First- and Second-Order Differentiators Based on Coupled-Line Directional Couplers

Abstract

First- and second-order differentiators based on coupled-line directional couplers are presented. The time-derivative effect of coupled-line directional couplers is shown theoretically by applying Kirchhoff's laws to infinitesimal electromagnetically coupled transmission line sections. The time-derivation is accurate under the condition Delta <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Tf</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> Gt 1/4, where Delta <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> is the pulse width and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is the coupler center frequency; the required coupler operation frequency is thus inversely proportional to the pulse duration. The first-order differentiator, which is a simple coupled-line coupler, and the second-order differentiator, which requires in addition an interconnecting delay line, are demonstrated experimentally, and compared to ideal (mathematical) derivators.