Digital to Analog Converter

A digital-to-analog converter (DAC, D/A, D2A or D-to-A) is a circuit that converts digital data (usually binary) into an analog signal (current or voltage). One important specification of a DAC is its resolution. It can be defined by the numbers of bits or its step size.

Digital to Analog Converter using the Summing Amplifier

The following diagram shows a 3 bit digital to analog converter implemented using a summing opamp amplifer.

Digital to Analog Converter Summing Amplifier

From the summing amplifier circuit (see equation 6), the output voltage is \begin{equation} V_{out} = -R ( {{V_{2}}\over R } + {{V_{1}}\over 2R } + {{V_{0}}\over 4R } ) \end{equation} Simplifying, we obtain \begin{equation} V_{out} = - {1 \over 4} ( 4V_{2} + 2V_{1} + V_{0} ) \end{equation}

Output Table
V2 V1 V0 Digital Value Vout
0 0 0 0 0
0 0 1 1 -0.25
0 1 0 2 -0.5
0 1 1 3 -0.75
1 0 0 4 -1.0
1 0 1 5 -1.25
1 1 0 6 -1.5
1 1 1 7 -1.75

From the table, we can conclude the following

  • The inputs can be thought of as a binary number, one that can run from 0 to 7.
  • V2 is the MSB (most significant bit) and V0 is the LSB (least signifcant bit).
  • The output is a voltage that is proportional to the binary number input.
  • The resolution of this DAC is 3 (the number of bits) or -0.25V (the step size).
  • To have more bits, add an additional resistor for each additional bit. Note the relationship between adjacent resistor values.

R-2R Binary Ladder Digital to Analog Converter

The R-2R Digital to Analog Converter uses only two resistance values R and 2R regardless of the number of bits of the converter compared to the summing amplifier implementation where each bit resistor has a different value. The circuit shown is a 3 bit DAC.

R-2R Binary Ladder Digital to Analog Converter

Depending on the state of bit B2, B1 or B0, the respective current I2, I1 or I0 is switched either to ground or to V- of the op amp. Thus \begin{equation} I_{out} = B_2I_2 + B_1I_1 + B_0I_0 \end{equation}

To analyse this circuit, first we observe that since the output is connected to V- through Rf, the opamp is in a negative feedback configuration. Thus \begin{equation} V_- = V_+ = 0 \end{equation}

Therefore the individual current values I2, I1, I0 are unaffected by the switch setting and the resistor network circuit can be redrawn to the following

R-2R Binary Ladder Resistor Network

Due to the nature of the resistance network and values, we can obtain the current values by inspection.

From ohms law, \begin{equation} I_0 = I_c \end{equation} From KCL at node V0, \begin{equation} \begin{split} I_b &= I_0 + I_c \\ &= 2I_0 \end{split} \label{5} \end{equation} From KVL, \begin{equation} \label{6} V_1 = I_bR + I_02R \end{equation} Substituting equation \ref{5} into equation \ref{6} \begin{equation} \label{8} V_1 = 4I_0R \end{equation} Again applying ohms law to I1 and using equation \ref{8}, we obtain \begin{equation} V_1 = I_12R = 4I_0R \end{equation} or \begin{equation} \label{10} I_1 = 2I_0 \end{equation} From KCL at node V1, and using equation \ref{5} and \ref{10} \begin{equation} \begin{split} I_a &= I_b + I_1 \\ &= 2I_1 \end{split} \end{equation} From KVL, \begin{equation} V_{ref} = I_aR + I_12R = 4I_1R \end{equation} We leave the reader to figure out how we obtain \begin{equation} I_2 = 2I_1 = 4I_0 = {V_{ref} \over 2R} \end{equation} Thus the R-2R network can be seen to be like a current source whose output depends on switch setting B2, B1, B0 that controls I2, I1, I0 respectively \begin{equation} I_{out} = {V_{ref} \over 8R}(4B_2 + 2B_1 + B_0) \end{equation}

Including the opamp which behaves like I-V converter, we obtain the voltage ouput (note the direction of the current source) \begin{equation} V_{out} = -{{V_{ref}R_f \over 8R}}(4B_2 + 2B_1 + B_0) \end{equation} Letting Vref = 1 and Rf = 2R, we obtain the following output table

Output Table
B2 B1 B0 Digital Value Vout
0 0 0 0 0
0 0 1 1 -0.25
0 1 0 2 -0.5
0 1 1 3 -0.75
1 0 0 4 -1.0
1 0 1 5 -1.25
1 1 0 6 -1.5
1 1 1 7 -1.75

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