# DAC Example Problem

In the previous chapter, we discussed the two types of DACs. This chapter discusses an example problem based on R-2R ladder DAC.

## Example

Let us find the value of analog output voltage of R-2R Ladder DAC for a binary input, $b_{2}b_{1}b_{0}$ = 100.

### Circuit Diagram and its Simplification

The circuit diagram of a 3-bit R-2R Ladder DAC when binary input, $b_{2}b_{1}b_{0}$ = 100 applied to it is shown in the following figure −

In the above circuit, there exists series and parallel combinations of resistors to the left of point A with respect to ground. So, we can replace that entire resistor network with a single resistor having resistance of $2R\Omega$.

The simplified circuit diagram is shown in the following figure −

We can replace the part of the network that is connected to the left of point B with respect to ground by using a Thevenin’s equivalent circuit. The modified circuit diagram is shown in the following figure −

In the above circuit, there exist a series combination of two resistors. Replace this combination with a single resistor. The final circuit diagram after simplification is shown in the following figure −

Now, the above circuit diagram looks like an inverting amplifier. It is having an input voltage of $-\frac{V_{R}}{2}$ volts, input resistance of $2R\Omega$ and feedback resistance of $2R\Omega$.

The output voltage of the circuit shown above will be −

$$V_{0}=-\frac{2R}{2R}\left(-\frac{V_{R}}{2}\right)$$

$$V_{0}=\frac{V_{R}}{2}$$

Therefore, the output voltage of 3-bit R-2R Ladder DAC is $\frac{V_{R}}{2}$ volts for a binary input, $b_{2}b_{1}b_{0}$ = 100.