Digital to Analog Converters

Digital to analog conversion:

The process of converting discrete time digital signal into the continuous analog signal is called Digital to analog conversion(DAC). The device used for this process are called Digital to analog converters. DAC converters produce output current or voltage proportional to the corresponding digital quantity(binary digit) applied to its input. In many applications , the digital output of the microcomputer is converted into analog quantity which is used for the control of relay , small motors ,actuators etc

Types of Digital to analog converters:

1. Weighted Resistors Network(WRN) DAC
2. R-2R Ladder Network

1.Weighted Resistors Network(WRN) DAC:

The Weighted Resistors Network DAC, which contains individual electrical components for each bit of the DAC connected to a summing point. These precise voltages or currents sum to the correct output value. This is one of the fastest conversion methods but suffers from poor accuracy because of the high precision required for each individual voltage or current. Such high-precision components are expensive, so this type of converter is usually limited to 8-bit resolution or less.

It consists of the following four major components.

1. n switches one for each bit applied to the input
2. a weighted resistor ladder network, where the resistance is inversely proportional to the numerical significance of the corresponding binary digital
3. a reference voltage V_ref
4. a summing amplifier that adds the current flowing in the resistive network to develop a signal that is proportional to the digital input.

The behavior of the circuit may be analyzed easily by using "Millman's theorem". It state that "the voltage appearing at any node in a resistive network is equal to the summation of the current entering the node (assuming the node voltage is zero) divided by the summation of the conductance connected to the mode".

Mathematically we can write,

Assume that the resistor R1 , R2 , R3 ....... Rn is binary weighted resistors, thus

A Resistor Ladder Network, can deliver a binary number say number of n bits

Each bit controls a switch S_i that is connected to V . when a_i= 1 , then the bit is ON, and when a_i= 0, then the bit is OFF.

The reference voltage source V_R is considered to have zero internal impedance. The resistor that is connected to the switches has a value such as to make the current flow proportion to the binary weight of the respective input. But the resistor in the MSB position has the value R, the next has the value 2R etc. The resistor of the LSB has the value of (2^n-1 ) R. The current flowing in the summing amplifier is given by,

Multiplying and dividing by (2^n-1 )R on numerator and denominator of right-hand side, we get,

Above relation shows that output voltage of the D/A converter is proportional to a number represented by the switch that is connected to V_R i.e. a_i = 1 Maximum current will flow when all a coefficient is 1, i.e.

When all the bits of the digital word have a value of 1, then the output current of D/A converter is termed the full-scale output current and is an important design parameter. On the other hand, if all switches are open i.e. all a coefficients are zero, then the output voltage (current) is zero. The maximum output voltage V_o= ­-IR_i depends on the feedback resistor R_f . As the operational amplifier is operated in the negative feedback mode for the purpose of summing so that it performs as an excellent current to voltage converter.

Advantages:

1.As only one resistor is used per it in the resistor network, thus it is an economical D/A converter

Disadvantages / Limitations:

1. Resistors used in the network have a wide range of values, so it is very difficult to ensure the absolute accuracy and stability of all the resistors.
2. It is very difficult to match the temperature coefficients of all the resistors. This factor is especially important in D/A converters operation over a wide temperature range.
3. As the switches represent finite impedance that is connected in series with the weighted resistors and their magnitudes and variations have to be taken into account in a D/A converter design.
4. When n is so large, the resistance corresponding to LBS can assume a large value, which may be comparable with the input resistance of the amplifier. This leads to erroneous results.

R-2R Ladder Network:

The R-2R LadderDAC which is a binary-weighted DAC that uses a repeating cascaded structure of resistor values R and 2R. This improves the precision due to the relative ease of producing equal valued-matched resistors (or current sources).In the case of a weighted resistor, DAC requires a wide range of resistance values and switches for each bit position if high accuracy conversion is required. A digital to analog converter with an R-­2R ladder network as shown in the figure below eliminates these complications at the expense of an additional resistor for each bit.

The operation of R­-2R ladder DAC is easily explained considering the weights of the different bits one at a time. This can be followed by superposition to construct analog output corresponding any digital input word. Let only the MSB is turned ON in the first case, and all other bits are OFF, a simplified equivalent circuit can be drawn as shown in the figure below.

The equivalent circuit with only switch S connected to the reference voltage is shown in figure below

By using principle of superposition, If all contributions are summed up then the resultant output is given as,

Note that a depends on whether the switch is at 0 or at V . Thus, the output of the DAC is proportional to the sum of the weights represented by those switches that are connected to V and the ratio of resistors in the R­ 2R ladder network. From this circuit, the voltage at node (n­-1) is given by,

As the input terminal of the operational amplifier is at virtual ground. If the feedback resistor R for the operational amplifier is taken as 3R, the corresponding output voltage due to the MSB alone is

Let next switch S_n-2 at V_R with all other switches at zero. Here, the current at node (n-2) divide equally to the right and to the left resulting in a voltage at node (n-2)

This voltage is attenuated by a factor of 2 at node (n-1) as

Output voltage due to node (n­-2) at the operational amplifier is

References

1.http://www.daenotes.com/electronics/

2.https://en.wikipedia.org/wiki/Digital-to-analog_converter