If you find an error in what I’ve made, then fork, fix lectures/l04_dac.md, commit, push and create a pull request. That way, we use the global brain power most efficiently, and avoid multiple humans spending time on discovering the same error.

Resistor based DACs
DAC errors
DAC complexity
Binary scaled DACs
- Binary coding
- Thermometer encoding
Current mode DACs
References

Processing of signals has shifted into the digital domain. But the real world is analog. In order to interact with the analog we need to convert the digital signals (discrete-value, discrete-time) back to analog signals (continuous value, continuous time).

The SI base units define the fundamental analog quantities as second, meter, kilogram, ampere, kelvin, mole and candela (</aic2026/a_refresher>). Assume that electronic circuits interact with the real world in terms of second and ampere.

Related to Ampere we have the derived units of charge (Ampere Seconds), Volt (W/A), Ohm (V/A), or indeed Simens (1/$\Omega$).

As such, to create a digital to analog converter, we somehow have to create a circuit that has a function of

\[I_{out} = D_{in} \times I_{ref}\text{ [I]}\] \[t_{out} = D_{in} \times t_{ref}\text{ [s]}\] \[Q_{out} = D_{in} \times Q_{ref}\text{ [C]}\] \[V_{out} = D_{in} \times V_{ref}\text{ [V]}\] \[R_{out} = D_{in} \times R_{ref}\text{ [}\Omega\text{]}\]

The digital value is dimensionless, as such, there must be a reference value

Digital to analog conversion can be indirect through the relations between voltage, resistance, current, time, inductance and capacitance.

\[V = R I\] \[Q = C V\] \[dt = \frac{C dV}{I}\] \[dt = \frac{L dI}{V}\]

Resistor based DACs

\[I_{ref} = \frac{V_{ref}}{3 R}\] \[V_{out} = D_{in} R I_{ref} = \frac{D_{in} R V_{ref}}{3 R} \text{ [V]}\] \[V_{out} = \frac{b_0 2 R V_{ref}}{3 R} + \frac{\overline{b_0} R V_{ref}}{3 R}\] \[V_{out} = \begin{cases} \frac{2}{3} V_{ref}, & b_0 = 1 \\ \frac{1}{3} V_{ref}, & b_0 = 0 \end{cases}\]

\[I_{ref} = \frac{V_{ref}}{2 R}\] \[V_{out} = \frac{b_0 2 R V_{ref}}{2 R} + \frac{\overline{b_0} R V_{ref}}{2 R}\] \[V_{out} = \begin{cases} V_{ref}, & b_0 = 1 \\ \frac{1}{2} V_{ref}, & b_0 = 0 \end{cases}\]

DAC errors

Digital to analog converters do not add quantization error. The quantization error is already in the digital word.

\[V_{out} = a_1^1 D_{in}^1 + B + \left( a_n^n D_{in}^n + ... a_2^2 D_{in}^2\right)\]

DAC output will contain gain errors, offset errors, and non-linear components

\[DNL[k] = \frac{V[k+1] - V[k]}{V_{LSB}} - 1\] \[INL[k] = \frac{V[k] - V_{ideal}[k]}{V_{LSB}}\]

DAC complexity

As number of resistors grow, the switches grow as

\[\sum_{n=1}^{N} 2^n = 2^{N+1} - 2\]

Use a matrix with R rows and C columns. Need R + C switches, or

\[2^{N} + 2^{N/2}\]

Switches in a 10-bit digital to analog converter.

\[\begin{array}{ll} \text{Tree: } &2^{N+1}-2 = 2046 \\ \text{Matrix : } &2^{N} + 2^{N/2} = 1056 \\ \text{6b Matrix + 4b Tree: } & 2^{M+1} - 2 + 2^{N} + 2^{N/2} = 80 \end{array}\]

Large number of bits, will be large number of resistors and switches.

Binary scaled DACs

\[R_{in} = 2R || 2R = R\]

\[R_{in} = R + R = 2R\] \[I_{0} = \frac{V_0}{2R} = \frac{V_1}{4R}\]

\[R_{in} = 2R || 2R = R\] \[I_{0} = \frac{V_0}{2R} = \frac{V_1}{4R}\] \[I_{1} = \frac{V_1}{2R}\]

\[I_{RF} = I_1b_1 + I_0b_0 = \frac{V_{REF}}{2R}b_1 + \frac{V_{REF}}{4R}b_0\] \[V_{O} = \left(\frac{V_{REF}}{2R}b_1 + \frac{V_{REF}}{4R}b_0\right)R_{F0}\]