ECE Spring Prof. David R. Jackson ECE Dept. Notes 1

Transcription

1 ECE 6345 ping 5 Po. David. Jackson ECE Dept. Notes

2 Oveview In this set o notes we discuss the CAD model o the micostip antenna. Discuss complex esonance equency Deive omula o Q Deive omula o input impedance Deive omula o impedance bandwidth

3 CAD Model o Micostip Antennas L p (pobe inductance) Pobe-ed patch antenna L C Tank (LC) cicuit The cicuit model is justiied om the eigenunction method in the cavity model, discussed late. 3

4 Tank Cicuit: complex esonance equency L C + - V G Y Y Y Y Tansvese esonance Equation (TE): The complex esonance equency is denoted as ω. 4

5 esonance Fequency (cont.) TE: G jω C+ j ω L ( ) ( ) ( ) jω LG + ω LC ω LC + ω jlg + ω 4 jlg L G LC ± + LC G, ω LC so choose + sign 5

6 esonance Fequency (cont.) ω j + C LC 4 L C Denote: ω ω + jω ω L 4 LC ω C C 6

7 esonance Fequency (cont.) ω L 4 LC ω C C Assume >> L C (a good esonato) We then have: ω LC ω C 7

8 Natual esponse (no souce) + L C V The complex esonance equency is ω. - ω ω + jω In the time domain: vt ( ) e ( j t Ve ω ) (Take V ) so ( jω ) t ω t vt ( ) e e e 8

9 Natual esponse (cont.) ω t vt ( ) e cos ω t ( ) vt () e ω t t T π ω 9

10 i(t) L C toed Enegy + - v(t) (Take V ) Fo the capacito: ( ) Ce U E t Cv t ω t () cos ( ω t) Fo the inducto: I V jω L jω L jω L Theeoe, i( t) e Ie e e e e sin t ω L j ω L ( jω ) t ω t + jω t ω t ( ω )

11 toed Enegy (cont.) i(t) L C + - v(t) ( ) () UH t Li t ω L Ce sin ( ω t) ( ω ) ω t L e sin t ω t Note: U E( t) U H ( t) Ce ω < >< > 4 t Also, note that U ( t) U E( t) + U H ( t) Ce ω t

12 toed Enegy (cont.) ( ) ( ) ( ) + ( ) ( ) U t U t U t U t U t E H E Hence U t Ce U e ( ) ω t () ω t U () t t

13 Q o Cavity Q U π T U D T U D enegy dissipated pe cycle (peiod T) π U Q T U T D / T o Q ω U AVE PD AVE P D aveage powe dissipated (this includes adiation loss) 3

14 Q o Cavity (cont.) Q ω t ω t Ce C e ω ω ( ω ) ω t < Gv( t) > G< e cos t > ω ω t ω C G ω C ω t Ce Note : Ge ω t e cos ( ω t) ( ω t) < > ω t < > e ω t e cos 4

15 Q o Cavity (cont.) We then have C Q ω C ω L L ecall that ω LC ω C Hence ω ω + j LC C 5

16 Q o Cavity (cont.) Hence j Q ω ω + j LC C L j C ω ω ω Q ω ω so

17 Q o Cavity (Cont.) We can thus wite ω t Q ( ) cos vt e ω t ( ) U ( t) U () e ω Q t 7

18 Input Impedance L j C j G Y LC ω ω L C j L j C j L j C j G Z LC ω ω ω ω ω ω L C The pobe inductance is neglected hee. 8

19 Input Impedance (cont.) o Z LC ω ω + j ( ω C) ω ω L ω ω ω + j Q Q ω ω Then we have: Z CL ω ω Deine whee + jq ω π (eal esonance equency) 9

20 Input Impedance (cont.) F Deine: F ( ) + ( )( ) ( ) o Hence, we have Z LC + jqf + jq ( )

21 Input Impedance (cont.) ZLC + jqf We then have Deine: x QF Q Q ( ) Z LC Z LC x Z X + jx + x + x LC LC LC

22 Input Impedance (cont.) Z LC LC X LC Z LC LC X LC x

23 election Coeicient Z Z L C Z Z Z LC Γ Z + Z Z + LC YLC ( + jx) + Y + ( + jx) LC jx + jx LC LC 3

24 Bandwidth Γ 4 x + Γ + x Γ W Bandwidth deinition is based on W < The value is oten chosen as..) ΓΓ ΓΓ. x. x x 4

25 Bandwidth (cont.) Factional bandwidth: BW ecall that x QF Q We can solve o in tems o x: x Q x Q so x Q x ± + Q 4 5

26 Bandwidth (cont.) To detemine coect sign, enoce that x, (o choose the plus sign.) Hence x Q x + + Q 4 Theeoe x Q x Q x x Q Q 6

27 Bandwidth (cont.) Hence, BW x Q Now we need to solve o x : +Γ Γ so Γ + Also, x x x Γ Γ 4+ x 4+ x 4+ x 7

28 Bandwidth (cont.) Theeoe x 4 + x + so x 4+ x + A Thus we have 4A+ xa x o ( ) x A A 4 8

29 ( ) ( ) 4 A x A Bandwidth (cont.) The solution is: 9

30 Bandwidth (cont.) Hence, x We then have BW Q Fo we have: BW Q 3

31 Bandwidth (cont.) Note: W. Γ 3 log 9.5 db [ ] [ db]

32 Complete Model Z L p L C X ωl ω L p p p Z jx + in p Z LC X p ( BW ) (This will be deived in a HW poblem.) es X p es X 3

33 Complete Model (cont.) Deine X X / p p In tems o the nomalized vaiable x, the esonance equency es whee the input impedance is puely eal, coesponds to x es 4X X p p (This will be deived in a HW poblem.) I X p << then x es X p (This ollows om a binomial expansion o the squae-oot tem in the numeato.) 33

34 Complete Model (cont.) At the esonance equency, the input esistance is then es in + x es es in X p + es in X p X es 34

35 Complete Model (cont.) es in X p + Note that the pobe eactance changes the input esistance at esonance. Given a speciied value o the input esistance at esonance (e.g., in es 5 Ω), we wish to solve o the coesponding value o. Note that the CAD omula o esonant input esistance (in the shot-couse notes) gives us the value o in tems o the eed location. 35

36 Complete Model (cont.) To solve o, use Zeo iteation: X es es p in + in es in and solve iteatively: ( ) + i es es in in X p ( i ) ( ) Fist iteation: X es p in + es in econd iteation: X es es p in + in X es p in + es in 36