ECE Spring Prof. David R. Jackson ECE Dept. Notes 2

Transcription

1 ECE 6345 Sping 05 Po. David R. Jackson ECE Dept. Notes

2 Oveview This set o notes teats cicula polaization, obtained b using a single eed. L W 0 0 W ( 0, 0 ) L

3 Oveview Goals: Find the optimum dimensions o the CP patch Find the input impedance o the CP patch Find the patten (aial-atio) bandwidth Find the impedance bandwidth o CP patch 3

4 mplitude o Patch Cuents L h W (, ) I 0 0 ε w I [ ] Fist Step: Find the patch cuents ( and diections), and elate them to the input impedance o the patch. 4

5 mplitude o Patch Cuents (cont.) h W (, ) I 0 0 L ε w I [ ] -diected cuent mode (,0): -diected cuent mode (0,): J J s s π ˆ sin L ˆ π sin W 5

6 mplitude o Patch Cuents (cont.) The mode (TM 0 ): J nˆ H zˆ H s so J s H H ˆ π sin L To ind E, use H jωε E E z H H π π cos jεε 0 jεε 0 L L 6

7 mplitude o Patch Cuents (cont.) Z V V he h E E Z Z I in z z z in in Fo the (,0) mode we have Z in jh π π 0 cos ωε 0ε L L o Z ωε ε L in 0 π 0 jπ h cos L simila deivation holds o the mode. 7

8 mplitude o Patch Cuents (cont.) The mode (TM 0 ): Z ωε ε W in 0 π 0 jπ h cos W The patch cuent amplitudes can then be witten as: ( ) Z ( ) Z in in whee ( ) 0 ωε ε L π 0 jπ h cos L ( ) 0 cos W ωε ε W π 0 jπ h 8

9 mplitude o Patch Cuents (cont.) ( ) 0 ωε ε L π 0 jπ h cos L ( ) 0 ωε ε W π 0 jπ h cos W ssume L W 0 0 W Then ( ) ( ) ( 0, 0 ) L 9

10 mplitude o Patch Cuents (cont.) Because o the neal equal dimensions and the eed along the diagonal, we also have R R R (The esistance in the cicuit model o eithe mode is the same.) R i esonant input esistance o the mode i, when ecited b itsel (R onl depends on 0, R onl depends on 0 ). We then have: whee Z Z in in R ωε ε 0 R L π 0 jπ h cos L Reminde: The ba denotes impedances that ae nomalized b R (eithe R o R ). 0

11 mplitude o Patch Cuents (cont.) The coeicient can be witten as so R ωε ε L 0 π 0 jπ h cos L π R cos 0 edge L ωε0εl π 0 jπ h cos L π ωε ε L R L jπ h 0 0 edge cos

12 Cicula Polaization Condition Let L W δ ( 0, 0 ) L amplitude o mode amplitude o mode W 0 0 The CP condition is ± - o RHCP + o LHCP j

13 Cicula Polaization Condition (cont.) The equenc CP is deined as the equenc o which we get CP at boadside. CP We have: + j Q ( ) + j Q ( ) whee esonance equenc o (,0) mode esonance equenc o (0,) mode 3

14 Q Q j j + Choose: Then we have (LHCP) Cicula Polaization Condition (cont.) j e e e j j j j j π π π 4 + j LHCP: CP at

15 Cicula Polaization Condition (cont.) The equenc conditions o CP can be witten as: CP + Q Q Q CP + Q Q Q CP CP so + CP o + ( ) CP 5

16 Cicula Polaization Condition (cont.) lso, Q Q Q CP CP Let Then we have CP Q CP Q CP CP Q 6

17 Cicula Polaization Condition (cont.) Summa o equencies CP Q + CP Q (LHCP) CP + Q CP Q (RHCP) CP equenc o which we get CP at boadside. 7

18 Patch Dimensions o CP W L ε PMC L ε L PMC ( h,ε W ) L, (Hammestad omula) 8

19 Phsical Dimensions o CP (cont.) ( h,ε W ) L, e L L+ L k 0 ε L e π (esonance condition) Let k k π µε π µε (known wavenumbes) 9

20 Phsical Dimensions o CP (cont.) k L ε π k L+ L ε π e ( ) 0 0 Similal, we have ( ) k W + W ε π 0 Hence π L L h, ε, W k ε 0 0 ( ) π W W h, ε, L k ε ( ) Note: Fo W, we use the same omula as L, but eplace W L. 0

21 Phsical Dimensions o CP (cont.) Since the patch is neal squae, the two inging etensions ae neal equal ( L W). Hence we have L W π L k ε 0 π L k ε 0 whee k k π µε π µε (known wavenumbes) Note: The inging length L depends on W, so these equations must be solved numeicall (using, e.g., iteation).

22 Hammestad s Fomula W e L 0.4h h ε + W ε e 0.58 h ε e ε ε h W

23 Input Impedance o CP Patch R R Zin ( ) Zin ( ) + Zin ( ) + + jq + ( ) ( ) j Q t CP : / ( Q) and / ( Q) (LHCP) so Z in ( ) 0 R R + j + j R( + j) + R( j) R( + j) + R( j) ( j)( + j) o Z in R Recall: R R R o TM 0 mode The CP equenc CP is also the esonance equenc whee the input impedance is eal (i we neglect the pobe inductance). 3

24 Input Impedance o CP Patch Hence, at the esonance (CP) equenc CP we have Z in cos π 0 Redge L R edge R edge o TM 0 mode Note: We have a CD omula o R edge. W The ed position 0 can be chosen to give the desied input esistance at the esonance equenc CP. ( 0, 0 ) L 4

25 CP (ial Ratio) Bandwidth We now eamine the equenc dependence o the tem /. + j Q ( ) + j Q ( ) + + jq( jq( ) ) whee + CP Q CP Q (LHCP) 5

26 CP Bandwidth (cont.) Deine CP This is the atio o the opeating equenc to the CP equenc. Then + Q Similal, Q 6

27 CP Bandwidth (cont.) Hence jq jq Q jq jq Q Note: + j j j Let Q ( ) Then + j ( + ) + j( + ) j ( ) j( ) 7

28 CP Bandwidth (cont.) + j( + ) j( ) Q ( ) CP E () t B τ R B 8

29 CP Bandwidth (cont.) Fom ECE 6340: R cotξ whee sin ξ sin( γ)sinφ γ tan φ ag In ou case, γ tan + ( + ) + ( ) φ + + tan ( ) tan ( ) 9

30 CP Bandwidth (cont.) Set R R 3 db ( ) Fom a numeical solution: ±

31 CP Bandwidth (cont.) Hence ( ) Q ± so + ± Q Q Q Theeoe, + + R CP + BW CP Q Hence, we have BW R CP Q 3

32 Impedance Bandwidth Z in ( ) R R + + jq + ( ) j Q( ) R R + + jq + jq Q + Q R R + + j ( + ) j ( ) R R + + j( + ) j( ) ( o ) whee ( ) Q Note: t 0 we have Z in R R j j R 3

33 Impedance Bandwidth (cont.) Z in Zin + R + j( + ) j( ) Zin Z0 Zin R Zin Γ Z + Z Z + R Z + in 0 in in S SWR + Γ Γ Set S S0 (bandwidth limits) 0 ± (deivation omitted) 33

34 Impedance Bandwidth (cont.) Hence ( ) Q ± so ± Q The band edges (in nomalized equenc) ae then + + Q Q 34

35 Impedance Bandwidth (cont.) + imp CP + BW CP Hence BW imp CP Q Hence BW imp CP Q 35

36 Summa CP antenna BW BW imp CP R CP Q.44 Q Q Linea antenna BW imp Lin Q Q 36