ECE Spring Prof. David R. Jackson ECE Dept.

Transcription

1 ECE 6341 Sprng 016 Prof. Da R. Jackon ECE Dept. Note Note

2 Oerew n th et of note we ere the SD formlaton ng a more mathematcal, bt more general, approach (we rectly Forer tranform Maxwell eqaton). Th allow for all poble type of orce (horontal, ertcal, electrc, an magnetc) to be treate n one eraton.

3 General SD Metho Start wth Ampere law: = = H J jωε E = + t ˆ where ˆ ˆ t = x + y x y Ame a D patal tranform: ( ) ˆ( ) = xˆ jk + y jk t x y ( ˆ ˆ ) x yky = j xk + = = jk t jk ˆ t 3

4 General SD Metho (cont.) Hence we hae jk ˆ ˆ t + H = J + jωε E Next, repreent the fel a Note that ˆ ˆ = ˆ ˆ ˆ = ˆ ˆ ˆ = ˆ H = H ˆ + H ˆ t ˆ = ˆ H ˆ + ˆ ˆ ( ) ( H ) y φ û x Take the ˆ, ˆ, ˆ component of the tranforme Ampere eqaton: 4

5 General SD Metho (cont.) ) ˆ jk H = J + jωε E t H ˆ ) = J + jωε E H ˆ ) jk H + = J + jωε E t Examne the fel: (gnore eqaton) ( E,, ) H E ˆ jkth = J + jωε E (1) H = J + jωε E () 5

6 Fel We wh to elmnate E from Eq. (1). To o th, e Faraay law: = E M jωµ H jk ˆ t + ˆ E = M jωµ H Take the ˆ component of the tranforme Faraay Law: Recall: ˆ ˆ = ˆ ˆ ˆ = ˆ ˆ ˆ = ˆ E jk E + = M jωµ H t (3) 6

7 Fel (cont.) Sbttte E from (1) nto (3) to elmnate E jk H = J + jωε E t (1) E jk E + = M jωµ H t (3) ( ) E jk J jk H M j H jωε 1 t t + = ωµ or E k k t t + H + jωµ H = M + J jωε ωε 7

8 Fel (cont.) Note that kt 1 + jωµ = ω µε jωε jωε 1 = jωε ( kt ) ( kt k ) 1 = jωε k = jωε ( k kt ) Hence E k kt H = M + J (4) jωε ωε 8

9 Fel (cont.) Eqaton () an (4) are the fnal fel moelng eqaton: H = J jωε E E k k = + + ωε jωε t M J H 9

10 Fel (cont.) Defne N moelng eqaton: ( ) (,, ) x y V E k k ( ) (,, ) x y H k k We then hae H = J jωε E E k k = + + ωε jωε t M J H = jωε V J k k jωε ωε V t = + M + J 10

11 Telegrapher Eqaton L + C Allow for trbte orce + = L + t 0 o ( ) ( ) = L t 11

12 Telegrapher Eqaton (cont.) Hence, n the phaor oman, V = jωl + V Alo, = ( C ) + 0 o = C t + t Hence, we hae = jωcv + 1

13 Telegrapher Eqaton (cont.) Compare fel eqaton for fel wth TL eqaton: jωε V J = = jωcv + k k jωε ωε V t = + M + J V = jωl + V 13

14 Telegrapher Eqaton (cont.) We then make the followng entfcaton: C ωl = = ε k ωε Hence TL k k = ω LC = ω ε = k ωε Z L k 1 k C ω ε ε ωε TL 0 = = = o k Z TL TL 0 = = k k ωε 14

15 Sorce: For the orce we hae, for the cae: = J k V = M + J ωε t 15

16 Sorce: (cont.) Specal cae: planar horontal rface crrent orce: Ame ( ) = ( ) J xy,, J xy, δ ( ) ( ) = ( ) M xy,, M xy, δ ( ) Then we hae ( ) = J = J δ ( ) V = M = M δ Thee correpon to lmpe crrent an oltage orce: V = J = M lmpe parallel crrent orce lmpe ere oltage orce 16

17 Sorce: (cont.) For a ertcal electrc crrent: Ame ( ) J xy,, = J ( xy, ) δ ( ) planar ertcal crrent trbton = 0 17

18 Sorce: (cont.) ( ) J xy,, = J ( xy, ) δ ( ) kt kt V = J ( kx, ky) = J ( kx, ky) δ ωε ωε ( ) Th correpon to a lmpe ere oltage orce: kt V = J ( kx, ky ) ωε 18

19 Sorce: (cont.) Specal cae: ertcal electrc pole ( ) ( ) ( ) J xy,, = δ xδ yδ( ) ( ) ( ) J ( xy, ) = δ xδ y J ( k, k ) = 1 x y Hence V k ωε t = 19

20 Fel Ue alty: E H J M H E M J ε µ H = J jωε E E k k = + + ωε jωε t M J H E M jωµ H H k k =+ + ωµ jωµ = t J M E 0

21 (cont.) Defne: ( ) (,, ) x y V E k k ( ) (,, ) x y H k k V jωµ M = k k = V + J + M jωµ ωµ t V = jωl + V = jωcv + We then entfy L ωc = = µ k ωµ k Z TL 0 = = k ωµ k 1

22 (cont.) For the orce, we hae V = M k J = + M ωµ t Specal cae of planar horontal rface crrent: V = M = + J lmpe ere oltage orce lmpe parallel crrent orce

23 Sorce: (cont.) For a ertcal magnetc crrent: Ame ( ) M xy,, = M ( xy, ) δ ( ) Th correpon to a lmpe parallel crrent orce: Then we hae k = M k k ω (, ) t x y 3

24 Sorce: (cont.) Specal cae: ertcal magnetc pole ( ) ( ) M xy,, = δ( x) δ yδ( ) ( ) M ( xy, ) = δ( x) δ y M k, k = 1 ( ) x y Hence k ω t = 4

25 Smmary Relt for 3D (olmetrc) orce V V = E = H = E = H V V Horontal = J = M = + J = M V Vertcal k J ωε t = k M ωµ t = Dtrbte orce: ether parallel crrent orce or ere oltage orce 5

26 Smmary Relt for D (planar) orce V V = E = H = E = H V V Horontal = J = M = + J = M V Vertcal k J ωε t = k M ωµ t = Lmpe orce: ether parallel crrent orce or ere oltage orce 6

27 Smmary (cont.) N Moel / / + V / / + V / + - / V ( ) = ( ) ( ) = ( ) V V / / / / / / ( ) = ( ) ( ) = ( ) V V V / / / V / / / 7

28 Smmary (cont.) Mchalk fncton / + / V / + / V 1 [A] [V] 8