By using a circuit built with a single analog multiplier and five precision resistors, an output voltage (Vo) 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:
Vo = a + b Vx + Vx2
The input terminal of IC1 are connected to create a positive square term and present the Vx 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:
Vo = 0.4 V + 0.5Vx + 0.05 Vx2
The reference VREF would typically be selected as a stable dc voltage with the same polarity as the offset term a in the quadratic. In this example, VREF 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 = bVREF 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 (Vx) 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 VREF 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.