Op-Amp Circuit Analysis · USAPhO 2016 #A2

Physics · electromagnetism

Problem

A student designs a simple integrated circuit device that has two inputs,

V_a

and

V_b

, and two outputs,

V_o

and

V_g

. The inputs are effectively connected internally to a single resistor with effectively infinite resistance. The outputs are effectively connected internally to a perfect source of emf

E

. The integrated circuit is configured so that

E = G(V_a-V_b)

, where

G

is a very large number somewhere between

10^7

and

10^9

. The circuits below are chosen so that the precise value of

G

is unimportant. [Figure: Left shows internal schematic with Va and Vb inputs connected to terminals a and b, and Vo and Vg outputs connected to terminals o and g. Right shows the symbol used in circuit diagrams.] a. Consider the following circuit.

R_1 = 8.2

kΩ and

R_2 = 560

Ω are two resistors. Terminal

g

and the negative side of

V_{in}

are connected to ground, so both are at a potential of 0 volts. Determine the ratio

V_{out}/V_{in}

. [Figure: Circuit diagram showing op-amp with R1 connected from terminal o to terminal a, R2 connected from terminal a to ground, Vin connected to terminal b, and Vout measured at terminal o.] b. Consider the following circuit. All four resistors have identical resistance

R

. Determine

V_{out}

in terms of any or all of

V_1

V_2

, and

R

. [Figure: Circuit diagram showing op-amp with four resistors R in a configuration connecting V1 and V2 to the inputs and Vout at the output.] c. Consider the following circuit. The circuit has a capacitor

C

and a resistor

R

with time constant

RC = \tau

. The source on the left provides variable, but bounded voltage. Assume

V_{in}

is a function of time. Determine

V_{out}

as a function of

V_{in}

, and any or all of time

t

and

\tau

. [Figure: Circuit diagram showing op-amp with resistor R and capacitor C in an integrator configuration.]

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