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Binary Weighted Resistor DAC consists of an inverting amplifier op-amp and a string of weighted resistors to distinguish each bit starting from LSB to MSB position. Each resistor represents a digital bit to be converted into analog form.

The maximum information in the digital real world is in analog form. For example, getting data from sensors, audio applications, multimedia, everything is an analog form. Therefore, we use DAC and ADC to convert data from one form to another. We digitize these signals for fast computing which saves us time and can process complex data in a few microseconds. But the real world is unable to understand these combinations of binary bits and needs to be converted back into analog information with the help of DAC.

## What is a DAC?

A DAC is an electronic circuit that takes in digital data as input and transforms it into an output analog signal. It is actually the output voltage proportional to the binary code given at DAC input and then used to drive various circuits. They have vast applications but let us take a simple example of an audio system to showcase its importance.

The DAC in the audio system takes in the audio binary data. It processes and converts all the 0s and 1s to a continuous analog signal which is then transmitted to an amplifier. Thus the sound we hear from the speaker is the amplified analog signal.

There are different types of DACs implemented with a special technique to generate analog output but this post will overview Binary Weighted Resistor digital to analog converter.

## Binary Weighted Resistor DAC

It is the type of DAC that transforms a particular binary code into an equivalent analog signal. If the binary code given at the input terminal is altered continuously, the output will change as well. This type consists of weighted resistors whose values are kept as the multiples of two and an inverted summing operational amplifier which results in an output signal with 180 degrees phase shift. The reference voltage is either generated internally or is provided to the DAC converter to decide the maximum output voltage of the converter.

Before discussing the transfer function of the binary-weighted resistor DAC, we should be familiar with resolution and full-scale output voltage.

### DAC Resolution

The resolution of ADC is given by the number of bits. It depicts the number of output levels a DAC can generate. The general formula of resolution is:

Resolution=2N

Where N represents the number of bits. For a 4-bit DAC, the resolution will be 16.

### Step size

It is the smallest change that a DAC can produce in the output or we can say that it is a difference between two consecutive voltage levels of DAC. It is determined by dividing the reference voltage by 2 raised to the number of bits.

Step size= Vref/2N

The reference voltage is also multiplied by a factor i.e is the internal gain of the ADC that depends on the design of the DAC.

The resolution and step size are inversely related to each other. Increasing the resolution decreases the step size between the output levels that lead to smooth and precise analog output.

### Full-scale Output Voltage

The maximum voltage that we can attain from a DAC is called full-scale output voltage. FSO voltage depends entirely on the number of bits and the reference voltage given to the digital to analog converter. It is obtained through

FSO=(2N-1) x Vref/2N

## Binary Weighted Resistor DAC Working

The configuration uses a summing amplifier whose output voltage is proportional to the sum of the voltages applied at the input.

Vout = – {(Rf /R0) V0 + (Rf /R1) V1 + (Rf /R2) V2 +… + (Rf/Rn-1) Vn-1}

### Circuit

If resistors with precise values are connected to each of the input voltages to scale the gain, it starts working as a DAC. All the input voltages are connected to the same reference voltage so the formula becomes

Vout = – {(Rf /R0) Vref + (Rf /R1) Vref + (Rf /R2) Vref +… + (Rf/Rn-1) Vref} Vout = – Vref {(Rf /R0) + (Rf /R1) + (Rf /R2) +… + (Rf/Rn-1) }

In the circuitry, the binary input is applied through digital switches that alter between the reference voltage and ground. If we want the input bit to be 1, the switch should be connected to Vref and for digital input 0, it is connected to the ground. In the equation given below, B0 is the least significant bit and Bn-1 is the most significant bit.

Vout = – Vref { B0 (Rf /R0) + B1 (Rf /R1)+ B2 (Rf /R2) +… + Bn-1 (Rf/Rn-1) }

For the compactness, Rf = R whereas the weighted resistors i.e R0, R1, R2,…, Rn-1 are scaled with the scaling factor 2(N-1)-n to give the output gain according to the weight of each bit.

Rn = 2^{(N-1)-n}R

where N represents the number of bits and n is the bit position. So, the generalized output of the binary weighted resistor DAC is given out as

Vout = – Vref { B0 (R/2^{(N-1)}R) + B1 (R /2^{(N-2)}R) + B2 (R /2^{(N-3)}R) +… + BN-2 (R /2^{1}R) +BN-1 (R /2^{0}R) }

Vout = – Vref { B0 (1/2^{(N-1)}) + B1 (1 /2^{(N-2)}) + B2 (1/2^{(N-3)}) +… + BN-2 (1 /2^{(1)}) +BN-1 (1 /2^{(0)}) }

For a 4 bit binary DAC the output voltage equation is

Vout = – Vref { B0 (1/2^{3}) + B1 (1 /2^{2}) + B2 (1/2^{1}) + B3 (1/2^{0}) } Vout = – Vref { B0 (1/8) + B1 (1 /4) + B2 (1/2) + B3 }

## Binary Weighted Resistor DAC Example

Let us take the example with a 4-bit binary code. Assuming Vref = 5 V and code as 1011, the weighted output voltage is

Vout = – 5 { 1 (1/2^{3}) + 1 (1/2^{2}) + 0 (1/2^{1}) + 1 (1/2^{0}) } Vout= – 5 { (1/8) + (1/4) + 0 + 1 } Vout= – 6.875 V

Remember that B0 is taken as the LSB and B3 is the MSB.

As this is a 4-bit DAC, we can have 16 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 4-bit code.

Binary Inputs | Vout | |||

B3 | B2 | B1 | B0 | Volts |

0 | 0 | 0 | 0 | -0 |

0 | 0 | 0 | 1 | -0.625 |

0 | 0 | 1 | 0 | -1.25 |

0 | 0 | 1 | 1 | -1.875 |

0 | 1 | 0 | 0 | -2.50 |

0 | 1 | 0 | 1 | -3.125 |

0 | 1 | 1 | 0 | -3.75 |

0 | 1 | 1 | 1 | -4.375 |

1 | 0 | 0 | 0 | -5.00 |

1 | 0 | 0 | 1 | -5.625 |

1 | 0 | 1 | 0 | -6.25 |

1 | 0 | 1 | 1 | -6.875 |

1 | 1 | 0 | 0 | -7.50 |

1 | 1 | 0 | 1 | -8.125 |

1 | 1 | 1 | 0 | -8.750 |

1 | 1 | 1 | 1 | -9.375 |

The table shows LSB or the step size is measured to be -0.625 Volts while the MSB or full-scale voltage turns out as -9.375 Volts.

## Drawbacks

- The binary-weighted DAC has quite a large gap between LSB and MSB resistors values and requires a very precise value of resistors.
- It becomes impractical for higher-order DACs and is suitable for less resolution DACs.
- The stability of the device is resistor-dependent and is difficult to maintain an accurate resistance ratio with temperature variations.

## Advantages

- It has a simple assembly.
- It has a fast conversion speed.
- Simple conversion circuit

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