Reexamining the Impedance Matching Equation of Alfred Lopez: A new light and new understanding
This work reexamines the impedance matching equation of Alfred Lopez, shedding new light on his evaluations and providing a new understanding of this equation. Following the work of Fano and Wheeler, Lopez introduced an empirical closed-form solution for determining the relationship between the antenna quality factor (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i>) and its bandwidth (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i>) for any reflection coefficient (Γ) when an impedance-matching network is used. In this work, we first show that an infinite set of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a<sub>n</sub></i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b<sub>n</sub></i> coefficients satisfies Lopez’s equation, and this set of coefficients depends on the reflection coefficient Γ. Second, we show that when Γ > 1/3, the values of this infinite set of coefficients converge to the range of values provided by Lopez; this explains why Lopez presented calculations only when Γ > 1/3. Therefore, when Γ > 1/3, the values of coefficients <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a<sub>n</sub></i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b<sub>n</sub></i> are nearly independent of the reflection coefficient (i.e., for practical calculations, no dependency of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a<sub>n</sub></i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b<sub>n</sub></i> on Γ needs to be considered). However, we show that for Γ > 1/3, the values of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a<sub>n</sub></i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b<sub>n</sub></i> depend on Γ; in such cases, an infinite set of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a<sub>n</sub></i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b<sub>n</sub></i> coefficients must be used to evaluate the quality factor and bandwidth relationship that Bode, Fano, Wheeler, and others have studied.
Authors
- Stavros V. Georgakopoulos (ORCID: https://orcid.org/0000-0002-1626-6589)
- Constantinos L. Zekios (ORCID: https://orcid.org/0000-0002-3195-8612)
- Anastasios G. Koutinos (ORCID: https://orcid.org/0000-0002-8050-0085)
Institutions
- Florida International University (US)
Publication Details
- Journal
- IEEE Antennas and Propagation Magazine
- Published
- 2024-09-23
- DOI
- https://doi.org/10.1109/map.2024.3457308
- Primary Topic
- Experimental and Theoretical Physics Studies
- Type
- article
- Field-Weighted Citation Impact
- 0.00