ECE Spring Prof. David R. Jackson ECE Dept. Notes 41

Transcription

1 ECE 634 Sprng 6 Prof. Davd R. Jackson ECE Dept. Notes 4

2 Patch Antenna In ths set of notes we do the followng: Fnd the feld E produced by the patch current on the nterface Fnd the feld E z nsde the substrate Fnd the voltage between the patch and the ground plane Fnd the nput mpedance of the patch (when fed by a probe) L W y Assume that the patch current has the followng form: h ε r, µ r J s π, = cos W L ( y)

3 Calculate the Feld E Fnd E( y,,) L W y h ε r, µ r Domnant (,) mode: J s π, = cos W L ( y) 3

4 Feld E (cont.) Recall that E = G J s G k k zz kv zz kv zz TE, ;, = (, ) + (, ) y y kt In ths problem z = z = 4

5 Feld E (cont.) J s π, = cos W L ( y) L/ W/ y jk jky y J π s k, k y = cos e d e dy W L L/ W/ L cos k π L W J s k, ky = snc k y π kl 5

6 Feld E (cont.) z V : (,) k z I + V ( ) ( ) k = k k k z y k = k k k z y / / Z Z V (,) + k z [A] z = z = h Z k k k η k = = η Z = = z z z z ωε k ωε ε r k 6

7 Feld E (cont.) TE z TE V : (,) k z TE I + V TE ( ) ( ) k = k k k z y k = k k k z y / / TE Z TE V (,) + z = TE Z k z [A] z = h Z ωµ η ωµ = = Z = = η µ TE TE r kz kz k kz kz k ( / ) ( / ) 7

8 Feld E (cont.) At z = : V (,) = Z n = Y = Y + Y + n n n = Y jycot kzh Hence V V TE (,) (,) = = D D m e ( k ) t ( k ) t cot ( ) TE TE cot D k Y jy k h m t z D k Y jy k h e t z 8

9 Feld E (cont.) G k k zz kv zz kv zz TE, ;, = (, ) + (, ) y y kt V V TE (,) (,) = = D D m e ( k ) t ( k ) t = cot ( ) TE TE = cot D k Y jy k h m t z D k Y jy k h e t z Hence, we have: k k y E k, ky, = J s k, ky + kt Dm( kt) De( kt) 9

10 Feld E (cont.) Takng the nverse Fourer transform, we have E y,, = J k, k ( π ) + + s y kt k k y + D k D k m t e t e ( y ) j k + k y dk dk y L cos k π L W J s k, ky = snc k y π kl

11 Feld E z Fnd E z (,y,z) nsde the substrate (-h < z < ) L W y h ε r, µ r Domnant (,) mode: J s π, = cos W L ( y)

12 Feld E z (cont.) From Notes 4 we have: E k k z k I z (,, ) = z y t ωε ε r = ωε ε ( k ) ( ˆ t I z J s u) = ωε ε ωε ε r r r ( kt ) I ( z)( J s ) ( kt ) I ( z)( J s ) = ωε ε r ( k ) I ( z)( J s ) cosφ k kt

13 Feld E z (cont.) Hence, n the space doman we have j( k+ kyy) E y z k J I z e dk dk (,, ) = ( π ) z s y ωε ε r Note: Only z waves contrbute to the vertcal electrc feld. From Notes 4: I ( z) ( kz z+ h ) cos = Dm( kt) jz sn kzh 3

14 Voltage Fnd V (,y) between the patch and the ground plane. L W y h ε r, µ r Domnant (,) mode: J s π, = cos W L ( y) 4

15 Voltage (cont.) h (, ) = (,, ) = (,, ) V y E yzdz E yzdz ( π ) z Usng the result from the prevous calculaton for E z, we have: j( k+ kyy) V (, y) = ( k ) J F ( k ) e dk dk ωε ε r h z s t y where F kt I z dz h From Notes 4: F k t = Dm( kt) jz kz 5

16 Input Impedance Fnd the nput mpedance Z n (,y ) of the probe-fed patch antenna z h J z J s (, y ) L ε r I = [ A] The probe s vewed as an mpressed current. 6

17 Input Impedance (cont.) z h J z J s (, y ) L ε r I = [ A] E = y, S, S s the patch surface Set [ ] E, (, ) Js + E Jz = y S Ths s the Electrc Feld Integral Equaton (EFIE) 7

18 Input Impedance (cont.) Assume: (, ) = (, ) J y AB y B s s s π, = cos W L ( y) A s an unknown ampltude. The EFIE s then [ ] AE, (, ) Bs + E Jz = y S Pck a testng functon T (,y): or [ ] A T (, y) E Bs ds + T (, y) E J z ds = S S [ ] { } s z T (, y) A E B + E J ds = S 8

19 Galerkn s Method: T,y B,y (The testng functon s the same as the bass functon.) Hence, we have: s [ ] A Bs(, y) E Bs ds + Bs, y E J z ds = S Input Impedance (cont.) The soluton for the unknown ampltude coeffcent A s then S A (, ) s z S = = S B y E J ds (, ) [ ] B y E B ds s s J B z, B, B s s [ ] J z, Bs E J z Bs, y ds S B, B E B B (, y) ds s s s s S 9

20 Input Impedance (cont.) The nput mpedance s calculated as: (, y ) V From lnearty, we have (, ) Zn = = V y I (, ) Z = AV y n N where (, ) (, ) V y V y N A = From the last eample: jk ( + kyy) V (, y) = ( k ) B (, y) F ( k ) e dk dk ( π ) N s t y ωε ε r

21 Input Impedance (cont.) Net, we return to the calculaton of A : A (, ) s z S = = S B y E J ds (, ) [ ] B y E B ds s s J B z s, B, B s s From recprocty:,, N z s = s z = z,, = N, h J B B J E y z I dz V y From the formula for the feld E : Z B, B = G k, k B k, k dk dk ( π ) (, ) = (, ) Note : B k k B k k s y s y s s y s y y

22 Input Impedance (cont.) ( π ) Summary n N Z B, B = G k, k B k, k dk dk ( π ) (, ) Z = AV y A jk ( + kyy) V, y = k B k, k F ( k ) e dk dk N s y t y ωε ε r (, ) VN y = Z s s y s y y L cos k π L W B s ( k, ky ) = snc k y π kl F k t = Dm( kt) jz kz

23 Input Impedance (cont.) Convertng to polar coordnates, we have: π / Z = G k φ B k φ k dk dφ, (, ) t s t t t π π / N (, ) =, s y t π ωε ε r V y j k B k k F k sn cos k k y k dk dφ t t Note : jk jk y + jk jk y + jk + jk y jk + jk y e e e e e e + e e y y y y jk + jk jk yy + jk jk + jk y y e e e e e e = + jk yy ( sn ( ) ) sn ( ) + jk y y e j k e j k = + ( ky y ) ( jsn ( k ) ) = cos 3

24 Input Impedance (cont.) The path of ntegraton s shown below. π / Z = G k φ B k φ k dk dφ (, ) (, ) t s t t t π C π / j V y k B k k F k k k y k dk dφ (, ) = (, ) sn ( ) cos ( ) N s y t t t π ωε C ε r Im k t LR C h R β k k Re k t Note: The path must etend to nfnty. 4

25 Input Impedance (cont.) Improvement: Add probe reactance to account for the stored magnetc energy near the metal probe. Z probe jx p X ηµ h ln γ ln π ln µε = p r λ a / λ r r γ.577 ( Euler's constant) 5

26 Input Impedance (cont.) D. M. Pozar, Input mpedance and mutual couplng of rectangular mcrostrp antennas, IEEE Trans. Antennas Propagat., vol. AP-3. pp. 9-96, Nov. 98. [6] E. H. Newman and P. Tulyathan, Analyss of mcrostrp antennas usng moment methods, IEEE Trans. Antennas Propagat., vol. AP-9. pp , Jan