Contactless High-Sensitivity Thermal Relay

M. S. ROITMAN

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

The circuit described can be successfully used for measuring, stabilizing, and regulating temperature. The merits of the circuit are its great flexibility and the possibility of obtaining high sensitivity. The flexibility of the circuit is manifested above all in the possibility of adjusting the output characteristic from monotonic all the way to step-like.

Let us consider the operation of the circuit (Fig. 1).

Fig. 1.

A voltage is applied to the base of triode \(\Pi_1\) (Fig. 1):

\[ U_{b}=\frac{UR_1}{R_1+R_{T_1}}+Ri_3, \tag{1} \]

where \(R_{T_1}\) is the resistance of the thermistor (MMT);

\(i_3\) is the collector current of triode \(\Pi_3\) (this current, as will be shown below, is a function of temperature).

The collector current

\[ i_1=\frac{U_{b_1}}{h_{11}}\cdot \beta_1+I_{\mathrm{kn}_1}, \tag{2} \]

where \(h_{11}\) is the input resistance of triode \(\Pi_1\),
\(\beta_1\) is the current gain,
\(I_{\mathrm{kn}_1}\) is the initial collector current.

A voltage is applied to the base of triode \(\Pi_2\)

\[ U_{b_2}=\frac{UR_{T_2}}{R_{T_2}+R_2} \]

and the collector current

\[ i_2=\frac{U_{b_2}}{h'_{11}}\beta_2+I_{\mathrm{kn}_2}, \]

where \(h'_{11}\) is the input resistance of triode \(\Pi_2\),
\(\beta_2\) is the current gain,
\(I_{\mathrm{kn}_2}\) is the initial collector current of triode \(\Pi_2\).

The resistance of thermistors \(R_{T_1}\) and \(R_{T_2}\) will decrease as the temperature increases. It is easy to see that, when the temperature rises, the current \(i_1\) increases, while \(i_2\) decreases.

When the temperature decreases, \(i_1\) will decrease, while \(i_2\) will increase. It is also easy to see that, for \(i_2 > i_1\), triode \(P_2\) is in saturation, the voltage drop across it is small, and triode \(P_3\) is practically cut off. As the temperature increases, the current \(i_1\) rises, and when \(i_1 \simeq i_2\), the voltage across \(P_2\) will increase and triode \(P_3\) will begin to open, i.e., the current \(i_3\) will increase. The resistance \(R'_1\) is an element of positive feedback. For \(R'_1 = 0\), or below a certain critical value \(R'_{1\ \mathrm{crit.}}\), the dependence of \(i_3\) on temperature is monotonic; for \(R'_1 \geq R'_{1\ \mathrm{crit.}}\), it is step-like.

Let us first consider the operation of the circuit for \(R'_1 = 0\). The sensitivity of the circuit to temperature change is

\[ S_t=\frac{di_3}{dt}\simeq\left(\frac{di_1}{dt}-\frac{di_2}{dt}\right)\beta_3, \tag{3} \]

\[ \frac{di_1}{dt} = \frac{\partial U_{b_1}}{\partial R_{T_1}}\cdot\frac{\partial R_{T_1}}{\partial t} - \frac{\partial h_{11}}{\partial U_{b_1}}\cdot \frac{\partial U_{b_1}}{\partial R_{T_1}}\cdot \frac{\partial R_{T_1}}{\partial t} + \frac{\partial \beta_1}{\partial h_{11}}\cdot \frac{\partial h_{11}}{\partial U_{b_1}}\cdot \frac{\partial U_{b_1}}{\partial R_{T_1}}\cdot \frac{\partial R_{T_1}}{\partial t}, \]

\[ \frac{di_2}{dt} = \frac{\partial U_{b_2}}{\partial R_{T_2}}\cdot\frac{\partial R_{T_2}}{\partial t} - \frac{\partial h'_{11}}{\partial U_{b_2}}\cdot \frac{\partial U_{b_2}}{\partial R_{T_2}}\cdot \frac{\partial R_{T_2}}{\partial t} + \frac{\partial \beta_2}{\partial h'_{11}}\cdot \frac{\partial h'_{11}}{\partial U_{b_2}}\cdot \frac{\partial U_{b_2}}{\partial R_{T_2}}\cdot \frac{\partial R_{T_2}}{\partial t}. \]

Taking into account that

\[ R_T=R_{T0}e^{B/T};\qquad -\frac{B}{t^2}=\alpha_t; \]

\[ h=h_0 e^{qU_b/kT}, \]

where \(R_{T0}\) and \(B\) are constants depending on the physical properties of the thermistor material;

\[ \alpha_t \text{ is the temperature coefficient of resistance,} \]

\[ q \text{ is the electron charge,} \]

\[ k \text{ is Boltzmann’s constant,} \]

\[ T \text{ is the absolute temperature } \left(\frac{kT}{q}\simeq 0.26\ \mathrm{V}\ \text{at }25^\circ\mathrm{C}\right). \]

we obtain:

\[ \frac{di_1}{dt} = \frac{UR_1}{R_1+R_{T_1}}\cdot \frac{\beta_1}{h_{11}}\cdot \frac{\alpha_t\cdot R_{T_1}}{R_1+R_{T_1}} \left[ 1+ \frac{UR_1}{R_1+R_{T_1}}\cdot \left(1+\frac{\partial\beta_1}{\partial h_{11}}\right)\cdot \frac{q}{kT} \right], \tag{4} \]

\[ \frac{di_2}{dt} = \frac{-UR_{T_2}}{R_2+R_{T_2}}\cdot \frac{\beta_2}{h'_{11}}\cdot \frac{\alpha_t R_2}{R_2+R_{T_2}} \left[ 1+ \frac{UR_{T_2}}{R_2+R_{T_2}}\cdot \frac{q}{kT}\cdot \left(1+\frac{\partial\beta_2}{\partial h_{11}}\right) \right]. \tag{5} \]

Taking into account that \(R_{T_1}\gg R_1\) and \(R_2\gg R_{T_2}\), and accepting \(\frac{\partial\beta}{\partial h_{11}}\simeq 0\), we may simplify expressions (4) and (5):

\[ \frac{di_1}{dt} = U_{b_1}\frac{\beta_1}{h_{11}}\alpha_t \left(1-U_{b_1}\frac{q}{kT}\right) \]

and

\[ \frac{di_2}{dt} = -\,U_{b_2}\frac{\beta_2}{h'_{11}}\alpha_t \left(1+U_{b_2}\frac{q}{kT}\right). \]

Hence

\[ \frac{d i_1}{i_1} \approx \alpha_t \left(1+\frac{U_{\mathrm{б}_1}}{0.026}\right) dt; \qquad \frac{d i_2}{i_2} \approx -\alpha_t \left(1+\frac{U_{\mathrm{б}_2}}{0.026}\right) dt \]

\[ \text{and} \qquad \frac{d i_3}{i_3} \approx \alpha_t \left(2+\frac{U_{\mathrm{б}_1}+U_{\mathrm{б}_2}}{0.026}\right). \tag{6} \]

Expression (6) makes it possible to directly estimate the relative sensitivity of the circuit to a change in temperature. For example, at \(U_{\mathrm{б}_1} \approx U_{\mathrm{б}_2} \approx 130\ \text{mV}\) and \(\alpha_t = \dfrac{4\%}{1^\circ\mathrm{C}}\), a temperature change of \(1^\circ\mathrm{C}\) will lead to a relative change in the current \(i_3\) of 48%. Taking \(i_1=i_2=1\ \text{mA}\) and \(\beta_3=50\), we obtain an increment of the current \(i_3\) of 24 mA.

Fig. 2.

Figure 2 gives the experimentally obtained dependence \(U_{\mathrm{out}} = f(t)[R_3=1\ \text{k}\Omega]\) at \(R'_1=0\) (curve a) and at \(R'_1 > R'_{\mathrm{crit.}}\) (curve b).