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The modern analog computer is based on an electronic circuit known as an operational amplifier. Early operational amplifiers ("op amps" for short) used vacuum tubes, since that was the only available technology. Modern op amps are constructed as semiconductor integrated circuits. Either way, the general theory is the same. We will discuss the internal workings of op amps in a separate page. For the overall discussion of analog computer circuits and op amp behavior in such applications, we will make three assumptions about op amps:
Although these assumptions aren't really correct, they're close enough that the circuit works well, so long as the electronic components connected to the amplifier to control its operation have reasonable values. For a discussion on typical IC op amps and their realworld characteristics, follow this link. The figure to the right shows the basic circuit used in analog computers. The triangle represents our amplifier. For our discussions here, we'll assume standard IC amplifiers permitting a typical signal voltage range of ±10 volts. Associated with the amplifier are two resistors: an input resistor (R_{in}) and a feedback resistor (R_{f}). In addition, we will state that the amplifier inverts the signal. That is, a positive input signal will result in a negative output signal, and viceversa. With this combination of characteristics, we can use precision resistors and other components to accurately determine how the circuit will behave. Now, let's consider what will happen if some input voltage is applied to the V_{in} connection. If no current flows through R_{in}, there will be no voltage drop across this resistor, and the applied input voltage will appear at the input to the amplifier itself. This will be amplified and inverted by the amplifier, which will try to produce an infinite but opposite output voltage (remember #1 above). Obviously, this can't happen. That inverted output voltage will produce a voltage drop across both R_{in} and R_{f}, causing current to flow through both resistors. None of this current will be accepted or used by the amplifier itself (#2 above), so the current flowing through R_{f} must be the same as the current flowing through R_{in}. To determine the current through the two resistors, we must determine the output voltage and the voltage at the amplifier input, where both resistors are connected. Once again, refer to #1 above. With an infinite voltage gain, any voltage at the amplifier's input will cause an excessive output voltage. Therefore, the voltage at the junction must always be zero. The amplifier output will provide whatever voltage is required to maintain that condition, and keep the currents through R_{in} and R_{f} the same. This in turn means that so long as the circuit is operating within its bounds (output voltage within the range of ±10 volts), the junction of these components will be a virtual ground. Knowing this, we can use Ohm's Law to calculate the currents through the two resistors. Furthermore, since the two currents must be exactly the same, we can set them equal to each other and use that relationship to determine the output voltage of the amplifier:
Using a little algebra, we can solve this equation for V_{out}, or we can solve it for the ratio of V_{out}/V_{in} to get the voltage gain of the overall circuit:
From these equations, we can see that the voltage gain of the overall circuit is set entirely by the ratio of R_{f}/R_{in}. This is the secret of the operational amplifier: it uses extremely high gain combined with a lot of negative feedback in order to achieve accurate and predictable results. If we use precision resistors, we can obtain precise and measurable results.

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