On Resistive Ladder Networks for Use in Ultra-Low Frequency Active-RC Filters

摘要

It is known that the transfer resistance of a resistive ladder can be many times the sum of the actual resistances used to make it. This fact has recently been utilized in constructing ultra-low frequency active-RC filters for biomedical applications, thus saving a significant amount of silicon area in IC implementation. This paper contains an investigation of four kinds of such ladders, viz., (i) ({L}_{1}): the (R-2R) ladder, as commonly used in data converters, (ii) ({L}_{2}): the (R{-}alpha R) ladder, which is a generalization of ({L}_{1}), (iii) ({L}_{3}): the arithmetic progression ladder, in which the series resistances as well as the shunt conductances increase from input to output in arithmetic progression, and (iv) ({L}_{4}): the geometric progression ladder, in which the series resistances as well as the shunt conductances increase from input to output in geometric progression. While ({L}_{1}) is analyzed by inspection, ({L}_{2}) is shown to obey a linear second-order difference equation with constant coefficients, yielding an explicit and elegant expression for the transfer resistance. ({L}_{3}) and ({L}_{4}) also obey such a difference equation but not with constant coefficients, and as such, are not amenable to explicit solution. Theses are analyzed here by using the step-by-step ladder analysis method, starting from the output end, and the results for one-, two-, and three-section ladders are given. The four types of ladders are compared on the basis of a specified transfer resistance. It is shown that ({L}_{3}) and ({L}_{4}) have several advantages over ({L}_{1}) and ({L}_{2}). However, besides the area saving factor, the choice for a given situation will depend on several factors, viz., the basic resistance, the total resistance used, the number of resistors, the spread of resistors, and the ease of fabricating the resistors, in addition to other possible technological factors.