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
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
Secton 6.3 Seres-Parallel Combnatons. Inductors n seres-parallel. Capactors n seres-parallel 9
30 Inductors n seres n eq eq d d d d d 3 3 3
3 Inductors n parallel n eq t t eq t t t t t t t d t d t d t d 0 3 0 3 3 0 0 3 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0
3 Capactors n seres n eq C C = ) ( ) ( 0 0 3 3 0 0 t d C t d C t t eq t t
33 Capactors n parallel n C eq C = 3 3 d C d C eq
Secton 6.4 6.5 Mutual Inductance. Physcs. - relaton and dot conenton 3. Energy 34
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
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
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 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).
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
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
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
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
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
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
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
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