0195136047.pdf

5.4 APPLICATIONS OF OPERATIONAL AMPLIFIERS 247

If all resistorsRnare the same, sayRn=R 1 , for alln, then

vo=−

Rf

R 1

∑N

n= 1

vin, if allRn=R 1 (5.4.17)

which corresponds to an inverting summing amplifier with gain.

Noninverting Summing Amplifier

The circuit of Figure 5.4.2 is generalized for multiple inputs, as shown in Figure 5.4.4. With the
ideal op-amp assumption, based on Equation (5.4.14), the noninverting gain is
vo
v 2

= 1 +

Rf

Rd

(5.4.18)

By superposition, the outputvois the sum of the responses taken individually. For inputm,

v 2 =vim

[R 1 ‖R 2 ‖···‖RM]withoutRm

Rm+[R 1 ‖R 2 ‖···‖RM]withoutRm

=

vim

Rm

[R 1 ‖R 2 ‖···‖RM] (5.4.19)

where[R 1 ‖R 2 ‖···‖RM]represents the resistance of all resistorsR 1 ,R 2 ,...,RMin parallel.
Hence, due to inputvim,

vom=

(

1 +

Rf

Rd

)

[R 1 ‖R 2 ‖···‖RM]

Rm

vim (5.4.20)

The total output voltage is

vo=

∑M

m= 1

vom=

(

1 +

Rf

Rd

)

[R 1 ‖R 2 ‖···‖RM]

∑M

m= 1

vim

Rm

(5.4.21)

For the special case when all these resistors are equal, i.e., whenRm=Rfor allm, then it follows

vo=

(

1 +

Rf

Rd

)

1

M

∑M

m= 1

vim (5.4.22)

which can be interpreted in the following ways:

1. A single-input noninverting amplifier gain 1+Rf/Rdthat has an input equal to the average
   ofMinputs;

−

+

vo

vi 1

vi 2

viM

2 3

1

Rd

RM

Rf

R 2

R 1

Figure 5.4.4Noninverting summing amplifier.