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Characterizing Optimal Multiport Matching Transformers

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Abstract

Multiport transformers offer enhanced matching capabilities for multiport loads. These transformers are analyzed to characterize wideband matching designs. An eigenvalue decomposition of the multiport load produces a geometric description of the sublevel sets of the matching transformers and identifies those frequency bands where multiport transformers are effective. A multiport antenna provides a concrete design example.

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Notes

  1. Two books on the scattering formalism are Baher [3] and Balabanian and Bickart [4]. Baher omits the factor of 1/2 but carries this rescaling into the power definitions. Most other books use the power-wave normalization [9]: \(\mathbf {a}=R_{0}^{-1/2} \{ \mathbf {v}+ Z_{0} \mathbf {i}\}/2\), where the normalizing matrix Z 0=R 0+jX 0 is diagonal.

  2. Although the lossless 2-ports are a subset of the passive 2-ports, Ball and Helton show that greater gain cannot be achieved by matching over the larger class of passive 2-ports [5]. Consequently, matching can be restricted to the lossless 2-ports.

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Acknowledgements

The expertise and diligence of the reviewers in their careful reading of this paper is gratefully acknowledged.

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Correspondence to Jeffery C. Allen.

Appendix: Mapping Inequalities to Coordinate Bounds

Appendix: Mapping Inequalities to Coordinate Bounds

A single linear inequality v T xv 0 determines a half space of ℝN. This half space can be covered by orthants of the form xp where each orthant has its vertex located on the half-space’s boundary.

Lemma 3

Let \(\mathcal{N}(\mathbf {p})\) denote the orthant \(\mathcal{N}(\mathbf {p}) := \{ \mathbf {x}\in\mathbb{R}^{N} : \mathbf {x}\le \mathbf {p}\}\). If 0<v∈ℝN, the half-space \(\mathcal{H}(\mathbf {v},v_{0}) := \{ \mathbf {x}\in\mathbb{R}^{N} : \mathbf {v}^{T}\mathbf {x}\le v_{0} \}\), is the union of the orthants:

$$\mathcal{H}(\mathbf {v},v_0) = \bigcup \bigl\{ \mathcal{N}(\mathbf {p}) :\mathbf {v}^T \mathbf {p}= v_0 \bigr\} .$$
(70)

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Allen, J.C., Rockway, J.W. Characterizing Optimal Multiport Matching Transformers. Circuits Syst Signal Process 31, 1513–1534 (2012). https://doi.org/10.1007/s00034-011-9387-5

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