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Small Antenna Handbook by Robert E. Collin, Robert C. Hansen

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Chapter 2

Bandwidth and Matching

2.1 Introduction

Chapter 1 was concerned with ESA values of Q. In this chapter, the emphasis is on impedance matching limitations and the relation between Q and bandwidth (BW). Key names of the major contributors are Foster (reactance theorem), Fano (matching limitations), and Moreno (loss magnification). Finally, a comparison is made between a matched short whip antenna and a whip with a high-impedance preamp.

In relating bandwidths for different VSWR, a convenient formula is

(2.1) equation

For example,

(2.2) equation

2.2 Foster's Reactance Theorem and Smith Chart

In 1924, R. M. Foster published a paper showing that a lossless reactance always has a positive slope of reactance or susceptance with frequency (Foster, 1924). This has become known as Foster's reactance theorem. Such networks have their poles and zeroes on the real axis, and they alternate. See also Bode (1945) and Guillemin (1935). Because all antennas have a virtual loss due to the radiation resistance, Foster's theorem should not apply to antennas. There has been some controversy regarding whether Foster's reactance theorem applies to antennas, and in particular, to electrically small antennas. Best (2004) claims that it does not apply near antiresonance, where the reactance is rapidly changing sign. ...

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