Weighted-Resistor D/A Converter

A binary weighted resistor ladder D/A converter is shown in figure 1. 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 are inversely proportional to the numerical significance of the corresponding binary digital
3. a reference voltage V_ref and
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 ....... R_n are binary weighted resistors, thus
R₁ = R
R₂ = 2R
R₃ = 4R
..............
.............
.............
R_n = (2^n-1) R

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

Each bit controls a switch s_i that is connected to V_ref.
when       a_i = 1 , then bit is ON, and when a_i = 0, then bit is OFF.
The reference voltage source V_R is considered to have zero internal impedance. The resistor that are connected to the switches have 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 have the value of
(2^n-1) R.
The current flowing in the summing amplifier is

Multiplying and dividing by (2)^n-1 R

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

When all the bits of digital word have 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_i coefficients are zero, then the output voltage (current) is zero.
The maximum output voltage V_o = -R_iI 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

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 specially important in D/A converters operation over a wide temperature range.
3. 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.
4. As the switches represent finite impedance that are connected in series with the weighted resistors and their magnitudes and variations have to be taken in to account in a D/A converter design.