Op Amps 
Written by Bryce Ringwood  
Operational amplifiers (Op Amps) had their origins in analog computers. Nowadays, they have innumerable other applications and range in price from a few cents to thousands of rands. My second introduction to op amps was when I had to measure small rock deformations in the tunnels of the Drakensberg Pumped Storage Scheme. Here, the op amps had to amplify (accurately) the miserably small voltage from strain gauges attached to rock anchors. These amplifiers had to have stable gain (over a period of many months), they weren't allowed to "drift"  that is wander off the correct measured voltage due to some internal temperature or other effect and they had to amplify by a correctly set amount. Precision operational amplifiers have open loop gains of over a million, have low offset drift and may have a huge bandwidth. What does all this mean ? If we look at the ubiquitous "741" op amp, we will see that it has an open loop gain of perhaps 20 000. That means that if you were to apply a small voltage of 10 microvolts across the + and  terminals (grounding the  terminal), then you would measure 20 000* 10 / 1000000= 0.2 volts between output and ground. Except that won't really fly. There will be an offset voltage developed across the input terminals, which will also be amplified. Then again, the gain of 20 000 is just 'typical'. Finally  how was that 10 microvolts generated? By using a battery and potential divider ? There will be bias currents flowing in those resistors  net result  its anyone's guess what you will measure. Op amps are never used like this. They are always used in a closed loop circuit with feedback elements between the output and input pins. Let's look at the "perfect" op amp.
The starred conclusions assume the op amp is used in a closed loop circuit, in other words there are feed back elements, such as resistors between the output and input termials. Let's look at the inverting amplifier as a simple example of this. The Inverting Amplifier
\textstyle I_{1} = {\displaystyle {E_{in}}\over \displaystyle R_s}
\textstyle I_2 = {\displaystyle {E_{out}} \over \displaystyle R_f}
Since no current flows into the amplifier, \textstyle I_1 = I_2, and thus the voltage gain is
\textstyle {{\displaystyle E_{out}} \over {\displaystyle E_{in}}} =  {{\displaystyle R_f} \over {\displaystyle R_s}}
Note that if the two resistors are equal, then the amplifier simply inverts the input voltage. In analog computers, the source resistor is often replaced with a number of equal resistors, each taking a different value for E_{in. }The output in this case is simply the negative of the sum of the inputs. I'm going to leave you to figure that one out. The NonInverting AmplifierI don't remember this being used in the analog computer. Here it is anyway.
The only thing to notice here is that the negative input is at the tap of the potential divider formed by R_{f }and R_{s.}. Because "The Differential Input Voltage is Zero" rule 2, then the voltage at this point must be equal to E_{in}.
\textstyle E_{in} = {\displaystyle {E_{out}} {{\displaystyle R_s} \over {\displaystyle {R_f }+{R_s}}}}
\textstyle E_{out} = {\displaystyle {E_{in}} ( 1 + {{\displaystyle R_f} \over {\displaystyle {R_s}}})}
You can remove R_{s }to make a buffer amplifier with a gain of 1. Almost any value of R_{f} can be used. Differential AmplifierAgain, not a circuit used in analog computers as a computing element  at least, not the machine I was used to programming. The differential amplifier can be used with things like straingauge bridges, and other transducers with a bridge output, however, since time immemorial, there have been instrumentation amplifiers, sometimes made from a combination of opamps offering better performance.
The formula for E_{out} is prettywell what you would guess it to be: .
\textstyle E_{out} = {{\displaystyle R_f} \over {\displaystyle {R_s}}}(E_{in_2}  E_{in_1} )
See if you can work out how to prove it by looking at the output voltages arising from the inverting and noninverting inputs separately. IntegratorThis is perhaps the primary analog computing element. What it does is to take a function that varies with time and integrate it. Imagine for example the simplest possible function  a straight line parallel to the xaxis with a value of one volt. Apply this to the integrator, and it will generate a ramp voltage of 1 volt per second until the op amp overloads. The integrator simply works out the area under the curve representing the input function. The curve can be as complicated as you like. Here's how it works.
Why "Basic" ?  Well, as we shall see, there are some bits missing that will make it awkward to use in practice. As in the case of the invering amplifier, we look at the currents flowing in R_{s }and C.
\textstyle I_{1} = {\displaystyle {E_{in}}\over \displaystyle R_s}
\textstyle I_2 = C{\displaystyle {dE_{out}} \over \displaystyle dt}
Rearranging: \textstyle
E_{out} ={{{\displaystyle1 \over \displaystyle {RC} }\displaystyle \int^t_0 {E_{in} dt } } + constant}
InaccuraciesConsider the differential amplifier. If we short the inputs to ground (both E_{in} = 0 Volts), we should expect that we would get a zero voltage output, and that it would remain at zero. If you try this with a popular 741, you will see that there is, in fact, an output voltage and that it does not remain particularly steady. This is a function of the opamp circuitry, and the more you pay  the less the offset drift with temperature. Other errors arise from the fact that the realworld opamps do NOT have infinite gain and nor do they have infinite input resistance. Some opamps (particularly old 741s) exhibit noise at the input. This can be the normal noise, or it can also be "popcorn noise"  random bursts of noise. For this reason, my analog computer project (next) declined the use of 741s, in favour of some OP07 amplifiers, which I had to desolder from an earlier project. The results were better than I could have imagined. Op Amps work by using carefully constructed DC amplifier circuits nowadays. For extreme accuracy, it is possible to purchase chopper stabilised op amps. These use a separate AC and DC channel. The input voltage is chopped to form an AC voltage, then amplified, then demodulated to DC in order to provide an error voltage to stabilise the DC amplifier. Fiendishly complicated, I'm sure you'll agree. The Datel AM4902A (pictured) is a chopper stabilised opamp of the 1980s era. With an open loop gain of $5\times 10^8$ it would have made a pretty good integrator. The input offset drift is just about nothing, and it had an input impedance of 100 Meg.Almost the "ideal" amplifier, it cost a fortune. I was going to use it, but OP07s became available, which were good enough for what I wanted at a fraction of the price. Also pictured is a valvetype strain gauge amplifier using a chopper. More modern (and cheaply available) chopper stabilised amps such as the LT1150 and ICL7650/2/3 are to be found ain RS Electronics catalogue. One thing to beware of is noise generated by the chopper circuitry. In SummaryThe above is a very short introduction to opamps, so you can play with them without injuring yourself. You can use them for all sorts of projects  to make oscillators, headphone amps, there are even some you can use in radio circuits as mixers and i.f. amps. There are hundreds of different types, each with a specific niche. Download a few datasheets and you will see many applications. Maybe they can be used in an experimental homodyne receiver, perhaps with no inductors. There's a thought.
Tektronix Chopper Stabilised Valve Amp I didn't mention comparators close cousin to op amps because they won't be used in the analog computer described next. Comparators normally have a logical voltage output and are used to compare the dc voltage on the positive and negative pins. Further readingVisit the analogmuseum.org website. Try to get a copy of "Horowitz and Hill: The Art of Electronics".
