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.

- String resistor ladder network (analog to digital conversion, or ADC)
- R–2R resistor ladder network (digital to analog conversion)
- Accuracy of R–2R resistor ladders
- Resistor ladder with unequal rungs
- See also
- References
- External links

An R–2R ladder is a simple and inexpensive way to perform digital-to-analog conversion, using repetitive arrangements of precise resistor networks in a ladder-like configuration. A string resistor ladder implements the non-repetitive reference 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 (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 resistors are required, while the R–2R resistor ladder only increases linearly with the number of bits, as it needs only resistors. - Advantage: higher impedance values can be reached using the same number of components.

A basic R–2R resistor ladder network is shown in Figure 1. Bit a_{n−1} (most significant bit, MSB) through bit a_{0} (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.^{ [1] }

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:

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 = 00000_{2}) and the maximum (VAL = 31 = 11111_{2}):

with steps (corresponding to VAL = 1 = 00001_{2})

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 01111_{2} to 10000_{2}. 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.

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 R/32 (i.e., much better than 3%). Further, to avoid problems at the 10000_{2}-to-01111_{2} 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 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.^{ [2] } On a printed circuit board, using discrete components, resistors of 1% accuracy would suffice for a 5-bit circuit, however with bit counts beyond this the cost of ever increasing precision resistors becomes prohibitive. For a 10-bit converter, even using 0.1% precision resistors would not guarantee monotonicity of output. This being said, high resolution R-2R ladders formed from discrete components are sometimes used, the nonlinearity being corrected in software. One example of such approach can be seen in the Korad 3005 power supply.

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-equivalent resistance of the lower-significance rungs. Figure 2 shows a linear 4-bit DAC with unequal resistors.

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.

An **operational amplifier** is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op-amp produces an output potential that is typically hundreds of thousands of times larger than the potential difference between its input terminals. Operational amplifiers had their origins in analog computers, where they were used to perform mathematical operations in many linear, non-linear, and frequency-dependent circuits.

In electronics, an **analog-to-digital converter** is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an input analog voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.

In electronics, a **comparator** is a device that compares two voltages or currents and outputs a digital signal indicating which is larger. It has two analog input terminals and and one binary digital output . The output is ideally

In electronics, a **digital-to-analog converter** is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function.

In electronics, a **Schmitt trigger** is a comparator circuit with hysteresis implemented by applying positive feedback to the noninverting input of a comparator or differential amplifier. It is an active circuit which converts an analog input signal to a digital output signal. The circuit is named a "trigger" because the output retains its value until the input changes sufficiently to trigger a change. In the non-inverting configuration, when the input is higher than a chosen threshold, the output is high. When the input is below a different (lower) chosen threshold the output is low, and when the input is between the two levels the output retains its value. This dual threshold action is called *hysteresis* and implies that the Schmitt trigger possesses memory and can act as a bistable multivibrator. There is a close relation between the two kinds of circuits: a Schmitt trigger can be converted into a latch and a latch can be converted into a Schmitt trigger.

In electronics, a **voltage divider ** is a passive linear circuit that produces an output voltage (*V*_{out}) that is a fraction of its input voltage (*V*_{in}). **Voltage division** is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

**Delta-sigma** modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into lower bit-count, higher-frequency digital signals as part of the process to convert digital signals into analog as part of a digital-to-analog converter (DAC).

This article illustrates some typical **operational amplifier applications**. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal features as shown in the diagram, but here a simplified schematic notation is used, many details such as device selection and power supply connections are not shown. Operational amplifiers are optimised for use with negative feedback, and this article discusses only negative-feedback applications. When positive feedback is required, a comparator is usually more appropriate. See Comparator applications for further information.

An **open collector** is a common type of output found on many integrated circuits (IC), which behaves like a switch that is either connected to ground or disconnected. Instead of outputting a signal of a specific voltage or current, the output signal is applied to the base of an internal NPN transistor whose collector is externalized (open) on a pin of the IC. The emitter of the transistor is connected internally to the ground pin. If the output device is a MOSFET the output is called **open drain** and it functions in a similar way. For example, the I²C bus is based on this concept.

A **flash ADC** is a type of analog-to-digital converter that uses a linear voltage ladder with a comparator at each "rung" of the ladder to compare the input voltage to successive reference voltages. Often these reference ladders are constructed of many resistors; however, modern implementations show that capacitive voltage division is also possible. The output of these comparators is generally fed into a digital encoder, which converts the inputs into a binary value.

A **switched capacitor** (**SC**) is an electronic circuit element implementing a filter. It works by moving charges into and out of capacitors when switches are opened and closed. Usually, non-overlapping signals are used to control the switches, so that not all switches are closed simultaneously. Filters implemented with these elements are termed "switched-capacitor filters", and depend only on the ratios between capacitances. This makes them much more suitable for use within integrated circuits, where accurately specified resistors and capacitors are not economical to construct.

**Verilog-AMS** is a derivative of the Verilog hardware description language that includes analog and mixed-signal extensions (AMS) in order to define the behavior of analog and mixed-signal systems. It extends the event-based simulator loops of Verilog/SystemVerilog/VHDL, by a continuous-time simulator, which solves the differential equations in analog-domain. Both domains are coupled: analog events can trigger digital actions and vice versa.

A **successive approximation ADC** is a type of analog-to-digital converter that converts a continuous analog waveform into a discrete digital representation via a binary search through all possible quantization levels before finally converging upon a digital output for each conversion.

An **active load** or **dynamic load** is a component or a circuit that functions as a current-stable nonlinear resistor.

A **clamper** is an electronic circuit that fixes either the positive or the negative peak excursions of a signal to a defined value by shifting its DC value. The clamper does not restrict the peak-to-peak excursion of the signal, it moves the whole signal up or down so as to place the peaks at the reference level. A **diode clamp** consists of a diode, which conducts electric current in only one direction and prevents the signal exceeding the reference value; and a capacitor, which provides a DC offset from the stored charge. The capacitor forms a time constant with the resistor load, which determines the range of frequencies over which the clamper will be effective.

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. To be referred to as *electronic*, rather than *electrical*, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.

A **digital potentiometer** is a digitally-controlled electronic component that mimics the analog functions of a potentiometer. It is often used for trimming and scaling analog signals by microcontrollers.

An **integrating ADC** is a type of analog-to-digital converter that converts an unknown input voltage into a digital representation through the use of an integrator. In its basic implementation, the dual-slope converter, the unknown input voltage is applied to the input of the integrator and allowed to ramp for a fixed time period. Then a known reference voltage of opposite polarity is applied to the integrator and is allowed to ramp until the integrator output returns to zero. The input voltage is computed as a function of the reference voltage, the constant run-up time period, and the measured run-down time period. The run-down time measurement is usually made in units of the converter's clock, so longer integration times allow for higher resolutions. Likewise, the speed of the converter can be improved by sacrificing resolution.

The **Kelvin-Varley voltage divider**, named after its inventors William Thomson, 1st Baron Kelvin and Cromwell Fleetwood Varley, is an electronic circuit used to divide voltages, i.e. to generate an output voltage as a precision ratio of an input voltage, with several decades of resolution. In effect, the **Kelvin–Varley divider** is an electromechanical precision digital-to-analog converter.

A **logarithmic resistor ladder** is an electronic circuit composed of a series of resistors and switches, designed to create an attenuation from an input to an output signal, where the logarithm of the attenuation ratio is proportional to a digital code word that represents the state of the switches.

Wikimedia Commons has media related to . Mixed analog and digital circuits |

- ECE209: DAC Lecture Notes - Ohio State University
- EE247: D/A Converters - Berkeley University of California
- Simplified DAC/ADC Lecture Notes - University of Michigan
- Digital to Analog Converters (slides) - Georgia Tech

- Tutorial MT-014: String DACs and Fully-Decoded DACs - Analog Devices
- Tutorial MT-015: Binary DACs - Analog Devices
- Tutorial MT-016: Segmented DACs - Analog Devices
- Tutorial MT-018: Intentionally Nonlinear DACs - Analog Devices

- R2R Resistor Ladder Networks - BT Technologies
- R/2R Ladder Networks Application Note - TT Electronics

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