Resistor Ladder

Two Resistor Ladder configurations are known, a string resistor ladder and a R-2R ladder.A R-2R Ladder is the most simple and inexpensive way to perform digital-to-analog conversion, using repetitive arrangements of precision resistor networks in a ladder-like configuration. A string resistor ladder implementsthe non-repetitive reference network.

tring resistor ladder network

A string of many, often equally dimensioned, resistors connected between two reference voltages is a resistor string ladder network. The resistors act as voltage dividers between the referenced voltages. 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. Often a voltage is converted to a "current" enabling the possibility to use a R-2R ladder network.

* Disadvantage: for a n-bits ADC the number of resistors grows exponentially, as $2^n-1$ resistors are required, while the R-2R resistor ladder only increases linearly with the number of bits as it needs only $2n$ resistors.
* Advantage: Higher impedance values can be reached using the same number of components.

R-2R resistor ladder network

A basic R-2R resistor ladder network is shown in Figure 1. The digital inputs or bits range from the most significant bit (MSB) to the least significant bit (LSB). The bits are switched between either 0V or VREF and depending on the state and location of the bits VOUT will vary between 0V and VREF minus one LSB's voltage. Because VREF is a digital signal, it is often +5V, as this is a logic 1.

For a digital value VAL, of a R-2R DAC of N bits of 0V/VREF, the output voltage VOUT is:

VOUT = VREF x VAL / 2^N

Therefore, for VREF = 5V (typical TTL logic 1 voltage), N = 2 and VAL = 3 (maximum absolute number you can represent with 2 bit), we have

VOUT = 5 x 3 / 4 = 3.75

You would expect VOUT to be able to reach VREF, but it doesn't, and there's a reason: 0V counts as a "digital step", otherwise we would have an extra step above the maximum number of steps allowed by the number of bits.

The MSB causes the greatest change in output voltage and the LSB causes the smallest. The R-2R ladder is inexpensive and relatively easy to manufacture since only two resistor values are required. 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. But a very small inaccuracy in the more significant resistors will entirely overwhelm the effect of the less significant digits changing, giving non-monotonic behavior at major crossings, such as 01111 -> 10000. Also, there will be (decoding) voltage spikes at such major crossings even with perfect resistor values. These can be filtered, with capacitance at the output node for instance; that will of course reduce the bandwidth. Also, the 2R resistance is augmented by the digital output impedances. For these reasons and doubtless others, this kind of DAC is only used with caution. In mass production the cost of a DAC chip would be small as opposed to the problems of installing and soldering numerous small parts.

Accuracy of resistor ladders

On a printed circuit board, high performance R-2R networks may be printed directly onto a single substrate using a single film, making the resistors share similar electrical characteristics, and may be laser trimmed to provide increased precision. With on-chip resistor ladders analog-to-digital converter achieving 14 bits accuracy have been demonstrated.

External links

* [http://www.bitechnologies.com/pdfs/resistorladder.pdf BI Technologies R2R Resistor Ladder Networks] (this reference needs to be re-checked for correctness concerning the variation of the output voltage on the R-2R ladder)
* [http://www.ikalogic.com/dac08.php Digital to Analog converters(DAC), using R/2R networks]