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Matching networks: automated Darlington synthesis of immittance functions

Matching networks: automated Darlington synthesis of immittance functions

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Real frequency matching network design techniques require the “Darlington synthesis” of positive real immittance functions. Therefore, this chapter is devoted to “automated Darlington synthesis” to construct lossless matching networks with lumped and distributed elements. In this chapter, several high precision lowpass/bandpass/high-pass LC (Inductor L, Capacitor C) ladder synthesis algorithms are presented. Novel algorithms directly work on the driving-point input immittance or bounded-real reflection functions which describe any kind of cascade-connection LC ladders, perhaps with Brune and C-sections in resistive termination. The crux of the idea is that, at each step of the proposed synthesis methods, a simple pole at DC, or at infinity, or at a finite frequency is removed; then, the remaining immittance function is corrected using what we call is the parametric method. At this point, parametric method warrants the exact LC ladder nature of the remaining immittance function. Thus, at the end of synthesis process, an LC ladder is obtained with high numerical precision.

Chapter Contents:

  • 4.1 High-precision lowpass ladder synthesis via parametric approach
  • 4.1.1 Lowpass LC ladder form
  • 4.1.2 Parametric representation of an immittance function
  • 4.1.3 Warranted ladder network synthesis via parametric synthetic division
  • 4.1.4 Lowpass LC ladder network synthesis
  • 4.1.5 Algorithm: guaranteed synthesis of a lowpass LC ladder from a given minimum driving-point immittance function F(p) = a(p)=a(p) using MATLAB
  • 4.2 LC ladder forms of bandpass structures
  • 4.2.1 Generation of a minimum function via parametric approach for a bandpass LC ladder network
  • 4.2.2 Extraction of a transmission zero at DC
  • 4.2.3 Extraction of a pole at infinity
  • 4.2.4 Bandpass LC ladder synthesis algorithm by means of case studies
  • 4.2.4.1 Case Study 1
  • 4.2.5 General rules for bandpass LC ladder synthesis
  • 4.2.5.1 Case Study 2: An alternative circuit topology
  • 4.2.6 A general synthesis function on MATLAB
  • 4.2.6.1 Case Study 3: Generation of complete circuit topology
  • 4.2.7 Assessment of the numerical error accumulated due to numerical computations
  • 4.2.7.1 Case Study 4: Evaluation of relative errors accumulated in synthesis process
  • 4.3 Computer-aided Darlington synthesis of an immittance functions with transmission zeros at DC and infinity, at finite frequencies and in RHP
  • 4.3.1 Brune section extraction using impedance-based approach
  • 4.3.2 MATLAB implementation of the new synthesis algorithm
  • 4.3.3 Synthesis via chain matrix method
  • 4.3.4 Algorithm: impedance synthesis via chain matrix approach
  • 4.3.5 Real and complex transmission zeros
  • 4.3.6 Impedance correction via parametric approach
  • 4.3.7 Assessment of the synthesis error
  • 4.3.8 Examples
  • 4.4 Reflectance-based impedance generation and its synthesis
  • 4.4.1 Simplified real frequency technique
  • 4.4.2 Generation of driving-point input impedance from a realizable reflectance
  • 4.4.3 Synthesis of driving-point impedance zin(p) = a(p)/b(p)
  • 4.4.3.1 Case 1
  • 4.4.3.2 Case 2
  • 4.4.3.3 Case 3
  • 4.4.3.4 Case 4
  • 4.4.3.5 Case 5
  • 4.4.3.6 Case 6
  • 4.4.4 Examples
  • 4.5 High precision synthesis of a Richards immittance via parametric approach
  • 4.5.1 Description of lossless two-ports in terms of Richards variable
  • 4.5.2 Generation of a Richards immittance via parametric method
  • 4.5.3 Properties of a Richards immittance function
  • 4.5.4 Parametric approach in Richards domain
  • 4.5.5 Cascade connection of k-unit elements
  • 4.5.6 UE extractions employing the chain parameters
  • 4.5.7 Correction of the Richard impedance after each extraction
  • 4.5.8 Numerical error assessment of the new synthesis software package
  • 4.5.9 Algorithm: Richards high-precision synthesis
  • 4.5.10 Integration of new Richards synthesis tool with real frequency matching algorithm
  • 4.5.11 Alternative design
  • 4.5.12 Conclusion
  • 4.6 Practical design of matching networks with mixed lumped and distributed elements
  • 4.6.1 Almost equivalent transmission line model of a CLCPI section
  • 4.6.2 Physical model of an inductor using ideal parallel plate transmission line
  • Appendix Computation of the element values of CT-TRL-CT from the given lumped element C-L-C PI section
  • References
  • MATLAB® program lists

Inspec keywords: network synthesis; lumped parameter networks; electric immittance; ladder networks; LC circuits; distributed parameter networks

Other keywords: lossless matching networks; bounded-real reflection functions; high-pass ladder synthesis algorithm; lowpass ladder synthesis algorithm; bandpass ladder synthesis algorithm; LC ladder synthesis algorithm; frequency matching network design; distributed elements; cascade-connection LC ladders; automated Darlington synthesis; driving-point input immittance; parametric method; immittance functions; lumped elements; resistive termination

Subjects: Analogue circuit design, modelling and testing; Distributed linear networks; Filters and other networks; Lumped linear networks

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