"Chapter 1 - The Op Amp's Place in the World" - HTL Wien 10

Simultaneous Equations
4.3 Simultaneous Equations
4-8
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).
y mx b
(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.
V OUT mV IN b
V OUT mV IN b
V OUT mV IN b
V OUT mV IN b
(4–13)
(4–14)
(4–15)
(4–16)
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.
1 m(0.1) b
(4–17)
4 m(0.2) b
Multiply Equation 4–17 by 2 and subtract it from Equation 4–18.
2 m(0.2) 2b
b 2
(4–18)
(4–19)
(4–20)
After algebraic manipulation of Equation 4–17, substitute Equation 4–20 into Equation
4–17 to obtain Equation 4–21.