Exercise Solutions

Easy:

1. The basic principle of a Weighted Resistor DAC is to sum currents (or voltages) that are proportional to the binary weight of each digital input bit. Each digital input bit controls a switch, connecting a resistor (whose value is inversely proportional to the bit's weight) to a summing junction, typically the input of an Op-Amp. The Op-Amp then converts this sum of weighted currents into a proportional analog output voltage.
2. A Weighted Resistor DAC requires a wide range of precise resistor values that are binary weighted. For example, if the MSB resistor is 'R', the next bit would use 'R/2', then 'R/4', 'R/8', and so on, down to 'R/$2^{N-1}$' for the LSB. So, values could be R, R/2, R/4, etc.

Medium:

1. Given: N = 3 bits, $V_{REF} = 5V$, $R_f = R_0$. Digital input is $D_2D_1D_0$.
   The formula (assuming $R_f = R_0$ and inverting Op-Amp) is:
   $V_{out} = -V_{REF} \times \left(D_2 + \frac{D_1}{2} + \frac{D_0}{4}\right)$ a) For digital input "110" ($D_2=1, D_1=1, D_0=0$):
   $V_{out} = -5V \times \left(1 + \frac{1}{2} + \frac{0}{4}\right)$
   $V_{out} = -5V \times (1 + 0.5 + 0)$
   $V_{out} = -5V \times (1.5)$
   $V_{out} = \textbf{-7.5V}$ b) For digital input "001" ($D_2=0, D_1=0, D_0=1$):
   $V_{out} = -5V \times \left(0 + \frac{0}{2} + \frac{1}{4}\right)$
   $V_{out} = -5V \times (0 + 0 + 0.25)$
   $V_{out} = -5V \times (0.25)$
   $V_{out} = \textbf{-1.25V}$
2. Matching resistor values accurately becomes significantly more challenging for high-resolution Weighted Resistor DACs because the range of required resistor values grows exponentially with the number of bits. For an 8-bit DAC, the ratio between the largest (MSB) and smallest (LSB) resistor is $2^{8-1} = 2^7 = 128$. For higher resolutions (e.g., 10-bit with a 512:1 ratio, or 12-bit with a 2048:1 ratio), fabricating and matching resistors over such a vast range with high precision is extremely difficult. The main implication of this challenge on the DAC's performance is a degradation in linearity and accuracy. Inaccurate resistor matching leads to Integral Non-Linearity (INL) and Differential Non-Linearity (DNL) errors, potentially causing the DAC to be non-monotonic (where the output does not always increase with increasing digital input), which is unacceptable in many applications.

Hard:

1. The primary advantage of the R-2R Ladder DAC over the Weighted Resistor DAC, from the perspective of integrated circuit fabrication and accuracy, is its reliance on only two precise resistor values (R and 2R). Impact on Fabrication: In IC manufacturing, it is far easier to fabricate multiple instances of resistors with precisely controlled ratios (like 1:2) than it is to produce a wide range of absolute resistor values that also maintain precise absolute values and relative ratios over large differences (e.g., R, R/2, R/4, ..., R/256). Even if the absolute values of R and 2R resistors in an R-2R ladder vary slightly due to manufacturing variations, their ratio can be very accurately maintained across the chip. Impact on Accuracy and Linearity: Because the R-2R DAC's operation fundamentally depends on the accurate ratio of R and 2R, and this ratio is easily maintained in fabrication, R-2R DACs inherently offer much better accuracy and linearity for higher bit resolutions. The linearity of a DAC is highly dependent on the precision of its internal resistor network. The Weighted Resistor DAC's requirement for a wide range of precisely matched absolute resistor values makes it extremely challenging to achieve good linearity for DACs with more than a few bits of resolution, leading to significant INL/DNL and potential non-monotonicity.

Quiz Answers

1. c) R/2
   • The values are binary weighted as R, R/2, R/4, etc.
2. c) Inverting Summing Amplifier
   • This configuration is used to sum the currents flowing through the weighted resistors.
3. False.
   • The Weighted Resistor DAC is not preferred for high-resolution applications due to the complexity of its component requirements (wide range of precise resistor values) and the difficulty in achieving good resistor matching, which impacts linearity. The R-2R DAC is generally preferred for high resolution.
4. b) The difficulty in precisely matching a wide range of resistor values.
   • This is the fundamental challenge that limits the practical resolution and accuracy of Weighted Resistor DACs.