Simultaneous Equations

Taking an orderly path to developing a circuit that works the first time starts here; follow these steps until the equation of the op amp is determined. Use the specifications given for the circuit coupled with simultaneous equations to determine what form the op amp equation must have. Go to the section that illustrates that equation form (called a case), solve the equation to determine the resistor values, and you have a working solution.

A linear op amp transfer function is limited to the equation of a straight line (Equation 4–12).

The equation of a straight line has four possible solutions depending upon the sign of m, the slope, and b, the intercept; thus simultaneous equations yield solutions in four forms.

Four circuits must be developed; one for each form of the equation of a straight line. The four equations, cases, or forms of a straight line are given in Equations 4–13 through 4–16, where electronic terminology has been substituted for math terminology.

Given a set of two data points for VOUT and VIN, simultaneous equations are solved to determine m and b for the equation that satisfies the given data. The sign of m and b determines the type of circuit required to implement the solution. The given data is derived from the specifications; i. e., a sensor output signal ranging from 0.1 V to 0.2 V must be interfaced into an analog-to-digital converter that has an input voltage range of 1 V to 4 V.

These data points (VOUT = 1 V @ VIN = 0.1 V, VOUT = 4 V @ VIN = 0.2 V) are inserted into Equation 4–13, as shown in Equations 4–17 and 4–18, to obtain m and b for the specifications.

Multiply Equation 4–17 by 2 and subtract it from Equation 4–18.

After algebraic manipulation of Equation 4–17, substitute Equation 4–20 into Equation 4–17 to obtain Equation 4–21.

Now m and b are substituted back into Equation 4–13 yielding Equation 4–22.

Notice, although Equation 4–13 was the starting point, the form of Equation 4–22 is identical to the format of Equation 4–14. The specifications or given data determine the sign of m and b, and starting with Equation 4–13, the final equation form is discovered after m and b are calculated. The next step required to complete the problem solution is to develop a circuit that has an m = 30 and b = –2. Circuits were developed for Equations 4–13 through 4–16, and they are given under the headings Case 1 through Case 4 respectively.

There are different circuits that will yield the same equations, but these circuits were selected because they do not require negative references.