# Engineering: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 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:

- [math]\displaystyle{ \mathrm{CodeValue} = \sum_{i=1}^N s_i \cdot 2^{i-1} }[/math]

where [math]\displaystyle{ s_i }[/math] 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:

- [math]\displaystyle{ V_{out} = V_{in} \cdot c \cdot (\mathrm{CodeValue} + d ) }[/math]

where [math]\displaystyle{ c }[/math] and [math]\displaystyle{ d }[/math] are design constants and where [math]\displaystyle{ V_{in} }[/math] 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:

- [math]\displaystyle{ \log (V_{out} / V_{in}) = c \cdot \mathrm{CodeValue} }[/math]

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

- [math]\displaystyle{ V_{out} = V_{in} \cdot \alpha ^ \mathrm{CodeValue} }[/math]

where [math]\displaystyle{ c = \log(\alpha) \, . }[/math]

## 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:

- [math]\displaystyle{ 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 equivalently express their attenuation in decibels:

- [math]\displaystyle{ dB(Ratio_i) = 20 \log_{10} \alpha^{2^{i-1}} = 2^{i-1} \cdot 20 \cdot \log_{10} \alpha }[/math] for [math]\displaystyle{ i = 1 .. N }[/math] and [math]\displaystyle{ sw_i = 1 }[/math]

This reveals a basic property: [math]\displaystyle{ dB(Ratio_{i+1}) = 2 \cdot dB(Ratio_i) }[/math]

To show that this [math]\displaystyle{ Ratio_i }[/math] satisfies the overall intention:

- [math]\displaystyle{ \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]

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:

- [math]\displaystyle{ \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]\displaystyle{ \alpha }[/math] 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:

- [math]\displaystyle{ \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]\displaystyle{ \alpha }[/math] 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]}

## See also

## References

- ↑ "Multiplying DACs, flexible building blocks". Analog Devices inc.. 2010. http://www.analog.com/static/imported-files/overviews/AnalogMultiplyingDACs.pdf. Retrieved 29 March 2012.
- ↑ "LOGDAC CMOS Logarithmic D/A Converter AD7118". Analog Devices Inc.. http://www.analog.com/media/en/technical-documentation/obsolete-data-sheets/1257410AD7118.pdf.
- ↑ Burton, David P., "Signal-controllable attenuator employing a digital-to-analog converter", US patent 4521764, issued 4 June 1985

## External links

- Online calculator to configure logarithmic ladder networks

Original source: https://en.wikipedia.org/wiki/Logarithmic resistor ladder.
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