Analog-Digital: REVIEW of Inverting Amplifier

An operational amplifier, better known as an op-amp, is a versatile circuit component that can be configured to perform different operations on the voltages at its two input terminals. Op-amps have a positive and negative terminal, which serve as inputs to the op-amps. The op-amp can be configured to serve variety of purposes. In a D/A, it is typically configured as an inverting amplifier shown below. In this configuration, the positive terminal is typically grounded, an input voltage is connected to a resistor which in turn connects to the negative terminal, and a resistor connects the negative terminal to the output of the op-amp. 

In order to understand these components at an idealized level, we have to establish two rules of ideal op-amps configured as an inverting amplifier:

Note that since the positive terminal is grounded, Vb is 0V. Since Va has the same voltage as Vb, Va is also 0V. Also, because there is no current going into the op-amp, Ri and Ro are effectively in series. Since this is the case, the current going through Ri (labeled Ii) and the current going through Ro (labeled Io) must be the same. By Ohm’s Law, the voltage across Ri (Vi - Va) divided by Ri’s resistance will yield Ii. In other words, Ii = (Vi - Va) / Ri, and since Va = 0, Ii = Vi / Ri. Similarly, the voltage across Ro (Va - Vo) divided by Ro’s resistance will yield Io. So Io = (Va - Vo) / Ro = -Vo / Ro. Since Ii = Io, Vi / Ri = -Vo / Ro. Solving for Vo gives us the following equation: Vo = -Ro / Ri * Vi. What the equation effectively tells us is that Vo is directly proportional to Vi based on the ratio between Ro and Ri. Although the output voltage is negative, we are only interested in the magnitudes of the voltages (as the negative sign can be easily remedied). With this observation, however, we can implement a system that utilizes multiple inputs and subsequently changes the output voltage.