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:

$\mathrm{CodeValue} = \sum_{i=1}^N s_i \cdot 2^{i-1}$ where $s_i$ represents a value 0 or 1 depending on the state of the i^th switch.

For a conventional DAC or R-2R network, the output signal value (its voltage) would be:

$V_{out} = a \cdot (\mathrm{CodeValue} + b ) \cdot V_{in}$ where $a$ and $b$ are design constants and where $V_{in}$ typically is a constant reference voltage.

(DA-converters that are designed to handle a variable input voltage are termed multiplying DAC.^[1])

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

$\log (V_{out} / V_{in}) = a \cdot (\mathrm{CodeValue} + b )$ where $V_{in}$ is a variable input signal.

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_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 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:

$\log (V_{out}/V_{in}) = \log (\prod_{i=1}^N Ratio_i) = \sum_{i=1}^N \log (Ratio_i) = a \cdot (CodeValue + b )$

for $b = 0$ and $a = \log(\alpha)$

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.

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)

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^[2] 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.,^[3] protected through a 1981 patent filing.^[4]

See also

• Resistor ladder
• Digital-to-analog converter
• Attenuator circuits

References

1. "Multiplying DACs, flexible building blocks" (PDF). Analog Devices inc. 2010. Retrieved 29 March 2012.
2. US patent 3108266, Gordon, B. M., "Signal Conversion Apparatus", issued 22 October 1963 
3. "LOGDAC CMOS Logarithmic D/A Converter AD7118" (PDF). Analog Devices Inc. Archived from the original (PDF) on 25 August 2015. Retrieved 25 August 2015.
4. 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