# Temperature-to-period circuit provides linearization of thermistor response

Designers often use thermistors rather than other temperature sensors because thermistors offer high sensitivity, compactness, low cost, and small time constants. But most thermistors’ resistance-versus-temperature 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.

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 buffer-amplifier 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.

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}.

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