Arithmetic Circuits#

Example: Adder-Subtractor via Op-Amp

An adder can be implemented using specially chosen Ohmic resistors and an ideal op-amp:

As long as the ideal op-amp is operated in its linear region and the resistors are chosen such that \(\frac{\sum G_k + G_0}{G_0} = \frac{\sum G'_k + G'_0}{G'_0}\), the above circuit behaves like a generalized summing amplifier (or adder-subtractor), producing an output voltage \(v_{\text{out}}\) that is a weighted difference of the input voltages \(v_1, \dotsc, v_n, v_1', \dotsc, v_m'\):

To ensure that the ideal op-amp is indeed operated in its linear region, we need the calculated output voltage to stay within the supply limits, \(v_{\text{out}} \in [-V_{\text{sat}}; +V_{\text{sat}}]\).

We can see this by analyzing the network.

Linear region:

When the ideal op-amp is operated in its linear region, we have \(v_+ = v_-\). We apply Kirchhoff's current law to the input terminals.

At the non-inverting terminal (\(+\)), the sum of currents leaving the node is zero:

At the inverting terminal (\(-\)), the sum of currents leaving the node is also zero:

By equating \(v_{+} = v_{-}\), we have the following:

Since \(\frac{\sum G_k + G_0}{G_0} = \frac{\sum G'_k + G'_0}{G'_0}\), we arrive at the expression provided in the diagram:

Saturation regions:

When the ideal op-amp is operated outside its linear region, the differential voltage \(v_d = v_+ - v_-\) is non-zero. The voltage \(v_+\) is still determined solely by the input divider network:

However, the voltage \(v_-\) depends on the clamped output voltage (\(v_{\text{out}} = \pm V_{\text{sat}}\)):

When the ideal op-amp is operated in its negative saturation region, we have \(v_d \lt 0\) and \(v_{\text{out}} = -V_{\text{sat}}\). This implies \(v_+ \lt v_-\), meaning the weighted sum of the inverting inputs dominates the non-inverting inputs.

When the ideal op-amp is operated in its positive saturation region, we have \(v_d \gt 0\) and \(v_{\text{out}} = +V_{\text{sat}}\). This implies \(v_+ \gt v_-\), meaning the weighted sum of the non-inverting inputs dominates.