# 9. Differential Summing

We have seen that an op-amp can be configured to produce an output that is a weighted sum of multiple inputs. If the sum includes both positive and negative signs, D*ifferential Summing *results.

The op-amp configuration of Figure 40 produces an output voltage, *v _{out}*, given by

where

We choose the resistors to achieve bias balance, as follows:

The input resistance for each inverting input, *v _{j}*, is

*R*.

_{j}If the inverting and non-inverting terminals each have only one input, the result is a *differencing amplifier. *This is illustrated in Figure 41.

The output voltage for this configuration is

Note that RA = Ra [see Equation 91]. To achieve bias balance, we choose

The input resistance for the *v _{a}* terminal is

*R*. The input resistance for the

_{A}*v*terminal is , where

_{1}*R*is found from Equation (56) to be

_{in}Then *R _{in}*(at

*V*) is approximately

_{1}*R*+

_{1}*R*.

_{2}*R*is found in Equation (52) and is equal to

_{out}To achieve unit-gain differencing, where the output is given by *v _{out}* =

*v*–

_{1}*v*we set

_{a}*R*=

_{A}*R*=

_{F}*R*=

_{1}*R*If a 741 op-amp is used, a typical value for these four resistors is 10 kΩ.

_{x}.The input resistance into *v _{out}* is then 10 kΩ and into the

*v*terminal the resistance is 20 kΩ.

_{1}Suppose that equal-gain differencing is desired but that the gains need not be unity. We then set *R _{1}* =

*R*and

_{A}*R*=

_{x}*R*. The output voltage is then

_{F}The input resistance into the *v _{a}* terminal is , and into the non-inverting terminal it is , which is approximately

*R*+

_{A }*R*since

_{F}*R*>>

_{in }*R*. Values for input resistance and

_{F}*R*are easily determined with the use of Equations (95) and (96).

_{out}A useful modification of the differencing amplifier configuration of Figure 41 is the *sign switcher *as shown in Figure 42.

With the switch in the position shown in Figure 42(a), *v _{out }*= –

*v*, and for the opposite switch position,

_{in}*v*= 2

_{out}*v*–

_{in}*v*=

_{in}*v*. The input resistance is 10 kΩ for each position, and the bias is balanced in each position. Figure 42(a) shows a single-pole double-throw switch. Sign switching can also be accomplished with a single-throw switch as shown in Figure 42(b). However, for this implementation, the bias and input resistances are not equal in the two switch positions.

_{in}__Example__

Determine the output resistance and the input resistance at each of the inputs of a multiple input 741 amplifier which has an output of

Assume that the value of the resistance at the + and – terminals is 10 kΩ.

*Solution:* Using the larger of *X* and (*Y*+1) yields the multiplying factor for determining *R _{F}*.

The circuit is shown in Figure 43.