Measurement and process control loops often use 8-bit
microcontrollers. The devices are inexpensive and widely available,
and they can be programmed in many popular high-level
languages—like C and Basic. However, if the loop requires a nonlinear
sensor, the designer faces the added challenge of having to
develop a software linearization algorithm.
One solution is to design hardware to perform the required
curve fitting before the sensor’s output signal is applied to the
microcontroller’s analog-to-digital converter (ADC) input. This is
an especially attractive solution if processor memory is limited and
cost and component count can be kept low. An added benefit is a
small printed-circuit board (PCB) footprint.
An example is the circuit shown in the figure, which uses a
type-E thermocouple to measure temperatures that are expected
to vary from 0°C to 500°C. The circuit’s output ranges from 0 to
5 V—the full span of the ADC—and can be expressed as a linear
system equation:
The thermocouple amplifier section consists of a thermocouple
cold-junction compensator (IC1), an op amp (IC2), and accompanying
circuitry. IC1 tracks ambient temperature (TR) and adds
a correction voltage (VC) to cancel the cold-junction voltage
(VR) created when chromel-constantan thermocouple leads are
attached to the copper pads of a PCB. IC2 then amplifies the thermocouple’s
temperature-dependent signal (VM) by 175 to produce
VT prior to linearization. A gain of 175 eliminates the need for
additional amplification during curve fitting.
Low-pass filter C1-R2 has a pole at approximately 6 Hz to
remove power-supply noise.
You can use an Excel spreadsheet to create the nonlinear mathematical
relationship between the output of the thermocouple
amplifier (VT) and the input to the ADC (VO). The table shows
11 temperature entries for TM (0°C to 500°C in 50°C steps) with
their corresponding thermoelectric voltages, VM. The values of VM
were derived from a standard type-E thermocouple reference table.
Also shown are VT and VO, which are charted using the (XY) Scatter
feature of the software. The equation for VO is created using the
software’s trendline feature:
An analog multiplier (IC3) and five 1% resistors implement this
equation, a second-order polynomial of the form:
The four (X and Y) inputs of IC3 are wired to create a negative
square term that’s scaled at the chip’s output by an internal
scale factor of 0.1 V. Comparing terms, we note that c must equal
0.0165. Since R3 and R4 form a voltage divider to attenuate VT, we
can express c as follows:
You can then solve for R3 by substituting c = 0.0165 and selecting
a value for R4, which for this design was chosen to be 10 k?.
Resistors R5, R6, and R7 form a passive adder to create the offset
term, a, and the linear coefficient, b, in Equation 3. The output of
the passive adder is applied directly to the Z-input of IC3, which
adds the offset and linear terms to the square term. Again comparing
terms, a (Equation 3) equals 0.0342 V
(Equation 2). We can express this as:
To design this part of the passive adder, we
chose a stable 2.500-V reference for VREF,
selected R5 to be 10 k?, and solved for R6.
Finally, compare b (Equation 3) with the
corresponding value of 0.8703 (Equation 2)
and express the linear coefficient b as:
This equation is solved to determine the
final component, R7:
We evaluated the circuit by replacing the
thermocouple with a low-impedance voltage
source to simulate VM. IC3’s output
voltage (VO) exhibited a worst-case error
of about 3.4°C at 0°C (VO = 0.034 V) and
an error of -1.8°C at 500°C (VO = 4.982 V)
at full scale. At 250°C, a mid-range error of 0.2°C (VO = 2.502 V) was recorded.
Reference:
“IC Generates Second-Order Polynomial,” Electronic Design,
Aug. 5, 1993, p . 83.
See associated table