Op Amps. Find an expression for the gain.?

Question

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I'm not sure what that wiper is but it looks like an inverting op amp which has a default gain of -R2/R1. The wiper there is throwing me off so I'm not sure how to approach this problem.

Answer 1

The wiper is an electrical contact that touches the resistive material in the potentiometer (POT). k=0 represents the wiper touching the bottom of the resistor, so its output would be ground. k=1 represents the wiper touching the top of the resistor, so it would be sampling the voltage vI. With k=.5, the wiper is touching the midpoint of the resistive strip, and assuming it is a linear POT so that the resistance varies linearly with position, the wiper would be sampling a voltage of VI/2, as long as there is no current flowing out of or into the wiper terminal. Since this assumed to be an ideal op amp, there is no such current flow.



A key feature of an ideal negative-feedback amplifier is that it will adjust its output (Vo) so that the V- input matches the V+ input. A negative-feedback amplifier is one that connects the output Vo via some impedance back to the v- input, in this case through R2.



The POT acts as a voltage divider, so:

V+ = kR/R x vI = k VI



Since V- = V+:

V- = kVI



The current through R1 is:

IR1 = (VI - V-)/R1 = (VI-kVI)/R1 = (1-k)VI/R1



The output voltage is given by:

Vo=(V-) - VR2

Since there is no current into V-, the current through R1 also passes through R2. So Vr2 can be calculated as:

VR2= IR1 x R2 = (1-k)VIR2/R1

Substituting VR2 into the equation for Vo:

Vo=(V-)-(1-k)VIR2/R1

Substituting V- into the above equation:

Vo=kVI-(1-k)VIR2/R1

Vo=VI[k-(1-k)R2/R1]

Since gain A = Vo/Vi, rearrange above equation:

A=Vo/VI=[k-(1-k)R2/R1]



General answer:

A = k - (1-k)R2/R1



Since R2=R1

A = k -(1-k)



Specific answer:

A = 2k-1



For k=0, A=-1

For k=1, A=1

For k=.5, A=0



Interesting circuit.

Answer 2

With the negative feedback, the opamp will force the output such that the negative input is at the same potential as the positive input (assuming a perfect opamp --which this problem probably does).



When the wiper is at ground (k = 0%), it acts just like a normal inverting amp with a gain of -1 (as you figured out).



When the wiper is at any other value k>0, the voltage at the positive input is k*Vi (do the voltage divider equation to figure out why).



So, if the voltage at the positive input is at k * Vi and the opamp circuit negative feedback does what it can to force the negative input to also be k * Vi, I think you can figure out the rest.



.

Answer 3

They are asking you what the output will be with 50%, 100% and 0% of the input voltage. Express the voltage (vo) in terms of the input (vi).