) ( ) The pole-zero matching method. European School of Antennas. Rigorous multiport network. Loss less two port network. Two port network.
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1 European School of Antennas Rigorous multiport networ The pole-ero matching metho Stefano aci Universit of Siena, Dept. Of nformation Engineering Via Roma 5, -53 Siena, tal ( ( ( FW, FW, ( ( ( FW, ( ( FW, Presente at the European school of Antennas Antenna Center of Ecellence April 9, 5 Contributors from Universit of Siena:,. Caiao, F. Caminita, A. Cucini,. Ettorre,. Nannetti = (, ; F SS Loss less two port networ S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 Two port networ Cut-off region of the higer orer moes in the κ iagram, H H H H (, ; = Q Q Q Q H H H H (, ; = p P P p p P P p F SS Dipole tpe Slot tpe = = ( j ( cot ( h = ( j ( cot ( h = - et = et = = c = (Light cone c = c c = S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 Light cone comment E E E E = c = (Light cone c = c c = ( H ( H (, ; = H o = Q Q o o Through the inverse of the o matri, incorporates the information from all the higher-orer - an -FWs associate with the iscontinuit. This implies that the information on the overall ispersion euation in the cut-off region of higher-orer moes (Fig. 5b can be obtaine irectl from. Outsie of this region, a moel with more ports nees to recover the ispersion properties. S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4
2 Diagonaliation Diagonaliation of the o matri α E E E E ( ( ( ( = ( ( = (, (, (, tan ( ij ( ij = (,, Rotation matri cosα sinα R ( α = sinα cosα ( ( ( cosα sinα (, ; cosα sinα = ( ( ( sin cos (, ; sin cos α α α α E E E E Unwrappe angle r = 4.5 α u [egrees] f [GH] 3 X Γ.7 X S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 Eigenvalue construction properties of the eigenvalues ( ( ( (, ; cosα sinα cosα sinα ( = (, ; ( ( sin cos α α sin cos α α (, (, (, (, ; tan ( ( α = α = (, i j Since, in the absence of losses, are purel reactive in the cutoff region of higher-orer moes (Fig. 5b; this implies a real value of α for an wavenumber (also when / c <. R The limit an, implies α an. Thus, the an transmission line networs are ecouple. et = ( ( ( ( ( ( α ( ( sin =, S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 Decoupling of an T lines in the principal planes Valiit region of the single moe assumption in the κ iagram 3 Decoupling between the an ports also occurs when the irection of wave propagation is along an plane of geometrical smmetr of the. The planes of smmetr efine the contour of the irreucible Brillouin region (BR. As a conseuence, the ominant an moes are ecouple along the segments of the BR contour which converge in the origin. E φ E E E E / = c / c / c / = / E E S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4
3 Foster s reactance theorem. Properties of (no losses Respects the properties of the LC riving point impeance function ii s an eros lie on the real w ais, are simple an alternate. iii A pole or a ero must be at w=. iv The poles are smmetricall isplace w.r.t the origin. m p (, ; ( (, ; ( = R( α R( α (, ; ( ( p ( ( Re DPOLES Lp=.4nH Cp=.45pF Ls=.39nH Cs=.7pF SLOTS L=.35 nh C=.79 pf L= 4.5 nh C=.77 pf L w D L = cm, W =.5 cm D =.cm t=,5 cm D w t L L = cm, W =.5 cm D =.cm t=,5 cm s( =,f Freuenc (GH (f, ero ero 7.7GH 34.GH pole ero pole ero 3.9GH pole Lp Ls Cs Cp C C L L S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 f S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 ( ( ( ( (, p p(,.. j C (, (, (,.. (,, = Rational approimation of the eigenvalues ( ( ( ( i (, p p(,.. j L (, (, (,.. (,, = Step : full wave analsis for plane wave incience Step : atching of poles an eros using the properties of (, ; Difficulties:. niviuation of poles an eros. Sufficient number of pole an eros to have goo accurac 3. Reconstruction of the pole an ero epenence on the wavenumber S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 Phase of Γ (= Phase of Γ (=. Sufficient number of pole an eros to have a goo accurac pol e pol e mm mm mm mm ero h=.95 mm 5 5 (a ero To investigate the range =(, ma one shoul inclue the pole-ero pair internal to W an one more pair ero-pole, the closest to ma. -9 ero S. aci - Universit of Siena (b - ACE Short course on artificial surfaces- Chalmers Universit-4 3. Reconstruction of the pole an ero ispersion iagram s an eros in - plane = 4.3 r h =.9 cm p (, (, j = Ap B j Cp j pj = Dp E j Fp j ( -X; (X-; Γ (, (, j = A B j j C j j = D E j j F j p ( κ, κ = G H κ L κ j (- Γ ( κ, κ = G H κ L κ j =. cm W =.cm L =. cm =.cm ( p p ( (, ( (,,, = jc (, ( (, ( (, To have information on the ispersion iagram of pole an eros, we nee few constant S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4
4 3. Reconstruction of reflection coefficient phase 3. Reconstruction of the ispersion iagram, (,, E, (,, T E,,,, A / (, (,, E, E, cc,, E (,, = / CC ero 4 4 r =.,9 r =., S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 r =.,9 r =., CC ero i,, S. aci - Universit of Siena - ACE Short erocourse i on, artificial surfaces- Chalmers Universit-4 H Plane inc E Γ (, = Γ (, h r w = sin c l =.5mm =.54mm w=.35mm l=.39mm h=.75mm r =.,,,, 3. Reconstruction of the ispersion iagram A / (, (,, E, E, cc,, E (,, = / CC = 4 = 4 9 = 4 = = 9 = PC S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 r =.,9 r =., CC ero i,, S. aci - Universit of Siena - ACE Short erocourse i on, artificial surfaces- Chalmers Universit-4 Lea waves Printe ipoles LW LW r =.,9 mm 4 LW LW SW Light line Dielectric light line SW SW LW m( Re( 4 (bare slab (bare slab Μ 4 X Γ Γ or X Hbri moes or Γ S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4
5 Dipole tpe Dipole vs slot tpe r = S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4 S. aci - Universit of Siena - ACE Short course on artificial surfaces- Chalmers Universit-4
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