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A logarithmic resistor ladder is an
electronic circuit An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. It is a type of electrical ...
composed of a series of
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active ...
s and
switch In electrical engineering, a switch is an electrical component that can disconnect or connect the conducting path in an electrical circuit, interrupting the electric current or diverting it from one conductor to another. The most common type of ...
es, designed to create an
attenuation In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable at ...
from an input to an output signal, where the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
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 converter In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. There are several DAC archite ...
s in general, and traditional R-2R Ladder networks specifically. Logarithmic attenuation is desired in situations where a large
dynamic range Dynamic range (abbreviated DR, DNR, or DYR) is the ratio between the largest and smallest values that a certain quantity can assume. It is often used in the context of signals, like sound and light. It is measured either as a ratio or as a base- ...
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 converter In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. There are several DAC archite ...
s, a binary word is applied to the ladder network, whose ''N'' bits are treated as representing an integer value according to the relation: :\mathrm = \sum_^N s_i \cdot 2^ where s_i represents a value 0 or 1 depending on the state of the ''ith'' switch. For a conventional DAC or R-2R network, the output signal value (its voltage) would be: :V_ = a \cdot (\mathrm + b ) \cdot V_ where a and b are design constants and where V_ typically is a constant reference voltage. (DA-converters that are designed to handle a variable input voltage are termed multiplying DAC.) In contrast, the logarithmic ladder network discussed in this article creates a behavior as: :\log (V_ / V_) = a \cdot (\mathrm + b ) where V_ 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 ''ratioi'' as: :Ratio_i = \text\; sw_i \;\text\; \alpha^ \;\text\; 1 For logarithmic scaled attenuators, it is common practice to express their attenuation in
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
s: :dB(Ratio_i) = 20 \log_ \alpha^ = 2^ \cdot 20 \cdot \log_ \alpha for i = 1 .. N and sw_i = 1 This reveals a basic property: dB(Ratio_) = 2 \cdot dB(Ratio_i) To show that this Ratio_i satisfies the overall intention: :\log (V_/V_) = \log (\prod_^N Ratio_i) = \sum_^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 2N 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 Rload. This leads to: :\begin R_ = (R_ \cdot R_) / (R_ + R_) \\ R_ + R_ = R_ \\ R_ / (R_ + R_) = Ratio_i \end With these equations, all resistor values of the circuit diagram follow easily after choosing values for N, \alpha and Rload. (The value of Rsource 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 Rsource. This leads to: :\begin R_ = R_ + R_ \\ R_ \cdot R_ / (R_ + R_) = R_ \\ R_ / (R_ + R_) = Ratio_i \end Again, all resistor values of the circuit diagram follow easily after choosing values for N, \alpha and Rsource. (The value of Rload does not influence the logarithmic behavior). For example, with a Rload of 1 kΩ, and 1 dB attenuation, the resistor values would be: Ra = 108.7 Ω, Rb = 8195.5 Ω. The next step (2 dB) would use: Ra = 369.0 Ω, Rb = 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.


Background

R-2R ladder networks used for Digital-to-Analog conversion are rather old. A historic description is in a patent 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., protected through a 1981 patent filing.


See also

*
Resistor ladder A resistor ladder is an electrical circuit made from repeating units of resistors. Two configurations are discussed below, a string resistor ladder and an R-2R ladder. An R–2R ladder is a simple and inexpensive way to perform digital-to-analog ...
*
Digital-to-analog converter In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. There are several DAC archite ...
* Attenuator circuits


References

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External links


Online calculator
to configure logarithmic ladder networks Analog circuits