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 (Q) and its bandwidth (B) for any reflection coefficient (Γ) when an impedance-matching network is used. In this work, we first show that an infinite set of a_n and b_n 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 a_n, b_n are nearly independent of the reflection coefficient (i.e., for practical calculations, no dependency of the a_n, b_n on Γ needs to be considered). However, we show that for Γ > 1/3, the values of a_n and b_n depend on Γ; in such cases, an infinite set of a_n and b_n coefficients must be used to evaluate the quality factor and bandwidth relationship that Bode, Fano, Wheeler, and others have studied.