# Back to the DAC

The questions below are due on Wednesday February 26, 2020; 11:59:00 PM.

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In Lab 2 we built a DAC using a resistive ladder. In this problem we'll revisit that circuit and examine its performance using some of the new tools we have learned.

###Variability

In lab we used discrete resistors (out of the cabinet) to build the resistor network. Standard resistors have a 5% tolerance, which means a resistor of nominal value R can vary from between R-0.05R to R+0.05R. How does this affect our DAC?

Consider the topology of our DAC, which is provided again below:

Assume that the resistors R can vary from R+\Delta R to R-\Delta R, and the 2R resistors can vary from 2R+2\Delta R to 2R-2\Delta R. We want to determine under what conditions the output voltage V_O will be minimal.

A good way to approach this question is to recast the circuit as a Thevenin equivalent to the left of port A-A', as show below:

where here R_L is nominally 2R, and R_{TH} must be determined from the original circuit. Under what conditions of R_{TH} and R_L will the output voltage V_O be minimal?

Enter expressions for R_{TH} and R_L that provide the minimum V_O under the conditions that the resistors R and 2R can vary as above. Enter \Delta R as DR.

 R_{TH} = R_L =

###Thevenin equivalent

Now, going back to the situation where the resistor values are exactly R and 2R, determine the Thevenin equivalent of the entire circuit (now including the right-most 2R), looking into port B-B':

In particular, determine expressions for V_{TH} and R_{TH} that are correct under all combinations of the voltage sources being zero or non-zero.

As always, enter V_1 as V1, etc.

 R_{TH} = V_{TH} =