Logarithmic resistor ladder
A logarithmic resistor ladder is an electronic circuit composed of a series of resistors and switches, designed to create an attenuation from an input to an output signal, where the logarithm of the attenuation ratio is proportional to a digital code word that represents the state of the switches. The logarithmic behavior of the circuit is its main differentiator in comparison with digital-to-analog converters in general, and traditional R-2R Ladder networks specifically. Logarithmic attenuation is desired in situations where a large dynamic range needs to be handled. The circuit described in this article is applied  in audio devices, since human perception of sound level is properly expressed on a logarithmic scale. Logarithmic input/output behavior. As in digital-to-analog converters, a binary word is applied to the ladder network, whose "N" bits are treated as representing an integer value according to the relation: For a conventional DAC or R-2R network, the output signal value (its voltage) would be: (DA-converters that are designed to handle a variable input voltage are termed multiplying DAC.) In contrast, the logarithmic ladder network discussed in this article creates a behavior as: Circuit implementation. This example circuit is composed  of 4 stages, numbered 1 to 4, and an additional leading "Rsource" and trailing "Rload". Each stage "i" has a designed input-to-output voltage attenuation "ratio" as: For logarithmic scaled attenuators, it is common practice to express their attenuation in decibels: This reveals a basic property: formula_13 To show that this formula_14 satisfies the overall intention: The different stages 1 .. N should function independently of each other, as to obtain 2 different states with a composable behavior. To achieve an attenuation of each stage that is independent of its surrounding stages, either one of two design choices is to  be implemented: constant input resistance or constant output resistance. Constant input resistance. The input resistance of any stage shall be independent of its on/off switch position, and must be equal to R. This leads to: With these equations, all resistor values of the circuit diagram follow easily after choosing values for N, formula_19 and R. (The value of R does not influence the logarithmic behavior) Constant output resistance. The output resistance of any stage shall be independent of its on/off switch position, and must be equal to R. This leads to: Again, all resistor values of the circuit  diagram follow easily after choosing values for N, formula_19 and R. (The value of R does not influence the logarithmic behavior) Background. R-2R ladder networks used for Digital-to-Analog conversion are rather old. A historic description is in a patent filed in 1955. Multiplying DA-converters with logarithmic behavior were not known for a long time after that. An initial approach was to map the logarithmic code to a much longer code word, which could be applied to the classical (linear) R-2R based DA-converter. Lengthening the codeword is needed in that approach to achieve sufficient dynamic range. This  approach was implemented in a device from Analog Devices Inc., protected through a 1981 patent filing.