The binary-weighted DAC is appropriate for DACs with low resolving power. This is because it requires a wide range of precise resistors to perform error-free operations for high-order DACs. It is impossible to maintain the accuracy of the weighted DACs and is expensive. This leads to the R-2R ladder technique, which implements only two resistors for DAC functionality for every digital bit.

This post is a brief guide on R-2R DAC and shows why the latter is better than the binary-weighted DAC that we have discussed in previous articles.

## R-2R Ladder DAC

The R-2R configuration is a simple arrangement that consists of parallel and series resistors connected in cascaded form to an operational amplifier. We can use an operational amplifier in inverting or non-inverting form, depending on the polarity of the output voltage that we want to get from the DAC. R-2R ladder resistors act as voltage dividers along with the entire network, with the output voltage dependent on the input voltages.

### R-2R Ladder DAC Components

The ladder arrangement consists of two resistors, i.e., a base resistor R and a 2R resistor, which is twice the value of the base resistor. This feature helps to maintain a precise output analog signal without using a wide range of resistor values.

A pair of R and 2R is used for one input bit. The digital inputs are provided through binary switches connected to Vref for input 1 and GND for input 0.

### 3-bit R-2R Ladder DAC Circuit Diagram

Let us understand the concept through a 3-bit R-2R ladder DAC. The following diagram shows the R-2R 3-bit ladder DAC. The leftmost side of the circuitry has the least significant bit, i.e., B0, whereas B2, which is the most significant bit, connects to the amplifier. The binary inputs are given through the binary switches. So, when we need a high bit, simply connect the relevant bit to the reference voltage, and when we require a low bit, the switch connects to the ground potential.

## R-2R Ladder DAC Analysis with Thevenin Theorem

The circuit is simplified to obtain the voltage contribution of each bit. It can be accomplished using Thevenin’s theorem.

Thevenin’s theorem is a technique through which we can obtain an equivalent circuit of the concerned resistance network. A Thevenin circuit consists of a Thevenin resistance and a Thevenin voltage that we can replace in the circuit, and it still works the same as the original resistance network.

### Working of Thevenin Theorem

As we need a Thevenin resistance and a Thevenin voltage for substitution, we first calculate RTh by short-circuiting all the voltage sources and replacing the current sources with open circuits. VTH is the no-load output voltage and is entirely dependent on the position of the input switches. Now, replace the original circuit with the Thevenin circuit. Hence, we obtain the total output voltage of a 3-bit R-2R ladder network by considering only one high bit at a time and summing the individual voltages of each bit using superposition to obtain the transfer function of the DAC.

#### When LSB is high

Let us first consider the binary code 001. Its VTH and RTH will be calculated in three stages.

The first stage measures the VTh and RTh of the dotted block. The dotted block is separately shown in the right figure. We can see that it is just a voltage divider circuit. So, Vth is calculated using the formula

VTh= 2R x Vref / 2R+2R

VTh=Vref / 2

For measuring the Thevenin resistance, short circuit the reference voltage. Two resistances, 2R and 2R, become parallel to each other. So,

RTh = 2R || 2R

RTh = R

##### First Equivalent Circuit

Below is the equivalent circuit of the original after simplifying the first stage. The Thevenin equivalent of the first stage is connected in series to the rest of the circuit.

Now, we calculate the Thevenin circuit of the second stage. We will solve the dotted block in the second stage. Two resistors of the same value, i.e., R, connect in series. So we replace them by the equivalent resistance 2R in the diagram below. The circuit is again configured to be a voltage divider with a reference voltage of Vref/2. So,

VTh = (2R x Vref/2) / 2R+2R

VTh = Vref / 4

Again, for the Thevenin resistance, we consider the voltage source of this block to be zero. It gives the same Thevenin as the previous because of the exact arrangement, and we will replace the concerned portion with equivalent Thevenin values.

RTh = 2R || 2R

RTh = R

##### Second Equivalent Circuit

This is the resultant circuit, which will be solved in the third stage. The VTh and RTh are as follows:

VTh= (2R x Vref/4) / 2R+2R

VTh=Vref / 8

RTh=(R+R) || 2R

RTh=R

The solution depicts that whenever only B0 is connected to the reference voltage and B2 = B1 = 0, the output voltage of the DAC would be Vref/8.

#### When only bit B1 is high

The binary code is 010. It will be solved in a similar fashion. The first stage minimizes the dotted block. The combination of two 2R resistors is in series with R. The resultant resistance is

(2R||2R)+R=2R

The transformation is as follows:

After transforming, the Thevenin equivalents are measured. VTh and RTh turn out to be

VTh= (2R x Vref) / 2R+2R

VTh=Vref / 2

RTh=(R+R) || 2R

RTh=R

It is substituted, and then the Thevenin equivalents are found for the third time. The left figure is the reduced figure after calculating the second Thevenin value. After shorting the voltage source, Rth is

RTh=(R+R) || 2R

RTh=R

And VTh,

VTh= (2R x Vref/2) / 2R+2R

VTh=Vref / 4

Hence, when B1 is 1 and the other two bits are grounded, the output voltage of the network is Vref/4.

#### When MSB Bit is high

Starting from the left side of the circuit, the first three resistors, i.e., 2R, 2R, and R, result in the equivalent resistance 2R

(2R||2R)+R=2R

So, now the circuit is reduces to the figure on the right.

(2R||2R)+R=2R

The right figure minimizes and becomes the left figure shown below. Now, measure the Thevenin VTh and RTh, and the final Thevenin voltage is Vref/2.

VTh= (2R x Vref) / 2R+2R

VTh=Vref / 2

RTh=(R+R) || 2R

RTh=R

Vref/2 is the output voltage of the R-2R Ladder network when MSB is high and the remaining bits are 0.

#### When all three bits are high

When all 3 bits connects to the reference voltage, the output voltage will be the sum or superposition of all three voltages.

Vr-2r = (Vref / 2)+(Vref / 4)+(Vref / 8)

Vr-2r = 7Vref / 8

## R-2R Ladder DAC Output Voltage and Transfer Equations

This leads us to the general Vr-2r output voltage equation.

Vr-2r = Vref{B0/2^{(N)} + B0/2^{(N-1)} + B0/2^{(N-2)} + … + B0/2^{2} + B0/2^{1} }

Where N is the number of bits. Vr-2r is applied to the inverting operational amplifier, and the output voltage is measured. The output would be 180 degrees out of phase with the input Vr-2r. The following is the general output voltage equation for R-2R DAC:

Vout = -(Rf/R) x Vr-2r

Vout = -(Rf/R){B0/2^{(N)} + B0/2^{(N-1)} + B0/2^{(N-2)} + … + B0/2^{2} + B0/2^{1} }Vref

The gain of the DAC is decided by the Rf/R factor. For unity gain and in-phase output, we can use the buffer amplifier to perform the functions.

### Example

For example, if the DAC generates voltage output by 101 binary code with Rf = 4 ohms, R = 2 ohms, and Vref = 5 V will be as follows:

Vout = -(4/2){1/2^{(3)} + 0/2^{(2)} + 1/2^{1} }(5)

Vout = -6.25V

As this is a 3-bit DAC, it can have 8 different combinations of binary code, each producing a specific output voltage limited to the reference voltage. The table below provides the output voltage corresponding to every possible 3-bit combination.

Binary Inputs | Vout, Rf=4ohms, R=2ohms | ||
---|---|---|---|

B2 | B1 | B0 | Volts |

0 | 0 | 0 | -0 |

0 | 0 | 1 | -1.25 |

0 | 1 | 0 | -2.5 |

0 | 1 | 1 | -3.75 |

1 | 0 | 0 | -5.0 |

1 | 0 | 1 | -6.25 |

1 | 1 | 0 | -7.5 |

1 | 1 | 1 | -8.75 |

The table shows the step size is measured to be -1.25 volts, while the full-scale voltage is -8.75 volts.

## Advantages

- It can be fabricated easily.
- The configuration requires only two resistors, with one being twice the other in value, instead of a wide range of resistors.
- The output resistance remains the same despite the number of bits.
- Increasing the resolution does not degrade the performance.

## Disadvantages

- It has a slow conversion speed.

## Conclusion

In conclusion, this tutorial provides an in-depth overview of the R-2R Ladder DAC. It covers the introduction, the circuit, and its components, along with the analysis via the Thevenin theorem. This tutorial also discusses equations and output voltages with the help of examples. At last, we can see the advantages, disadvantages of the R-2R Ladder DAC. Hopefully, this was helpful in expanding your knowledge of DAC.

You may also like to read:

- DAC7715 Digital to Analog Converter
- DAC STM32F4 Discovery Board
- PT8211 16-Bit DAC
- DAC0808 8-bit Digital to Analog Converter
- How to use DAC Module of Pic Microcontroller
- DAC0832 8-BIT DIGITAL TO ANALOG CONVERTER IC

This concludes today’s article. If you face any issues or difficulties, let us know in the comment section below.