I. INTRODUCTION

Temperature sensors are being widely used in advancing technology for various applications in semiconductor, automobile, aviation, environmental fields and so on. Until now, many types of temperature sensors have mainly been studied and used such as thermocouples, thermistors, resistance temperature detectors (RTDs), infrared temperature sensors, and the like. In the case of thermocouples, it is possible to measure over a wide temperature range, but the accuracy and stability are low. Thermistors have better accuracy than other types of sensors, but their temperature measurement range is limited. RTDs are stable over a limited temperature range, but have a narrow temperature measurement range. Besides, RTDs are expensive, and measurement inaccuracy occurs due to self-heating. For infrared temperature sensors, they can measure temperature without contact, even at high temperatures. However, they are sensitive to surface radiation with low accuracy.

A diode, a type of semiconductor device, is also being researched as a temperature measuring device. The range of use of semiconductor temperature sensors including diodes is about -50 degrees to +150 degrees, which is wider than other types of temperature sensors except for thermocouple temperature sensors. A diode temperature sensor is a device that uses the temperature dependency of the forward voltage of the diode terminals. As the temperature rises, the diode terminal voltage exhibits a characteristic of about -1 to -2 $\mathrm{mV} /{ }^{\circ} \mathrm{C}$. The previous studies using diodes as temperature sensors lack compatibility. They proposed a system that is only applicable to the particular type of diode used in their studies. In addition, complex and massive conditioning circuits with multiple op-amps, BJTs, diodes were used to implement semiconductor temperature sensor systems (^(1-8)), increasing system volume and cost. Furthermore, in previous studies, the sensitivity before the conditioning circuit depends on the sensitivity of the diode itself, and the sensitivity of the diode is regarded as a constant with respect to temperature change. On the other hand, in this study, we prove that the temperature sensitivity of the diode voltage is not constant, and in order to increase the accuracy of the temperature sensitivity of the diode voltage, the rate of change of the temperature sensitivity of the diode voltage for temperature change (2nd derivative of the temperature of the diode voltage) is newly introduced. In addition, in order to increase the temperature sensitivity of the entire system, using the voltage ratio between the diode and the resistor is proposed instead of using only the sensitivity of the diode itself. Also, a simple voltage source method is used in this study to reduce the complexity and volume of the system due to the implementation of the current source used in the most previous studies. Another advantage of this study is that the temperature sensitivity of the system can be controlled by adjusting the value of the applied voltage source, and the compatibility problem can be solved simply by the reference and measured values with single calibration point regardless of the type of diode used. It is a great solution for flexible and low-cost applications.

II. Theories

The output of diodes used as temperature sensors is the diode voltage $\left(V_{D}\right)$. Temperature curve of $V_{D}$ is pretty linear at least over the small region. That is around $0.5 \sim 0.8 \mathrm{~V}$ depending on the bias conditions. The theoretical equation for the diode current $\left(I_{D}\right)$ can be shown to be

where $I_{S}$ is the saturation current, $\eta$ is the emission coefficient, $V_{T}(=k T / q)$ is the thermal voltage, and $T$ is the absolute temperature. Note that the maximum average forward rectified current of diode is normally limited to $1 A$. As shown in Fig. 1, the diode has the temperature-dependent voltage-current characteristics. As the temperature increases, the diode voltage tends to decrease. In the following, we'll discuss why it decreases and how much it decreases with vary ing temperatures.

Through many mathematical processes from Eq. (1), the expression for temperature sensitivity of diode voltage $\left(s \equiv d V_{D} / d T\right)$ is given by

Note that Eq. (2) is regardless of ƞ. $V_{G0}$ (= $E_{G0}$/q) and $E_{G0}$ is the semiconductor bandgap voltage and bandgap energy at the reference temperature $T_{0 }$($T$ = 298$K$), respectively. For silicon, $E_{G0}$ = 1.12 $eV$ and, therefore, $V_{G0}$ = 1.12 $V$. Also, note that $E_{G}$ can be assumed to be constant with temperature albeit it’s not a constant because its effect on the sensitivity is negligible. In Eq. (2), we can find that $s$ is always negative and therefore, $V_{D}$ drops with increasing $T$.

III. System design and Analysis

In order to increase $s$, we propose the ($V_{R}$/$V_{D}$) approach, in which we use a single resistor and diode. The circuit diagram for the $V_{R}$/$V_{D}$ simulation and its experimental setup is presented in Fig. 3.

For $V_{R}$/$V_{D}$ calculation,

where $\overline{s}$ is the average $s$ over ${\Delta}$$T$ interval. Then,

where we use an approximation $1 /[1+f(x)] \cong 1-f(x)$ which is especially true for $|f(x)| <1$. Once $V_{D 0}$ is given, we can determine $s_{0}$ and $A$$_{0}$ in Eqs. (2) and (4), respectively. Also, $V_{R0}$ = $V_{S}$ - $V_{D0}$$_{\mathrm{.}}$ Now, we’ll apply the Method I in which $s=\overline{s}=s_{0}$, as explained earlier.

Assuming $\overline{s}$= $s_{0}$ in Eqs. (7) and (8) leads to

where 2$s_{0}^{2}\Delta T$ is relatively small compared to -$s_{0}$$V_{S}$ in magnitude because $s_{0}$ is generally in the range of about -1 to -2 $mV^{\circ}$C. Therefore, 2$s_{0}^{2}\Delta T$can be ignored as long as ${\Delta}$$T$ is not so big. Then, Eq. (9a) can be simplified to

Here, we define Eqs. (9a) and (9b) as Method I(detailed) and Method I, respectively, depending on whether the expression includes ${\Delta}$$T$ or not.

Then, we’ll apply the Method II in which $\overline{s}$= So $+A 0 \Delta T / 2$, as discussed earlier. Then, Eq. (8) becomes

Combining $\bar{s}=s_{0}+A_{0} \Delta T / 2$ and Eq. (7) yields (${\Delta}$$T$)$^{2}$=2/A$_{0}$[($V_{D}$-$V_{D}$$_{0}$)-$s_{0}$${\Delta}$$T$]. Then, through many steps, Eq. (11) can be expressed as

Assuming $\left|s_{0}\right|>>|A 0 \Delta T|$ over any small range of temperature in Eq. (12) leads to

Similarly, we define Eqs. (11a) and (11b) as Method II and Method II(simplified) respectively, depending on whether the expression includes ${\Delta}$$T$ or not. Also, for Method III, substitution Eq. (5) into Eq. (12a) leads to

In Table 2, we summarize the temperature sensitivity of $V_{R}$/$V_{D}$ for each method.

where $\Delta T$ expressions can be obtained by exploiting the fact that $d\left(V_{R} / V_{D}\right) / d T=\left(V_{R} / V_{D}\right) / \Delta T$. As an example, for Method $I$ (detailed), $\Delta T$ can be calculated by solving quadratic equation of $\Delta \mathrm{T}$ and it is given by

For the purpose of comparing temperature predictions, we have presented 5 Methods ($I$, $I$(d), $II$, $II$(s), $III$). The most accurate approach is Method III, which can be applied to a wider temperature range by considering the temperature-dependent characteristics of $s$ and $A$. However, the formula is very complex (Therefore, we’ll not mention Method III thereafter). Note that Method I and Method II(s) are easy to use. However, we may pick Method I and $II$ as the main prediction methods. Note that Method I is the simplest while Method II is most accurate. For small ${\Delta}$$T$, we can use Method I for simple and quick calculations without significant errors. However, for large ${\Delta}$$T$, Method II must be used for precise temperature measurement.

In Fig. 4, we show the $V_{D}$ and $V_{R}$/$V_{D}$ simulation results for $V_{S =}$ 1.8, 3.3, and 5 $V$. All the plots appear to be linear over the whole temperature range (-25~75 $^{\mathrm{o}}$C). Here, the reference temperature $T_{0}$ = 25 $^{\mathrm{o}}$C, and the temperature range is 50 degrees in both directions from $T_{0 }$(${\Delta}$$T$ = ${\pm}$ 50). Note that the temperature sensitivity prediction of $V_{R}$/$V_{D}$ for the whole temperature range looks linear as shown in Fig. 4(b). In the simulation, the 1N4002 diode and 100 Ω resistor are used. However, for any single or discrete measurement with respect to $T_{0}$, the prediction gets worse as $T_{m}$ deviates from $T_{0}$, especially for Method I.

In addition, it is observed that the temperature sensitivity of $V_{R}$/$V_{D}$ increases with $V_{S}$, as can be checked in Eqs. (9)-(12). Therefore, we can easily change the temperature sensitivity by controlling $V_{S}$. That is one of the main points of this study.

We show the prediction temperature using $V_{R}$/$V_{D}$ approach for $V_{S =}$ 1.8 $V$, as plotted in Fig. 5. The reference temperature $T_{0}$ is 25 $^{\mathrm{o}}$C and the prediction near $T_{0}$ is very accurate. However, temperature prediction gets worse as it deviates from the reference temperature. It can be clearly seen that the magnitude of ${\Delta}$$T$ prediction is higher in the regions above $T_{0}$, whereas the magnitude of ${\Delta}$$T$ prediction is lower in the regions below $T_{0}$. It is because $\overline{s}$, as can be seen in Table 1(a), is always negative and more decreases (= large sensitivity in magnitude) with increasing temperatures. Since the approximation theory is applied in $d$($V_{R}$/$V_{D}$)/$dT$ formula derivation processes, the exact ${\Delta}$$T$ predictions are not possible especially when ${\Delta}$$T$ is large. As shown in Fig. 5, Method II appears to be the most accurate method close to the actual value.

IV. Measurement Results

In Fig. 6, our temperature measurement setup is presented. A microprocessor-based temperature controller HSC-40 is used to monitor the temperature inside the chamber using a P-type thermocouple. The K-type GILTRON thermocouple (-200 ~ 200$^{\circ}$C) is additionally used to check the accurate temperature inside the chamber.

Note that $V_{D0}$ must be carefully measured prior to the temperature prediction. In order to utilize such diode circuits to measure temperature, one must obtain accurate values for $V_{D0}$ of the diode in advance.

The $V_{D }$values of experimental measurements, for $V_{S }$ = 1.8 $V$ and $R$ = 100 Ω, appear in Table 3. The graphs of $V_{D}$, $V_{R}$/$V_{D}$, and $T$ prediction are plotted in Fig. 7. Note that the simulation results agree well with the experimental results. For example, at 100 $^{\circ}$C, the temperature prediction error using Method II is less than 1 $^{\circ}$C.

V. Conclusions

In this study, we derive the diode voltage-temperature relational equation. Then, in order to increase temperature sensitivity, we propose the ($V_{R}$/$V_{D}$) approach as a function of $V_{S}$ and $V_{D0}$, whose sensitivity is accurate over any small range of temperature with respect to the reference temperature. However, over the temperature ranges far from the reference, the modified equation must be used to reduce the error in estimation temperature. Also, by controlling $V_{s}$ for a specified $V_{D0}$, the temperature sensitivity can be increased by several times in magnitude (e.g., 8.7 times, $s$ = 12.3 $mV^{\circ}$C) compared to the typical 1N 4002 diode ($s$ = -1.42 $mV^{\circ}$C) for $V_{s}$ = 5.0 $V$, $R$ = 100 Ω, and $V_{D0}$ = 0.761 $V$. This work presents a diode temperature-measurement system with simplicity, high-sensitivity, and sensitivity-controllability.