Short note: Digital to analog conversion

A digital-to-analog converter (DAC) converts an n-bit binary word from the computer into a proportional analog voltage or current. In a digital control system it sits at the controller output, followed by a hold circuit and the actuator.

Output relation

$$V_o = V_{ref}\left(\frac{b_{n-1}}{2} + \frac{b_{n-2}}{4} + \cdots + \frac{b_0}{2^n}\right)$$

Resolution (one LSB) $= V_{ref}/2^n$.

Example: 4-bit DAC, $V_{ref} = 16$ V, input 1011: $V_o = 16(1/2 + 0 + 1/8 + 1/16) = 11$ V.

Common types

1. Weighted-resistor DAC: resistors $R, 2R, 4R, \dots, 2^{n-1}R$ feed an op-amp summer. It is simple but needs a wide range of precise resistor values.
2. R-2R ladder DAC: uses only two resistor values, $R$ in series and $2R$ in each leg; the contribution of each bit is halved at each node. It is the most widely used type.

 Vo <--+--R--+--R--+--R--+-----+
       |     |     |     |     |
      2R    2R    2R    2R    2R
       |     |     |     |     |
      b3    b2    b1    b0    GND
 (each bit switch goes to Vref for 1 or to GND for 0;
  Vo is buffered by an op-amp)

Specifications

• Resolution: number of bits, or the smallest output step.
• Accuracy and linearity: deviation from the ideal straight line.
• Settling time: time to reach the final value within ±½ LSB.
• Monotonicity: the output must not fall when the code increases.

The DAC output is held constant between updates by a latch/hold (zero order hold), giving a staircase signal.