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 binary number that represents the state of the switches.
The logarithmic behavior of the circuit is its main differentiator in comparison with digital-to-analog converters (DACs) 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.
As in digital-to-analog converters, a binary number is applied to the ladder network, whose N bits are treated as representing an integer value: where
For comparison, recall a conventional linear DAC or R-2R network produces an output voltage signal of: where
[1]) In contrast, the logarithmic ladder network discussed in this article creates a behavior as: which can also be expressed as
satisfies the overall intention: The different stages 1 .. N should function independently of each other, as to obtain 2N 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.
The input resistance of any stage shall be independent of its on/off switch position, and must be equal to Rload.
The next step (2 dB) would use: Ra = 369.0 Ω, Rb = 1709.7 Ω. R-2R ladder networks used for linear digital-to-analog conversion are old (Resistor ladder § History mentions a 1953 article and a 1955 patent).
Multiplying DACs with logarithmic behavior were not known for a long time after that.
Lengthening the codeword is needed in that approach to achieve sufficient dynamic range.