Resistor ladder

Each tap of the string generates a different voltage, which can be compared with another voltage: this is the basic principle of a flash ADC (analog-to-digital converter). Often a voltage is converted to a current, enabling the possibility to use an R–2R ladder network.

• Disadvantage: for an n-bit ADC, the number of resistors grows exponentially, as ${\displaystyle 2^{n}}$ resistors are required, while the R–2R resistor ladder only increases linearly with the number of bits, as it needs only ${\displaystyle 2n}$ resistors.
• Advantage: higher impedance values can be reached using the same number of components.

A string resistor can function as a DAC by having the bits of the binary number control electronic switches connected to each tap.[4]

A voltage mode R–2R resistor ladder network is shown in Figure 1. Bit a_n−1 (most significant bit, MSB) through bit a₀ (least significant bit, LSB) are driven from digital logic gates. Ideally, the bit inputs are switched between V = 0 (logic 0) and V = V_ref (logic 1). The R–2R network causes these digital bits to be weighted in their contribution to the output voltage V_out. Depending on which bits are set to 1 and which to 0, the output voltage (V_out) will have a corresponding stepped value between 0 and V_ref minus the value of the minimal step, corresponding to bit 0. The actual value of V_ref (and the voltage of logic 0) will depend on the type of technology used to generate the digital signals.[7]

For a digital value VAL, of a R–2R DAC with N bits and 0 V/V_ref logic levels, the output voltage V_out is:

${\displaystyle V_{o}=V_{ref}\times {\tfrac {\text{Value}}{2^{N}}}}$

For example, if N = 5 (hence 2^N = 32) and V_ref = 3.3 V (typical CMOS logic 1 voltage), then V_out will vary between 0 volts (VAL = 0 = 00000₂) and the maximum (VAL = 31 = 11111₂):

${\displaystyle V_{o}=3.3\,{\text{V}}\times {\tfrac {31}{2^{5}}}=3.196875\,{\text{V}}}$

with steps (corresponding to VAL = 1 = 00001₂)

${\displaystyle \Delta V_{o}=3.3\,{\text{V}}\times {\tfrac {1}{2^{5}}}=0.103125\,{\text{V}}}$

The R–2R ladder is inexpensive and relatively easy to manufacture, since only two resistor values are required (or even one, if R is made by placing a pair of 2R in parallel, or if 2R is made by placing a pair of R in series). It is fast and has fixed output impedance R. The R–2R ladder operates as a string of current dividers, whose output accuracy is solely dependent on how well each resistor is matched to the others. Small inaccuracies in the MSB resistors can entirely overwhelm the contribution of the LSB resistors. This may result in non-monotonic behavior at major crossings, such as from 01111₂ to 10000₂.

Depending on the type of logic gates used and design of the logic circuits, there may be transitional voltage spikes at such major crossings even with perfect resistor values. These can be filtered with capacitance at the output node (the consequent reduction in bandwidth may be significant in some applications). Finally, the 2R resistance is in series with the digital-output impedance. High-output-impedance gates (e.g., LVDS) may be unsuitable in some cases. For all of the above reasons (and doubtless others), this type of DAC tends to be restricted to a relatively small number of bits; although integrated circuits may push the number of bits to 14 or even more, 8 bits or fewer is more typical.

The R–2R DAC described above directly outputs a voltage and so is called voltage mode (or sometimes normal mode).

Since the output impedance is independent of digital code, the analog output may equally-well be taken as a current into a virtual ground, a configuration called current mode (or sometimes inverted mode). Using current mode, the gain of the DAC may be adjusted with a series resistor at the reference voltage terminal.[8] The current for all bits pass through an equivalent resistance of 2R to ground. The less significant the bit, the more resistors its signal must pass through. At each node each bit's current is divided by two.[9]

Resistors used with the more significant bits must be proportionally more accurate than those used with the less significant bits; for example, in the R–2R network discussed above, inaccuracies in the bit-4 (MSB) resistors must be insignificant compared to 1⁄32 (~3.1%) of R. Further, to avoid problems at the 10000₂-to-01111₂ transition, the sum of the inaccuracies in the lower bits must also be significantly less than that. The required accuracy doubles with each additional bit: for 8 bits, the accuracy required will be better than 1⁄256 (~0.4%).

However, variances for resistances when manufactured in a single component tend to be much lower than variances between components or between batches of manufacturing, and hence a resistor network can be purchased as a single component. And within integrated circuits, high-accuracy R–2R networks may be printed directly onto a single substrate using thin-film technology, ensuring the resistors share similar electrical characteristics. Even so, they must often be laser-trimmed to achieve the required precision. Such on-chip resistor ladders for digital-to-analog converters achieving 16-bit accuracy have been demonstrated.[10]