# Differential Temperature Transducer

An Iterative Method of Evaluating the Signal

The Differential Temperature Transducer has been used for a number of years to measure the change in temperature of closed fluid-streams.

The instrument is based on the thermopile, which is inherently a differential device; even when used as an absolute-temperature detector, a thermocouple or thermopile does so only by virtue of a reference temperature made available to it. One junction (the cold junction) serves as the "reference" for the other junction (the hot junction). Where "absolute" temperature is being measured, a standard reference cold-junction of 0°C (ice bath temperature) is used.

A difficulty, however, is that thermocouple characteristics are nonlinear. Figure 1 shows a typical E(mV) vs T(temperature) curve. The National Bureau of Standards (now NIST) has published tables of thermocouple characteristics which define the nonlinearities, and various linearizing schemes are used to translate the millivolt thermocouple signal into an accurate temperature reading.

In the Differential Temperature Transducer, another difficulty arises: the reference temperature is not a fixed value. If the low-temperature side is the "reference," then the sensitivity (mV/°C) can vary considerably, depending on the value of that low-temperature point. Another way to describe the problem is by recognizing that, depending on where on the thermocouple characteristic curve one is operating, the conversion factor (slope) for signal strength to differential temperature is a variable; the curve of E (output) vs Temperature (T) is not linear.

Some information must be available, in addition to the DeltaT signal itself, to define the correct sensitivity (S):

Note that S, the Seebeck coefficient for the particular thermocouple materials, is a function of both ΔT and T_{ch},where T_{ch} is a characteristic temperature, such as T_{low}, T_{high}, or T_{avg}.

The temperature difference to be determined (ΔT) is:

where ΔE is the measured signal, in millivolts, and S is an average, or effective, sensitivity over the range T_{low} to T_{high}.

Equations (1) and (2) indicate the problem: S, in general, is a stronger function of T_{ch} than of ΔT.

## THE ITERATIVE METHOD

To implement the linearization method herein, we assume an approximate value of T_{low} (or better yet, a measured value). Figure 1 depicts a thermocouple characteristic, taken from published data.

We wish to linearize the curve between two points, T_{low} and T_{high}. On Figure 1, T_{low} is point 1; T_{high} is point 2. The millivolt signal to be linearized (ΔE) corresponds to T_{high} - T_{low}.

The sensitivity, the slope of the characteristic curve, is evaluated at T_{low} by drawing a tangent to the curve at the point. Let us call this dE/dT_{1}. Using this sensitivity at point 1 and ΔE, the measured signal, we compute the first approximation (ΔT_{i}) to ΔT:

Clearly, ΔT_{i} exceeds ΔT_{true} because of the increasing slope of the characteristic curve with increasing temperature. If, then, a new characteristic temperature is computed, an average between T_{low} and an apparent T_{high}, using ΔT_{i}, is:

Point 3 on the curve corresponds to T_{avi} and, if a new tangent is drawn at point 3, a still better value of effective sensitivity is found:

The result of the computation is re-inserted into (4), giving yet another (and better) point of the curve. With each iteration, the slope (sensitivity) lies closer and closer to the chord 1-2, whose slope is the correct value.

This procedure is nothing more than a graphical version of the familiar Newton-Raphson iteration procedure used to solve for zeroes of a polynomial.

Conveniently, the NBS has provided curve-fitting information in the form of coefficients of an 8th-order polynomial that closely approximates the thermocouple characteristic curve for copper-Constantan (type T) over a range of 0° to 400°C (see Table 1):

It is a simple matter to differentiate (6) to arrive at a polynomial expression for dE/dT:

The following simple algorithm can quickly, within one or two iterations (depending on the accuracy required and the magnitude of ΔT), yield a satisfactory result:

For the first pass, let T_{1} = T_{low}; then return to (8) with T_{avi} until ΔT converges to a stable value.

Note: The Delta-T Differential Temperature Transducer's thermopile consists of 10 pairs of junctions. Hence the sensitivity coefficients (a_{1} ... a_{8}) used in the computation exceed the published values in NBS 125 by a factor of 10.

## RESULTS

For a typical case where T_{low} = 20°C is erroneously estimated at 25°C, then the error in ΔT is found to be 0.1°C after two iterations. At that portion of the curve, for small errors in T_{low}, the error in ΔT is given by:

The more accurate the knowledge of the reference, or low temperature, the more accurate will be the calculated ΔT that we measure. Fortunately, the results are not very sensitive to this estimate.

If the value of T_{low} is known to vary considerably, it might be convenient to provide a simple, inexpensive temperature probe at the fluid inlet to the Differential Temperature Transducer to measure the actual temperature. The Delta-T transducer can be provided with such probe incorporated within the housing if desired.