Temperaturetoperiod circuit provides linearization of thermistor response
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Designers often use thermistors rather than other temperature sensors because thermistors offer high sensitivity, compactness, low cost, and small time constants. But most thermistors' resistanceversustemperature characteristics are highly nonlinear and need correction for applications that require a linear response. Using a thermistor as a sensor, the simple circuit in Figure 1 provides a time period varying linearly with temperature with a nonlinearity error of less than 0.1K over a range as high as 30K. You can use a frequency counter to convert the period into a digital output. An approximation derived from Bosson's Law for thermistor resistance, R_{T}, as a function of temperature, θ, comprises R_{T}=AB^{–θ} (see sidebar "Exploring Bosson's Law and its equation"). This relationship closely represents an actual thermistor's behavior over a narrow temperature range.
Figure 1 This simple circuit linearizes a thermistor’s response and produces an output period that’s proportional to temperature.
You can connect a parallel resistance, R_{P}, of appropriate value across the thermistor and obtain an effective resistance that tracks fairly close to AB^{–θ }30K. In Figure 1, the network connected between terminals A and B provides an effective resistance of R_{AB }AB^{–θ}. JFET Q_{1} and resistance R_{S} form a current regulator that supplies a constant current sink, I_{S}, between terminals D and E.
Through bufferamplifier IC_{1}, the voltage across R_{4} excites the RC circuit comprising R_{1} and C_{1} in series, producing an exponentially decaying voltage across R_{1} when R_{2} is greater than R_{AB}. At the instant when the decaying voltage across R_{1} falls below the voltage across thermistor R_{T}, the output of comparator IC_{2} changes its state. The circuit oscillates, producing the voltage waveforms in Figure 2 at IC_{2}'s output. The period of oscillation, T, is T=2R_{1}C_{1}ln(R_{2}/R_{AB})2R_{1}C_{1}[ln(R_{2}/A)+θlnB]. This equation indicates that T varies linearly with thermistor temperature θ.
Figure 2 Waveforms show input to comparator IC_{2} (lower trace) and its output (upper trace). In the lower trace, IR_{2} represents the voltage across R_{2}.
You can easily vary the conversion sensitivity, ΔT/Δθ, by varying resistor R_{1}'s value. The current source comprising Q_{1} and R_{1} renders the output period, T, largely insensitive to variations in supply voltage and output load. You can vary the period, T, without affecting conversion sensitivity by varying R_{2}. For a given temperature range, θ_{L} to θ_{H}, and conversion sensitivity, S_{C}, you can design the circuit as follows: Let θ_{C} represent the center temperature of the range. Measure the thermistor's resistance at temperatures θ_{L}, θ_{C}, and θ_{H}. Using the three resistance values R_{L}, R_{C}, and R_{H}, determine R_{P}, for which R_{AB} at θ_{C} represents the geometric mean of R_{AB} at θ_{L} and θ_{H}. For this value of R_{P}, you get R_{AB} exactly equal to AB^{–θ} at the three temperatures, θ_{L}, θ_{C}, and θ_{H}.
At other temperatures in the range, R_{AB} deviates from AB^{–θ}, causing a nonlinearity error that is appreciably less than 0.1K for most thermistors when the temperature range is 30K or less. You can easily compute R_{P} using: R_{P}=R_{C}[R_{C}(R_{L}+R_{H})–2R_{L}R_{H}]/(R_{L}R_{H}–R_{C}^{2}). Because temperaturetoperiodconversion sensitivity, S_{C}, is 2R_{1}C_{1}lnb, you can choose R_{1} and C_{1} such that R_{1}C_{1}=S_{C}[θ_{H}–θ_{C}]/ln(R_{AB} at θ_{L}/R_{AB} at θ_{H}) to obtain the required value of S_{C}. To get a specific output period, T_{L}, for the low temperature, θ_{L}, R_{2} should equal (R_{AB} at θ_{L})e^{Y}, in which Y represents (T_{L}/2R_{1}C_{1}). In practice, use a lower value for R_{2} because the nonzero response delay of IC_{2} causes an increase in the output period.
Next, set potentiometers R_{1} and R_{2} close to their calculated values. After you adjust R_{1} for the correct S_{C}, adjust R_{2} until T equals T_{L} for temperature θ_{L}. The two voltagedivider resistances, R_{3} and R_{4}, should be equal in value and of close tolerances. As a practical example, use a standard thermistor, such as a Yellow Springs Instruments 46004, to convert a temperature span of 20 to 50°C into periods of 5 to 20 msec. This thermistor exhibits resistances for R_{L}, R_{C}, and R_{H} of 2814, 1471, and 811.3Ω, respectively, at the low, midpoint, and high temperatures. Other parameters for the design include S_{C}=0.5 msec/K, θ_{L}=20°C, θ_{H}=50°C, θ_{C}=35°C, and T_{L}=5 msec.
Because only a fraction of current I_{S} is through the thermistor, I_{S} should be low to avoid selfheating effects. This design uses an I_{S} of approximately 0.48 mA, which introduces a selfheating error of less than 0.03K for a thermistor's dissipation constant of 10 mW/K. Figure 1 illustrates the values of the components in the example. All resistors are of 1% tolerance and 0.25W rating; use a polycarbonatedielectric capacitor for C_{1}.
Simulating various temperatures from 20 to 50°C by replacing the thermistor with standard, 2814 to 811.3Ω, 0.01%tolerance resistors produces T values of 5 to 20 msec with a maximum deviation from correct readings of less than 32 µsec, which corresponds to a maximum temperature error of less than 0.07K. Using an actual thermistor produces a maximum error of less than 0.1K for a thermistor dissipation constant of 10 mW/K or less.
Exploring Bosson's Law and its equation One cause of confusion you may encounter when reading the papers in the references involves authors’ use of different symbols to represent a variable or a constant while referring to a commonly known equation. Readers unfamiliar with Bosson’s Law and its accompanying equation can view the original paper (Reference A), in which the authors state Bosson’s Law as: R=A+[B/(T+θ)]. In the equation, quantities A, B, and θ represent certain thermistor constants, and temperature, T, is in kelvins. By taking the exponent of both sides, you get R=e^{A}×e^{[B/(T+θ)]}. If you replace the constant e^{A} with A, you can restate Bosson’s Equation as R=A×e^{[B/(T+θ)]}. Most authors have used this form to represent Bosson’s Equation only in their published papers. Here, T is temperature, and θ is a constant. Because T and θ come together in this equation as T+θ, you can alternatively use θ to represent temperature and T as a constant. The material in Reference B presents the equation as R=A×e^{[B/(TOθ)]} and denote it as Bosson’s Form. This equation uses T_{0} in place of T, and T_{0} is a constant. Reference B denotes the constant, T_{0}, as initial temperature and θ as temperature. With suitable assumptions, the literature derives the approximation R=AB^{–θ} from the Bosson Form. References A. Bosson, G, F Guttman, and LM Simmons, “A relationship between resistance and temperature of Thermistors,” Journal of Applied Physics , Volume 21, 1950, pg 1267. B. Yankov, IY, CI Gigov, and EA Yankov, “Linear temperaturetotime period converters using standard thermistors,” Measurement Science and Technology , Volume 1, No. 11, November 1990, pg 1168.

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