Inhomogeneous structure: Due to the fields within two guided-wave media, the microstrip does not support a pure TEM wave.

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1 Mirowave Filter Design Chp4. Transmission Lines and Components Prof. Tzong-Lin Wu Department of Eletrial Engineering National Taiwan University Mirostrip Lines Mirostrip Struture Inhomogeneous struture: Due to the fields within two guided-wave media, the mirostrip does not support a pure TEM wave. When the longitudinal omponents of the fields for the dominant mode of a mirostrip line is muh smaller than the transverse omponents, the quasi-tem approximation is appliable to failitate design.

2 Mirostrip Lines - Transmission Line Parameters Effetive Dieletri Constant (ε re ) and Charateristi Impedane(Z C ) ε re For thin ondutors (i.e., t 0), losed-form expression (error 1 % ): W/h 1: W/h 1: For thin ondutors (i.e., t 0), more aurate expressions: Effetive dieletri onstant (error 0.2 % ): Charateristi impedane (error 0.03 % ): Mirostrip Lines - Transmission Line Parameters Guided wavelength λ0 λg = ε Propagation onstant re or λ = g 300 f ( GHz) ε re mm β = 2π λ g Phase veloity ω υp = = β Eletrial length ε re Z, o β θ = βl

3 Mirostrip Lines - Transmission Line Parameters Losses Condutor loss Dieletri loss Radiation loss Dispersion ε re (f) Z o (f) Surfae Waves and higher-order modes Coupling between the quasi-tem mode and surfae wave mode beome signifiant when the frequeny is above f s s = tan Cutoff frequeny f of first higher-order modes in a mirostrip f f = The operating frequeny of a mirostrip line < Min (f s, f ) 1 ε 2πh ε 1 ε r r ( 2W + 0.8h) r Mirostrip Lines - Tx-Line Synthesis of transmission line eletrial or physial parameters

4 Coupled Lines Coupled line Struture The oupled line struture supports two quasi-tem modes: odd mode and even mode. Odd mode Eletrial wall Even mode Magneti Wall Eletri field Magneti field Coupled Lines Odd- and Even- Mode Effetive Dieletri Constant (ε re ) and Charateristi Impedane(Z C ) Odd mode Even mode Odd- and Even- Mode: The harateristi impedanes (Z o and Z e ) and effetive dieletri onstants (ε o re and ε e re) are obtained from the apaitanes (C o and C e ): Odd-Mode: Even-Mode: C a oand C a eare even- and odd-mode apaitanes for the oupled mirostrip line onfiguration with air as dieletri.

5 Coupled Lines Odd- and Even- Mode Effetive Dieletri Constant (ε re ) and Charateristi Impedane(Z C ) Odd- and Even- Mode Capaitanes: Odd mode Even mode Odd-Mode: Even-Mode: C p denotes the parallel plate apaitane between the strip and the ground plane: C f is the fringe apaitane as if for an unoupled single mirotrip line: C f aounts for the modifiation of fringe apaitane C f : C gd may be found from the orresponding oupled stripline geometry:, C ga an be modified from the apaitane of the orresponding oplanar strips:,, Disontinuities And Components Disontinuities Mirostrip disontinuities ommonly enountered in the layout of pratial filters inlude steps, open-ends, bends, gaps, and juntions. The effets of disontinuities an be aurately modeled by full-wave EM simulator or losed-form expressions and taken into aount in the filter designs. Steps in width: Open ends: Gaps: Bends:

6 Disontinuities Steps in width where Note : L wi for i = 1, 2 are the indutanes per unit length of the appropriate miriostrips, having widths W 1 and W 2, respetively. Z i and ε rei denote the harateristi impedane and effetive dieletri onstant orresponding to width W i, and h is the substrate thikness in mirometers. Disontinuities Open ends The fields do not stop abruptly but extend slightly further due to the effet of the fringing field. Closed-form expression: where The auray is better than 0.2 % for the range of 0.01 W/h 100 and ε r 128

7 Disontinuities Gaps where The auray is within 7 % for 0.5 W/h 2 and 2.5 ε r 15 Disontinuities Bends The auray on the apaitane is quoted as within 5% over the ranges of 2.5 ε r 15 and 0.1 W/h 5. The auray on the indutane is about 3 % for 0.5 W/h 2.

8 Components lumped indutors and apaitors Lumped indutors and apaitors The elements whose physial dimensions are muh smaller than the free spae wavelength λ 0 of the highest operating frequeny (smaller than 0.1 λ 0 ). Design of indutors High-impedane line Meander line Cirular spiral Square spiral Ciruit representation Initial design formula for straight-line indutor Components lumped indutors and apaitors Design of apaitors Interdigital apaitor Assuming the finger width W equals to the spae and empirial formula for apaitane is shown as follow Metal-insulator-metal (MIM) apaitor Estimation of apaitane and resistane is approximated by parallel-plate Ciruit representation

9 Components Quasilumped elements (1) Quasilumped elements Physial lengths are smaller than a quarter of guided wavelength λ g. High-impedane short line element λg l< 8 Derivation -Y 12 Y 11+Y 12 Y 22+Y 12 ( ) D AD BC osβl 1 osβl sinβl A B Y11 Y12 = 1 B B sinβl sinβl C D j sin βl os βl Y21 Y = = 22 1 A 1 osβl B B sinβ sinβ l l 1 1 indutive element: Y12 = = sinβl jx βl βl 2 βl βl 2 os sin os + sin βl βl osβl 1 2sin tan B apaitive element: Y11 + Y12 = = = sin Prof. = j T. L. Wu = j jz β β β β βl l l l l jz 2sin os 2sin os Components Quasilumped elements (2) Quasilumped elements Low-impedane short line element λg l< 8 Derivation ι Z, β osβl sinβl A B 1 C D = j sin βl os βl apaitive element: Z 12 Z 1 = = j sinβl jb ( AD BC) A osβl Z11 Z12 C C j sinβl j sinβl Z Z = = 1 D osβl C C j sinβ j sinβ l l sinβl B= Z osβl 1 βl x Z 11 Z12 = = tan = j x βl indutive element: = tan j sinβl

10 Components Quasilumped elements (3) Quasilumped elements Open- and short-iruited stubs (assuming the length L is smaller than a quarter of guided wavelength λ g ) C L λg l< 8 λg l< 8 Components Resonators

11 Loss Considerations for Mirostrip Resonators Unloaded quality fator Q u is served as a justifiation for whether or not the required insertion loss of a bandpass filter an be met. The total unloaded quality fator of a resonator an be found by adding ondutor, dieletri, and radiation loss together. EM simulator Quality fators Q and Q d for a mirostrip line or