# Thermopile Time Constants

Several methods can be used to determine the time constant of thermopile detectors depending on the specific waveform of the radiation used to excite the detector. In case the detector is subject to a step function of radiation, its response follows the function Vt = Vmax(1-e-t/τ), where Vt is the detector output at any time t. The time constant (τ) of the system is defined as the time when Vt is 63.2% of its maximum static value Vmax.

If the detector is subjected to sinusoidally modulated radiation, its frequency response follows the function Vd = Vs[1+(2πτ/T)2]–1/2, where Vs is the static amplitude of the output voltage generated by un-modulated radiation and Vd is the dynamic amplitude of the detector output voltage at any wave period T. To is the period at which the output amplitude Vd, decreases by 3 dB (0.707 Vs) from the static value, relates to the detector time constant by the expression τ = To/kπ, where k = 2 for sinusoidally modulated signals.

The waveform of chopper-modulated radiation closely corresponds to a square wave, and in this example, the coefficient k = 1.124. Jones (see reference below) found the same relationship between the time constant and square or sine wave modulated radiation.

## Thermopile Window/Filter

One may use a Red LED with either of the technique if the thermopile window/filter transmits in the visible spectrum. The appropriate coefficient should be applied according to the waveform used. At Dexter Research Center (DRC), the time constant is determined using two methods. 1) DRC uses a square wave modulated Red LED if the thermopile window/filter transmits in the visible spectrum. 2) DRC uses a chopped blackbody in case the thermopile window/filter does not transmit in the visible spectrum.

Measuring the approximate time constant directly with a modulated signal is relatively easy and fast. Adjust the peak-to-peak trace of the detector’s DC output to seven divisions on an oscilloscope using a very slow modulation frequency. Increase the frequency until the peak-to-peak trace spans five divisions (0.707 x 7 div. = 4.95 div.). This is about –3 dB of Vmax. The time constant can subsequently be determined from the wave period or from the frequency, again using the suitable coefficient for the waveform utilized.

Infrared Detectors: Eds. Hudson & Hudson; Benchmark Papers in Optics, V.2; Dowden, Hutchinson, & Ross, Inc.; Stoudsburg, PA; J. Wiley & Sons, 1975. Page 324. This information has been sourced, reviewed and adapted from materials provided by Dexter Research Center, Inc.

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