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Setting the Gain Coefficient

When we want to solve an algebraic equation, we typically find that any given unknown variable has a coefficient, or constant multiplier, associated with it. For example, consider a very simple equation: Y = 2X. In this equation, X is the unknown input value, and Y will be the result of the calculation. The number 2 is known as the coefficient applied to X.

In analog computers, the coefficient is set by adjusting the gain of the amplifier. The unknown input is applied as a voltage to the circuit input, and the output voltage, which will give the numerical answer to the equation for the current input voltage, can be directly measured. Keep in mind, of course, that the output voltage will also have the opposite polarity from the input voltage.

Analog circuit to solve the equation Y = 2X.

In the circuit to the right, The gain of the amplifier, and therefore the constant coefficient, is set by the input and feedback resistors. Remember that the effective gain of the circuit is Rf/Rin. Therefore, the gain of this particular circuit is 20k/10k = 2. Thus, any voltage X applied to the input will be doubled by the amplifier, producing a (negative) voltage Y at the output. If we must have the actual voltage Y, we can pass the -Y signal through an op amp with its gain set at -1. We could equally well invert the incoming X signal before applying it to the figure to the right.

Analog circuit to solve the equation Y = -3X.

Some equations include negatives directly. To the left is a circuit designed to solve the equation Y = -3X. As before, the constant coefficient (3 in this equation) is assigned as the gain of the amplifier.

Note that the coefficient does not have to be an integer — any desired coefficient can be set by selecting the proper ration of Rf/Rin. For example, if we exchanged the two resistors in the circuit to the left, we would be solving the equation Y = -X/3. Or, we could solve Y = -1.5X by replacing the 10k resistor to the left with a 20k resistor. Any basic linear equation can be solved using this type of circuit.

Resistors used in analog computers are not the typical standard resistor values. Rather, they are high-precision components intended and packaged for this application, and have values appropriate to this type of task.

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