R-2R ladder D/A converter

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 20R+21R+….+27R. So the largest resistor corresponding to bit b8 is 128 times the value of the smallest resistor correspond to b1. 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.

R-2R ladder D/A converter

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.

lader1

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.

lader2

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

lader3

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

lader4

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

VA = VR*2R/(R+R+2R)

      = VR/2

Thus when bit 1000 the output is VR/2. Similarly it can be found that using above process for bit 0100 the output will be VR/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

VA = VR/21 + VR/22 + VR/23 + ……. + VR/2n

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.

R-2R ladder D/A converter

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 = VR*(RF/R)[b1/21 + b2/22 + b3/23 + b4/24]

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

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