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Microstrip Lines Microwave Filter Design Chp4. Transmission Lines and Components Prof. Tzong-Lin Wu Department of Electrical Engineering National Taiwan University Prof. T. L. Wu Microstrip Lines Microstrip Structure Inhomogeneous structure: Due to the fields within two guided-wave media, the microstrip does not support a pure TEM wave. When the longitudinal components of the fields for the dominant mode of a microstrip line is much smaller than the transverse components, the quasi-TEM approximation is applicable to facilitate design. Prof. T. L. Wu Microstrip Lines - Transmission Line Parameters Effective Dielectric Constant (εre ) and Characteristic Impedance(Z C) εre For thin conductors (i.e., t → 0), closed-form expression (error ≤ 1 % ): W/h ≤ 1: W/h ≥ 1: For thin conductors (i.e., t → 0), more accurate expressions: Effective dielectric constant (error ≤ 0.2 % ): Characteristic impedance (error ≤ 0.03 % ): Prof. T. L. Wu Microstrip Lines - Transmission Line Parameters Guided wavelength λ0 300 λg = or λg = mm ε re f( GHz ) ε re Propagation constant 2π β = λg Phase velocity , Zo β ω c υ p = = β ε re Electrical length θ= β ℓ Prof. T. L. Wu Microstrip Lines - Transmission Line Parameters Losses Conductor loss Dielectric loss Radiation loss Dispersion εre (f) Zo(f) Surface Waves and higher-order modes Coupling between the quasi-TEM mode and surface wave mode become significant when the frequency is above fs −1 c tan ε r fs = 2πh ε r − 1 Cutoff frequency f c of first higher-order modes in a microstrip c fc = εr ()2W+ 0.8 h The operating frequency of a microstrip line < Min (fs, f c) Prof. T. L. Wu Microstrip Lines - Tx-Line Synthesis of transmission line – electrical or physical parameters Prof. T. L. Wu Coupled Lines Coupled line Structure The coupled line structure supports two quasi-TEM modes: odd mode and even mode. Odd mode Electrical wall Even mode Magnetic Wall Electric field Magnetic field Prof. T. L. Wu Coupled Lines – Odd- and Even- Mode Effective Dielectric Constant (εre ) and Characteristic Impedance(Z C) Odd mode Even mode Odd- and Even- Mode: The characteristic impedances (Zco and Zce ) and effective dielectric o e constants (ε re and ε re ) are obtained from the capacitances (Co and Ce): Odd-Mode: Even-Mode: a a C o and C e are even- and odd-mode capacitances for the coupled microstrip line configuration with air as dielectric. Prof. T. L. Wu Coupled Lines Odd mode – Odd- and Even- Mode Effective Dielectric Constant (εre ) and Characteristic Impedance(Z C) Odd- and Even- Mode Capacitances: Even mode Odd-Mode: Even-Mode: Cp denotes the parallel plate capacitance between the strip and the ground plane: Cf is the fringe capacitance as if for an uncoupled single microtrip line: ’ Cf accounts for the modification of fringe capacitance Cf : , Cgd may be found from the corresponding coupled stripline geometry: Cga can be modified from the capacitance of the corresponding coplanar strips: , , Prof. T. L. Wu Discontinuities And Components – Discontinuities Microstrip discontinuities commonly encountered in the layout of practical filters include steps, open-ends, bends, gaps, and junctions. The effects of discontinuities can be accurately modeled by full-wave EM simulator or closed-form expressions and taken into account in the filter designs. Steps in width: Open ends: Gaps: Bends: Prof. T. L. Wu Discontinuities – Steps in width where Note : L wi for i = 1, 2 are the inductances per unit length of the appropriate micriostrips, having widths W 1 and W 2, respectively. Zci and εrei denote the characteristic impedance and effective dielectric constant corresponding to width W , and h is the substrate thickness in i Prof. T. L. Wu micrometers. Discontinuities – Open ends The fields do not stop abruptly but extend slightly further due to the effect of the fringing field. Closed-form expression: where The accuracy is better than 0.2 % for the range of 0.01 ≤ W/h ≤ 100 and εr ≤ 128 Prof. T. L. Wu Discontinuities – Gaps where The accuracy is within 7 % for 0.5 ≤ W/h ≤ 2 and 2.5 ≤ εr ≤ 15 Prof. T. L. Wu Discontinuities – Bends The accuracy on the capacitance is quoted as within 5% over the ranges of 2.5 ≤ εr ≤ 15 and 0.1 ≤ W/h ≤ 5. The accuracy on the inductance is about 3 % for 0.5 ≤ W/h ≤ 2. Prof. T. L. Wu Components – lumped inductors and capacitors Lumped inductors and capacitors The elements whose physical dimensions are much smaller than the free space wavelength λ0 of the highest operating frequency (smaller than 0.1 λ0). Design of inductors High-impedance line Meander line Circuit representation Circular spiral Square spiral Initial design formula for straight-line inductor Prof. T. L. Wu Components – lumped inductors and capacitors Design of capacitors Interdigital capacitor Assuming the finger width W equals to the space and empirical formula for capacitance is shown as follow Metal-insulator-metal (MIM) capacitor Estimation of capacitance and resistance is approximated by parallel-plate Circuit representation Prof. T. L. Wu Components – Quasilumped elements (1) Quasilumped elements λλλ Physical lengths are smaller than a quarter of guided wavelength λg. l <<< g High-impedance short line element 8 Derivation -Y12 Y11 +Y12 Y22 +Y12 cosβℓ − 1 ℓ ℓ D −()AD − BC cosβjZ c sin β ℓ ℓ A B Y11 Y 12 B B jZcsinβ jZ c sin β = = = 1 ℓ ℓ Y Y −1 cos βℓ C D j sinβ cos β 21 22 −1 A Zc ℓ ℓ B B jZcsinβ jZ c sin β 1 1 −Y = = inductive element: 12 ℓ jZc sin β jx ℓ2 ℓ ℓ2 ℓ 2β β 2 β β 2 βℓ β ℓ cos− sin − cos + sin −2sin tan cosβℓ − 1 2 2 2 2 B capacitive element: YY+ = = =2 =j 2 = j 11 12 jZ sin βℓ ββℓℓ ββ ℓℓ Prof. T.Z L. Wu 2 c jZ2sin cos jZ 2sin cos c c 22c 22 Components – Quasilumped elements (2) Quasilumped elements λλλ l <<< g Low-impedance short line element 8 Derivation ι Zc, β ℓ AD− BC Zccos β Z c ℓ ℓ A () cosβjZ c sin β ℓ ℓ A B Z11 Z 12 C C jsinβ j sin β = = = 1 ℓ ℓ Z Z Z Z cos βℓ C D j sinβ cos β 21 22 1 D c c Zc C C jsinβℓ j sin β ℓ Zc 1 sin βℓ capacitive element: Z12 = = jsin βℓ jB B = Zc ℓ ℓ Zc cosβ− 1 β x x βℓ inductive element: ZZ11−= 12 = jZc tan = j = Z tan Prof. T. L. Wu j sinβℓ 2 2 2c 2 Components – Quasilumped elements (3) Quasilumped elements Open- and short-circuited stubs (assuming the length L is smaller than a quarter of guided wavelength λg) C L λ λ l < g l < g 8 8 Prof. T. L. Wu Components – Resonators Prof. T. L. Wu Loss Considerations for Microstrip Resonators Unloaded quality factor Q u is served as a justification for whether or not the required insertion loss of a bandpass filter can be met. The total unloaded quality factor of a resonator can be found by adding conductor, dielectric, and radiation loss together. EM simulator Quality factors Q c and Q d for a microstrip line or Prof. T. L. Wu.
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