Design of active RC filters with zero gain-sensitivity product

Abstract

A root locus with respect to the active parameter for any second-order active RC filter is circular with center on the real axis. It is shown that for any pole pair <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and pole frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_0</tex> , the circular root locus centered at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2Q \omega_0</tex> on the real axis and of radius <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{0}\sqrt{4Q^{2}-1}</tex> exhibits zero <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> -sensitivity and zero gain-sensitivity product (GSP) with respect to the active parameter. Furthermore, realization techniques to achieve zero <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> -sensitivity and zero GSP for prescribed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_0</tex> are outlined. Experimental results agree closely with the theoretical expectations.