0195136047.pdf

246 ANALOG BUILDING BLOCKS AND OPERATIONAL AMPLIFIERS

i 1 =

−v 1

R 1

=

−v 2

R 1

+

vo

AoR 1

(5.4.10)

i 2 =

v 1 −vo

R 2

=

v 2

R 2

−

vo

R 2

(

1

Ao

+ 1

)

(5.4.11)

Since the ideal op amp draws no current,

i 1 =i 2 (5.4.12)

Also, as seen from Figure 5.4.2,

v 2 =vi

In equating Equations (5.4.10) and (5.4.11), one gets

vo

vi

=

vo

v 2

=

(

1 +

R 2

R 1

)

1

1 +[(R 1 +R 2 )/AoR 1 ]

(5.4.13)

ForAo→∞, the ideal circuit gain is

vo

vi

= 1 +

R 2

R 1

(5.4.14)

(See Example 5.2.2.) Note that ifR 2 =0, for any nonzero value ofR 1 ,

vo

vi

= 1 (5.4.15)

which is the gain of an idealvoltage follower.The same result applies ifR 1 →∞(i.e., open

circuit) for any finiteR 2.

Inverting Summing Amplifier

The circuit of Figure 5.4.1 is extended by adding other input points, as shown in Figure 5.4.3.

Because of the virtual ground at terminal 1 (ideal op-amp assumption), the response of the circuit

to any one input is not affected by the presence of the otherN−1 inputs. Based on Equation

(5.4.7) one can write the total output as the sum of all responses,

vo=−

(

Rf

R 1

vi 1 +

Rf

R 2

vi 2 + ··· +

Rf

RN

viN

)

=−Rf

∑N

n= 1

vin

Rn

(5.4.16)

viN

−

+

vo

vi 1

vi 2

3

2

1

RN Rf

R 2

R 1 Figure 5.4.3Inverting summing amplifier.