By using a circuit built with a single analog multiplier and five precision
resistors, an output voltage (V_{o}) can be made to create a second-order
polynomial (*Fig. 1*).

The circuit comes in handy when a nonlinear sensor's output requires curve fitting before the signal is applied to the ADC input of a microcontroller. Thus, the microcontroller can be freed from the extra tasks involved in performing a software linearization algorithm.

Because all three polynomial terms are generated with a single IC package, component count is low, cost is reduced, and the circuit's footprints is small.

The circuit implements the following quadratic:

V_{o} = a + b V_{x} + V_{x}^{2}

The input terminal of IC1 are connected to create a positive square term and
present the V_{x} signal to the output with a 1/10-V scale factor.

Incorporating the voltage-divider network (resistors R_{3} and R_{4})
in the input signal path provides additional attenuation adjustment for the
coefficient (c) of the square term in the quadratic.

Then the passive adder (resistors R_{1}, R_{2} and R_{0})
is wired to IC1's internal summing circuit to generate the polynomial's other
two terms: the offset term (a) and the linear coefficient (b).

Figure 1 will create the example polynomial expression:

V_{o} = 0.4 V + 0.5V_{x} + 0.05 V_{x}^{2}

The reference V_{REF} would typically be selected as a stable dc voltage
with the same polarity as the offset term a in the quadratic. In this example,
V_{REF} is chosen to be +2.500 V—the output of a REF-03 voltage
reference IC.

To design the passive adder components, resistor R_{1} is chosen to
be 6.8 kΩ (6.81 k, 1%). The value of resistor R_{1}, which in this
example equals 21.25 kΩ (21.5 k, 1%), is obtained using the following expression:

R_{2} = bV_{REF} R_{1}/a

The circuit's equation constant (K), which equals 1, was derived and is given as:

K = (1 - b)/b

The value of resistor R_{0} is then found to be 10 kΩ, from the
expression:

R_{0} = R_{1}1R_{2}/K(R_{2} - R_{1})

The input voltage divider (R_{3} and R_{4}) modifies the square-term
coefficient according to the expression:

R_{4}/(R_{3} + R_{4}) = √10c

With R_{3} and R_{4} set to the values shown in *Figure
1*, the square-term coefficient is equal to the necessary 0.05 value.

The circuit's nonlinear output voltage can be seen on an XY plot as the input
(V_{x}) varies from 0- to 50-V peak (*Fig.
2*).

Since IC1 (the AD633JN) is a four-quadrant multiplier, its inputs can also
be wired to create a negative square term. The sign of the offset term (a) is
controlled by the selection of V_{REF} polarity.

Generating a linear term (b) greater than one or with a negative sign requires adding to the circuit an amplifier with the appropriate gain and phase relationship.