IC Generates Second-Order Polynomial

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_x²

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 R₃ and R₄) 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 R₁, R₂ and R₀) 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_x²

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 R₁ is chosen to be 6.8 kΩ (6.81 k, 1%). The value of resistor R₁, which in this example equals 21.25 kΩ (21.5 k, 1%), is obtained using the following expression:

R₂ = bV_REF R₁/a

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

K = (1 - b)/b

The value of resistor R₀ is then found to be 10 kΩ, from the expression:

R₀ = R₁1R₂/K(R₂ - R₁)

The input voltage divider (R₃ and R₄) modifies the square-term coefficient according to the expression:

R₄/(R₃ + R₄) = √10c

With R₃ and R₄ 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.