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DAC Again

The questions below are due on Wednesday March 04, 2020; 11:59:00 PM.
 
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The circuit shown below is another type of digital-to-analog converter. The voltage sources v_{DB0} through v_{DB7} represent the voltages supplied by the eight data bits from some digital system, DB0 through DB7. These voltages will be approximately 5V when the corresponding data bit is a logical high, and approximately 0V when the corresponding data bit is a logical low. The voltage v_{OFF}, which is set by a potentiometer, is an offset voltage that is used to center the output of the converter around 0V. Assume that the op-amp in the converter is ideal.

If only v_{DB0} and v_{DB6} are the only voltage sources "on", determine v_{DAC} as a function of known voltages, the resistors, and v_{OFF} and the circuit's resistors.

Voltage sources should be replaced with their values for this problem (5V on, 0V off). Enter v_{OFF} as vOFF.

v_{DAC}=

With v_{OFF} = 0 V, the output of the digital-to-analog converter should span the range of 0 V to −2.5 V. Thus, the output of the converter should be given by

v_{DAC} = −2.5 V \sum_{i=0}^7 \frac{2^i}{255}\text{DB}i

where each data bit \text{DB}i takes on the numerical value of 1 when high and 0 when low. In this manner, each successive data bit from DB0 to DB7 is given a voltage weighting twice that of the preceding data bit, making it possible for the converter to output voltages from 0 V to −2.5 V in steps of −2.5/255 V. Given this, what must be the relationships between the values of R_0 through R_7, and R_8?

In particular answer the following two questions:

What is the ratio R_8/R_1=

What is the ratio R_1/R_3=

The voltage rating of the piezo-electric speaker used in Lab 2 is approximately \pm 12.5V. Since the low-pass filter and power amplifier between the converter and the speaker will be designed to provide a total gain of 10 over the frequency range of interest, the output range of the analog-to-digital converter must be designed to match the speaker rating divided by 10. This is why the range is chosen to be 0 V to -2.5 V, with v_{OFF} = 0V. Note further that the output range of the converter is negative. This is because the converter is based upon the inverting op-amp amplifier configuration.

The role of v_{OFF} is to offset the output of the digital-to-analog converter so that it is centered around 0 V. That is, with DB0 through DB7 all low, v_{DAC} should be 1.25 V, and with DB0 through DB7 all high, v_{DAC} should be −1.25 V. Given this, what must be the value of v_{OFF}?

What must be the value of v_{OFF} be [in V]?