Logarithmic resistor ladder

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:

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

where ${\displaystyle s_{i}}$ 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:

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

where ${\displaystyle c}$ and ${\displaystyle d}$ are design constants and where ${\displaystyle V_{in}}$ 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:

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

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

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

where ${\displaystyle c=\log(\alpha )\,.}$

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:

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

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

${\displaystyle dB(Ratio_{i})=20\log _{10}\alpha ^{2^{i-1}}=2^{i-1}\cdot 20\cdot \log _{10}\alpha }$ for ${\displaystyle i=1..N}$ and ${\displaystyle sw_{i}=1}$

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

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

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

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.

• 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.

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]

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

• Online calculator to configure logarithmic ladder networks