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7.2: Drift Referred to the Input

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    58458
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    The most useful measure of the drift of an amplifier is a quantity called drift referred to the input, and unless specifically stated otherwise, this quantity is the one implied when the term drift is used. Drift referred to the input is defined with reference to Figure 7.1. This figure shows an amplifier with an assumed desired output voltage of zero for zero input voltage. The amplifier is initially balanced by making \(v_I = 0\), and adjusting some ampli­fier parameter (shown diagrammatically in Figure 7.1 as a variable resistor) until \(v_O = 0\). An external quantity, such as temperature, supply voltage, or time, is then changed and, if the amplifier is sensitive to this quantity, its output voltage changes. An input voltage is then applied to the ampli­fier, and \(v_I\) is adjusted until vo again equals zero. The drift referred to the input of the amplifier is equal to the value of \(v_I\) necessary to zero the output. The resultant magnitude is often normalized and specified, for example, as volts per degree Centigrade, volts per volt (of supply voltage), or volts per week. The minimum-detectable-signal aspect of this definition is self-evident.

    截屏2021-08-12 下午12.40.17.png
    Figure 7.1 System used to define drift referred to the input.

    In many situations we are concerned not only with the variability of the circuit as some external influencing factor is changed, but also with un­certainties that arise from the manufacturing process. In these cases, rather than initially balancing the circuit, the voltage that must be applied to its input to make its output zero may be specified as the offset referred to the input. The specifications related to drift and offset are at times combined by listing the maximum input offset that will result from manufacturing variations and over a range of operating conditions.

    There is a tendency to use an alternative (incorrect) definition of drift, which involves dividing the drift measured at the output of the amplifier by the amplifier gain. The difficulty in this approach arises since the gain is frequently dependent on the drift-stimulating variable.

    While alternative measurements of drift or offset may be equivalent in special cases, and are often used in the laboratory to simplify a measure­ment procedure, it is necessary to insure equivalence of other methods for each circuit. We shall normally use the original definitions for our calcu­lations.

    截屏2021-08-12 下午12.47.22.png
    Figure 7.2 Circuit illustrating drift calculation.

    Figure 7.2 shows a very simple amplifier, which will be used to illustrate drift calculations and to determine how the base-to-emitter voltage of a bipolar transistor changes with temperature. It is assumed that the drift of the circuit with respect to temperature is required, and that the initial temperature is \(300^{\circ}\ K\). It is further assumed that for the transistor used, \(i_C = 1\ mA\) at \(v_{BE} = 0.6\ V\) and \(T = 300^{\circ}\ K\). With \(v_I = 0\),these parameters show that it is necessary to adjust the potentiometer to its midposition to make \(v_O = 0\). The temperature is then changed to \(301^{\circ}\ K\), and it is observed the \(v_O\) is negative. (The amount is unimportant for our purposes.) In order to return \(v_O\) to zero (required by our definition of drift), it is necessary to return the transistor collector current to its original value. The change in \(v_{BE}\) required to restore collector current is identically equal to the required change in \(v_I\) and is therefore, by definition, the drift referred to the input of the amplifier. This discussion shows that drift for this circuit can be evalu­ated by determining how \(v_{BE}\) must vary with temperature to maintain con­stant collector current.

    Drift for the circuit shown in Figure 7.2 can be determined from the rela­tionship between transistor terminal variables and temperature. If ohmic drops are negligible and the collector current is large compared to the satu­ration current \(I_S\)(P. E. Gray et al., Physical Electronics and Models of Transistors, Wiley, New York, 1964.)

    \[i_C = I_S e^{qv_{BE}/kT} = AT^3 e^{qV_{go}/kT} e^{qV_{BE}/kT} = AT^3 e^{q(v_{BE} - V_{go})/kT}\label{eq7.2.1} \]

    where \(A\) is a constant dependent on transistor type and geometry, \(q\) is the charge on an electron, \(k\) is Boltzmann's constant, \(T\) is the temperature, and \(V_{go}\) is the width of the energy gap extrapolated to absolute zero divided by the electron charge (\(V_{go} = 1.205\) volts for silicon).(There is disagreement among authors concerning the exponent of \(T\) in Equation \(\ref{eq7.2.1}\), with somewhat lower values used in some developments. As we shall see, the quantity has rela­tively little effect on the final result. (The exponent appears only as a multiplying factor in the final term of Equation \(\ref{eq7.2.5}\) and as a coefficient in Equation \(\ref{eq7.2.8}\)). Furthermore, two similar tran­sistors should have closely matched values for this exponent, and the degree of match between a pair is the most important quantity in anticipated applications.) It is possible to verify the exponential dependence of collector current on base-to-emitter voltage experimentally over approximately nine decades of operating cur­rent for many modern transistors.

    Solving Equation \(\ref{eq7.2.1}\) for \(v_{BE}\) yields

    \[v_{BE} = \dfrac{kT}{q} \ln \dfrac{i_C}{AT^3} + V_{go}\label{eq7.2.2} \]

    The partial derivative of \(v_{BE}\) with respect to temperature at constant \(i_C\) is the desired relationship, and

    \[\left.\dfrac{\partial v_{BE}}{\partial T}\right|_{i_C = \text{const}} = \dfrac{k}{q} \ln \dfrac{i_C}{AT^3} - \dfrac{3k}{q}\label{eq7.2.3} \]

    However, from Equation \(\ref{eq7.2.2}\)

    \[\dfrac{k}{q} \ln \dfrac{i_C}{AT^3} = \dfrac{v_{BE} - V_{go}}{T}\label{eq7.2.4} \]

    Substituting Equation \(\ref{eq7.2.4}\) into Equation \(\ref{eq7.2.3}\) yields

    \[\left.\dfrac{\partial v_{BE}}{\partial T}\right|_{i_C = \text{const}} = \dfrac{v_{BE} - V_{go}}{T} - \dfrac{3k}{q}\label{eq7.2.5} \]

    The quantity \(v_{BE} - V_{go}/T\) is \(-2mV/ ^{\circ} C\) at \(T = 300 ^{\circ} K\) for the typical \(v_{BE}\) value of 0.6 volt. The term \(3k/q = 0.26 mV/ ^{\circ} C\); therefore to a good degree of approximation

    \[\left.\dfrac{\partial v_{BE}}{\partial T}\right|_{i_C = \text{const}} \simeq \dfrac{v_{BE} - V_{go}}{T}\label{eq7.2.6} \]

    The approximation of Equation \(\ref{eq7.2.6}\) links the two rule-of-thumb values of \(0.6 V\) and \(-2 mV/ ^{\circ} C\) for the magnitude and temperature dependence, respec­tively, of the forward voltage of a silicon junction.

    It is valuable to note two relationships that are exploited in the design of transistor d-c amplifiers. First, with no approximations beyond those implied by Equation \(\ref{eq7.2.1}\), it is possible to determine the required transistor base­ to-emitter voltage variation for constant collector current knowing only the voltage, the temperature, and the material used to fabricate the tran­sistor. Furthermore, if two silicon (or two germanium) transistors have identical base-to-emitter voltages at one temperature and at certain (not necessarily identical) operating currents, the temperature coefficients of the base-to-emitter voltages must be equal. Second, the base-to-emitter temperature coefficient at any one operating current is very nearly inde­pendent of temperature as shown by the following development. The vari­ation of temperature coefficient with temperature is found by differentiating Equation \(\ref{eq7.2.5}\) with respect to temperature, yielding

    \[\dfrac{\partial}{\partial T} \left [ \dfrac{\partial v_{BE}}{\partial T} \right ]_{i_C = \text{const}} = \dfrac{-(v_{BE} - V_{go}) + T(\partial v_{BE}/\partial T)}{T^2}\label{eq7.2.7} \]

    Substituting from \(\ref{eq7.2.5}\) for the \(\partial v_{BE}/\partial T\) term in Equation \(\ref{eq7.2.7}\), we obtain

    \[\dfrac{\partial}{\partial T} \left ( \dfrac{\partial v_{BE}}{\partial T} \right ) = -3k/qT\label{eq7.2.8} \]

    Evaluating Equation \(\ref{eq7.2.8}\) at \(300^{\circ} K\) shows that the magnitude of the change in base-to-emitter voltage temperature coefficient with temperature is less than \(1 \mu V/ ^{\circ} C\).(An interesting alternative development of this relationship is given in "An Exact Expression for the Thermal Variation of the Emitter Base Voltage of Bi-Polar Transistors," R. J. Widlar, National Semiconductor Corp., Technical Paper TP-1, March, 1967. )

    It is now possible to determine the drift referred to the input of our origi­nal amplifier. In order to return \(v_O\) in Figure 7.2 to zero at the elevated tem­perature, it is necessary to decrease \(i_C\) to its original value of \(1\ mA\), and this decrease requires a \(-2.26\ mV\) change in \(v_I\) (Equation \(\ref{eq7.2.5}\)). The drift re­ferred to the input of our amplifier is by definition \(-2.26\ mV/ ^{\circ} C\), and Equation \(\ref{eq7.2.8}\) insures that this drift is essentially constant over a wide range of temperatures.


    This page titled 7.2: Drift Referred to the Input is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James K. Roberge (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.