To overcome huge range of resistor used in weighted resistor D/A converter, R-2R ladder D/A converter is introduced. In my previous post I discussed about **weighted resistor D/A converter**. But the vital problem in weighted register D/A converter is use of huge range of different resistance. Suppose we have to design 8-bit weighted register D/A converter then we need the resistance value **2**^{0}R+2^{1}R+….+2^{7}R. So the largest resistor corresponding to bit b_{8} is 128 times the value of the smallest resistor correspond to b_{1}. But in case of R-2R ladder D/A converter, Resistors of only two value (R and 2R) are used. Now in bellow see the simple ladder network.

In ladder circuit the output voltage is also weighted sum of the corresponding digital input. Let take an example to understand how it works? As we can see the above network is a 4-bit ladder network so we take an example to convert analog signal correspond of **1000** digital bit. For **1000** bit we can see only MSB got 1 and rest all bits got 0. See the bellow picture to understand how it work if it got **1000**.

Now see at node1 (N1) resistor 2R connecting in b4 parallel with resistor 2R. And those 2R parallel 2R resistors make equivalent register of R shown in bellow diagram.

Now for N2 same thing happen B3 series with 2R and parallel with R + R resistors. It will also make equivalent resistor R at N3. See the bellow diagram

Repeating the same process we got equivalent of R resistor at N4.

Now at N4, if we calculate the output analog equivalent voltage then we will get

**V**_{A }= V_{R}*2R/(R+R+2R)

** = V**_{R}/2

Thus when bit 1000 the output is V_{R}/2. Similarly it can be found that using above process for bit 0100 the output will be V_{R}/4, for bit 0010 output will be VR/8 and for bit 0001 output will be VR/16.

By using superposition theorem we can find in any n-bit ladder network the output voltage will be

**V**_{A} = V_{R}/2^{1} + V_{R}/2^{2} + V_{R}/2^{3} + ……. + V_{R}/2^{n}

Where n is the total number of bits at the input.

Now see the practical circuit arrangement of 4-bit R-2R ladder D/A converter using op amp.

The inverting input terminal of the op amp work as a summing amplifier for the ladder inputs. So we can get out put voltage by bellow equation.

**V0 = V**_{R}*(R_{F}/R)[b1/2^{1} + b2/2^{2} + b3/2^{3} + b4/2^{4}]

## Hope above discussion will help to understand the working principle of R-2R ladder D/A converter.