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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.

Logarithmic input/output behavior

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:

\mathrm {CodeValue} =\sum _{i=1}^{N}s_{i}\cdot 2^{i-1}

where $s_{i}$ is 0 or 1 depending on the state of the i^th switch.

For comparison, recall a conventional linear DAC or R-2R network produces an output voltage signal of:

V_{out}=V_{in}\cdot c\cdot (\mathrm {CodeValue} +d)

where $c$ and $d$ are design constants and where $V_{in}$ typically is a constant reference voltage (or is a variable input voltage for a multiplying DAC.^[1])

In contrast, the logarithmic ladder network discussed in this article creates a behavior as:

\log(V_{out}/V_{in})=c\cdot \mathrm {CodeValue}

which can also be expressed as $V_{in}$ multiplied by some base $\alpha$ raised to the power of the code value:

V_{out}=V_{in}\cdot \alpha ^{\mathrm {CodeValue} }

where $c=\log(\alpha )\,.$

Circuit implementation

This example circuit is composed of 4 stages, numbered 1 to 4, and includes a source resistance R_source and load resistance R_load.

Each stage i has a designed input-to-output voltage attenuation Ratio_i as:

Ratio_{i}={\text{if}}\;sw_{i}\;{\text{then}}\;\alpha ^{2^{i-1}}\;{\text{else}}\;1

For logarithmic scaled attenuators, it is common practice to equivalently express their attenuation in decibels:

dB(Ratio_{i})=20\log _{10}\alpha ^{2^{i-1}}=2^{i-1}\cdot 20\cdot \log _{10}\alpha

for

i=1..N

and

sw_{i}=1

This reveals a basic property: $dB(Ratio_{i+1})=2\cdot dB(Ratio_{i})$

To show that this $Ratio_{i}$ satisfies the overall intention:

\log(V_{out}/V_{in})=\log(\prod _{i=1}^{N}Ratio_{i})=\sum _{i=1}^{N}\log(Ratio_{i})=\log(\alpha )\cdot CodeValue=c\cdot CodeValue

The different stages 1 .. N should function independently of each other, as to obtain 2^N 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. Because the stages operate independently, they can be inserted in the chain in any order.

Constant input resistance

The input resistance of any stage shall be independent of its on/off switch position, and must be equal to R_load.

This leads to:

{\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}}

With these equations, all resistor values of the circuit diagram follow easily after choosing values for N, $\alpha$ and R_load. (The value of R_source 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_source.

This leads to:

{\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}}

Again, all resistor values of the circuit diagram follow easily after choosing values for N, $\alpha$ and R_source. (The value of R_load does not influence the logarithmic behavior).

For example, with a R_load of 1 kΩ, and 1 dB attenuation, the resistor values would be: R_a = 108.7 Ω, R_b = 8195.5 Ω.

The next step (2 dB) would use: R_a = 369.0 Ω, R_b = 1709.7 Ω.

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 significantly 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 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. 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 DAC. Lengthening the codeword is needed in that approach to achieve sufficient dynamic range. This approach was implemented in a device from Analog Devices Inc.,^[2] protected through a 1981 patent filing.^[3]

References

^ "Multiplying DACs, flexible building blocks" (PDF). Analog Devices inc. 2010. Retrieved 29 March 2012.
^ "LOGDAC CMOS Logarithmic D/A Converter AD7118" (PDF). Analog Devices Inc. Archived from the original (PDF) on 25 August 2015. Retrieved 25 August 2015.
^ US patent 4521764, Burton, David P., "Signal-controllable attenuator employing a digital-to-analog converter", issued 4 June 1985

External links

Online calculator to configure logarithmic ladder networks

[1] "Multiplying DACs, flexible building blocks" (PDF). Analog Devices inc. 2010. Retrieved 29 March 2012.

[2] "LOGDAC CMOS Logarithmic D/A Converter AD7118" (PDF). Analog Devices Inc. Archived from the original (PDF) on 25 August 2015. Retrieved 25 August 2015.

[3] US patent 4521764, Burton, David P., "Signal-controllable attenuator employing a digital-to-analog converter", issued 4 June 1985

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@@ Line 1: / Line 1: @@
-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.
+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 [[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 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.
+The logarithmic behavior of the circuit is its main differentiator in comparison with [[digital-to-analog converter]]s (DACs) 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:
+As in [[digital-to-analog converter]]s, a binary number is applied to the ladder network, whose ''N'' bits are treated as representing an integer value:
-:<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.
+:<math>\mathrm{CodeValue} = \sum_{i=1}^N s_i \cdot 2^{i-1}</math>
+where <math>s_i</math> is 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.
+For comparison, recall a conventional linear [[digital-to-analog converter|DAC]] or [[Resistor ladder|R-2R network]] produces an output voltage signal of:
-(DA-converters that are designed to handle a '''variable''' input voltage are
+:<math>V_{out} = V_{in} \cdot c \cdot (\mathrm{CodeValue} + d )</math>
-termed '''multiplying DAC'''.<ref>{{cite web |url=http://www.analog.com/enwiki/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>)
+where <math>c</math> and <math>d</math> are design constants and where <math>V_{in}</math> typically is a '''constant reference''' voltage (or is a '''variable''' input voltage for a '''multiplying DAC'''.<ref>{{cite web |url=http://www.analog.com/enwiki/static/imported-files/overviews/AnalogMultiplyingDACs.pdf |title=Multiplying DACs, flexible building blocks |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.
+:<math>\log (V_{out} / V_{in}) = c \cdot \mathrm{CodeValue}</math>
+which can also be expressed as <math>V_{in}</math> multiplied by some base <math>\alpha</math> [[raised to the power of]] the code value:
+:<math>V_{out} = V_{in} \cdot \alpha ^ \mathrm{CodeValue}</math>
+where <math>c = \log(\alpha) \, .</math>
 == Circuit implementation ==
-[[File:Circuit diagram of logarithmic ladder DA converter, Mar 2012.gif|alt=Schematic diagram|Circuit diagram]]
+[[File:Circuit_diagram_of_logarithmic_ladder_DA_converter,_Mar_2012.png|alt=Schematic diagram|Circuit diagram]]
+This example circuit is composed of 4 stages, numbered 1 to 4, and includes a [[source resistance]] R<sub>source</sub> and [[load resistance]] R<sub>load</sub>.
+Each stage ''i'' has a designed input-to-output voltage attenuation ''Ratio<sub>i</sub>'' as:
-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:
+For logarithmic scaled attenuators, it is common practice to equivalently 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>
@@ Line 28: / Line 34: @@
 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>
+:<math>\log (V_{out}/V_{in}) = \log (\prod_{i=1}^N Ratio_i) = \sum_{i=1}^N \log (Ratio_i) = \log (\alpha) \cdot CodeValue = c \cdot CodeValue</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.
+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. Because the stages operate independently, they can be inserted in the chain in any order.
 === Constant input resistance ===
@@ Line 55: / Line 60: @@
 \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)
+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).
+For example, with a R<sub>load</sub> of 1 kΩ, and 1 dB attenuation, the resistor values would be:
+R<sub>a</sub>   = 108.7 Ω,
+R<sub>b</sub>  = 8195.5 Ω.
+The next step (2 dB) would use: R<sub>a</sub>   = 369.0 Ω,
+R<sub>b</sub>  = 1709.7 Ω.
 == 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.
+* Since the stages do not significantly 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.
+R-2R ladder networks used for ''linear'' digital-to-analog conversion are old ({{Slink|Resistor ladder|History}} mentions a 1953 article and a 1955 patent).
-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:
+Multiplying DACs 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 DAC. 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>{{cite web
+| url            = http://www.analog.com/media/en/technical-documentation/obsolete-data-sheets/1257410AD7118.pdf
-CMOS Logarithmic D/A Converter, [http://www.analog.com/enwiki/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>
+| title          = LOGDAC CMOS Logarithmic D/A Converter AD7118
+| publisher      = Analog Devices Inc.
+| archive-url    = https://web.archive.org/web/20150825062826/http://www.analog.com/media/en/technical-documentation/obsolete-data-sheets/1257410AD7118.pdf
+| archive-date   = 25 August 2015
+| access-date    = 25 August 2015
+}}</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 ==
@@ Line 78: / Line 96: @@
 * [http://www.vaneijndhoven.net/jos/attenuator-calculator/index.html Online calculator] to configure logarithmic ladder networks
-[[Category:Articles created via the Article Wizard]]
+[[Category:Analog circuits]]
-[[Category:Electrical components]]
-[[Category:Resistive components]]