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A '''logarithmic resistor ladder''' is an [[electronic circuit]] composed of a series of [[resistor]]s and [[switch]]es, 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.
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The logarithmic behavior of the circuit is its main differentiator in comparison with [[digital-to-analog converter]]s in general, and traditional [[Resistor ladder|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 [[Preamplifier|audio devices]], since human [[dynamic range#Audio|perception of sound level]] is properly expressed on a logarithmic scale.
 
== Logarithmic input/output behavior ==
As in [[digital-to-analog converter]]s, a [[Binary numeral system|binary word]] is applied to the ladder network, whose ''N'' bits are treated to represent an integer value according the relation:
:<math>\mathrm{CodeValue} = \sum_{i=1}^N s_i \cdot 2^{i-1}</math> where <math>s_i</math> represents a value 0 or 1 depending on the state of the ''i<sup>th</sup>'' switch.
 
For a conventional [[digital-to-analog converter|DAC]] or [[Resistor ladder|R-2R network]], the output signal value (its voltage) would be:
:<math>V_{out} = a \cdot (\mathrm{CodeValue} + b ) \cdot V_{in}</math> where <math>a</math> and <math>b</math> are design constants and where <math>V_{in}</math> typically is a '''constant reference''' voltage.
(DA-converters that are designed to handle a '''variable''' input voltage are
termed '''multiplying DAC'''.<ref>{{cite web |url=http://www.analog.com/static/imported-files/overviews/AnalogMultiplyingDACs.pdf |title=Multiplying DACs, flexible building blocks |format=PDF |year=2010 |publisher=Analog Devices inc. |accessdate=29 March 2012}}</ref>)
 
In contrast, the logarithmic ladder network discussed in this article creates a behavior as:
:<math>\log (V_{out} / V_{in}) = a \cdot (\mathrm{CodeValue} + b )</math> where <math>V_{in}</math> is a '''variable''' input signal.
 
== Circuit implementation ==
[[File:Circuit diagram of logarithmic ladder DA converter, Mar 2012.gif|alt=Schematic diagram|Circuit diagram]]
 
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<sub>i</sub>'' as:
:<math>Ratio_i = \text{if}\; sw_i \;\text{then}\; \alpha^{2^{i-1}} \;\text{else}\; 1</math>
 
For logarithmic scaled attenuators, it is common practice to express their attenuation in [[decibel]]s:
:<math>dB(Ratio_i) = 20 \log_{10} \alpha^{2^{i-1}} = 2^{i-1} \cdot 20 \cdot \log_{10} \alpha</math> for <math>i = 1 .. N</math> and <math>sw_i = 1</math>
 
This reveals a basic property: <math>dB(Ratio_{i+1}) = 2 \cdot dB(Ratio_i)</math>
 
To show that this <math>Ratio_i</math> satisfies the overall intention:
:<math>\log (V_{out}/V_{in}) = \log (\prod_{i=1}^N Ratio_i) = \sum_{i=1}^N \log (Ratio_i) = a \cdot (CodeValue + b )</math>
:for <math>b = 0</math> and <math>a = \log(\alpha)</math>
 
The different stages 1 .. N should function independently of each other, as to obtain 2<sup>N</sup> 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<sub>load</sub>.
 
This leads to:
:<math>\begin{cases}
R_{i,parr} = (R_{i,b} \cdot R_{load}) / (R_{i,b} + R_{load}) \\
R_{i,a} + R_{i,parr} = R_{load} \\
R_{i,parr} / (R_{i,a} + R_{i,parr}) = Ratio_i
\end{cases}</math>
 
With these equations, all resistor values of the circuit diagram follow easily after choosing values for N, <math>\alpha</math> and R<sub>load</sub>. (The value of R<sub>source</sub> 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<sub>source</sub>.
 
This leads to:
:<math>\begin{cases}
R_{i,ser} = R_{i,a} + R_{source} \\
R_{i,ser} \cdot R_{i,b} / (R_{i,ser} + R_{i,b}) = R_{source} \\
R_{i,b} / (R_{i,ser} + R_{i,b}) = Ratio_i
\end{cases}</math>
 
Again, all resistor values of the circuit diagram follow easily after choosing values for N, <math>\alpha</math> and R<sub>source</sub>. (The value of R<sub>load</sub> does not influence the logarithmic behavior)
 
== Circuit variations ==
* The circuit as depicted above, can also be applied in reverse direction. That correspondingly reverses the role of constant-input and constant-output resistance equations.
* Since the stages do not influence each other's attenuation, the stage order can be chosen arbitrarily. Such reordering can have a significant effect on the ''input'' resistance of the ''constant output resistance'' attenuator and vice versa.
 
== Background ==
R-2R ladder networks used for Digital-to-Analog conversion are rather old. A historic description is in a patent<ref>{{cite patent |country=US |number=3108266 |status=patent |title=Signal Conversion Apparatus |gdate=22 October 1963 |fdate=22 July 1955 |inventor=Gordon, B. M.}}</ref> 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.,<ref>LOGDAC:
CMOS Logarithmic D/A Converter, [http://www.analog.com/static/imported-files/data_sheets_obsolete/1257410AD7118.pdf "Analog Devices Inc. AD7118"]</ref> protected through a 1981 patent filing.<ref>{{cite patent |country=US |number=4521764 |status=patent |title=Signal-controllable attenuator employing a digital-to-analog converter |gdate=4 June 1985 |fdate=31 December 1981 |inventor=Burton, David P.}}</ref>
 
== See also ==
* [[Resistor ladder]]
* [[Digital-to-analog converter]]
* [[Attenuator (electronics)|Attenuator circuits]]
 
== References ==
{{reflist}}
 
== External links ==
* [http://www.vaneijndhoven.net/jos/attenuator-calculator/index.html Online calculator] to configure logarithmic ladder networks
 
[[Category:Articles created via the Article Wizard]]
[[Category:Electrical components]]
[[Category:Resistive components]]

Latest revision as of 13:34, 25 September 2014

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