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# Simple OP amp circuits with feedback

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Hi John

could you help me out with these? see attachment

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2 Questions and solutions involving simple OP amp circuits with feedback, one question concerns a 4 bit DAC utilising an opamp circuit, the other question analysis of potentiometer negative feedback op-amp circuit

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Op-amp circuits Questions & Solutions
3.

(a)
The output function V_out=8V_4+4V_3+2V_2+V_1 can be realised by the following Opamp circuit (an explanation below)

Circuit operation:
Consider input V_4 when switched to high input rail current i_4=V/R flows
Consider input V_3 when switched to high input rail current i_3=V/(R/2)=2 V/R flows
Consider input V_3 when switched to high input rail current i_2=V/(R/4)=4 V/R flows
Consider input V_3 when switched to high input rail current i_1=V/(R/8)=8 V/R flows
Total current is thus the sum of the branch currents so we can write

i=i_1+i_2+i_3+i_4=8 V/R+4 V/R+2 V/R+V/R

The feedback resistor provides negative feedback so the output voltage is given by Ohm's law as

V_O=-iR_F=-8 R_F/R V-4 R_F/R V-2 R_F/R V-R_F/R V

Finally add a second stage negative feedback Opamp circuit with unity gain given by ...