1 Introduction

For decades, P-N junction structures at forward bias has been used extensively for sensing ambient temperature. Nevertheless, P-N junctions are prone to non-desirable surface effects, as generation and recombination in the depletion layer.

Consequently, bipolar junction transistors (BJT), with collector to base short-circuit are preferred for temperature measurement applications, since these devices exhibit a current-voltage (I-V) characteristic, closer to a pure exponential function.

Often, due to their temperature coefficients, BJTs are used for the implementation of band-gap voltage references ^[1]. In this sense, Verster ^[2] proposed a methodology where two different collector current magnitudes, I₁ and I₂, are employed in a BJT circuit. The difference between both base-emitter voltages Δv_BE=v_BE1- v_BE2=(kT/q)ln(I₁/l₂) (where k is Boltzmann's constant, q is the electron charge and T is the absolute temperature) is used to calculate the absolute temperature T, with inaccuracies of the order of 3K to 4K.

Other approaches, suggest the use of BJTs as primary absolute thermometers. The acquisition of an absolute temperature value without any calibration step has many advantages from the manufacture and cost perspective. Felimban et al.^[3] introduced an absolute thermometer based on an emitter-base voltage characterization of a commercial BJT in the 77K-400K range. Using the short-circuit from collector to base as depicted in Fig. 1, the main component of BJT collector current is due to thermal diffusion of carriers. Therefore, the forward collector current is expressed by:

where I_s(T) is the inverse saturation current and β is q/kT.

Fig. 1 BJT biasing with shorted collector-base junction 

By using the BJT characterization data I_C vs. V_BE in a semilog-plot as done in ^[3], the absolute temperature can be computed, see Fig.2.

Fig. 2 Different I-V characteristics of commercial BJTs at 0°C 

2 Computation of the Absolute Temperature

From the I-V characterization data, equation (1) can be rearranged to the linear form using the natural logarithm function, this is:

where I'_C=ln(I_C), α=ln(I_s) and V_BE is the independent variable. Therefore, with the measured data (I_C, V_BE), taken to the form (2) and by using a linear fit, the absolute temperature can be extracted from the slope β.

However, this methodology present important drawbacks. First, only those data points related closely to (1) must be considered, i.e. data points close to a straight line in the semi-log plot. Second, slope computation is very sensitive to noise especially for small currents; this eventually will produce an error of several Kelvin in the temperature calculation. However, (2) can be used to compute I_s accurately through α as the axis intercept.

Recently, Mimila-Arroyo ^[4] proposed a different methodology using an auxiliary numerical operator exp[-β(T*)V_BE] where T* is a proposed temperature value. Again, from the BJT I-V characterization, the following expression can be formulated as the product of the collector current and the operator:

Consequently, by using (1) and (3) the following limit must be fulfilled:

Since I_S(T) has very little dependence with V_BE, it is expected that when T*=T, the I''_C vs. V_BE plot turns into a straight line parallel to the abscissa axis, see Fig. 3. Thus, when this graphical condition is satisfied, the absolute temperature T is obtained. This methodology reports temperature errors in the mK range.

Fig. 3 Linear plot of I''_C vs. V_BE for T*=T, the value of I_s=2.1129×10-14A was previously calculated using (2) at 273.15K 

3 Hybrid algorithm

Following the methodology in ^[4-5], T* must be adjusted manually until the graphical condition where a straight line parallel to the abscissa is obtained, in such case T is determined. Therefore, the proposed hybrid algorithm based on (2) and (4) is formulated in order to get a cost function, which allows numerically to find the absolute temperature in a fully automated fashion, the algorithm is as follows:

where (I_Ci, V_BE) is the ith I-V measurement pair. This function should have a global minimum when T*=T and can be find numerically by any optimization method.

4 Experimental Results

Several I-V measurements were conducted using commercial BJTs at different temperatures using Keithley 6487 picoammeter. A plot of the cost function f(T*) vs. T* with the 2N2222 and 2N3055 devices at 0°C is depicted in Fig. 4. As it can be seen following (5), the cost function is a continuous function with a global minimum close to the absolute temperature 273.15K.

Fig. 4 Plot of cost function for two commercial BJTs in the -50°C to 50°C range expressed in K 

The comparison of real and computed temperatures, following the methodology, are depicted in Table I and Table II. Several commercial BJTs were tested at different temperatures in the -20°C to 100°C range. Measurements were conducted in the certificated laboratory of metrology MetAs S.A. de C.V.

5 Discussion

The proposed methodology was developed with the aim of avoiding the use of derivative based methods, since these introduce inaccuracies due to noise presence. Although the experimental results are not as accurate as those reported in ^[4], the results are quite good as compared with other methodologies including ^[6], where more complex I-V models were considered.

As part of this work, the series resistance R_s of base-emitter junction were characterized following ^[7] for each transistor, Table I and Table II. This series resistance could be related with the high accuracy of 2N3055 and ZTX1048 as compared with the other transistors, probably, its I-V behavior is closer to (1) improving accuracy.

A secondary effect could be the package of the BJT which could help to dissipate the auto-heating effect in high currents. However, the 2N3055 with a low thermal resistance package does not show a lower absolute error over ZTX1048 as depicted in Fig. 5, in the -20°C to 100°C range. Further research of the series resistance role could help improving the accuracy.

Fig. 5 Measured absolute error of 2N3055 and ZTX10 in the -20°C to 100°C range 

6 Conclusions

A hybrid methodology for the calculation of the absolute temperature from the I-V characterization of a BJT is introduced. The proposed methodology can be implemented on any numeric platform and allows the use of any BJT as a primary thermometer in a certain range. The best experimental results was identified by employing the commercial BJT ZTX1048, showing the feasibility of the proposal methodology with temperature errors <0.07K in almost all the entire -20°C to 50°C range, see Fig. 5.