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This topic covers the basic transmission line equations, important conclusions, and power gains of amplifiers using S-parameters and full-wave analysis for microstrip structures.

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  1. topics • Basic Transmission Line Equations • Nine Power Gains of Amplifiers • Linear and Nonlinear Synthesis/Analysis • Full-Wave Analysis for Microstrip

  2. basic transmission line equations

  3. basic transmission line equations Important Conclusions from the above equations : • When ZL =ZO, ΓL = 0 and Zin = ZO • For an open transmission line ΓL = 1, Zin = -jZocotθ. Under the condition θ < π/2, the behavior of the input impedance is like that of a capacitance. Hence a short open-circuit transmission line can be used as a capacitance element. • For a shorted transmission line ΓL = -1, Zin = -jZotanθ. Under the condition θ < π/2, the behavior of the input impedance is like that of an inductance. Hence a short short-circuit transmission line can be used as an inductive element. • When an electrical length θ = π/2 (of physical length l = λ/4), the transmission line is called a quarter-wave transformer. The quarter-wave transformer has the following important property:

  4. basic transmission line equations For a Matched Lossless two-port Transmission Line with electrical length θ : S matrix ABCD matrix VSWR is the ratio of Vmax to Vmin. The relationship between VSWR and reflection coefficient is as follows:

  5. basic transmission line equations Shift in Reference Planes

  6. basic transmission line equations Shift in Reference Planes The S-matrices of the 50Ω transmission lines are represented by S1 and S2 : The S-matrix of the device can be represented by S1 and S2 and the measured matrix S’ as follows :

  7. nine power gains of amplifiers Power Gains of different amplifiers are determined using S- parameters to get the following results : Transducer power gain in 50-Ω system Transducer power gain for arbitrary ΓG and ΓL Unilateral transducer power gain Power gain with input conjugate matched

  8. nine power gains of amplifiers Available power gain with output conjugate matched Maximum available power gain Maximum unilateral transducer power gain Maximum stable power gain Unilateral power gain

  9. full wave analysis for microstrip METHOD FEATURES DISADVANTAGES Spectral Domain Method • One of the most popular methods for infinitesimally thin conductors on multilayer structures • Closed-form expressions for Fourier-transformed Green’s functions • Numerical efficiency • Cannot handle thick conductor structures • For tight coupling the number of basic functions becomes large; would involve convergent problems Finite Difference Method • Mathematical preprocessing is minimal • Can be applied to a wide range of structures • Numerically inefficient • Precautions must be taken when the method is applied to an open-region problem • Need layer computer storage for accurate solution

  10. full wave analysis for microstrip METHOD FEATURES DISADVANTAGES Finite Element Method • Similar to the finite difference method • Has variational features in the algorithm and is more flexible in the application • Developed to solve very large matrix equations • Numerically inefficient • Existence of so-called spurious (unphysical) zeros Mode- Matching Method • Typically applied to the problem of scattering at the waveguide discontinuity • Often used to solve enclosed planar structures, including metal thickness effects • Several different formulations possible, all theoretically equivalent; however, they may be different numerically • Precautions must be taken on relative convergence for some problems

  11. full wave analysis for microstrip METHOD FEATURES DISADVANTAGES Equivalent Waveguide Model • Very useful method for analysis of microstrip discontinuity problem

  12. linear and nonlinear synthesis Matching networks for single-frequency and wide-frequency bands (e.g., a 4:1 for complex loads and termination). Narrowband/wideband lumped and distributed filter synthesis. Oscillator synthesis from small- and large-signal S parameters. Parallel-series design, determination of all components, determination of efficiency, output power, phase noise, and other relevant data. Open and closed loop, PLL design, phase nosie determination, nonlinear switching, frequency lock phase lock. System analysis and optimization for noise figure IMD performance.

  13. linear and nonlinear analysis • When load impedance ZL equals ZO, the characteristic impedance of the transmission line, the load reflection coefficient ΓL = 0, and the input impedance equals the characteristic impedance of the transmission line, namely Zin = ZO. • For an open transmission line ΓL = 1, Zin = -jZOcotθ. Under the condition θ < π/2, the behavior of the input impedance is like that of a capacitance. Hence a short open-circuit transmission line can be used as a capacitance element. • For a shorted transmission line ΓL = -1, Zin = -jZOtanθ. Under the condition θ < π/2, the behavior of the input impedance is like that of an inductance. Hence a short short-circuit transmission line can be used as an inductive element. • When an electrical length θ = π/2 (of physical length l = λ/4), the transmission line is called a quarter-wave transformer. The quarter-wave transformer has the following important property: Zin = ZO2 /ZL.

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