Cursed circuits #3: true mathematics

Cursed circuits #3: true mathematics Op-amp arithmetics, explained in a more accessible way In the previous installments of Cursed Circuits , we looked at two switched capacitor circuits: the voltage halver and the capacitor lowpass filter . The basic non-inverting voltage amplifier. A three-way non-inverting summing amplifier. A simple difference amplifier (A - B). A logarithmic amplifier. V-I curve for 1N4148, normal (left) and log scale current (right). By author. Basic integrator. The single-supply, non-inverting integrator. In today’s episode, I’d like to talk about the abuse of operational amplifiers to do analog math. Analog computing at a scale is wildly impractical because errors tend to accumulate over time; nevertheless, individual techniques find a number of specialized uses, perhaps most prominently in analog-to-digital converters . So, let’s have a look at how it’s done! The following assumes familiarity with core concepts in electronic circuits and with the fundamentals of signal amplification. If you need a refresher, start with the two linked articles first. Op-amps at a glance Let’s start with a brief recap: operational amplifiers are to analog circuits what logic gates are to digital logic. They are simple but remarkably versatile building blocks that let you accomplish far more than appears possible at first blush. Unfortunately, in introductory texts, their operation is often explained in confusing ways. All that an op-amp does is taking two input voltages - V in- (“inverting input”) and V (“non-inverting input”) - and then outputting a voltage that’s equal to the difference between the two, amplified by a huge factor ( in+ A , often 100,000 or more) and then referenced to the midpoint of the supply ( OL V ). You can write it the following way: mid That’s all the chip does. Because the gain is massive, there is a very narrow linear region near V in- = V ; a difference greater than a couple of microvolts will send the output toward one of the supply rails. The chip doesn’t care about the in+ V or in- V it can’t “see” any external components you connect to it, and its internal gain can’t be changed. in+ To show the versatility of the component, we can have a quick look at the following circuit that you might be already familiar with - a non-inverting amplifier: One input of the op-amp is connected to the external signal source: V in+ = V . The other input is hooked up to a two-resistor voltage divider that straddles the ground and the output leg; the divider’s midpoint voltage is: signal As discussed earlier, the only way for the op-amp to output voltages other than 0 V or V supply is for V to be very close to in+ V . We can assume that we’re operating near that equilibrium point, combine the equations for the voltages on the two input legs, and write: in- Solving this for V out , we get: In other words, the output voltage is the input signal amplified by a factor...

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