Linear Computational Circuitry

We know that an opamp connected to a voltage divider with a voltage division ratio of \(\frac{1}{2}\) will have an overall voltage gain of 2, and that the same circuit with a voltage division ratio of \(\frac{2}{3}\) will have an overall voltage gain of 1.5, or \(\frac{3}{2}\):

There is definitely a mathematical pattern at work in these non-inverting opamp circuits: the overall voltage gain of the circuit is the mathematical inverse of the feedback network’s voltage gain.

Building on this concept, what do you think would be the overall function of the following opamp circuits?

The simple resistor network shown here is known as a passive averager. Describe what the word “passive” means in this context, and write an equation describing the output voltage (V_d) in terms of the input voltages (V_a, V_b, and V_c):

Hint: there is a network theorem that directly applies to this form of circuit, and it is known as Millman’s Theorem. Research this theorem and use it to generate your equation!

Add an op-amp circuit to the output of this passive averager network to produce a summer circuit: an operational circuit generating an output voltage equal to the sum of the four input voltages. Then, write an equation describing the whole circuit’s function.