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| A '''resistor ladder''' is an electrical circuit made of repeating units of [[resistors]]. Two configurations are discussed below, a string resistor ladder and a R-2R ladder.
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| An R-2R Ladder is a simple and inexpensive way to perform [[Digital-to-analog converter|digital-to-analog conversion]], using repetitive arrangements of precision [[resistor network]]s in a [[ladder]]-like configuration. A string resistor ladder implements the non-repetitive reference network.
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| ==String resistor ladder network (analog to digital conversion, or ADC)==
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| A string of many, often equally dimensioned, resistors connected between two reference voltages is a resistor string ladder network. The resistors act as [[voltage divider]]s 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]] (analog to digital converter). Often a voltage is converted to a ''current'' enabling the possibility to use a R-2R ladder network.
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| * Disadvantage: for a n-bits ADC the number of resistors grows [[exponential growth|exponentially]], as <math>\scriptstyle 2^n-1</math> resistors are required, while the R-2R resistor ladder only increases linearly with the number of bits as it needs only <math>\scriptstyle 2n</math> resistors.
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| * Advantage: Higher impedance values can be reached using the same number of components.
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| ==R-2R resistor ladder network (digital to analog conversion, or DAC)==
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| [[Image:r2r-ladder.png|thumb|420px|right|Figure 1: n-bit R-2R resistor ladder]]
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| A basic R-2R resistor ladder network is shown in Figure 1. Bit a<sub>n-1</sub> MSB (most significant bit) to Bit a<sub>0</sub> LSB (least significant bit) are driven from digital logic gates. Ideally, the bits are switched between 0 volts (logic ''0'') and V<sub>ref</sub> (logic ''1''). The R-2R network causes the digital bits to be weighted in their contribution to the output voltage V<sub>out</sub>. In this circuit 5 bits are shown (bits 4-0), giving (2<sup>5</sup>) or 32 possible analog voltage levels at the output. Depending on which bits are set to ''1'' and which to ''0'', the output voltage (''out'') will be a corresponding [[Quantization (signal processing)|stepped value]] between 0 volts and (V<sub>ref</sub> minus the value of the minimum step, Bit0). The actual value of V<sub>ref</sub> (and 0 volts) will depend on the type of technology used to generate the digital signals.<ref>http://www.interfacebus.com/voltage_threshold.html</ref>
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| For a digital value VAL, of a R-2R DAC of N bits of 0 V/V<sub>ref</sub>, the output voltage V<sub>out</sub> is:
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| :V<sub>out</sub> = V<sub>ref</sub> × VAL / 2<sup>N</sup>
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| In the example shown, N = 5 and hence 2<sup>N</sup> = 32. With V<sub>ref</sub> = 3.3 V (typical CMOS logic ''1'' voltage), V<sub>out</sub> will vary between 00000, VAL = 0 and 11111, VAL = 31.
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| Minimum (single step) VAL = 1, we have
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| :V<sub>out</sub> = 3.3 × 1 / 32 = 0.1 volts
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| Maximum output (11111) VAL = 31, we have
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| :V<sub>out</sub> = 3.3 × 31 / 2<sup>5</sup> = 3.2 volts
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| The R-2R ladder is inexpensive and relatively easy to manufacture since only two resistor values are required (or 1, 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 divider]]s whose output accuracy is solely dependent on how well each resistor is matched to the others. Small inaccuracies in the higher significant bit resistors can entirely overwhelm the contribution of the less significant bits. This may result in non-[[monotonic]] behavior at major crossings, such as from 01111 to 10000. 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 for instance (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.
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| === Accuracy of R-2R resistor ladders ===
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| Resistors used with the more significant bits must be proportionally more accurate than those used with the lower significant bits; for example, in the R-2R network shown above, inaccuracies in the Bit4 MSB resistors must be insignificant compared to R/32 (i.e., much better than 3%). Further, to avoid problems at the ''10000'' to ''01111'' transition, the sum of the inaccuracies in the lower bits must be significantly less than R/32. The required accuracy doubles with each additional bit—for 8 bits, the accuracy required will be better than 1/256 (0.4%). 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 [[electricity|electrical]] characteristics. Even so, they must often be [[laser trimming|laser trimmed]] to achieve the required precision. Such [[Integrated circuit|on-chip]] resistor ladders for [[digital-to-analog converter]]s achieving 14 bits accuracy have been demonstrated. On a printed circuit board, using discrete components, high precision resistors of 1% accuracy may be employed for a 5 bit circuit, however with bit counts beyond this the cost of ever increasing precision resistors becomes prohibitive. Even for a 5 bit circuit, to achieve high accuracy, it will be necessary to select matched resistors or to adjust individual resistors to a common value by adding high value resistors in parallel.
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| == Resistor Ladder with Unequal Rungs ==
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| [[Image:UnequalLadder.svg|thumb|336px|right|Figure 2: 4-bit linear R-2R DAC using unequal resistors]]
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| It is not necessary that each "rung" of the R-2R ladder use the same resistor values. It is only necessary that the 2R value matches the sum of the R value plus the [[Thévenin's theorem|Thévenin-equivalent]] resistance of the lower-significance rungs. Figure 2 shows a linear four-bit DAC with unequal resistors.
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| This allows a reasonably accurate DAC to be created from a heterogeneous collection of resistors by forming the DAC one bit at a time. At each stage, resistors for the "rung" and "leg" are chosen so that the rung value matches the leg value plus the equivalent resistance of the previous rungs. The rung and leg resistors can be formed by pairing other resistors in series or parallel in order to increase the number of available combinations. This process can be automated.
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| == See also ==
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| * [[Logarithmic resistor ladder]]
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| * [[Digital-to-analog converter]]
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| * [[Covox Speech Thing]]
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| * [[Voltage ladder]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| {{Commons category|Mixed analog and digital circuit diagrams}}
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| * [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)
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| * [http://www.ikalogic.com/dac08.php Digital to Analog converters(DAC), using R/2R networks]
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| * [http://www.ece.osu.edu/~pavlict/ece209/lab7_DAC/lab7_DAC_notes.pdf ECE 209: DAC Lecture Notes] — Gives short explanation of ''R''-''2R'' ladder with figure showing how currents are divided at each new branch. Dead Reference.
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| * [http://www.irctt.com/pdf/LADDERNETWORKS.pdf Discussion on R-2R ladders]
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| * [http://bretmulvey.com/R2RLadder Unequal Resistor R-2R Ladder Optimizer]
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| [[Category:Analog circuits]]
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