ON THE QUESTION OF THE STABILITY OF THRESHOLD DEVICES

V. I. Orlov, M. S. Roitman

(Presented by the scientific seminar of the AVTF)

In automatic-control and autometry systems, devices with a sharply nonlinear relationship between input and output signals are widely used (threshold devices, null indicators, electronic relays).

In works devoted to the analysis of instability of threshold devices (TD) [1, 2], the influence of the input circuits is not singled out separately or is not considered at all; namely, the influence of the thermal current of the collector junction \(I_{ko}\) and of the nonlinearity of the input resistance of the transistor. As practice has shown, these factors must be taken into account especially when designing threshold devices of increased sensitivity [3].

Fig. 1.            Fig. 2.

In systems using a TD, two cases may occur.

1. The reference-voltage source \(E_{\mathrm{op}}\) has a small internal resistance and is connected in series with the signal source \(E_c\) (Fig. 1).

2. The reference-voltage source has a comparatively large internal resistance and is connected in parallel with \(E_c\) relative to the input terminals of the transistor (Fig. 2).

For both cases we assume that the signal is applied to the base of the input transistor.

The first connection is used when several CUs have one reference voltage. The second connection circuit is more economical; its voltage transfer coefficient is more stable when the input resistance of the transistor changes, but this circuit requires individual adjustment and therefore is usually not used in systems where many identical CUs are required.

The input circuits of Figs. 1 and 2 are easily represented by a single equivalent circuit (Fig. 3), where \(E_b = E_c - E_{\text{op}}\) and \(R_b = R_1 + R_g\) (for the circuit of Fig. 1), or

\[ E_b = \frac{E_c \cdot R_1 - E_{\text{op}} \cdot R_g}{R_1 + R_g} \quad \text{and} \quad R_b = \frac{R_1 \cdot R_g}{R_1 + R_g} \]

for the circuit of Fig. 2). The current generator \(I_{b0}\) reflects the effect of the thermal current of the collector junction \(I_{k0}\) on the base circuit of the transistor. Taking \(I_{b0}\) into account, the voltage transfer coefficient of the input circuit is equal to

\[ K = \frac{U_{be}}{E_b} = \frac{R_{\text{in}}}{R_b + R_{\text{in}}} \left(1 + \frac{I_{b0} \cdot R_b}{E_b}\right). \tag{1} \]

Fig. 3.

According to formula (1), \(I_{k0}\) increases the transfer coefficient of the input circuit by the amount

\[ \frac{R_{\text{in}}}{R_b + R_{\text{in}}} \cdot \frac{I_{b0} \cdot R_b}{E_b}, \tag{1a} \]

and the temperature dependence of the current \(I_{k0}\) causes instability in the quantities \(R_{\text{in}}\), \(I_{b0}\), and, consequently, instability of the transfer coefficient of the input circuit.

The input resistance of the semiconductor triode, in turn, has a nonlinear dependence on the base current \(I_b\), while the base current is determined by the parameters of the input circuit, that is, by the quantities \(R_b\), \(E_b\), and \(I_{b0}\).

To calculate \(R_{\text{in}}\), let us introduce several designations:

\(I_{k0}\) — thermal current of the collector junction;

\(I_{kn}\) — collector current at \(R_b = 0\) and \(E_b = 0\);

\(I_{k\!ez}\) — collector current at \(R_b = \infty\) and \(E_b = 0\);

\[ I_{b0} = \frac{I_{k0}(I_{k\!ez} - I_{kn})}{(I_{k\!ez} - I_{k0})} \]

— current in the base circuit at \(E_b = 0\), \(R_b = 0\),

\[ \varphi_T = \frac{kT}{q} \]

— temperature potential [4].

As a result of solving the Ebers—Moll equations [1] using the parameters given above, one can obtain:

\[ R_{\text{in}} = \frac{\varphi_T}{I_{b0} - I_b} \ln \left[ \frac{(I_{b0} - I_b)(I_{k\!ez} - I_{k0})} {I_{k0}(I_{kn} - I_{k0})} + 1 \right]. \tag{2} \]

Let us estimate the input resistance of the triode for two characteristic points:

1. \(R_b = 0,\ E_b = 0\), that is, \(I_b = I_{b0}\)

\[ R_{\text{in}0} = \frac{\varphi_T (I_{k\!ez} - I_{k0})} {I_{k0}(I_{kn} - I_{k0})}. \]

1. \(R_b=\infty,\ E_b=0\), that is, \(I_b=0\)

\[ R_{\mathrm{in}\infty}=\frac{\varphi_T}{I_{b0}}\ln\left[\frac{(I_{\mathrm{co}e}-I_{\mathrm{co}})^2}{(I_{\mathrm{kn}}-I_{\mathrm{co}})\cdot I_{\mathrm{co}e}}\right]. \]

For approximate calculations of \(R_{\mathrm{in}}\) in the case of finite values of the resistance in the base circuit, the following procedure may be recommended:

1. In a regime close to the short-circuit regime \((R_b<10\ \mathrm{k})\), the current in the base circuit is determined as

\[ I_b=I_{b0}\frac{R_{\mathrm{in}0}}{R_{\mathrm{in}0}+R_b}, \]

and then its value is substituted into (2) and \(R_{\mathrm{in}}\) is calculated.

1. Similarly, for the case close to the open-circuit regime \((R_b>50\ \mathrm{k})\), the value of the current in the base circuit, equal to

\[ I_b=I_{b0}\frac{R_{\mathrm{in}\infty}}{R_{\mathrm{in}\infty}+R_b}, \]

is substituted into (2), and \(R_{\mathrm{in}}\) is calculated.

For the intermediate case, \(R_{\mathrm{in}}\) is determined as the mean value of the quantities calculated according to the two preceding points.

At \(E_b=0\), the resistance \(R_{\mathrm{in}}\) depends only on the magnitude \(R_b\) and the current \(I_{b0}\). As the input signal increases, \(R_{\mathrm{in}}\) decreases, which leads to a decrease in \(K\). The relative change in the transmission coefficient of the input circuit in this case is

\[ \frac{\Delta K}{K}=\frac{\Delta R_{\mathrm{in}}}{R_{\mathrm{in}}}\cdot\frac{R_b}{R_b+R_{\mathrm{in}}}. \tag{3} \]

Let us conditionally assume that the dead zone for the transistor lies within the limits from \(U_{be}=0\) to \(U_{be}=U_{be0}^{*}\), at which \(I_b=0\). Then, taking into account formulas (3) and (1), we obtain that, in order to cover the dead zone of the transistor, it is necessary to provide at the input a voltage

\[ E_{b0}=\frac{U_{be}}{K}\cdot\frac{\Delta K}{K} =\frac{U_{be0}}{K}\cdot\frac{\Delta R_{\mathrm{in}}}{R_{\mathrm{in}}}\cdot\frac{R_b}{R_b+R_{\mathrm{in}}}. \tag{4} \]

Moreover, \(E_{b0}\) can be obtained either by increasing \(E_c\), or by decreasing \(E_{\mathrm{op}}\). Usually, in order to increase the sensitivity of the amplifier, one proceeds by decreasing \(E_{\mathrm{op}}\).

In formula (4), \(\Delta R_{\mathrm{in}}\) for the transistor dead zone selected by us is

\[ \Delta R_{\mathrm{in}}=R_{\mathrm{in}\infty}-R_{\mathrm{in}}, \qquad \text{where} \]

\(R_{\mathrm{in}}\) is the input resistance at the given \(R_b\) and \(E_b=0\).

The dependence of the transmission coefficient of the input circuit \((K)\) on temperature \((t)\) causes instability of \(E_{b0}\). Let us write the expression \(\dfrac{dK}{dt}\) in general form as

\[ \frac{dK}{dt}=\frac{R_b}{R_b+R_{\mathrm{in}}} \left[ \frac{1}{R_b+R_{\mathrm{in}}} \left(1+\frac{I_{b0}\cdot R_b}{E_b}\right) \frac{dR_{\mathrm{in}}}{dt} + \frac{R_{\mathrm{in}}}{E_b}\frac{dI_{b0}}{dt} \right] \tag{5} \]

\[ \underline{\hspace{2.5cm}} \]

*) Depending on the particular amplifier circuit, this condition may be chosen differently.

To calculate by formula (5), it is necessary, in turn, to have expressions for \(\dfrac{dR_{\rm in}}{dt}\) and \(\dfrac{dI_{b0}}{dt}\), or, in finite increments, for \(\dfrac{\Delta R_{\rm in}(\Delta T)}{R_{\rm in}(T_0)}\) and \(\dfrac{\Delta I_{b0}(\Delta T)}{I_{b0}(T_0)}\). \(R_{\rm in}\) is determined as \(\dfrac{U_{be}\cdot R_b}{R_b I_{b0}-U_{be}}\). Since the voltage \(U_{be}\) is practically independent of temperature [5],

\[ \frac{dR_{\rm in}}{dt} = -\frac{R_{\rm in}\cdot R_b}{R_b\cdot I_{b0}-U_{be}}\cdot \frac{dI_{b0}}{dt} \quad \text{or} \quad \frac{\Delta R_{\rm in}(\Delta T)}{R_{\rm in}(T_0)} = -\frac{R_b+R_{\rm in}}{R_b}\cdot \frac{\Delta I_{b0}(\Delta T)}{I_{b0}(T_0)} . \tag{6} \]

The temperature dependence of the current \(I_{b0}\) is the same as for \(I_{k0}\). Using the well-known dependence of \(I_{k0}\) on temperature [1], we obtain

\[ \frac{\Delta I_{b0}(\Delta T)}{I_{b0}(T_0)} = e^{a\Delta T}a\Delta T, \tag{7} \]

where \(a=(0.07 \div 0.09)\dfrac{1}{^\circ K}\) for germanium triodes;

\(T_0\) is some initial value of the temperature on the absolute scale;

\(\Delta T\) is the temperature increment.

As a result, passing in formula (5) to finite increments and using (6) and (7), we have

\[ \frac{\Delta K(\Delta T)}{K(T_0)} = -\frac{E_b}{E_b+I_{b0}(T_0)\cdot R_b} e^{a\Delta T}\cdot a\Delta T, \tag{8} \]

The temperature instability of the magnitude of the dead zone \((\Delta E_{b0})\) is calculated by formula (4), taking (8) into account.

In conclusion it should be noted that all the calculation formulas have been obtained for threshold devices in which germanium alloy triodes are used. The use of silicon triodes for constructing sensitive threshold devices is inadvisable, since their dead zone and the absolute instability of its magnitude are much greater than in germanium triodes.

References

1. I. P. Stepanenko. Fundamentals of the Theory of Transistors and Transistor Circuits. Gosenergoizdat, 1963.

2. I. P. Stepanenko. Temperature and time drift of threshold voltages in a transistor trigger, in the collection of articles Semiconductor Devices and Their Application, edited by Ya. A. Fedotov, Sovetskoe Radio, issue 8, 1962.

3. M. S. Roitman, V. I. Orlov, A. S. Gofman. Electronic relays based on complex p-n systems. Proceedings of the Fifth All-Union Conference, 1963, Novosibirsk.

4. N. F. Nikolaevskii. Operating Parameters and Features of the Application of Transistors. Svyazizdat, 1963.

5. Ya. Budinskii. Transistor Switching Circuits. Svyaz, 1965.