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7.5: DRIFT CONTRIBUTIONS FROM THE SECOND STAGE

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    69437
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    Thus far the discussion has focused on single-stage direct-coupled ampli­fiers. No consideration has been given to situations that require a second stage either to provide greater voltage gain or to isolate a low-resistance load. The use of a second stage is mandatory in the design of operational amplifiers and thus must be investigated.

    There is a popular misconception that the dominant source of voltage drift for a d-c amplifier is always associated with its input stage. The argu­ment supporting this view is that drift arising in the second stage is divided by the gain of the first stage when referred to the input of the amplifier, and is negligible if the first-stage gain is high. This assumption is not always justified because of the extraordinarily low values of drift that can be achieved with a properly balanced first stage. Balancing techniques similar to those used for the input stage are not effective for the second stage, since its drift contribution is often attributable to variations in input current rather than in base-to-emitter voltage.

    Single-Ended Second Stage

    Figure 7.21 shows a differential first stage (with two matched transistors collectively labeled \(Q_1\)) driving a common-emitter PNP second stage. Two perturbation sources are shown, which will be used later to calculate drift. In addition to providing gain, the second stage shifts level so that the out­put voltage can swing both positive and negative with respect to ground. If the base resistance of all transistors is negligibly small, the voltage gain of this amplifier is

    \[\dfrac{v_o}{v_i} = \dfrac{-g_{m1} R_{L1} \beta_2 R_{L2}}{2(r_{\pi 2} + R_{L1})} \nonumber \]

    截屏2021-08-13 下午10.54.53.png
    Figure 7.21 Two-stage d-c amplifier.

    Drift referred to the input for this two-stage amplifier is calculated as before by determining how \(v_I\) must vary to keep vo equal to zero. Note that in order to maintain a fixed output voltage, it is necessary for \(i_{C2}\) to remain constant. There are a number of sources of drift for this amplifier. In this development only changes in \(i_{B2}\) and \(v_{EB2}\) that arise as the param­eters of \(Q_2\) vary are considered. These changes can be modeled by the per­turbation generators shown in Figure 7.21. If the changes are small compared to operating-point values, linear analysis methods can be used to deter­mine the drift referred to the input of the amplifier.

    \[v_I|_{i_{C2} = \text{const}} = \dfrac{-2 \Delta v_{EB2}}{g_{m1} R_{L1}} - \dfrac{2 \Delta i_{B2}}{g_{m1}}\label{eq7.5.2} \]

    The gain portion of the first term on the right of Equation \(\ref{eq7.5.2}\) can be ex­ pressed in terms of \(V\), the quiescent voltage across \(R_{L1}\). Similarly, the sec­ond term can be expressed in terms of \(I_{E1}\), \(I_{C2}\), the current gain of \(Q_2\), and its fractional change. These substitutions yield

    \[v_I|_{i_{C2} = \text{const}} = \dfrac{-2kT \Delta v_{EB2}}{qV} + \dfrac{4kT I_{C2} \Delta \beta_2}{q I_{E1} \beta_2^2} \simeq \dfrac{-\Delta v_{EB2}}{20V} + \dfrac{I_{C2} \Delta \beta_2}{10 I_{E1} \beta_2^2} \label{eq7.5.3} \]

    at room temperature.

    Typical values are used to illustrate magnitudes of these drift compo­nents with temperature. The voltage \(V\) is constrained by available supply voltages, and a value of 5 volts is assumed. The typical \(\Delta v_{EB}\) value of \(-2\ mV/ ^{\circ} C\) is used. A current gain of 300 coupled with a temperature co­efficient of 0.6% per degree Centigrade is assumed for \(Q_2\). Because the quiescent current level normally increases from the first stage to the second, a ratio of 5 is used for \(I_{C2}/I_{E1}\). Substituting these values into Equation \(\ref{eq7.5.3}\) shows that the drift attributable to changes in \(v_{EB2}\) is approximately 20 \(\mu V/ ^{\circ} C\), while the component arising from \(i_{B2}\) changes is \(10 \mu V/ ^{\circ} C\). These values contrast dramatically with the drift that can be obtained from a properly designed first stage, and indicate the dominant effect that the second stage can have on drift performance.

    The drift calculations of this section apply even if current gain only is required from the second stage. It is easy to show that the calculated values of drift are the same if an emitter follower is used in place of the grounded-emitter stage.

    The final term in Equation \(\ref{eq7.5.3}\) indicates the importance of changes in second-stage input current on drift performance. This term indicates that the drift performance deteriorates as the ratio of the quiescent operating current of the second stage to that of the first stage is increased. This result is one ex­ample of how certain design considerations (in particular, the desire to increase quiescent currents from the first to subsequent stages) must be compromised to achieve low drift performance.

    Differential Second Stage

    It is evident from the typical values calculated in the last section that unless care is taken in the design of the second stage of a d-c amplifier, this stage can easily dominate the drift performance of the circuit. One approach to the design of low-drift multistage d-c amplifiers is to use a differential second stage so that reflected drift is determined by differential rather than absolute changes in second-stage parameters.

    截屏2021-08-13 下午11.05.18.png
    Figure 7.22 Amplifier with two differential stages.

    Figure 7.22 shows a two-stage differential amplifier. Individual members of the first- and second-stage pairs are assumed matched. It is -further assumed that a single-ended output is desired, so one collector of the second-stage pair is grounded.

    Normally a resistor is used in place of the current source \(I_{E2}\). Since only differential input signals can be applied to the second stage, and therefore the common-emitter point of the second stage is incrementally grounded, the impedance connected to this point is irrelevant. However, the calcula­tions are somewhat more convenient if a current source is included.

    It is interesting to note that the voltage gain of this amplifier is identical to that of Figure 7.21. Since the common-emitter connection of the second stage is incrementally grounded for any possible input signal, no gain in­ crease results from the left-hand member of the PNP pair.

    Input drift attributable to second-stage differential base-to-emitter volt­age changes is generally negligible if any degree of match exists. The drift referred to the input of the second stage is equal to the ratio \(\Delta v_{BE2}/T\) per degree Centigrade (see Section 7.3.4). This value (typically on the order of 10 to 100 \(\mu V/^{\circ} C\)) is divided by the unloaded differential voltage gain of the first stage (twice the single-ended value calculated in the preceding section) when reflected to the input.

    The drift attributable to differential fractional changes in second-stage current gain is (assuming initially matched values for second-stage current gains)

    \[v_I |_{i_{C2} = \text{const}} = \dfrac{kTI_{E2}}{\beta_2 q I_{E1}} \left ( \dfrac{\Delta \beta_{2A} - \Delta \beta_{2B}}{\beta_2} \right ) \nonumber \]

    where the \(A\) and \(B\) subscripts indicate the two members of the second-stage pair. (The factor of four compared with the calculation of the last section occurs since each second-stage transistor is operating at \(I_{E2} /2\) and since the differential connection requires that only half the differential current change be offset at either side.) The quantity \((\Delta \beta_{2A} - \Delta \beta_{2B})/\beta_2\) is typically 0.1% per degree Centigrade for well-matched discrete components, and is often lower for integrated-circuit pairs. It is interesting to note that this component of drift dominates many amplifier designs, particularly if the current gains and the temperature coefficients of the second stage are not well matched, or if the operating. current level of the second stage is high relative to that of the first stage.

    The use of a Darlington second stage with its lower input current offers some improvement, since the higher voltage drift of the Darlington is tolerable in this stage. Another possibility is to adjust the relative collector currents of the second stage so that the differential change in second-stage base current with temperature is zero. Unfortunately, this adjustment is difficult to make.


    This page titled 7.5: DRIFT CONTRIBUTIONS FROM THE SECOND STAGE is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James K. Roberge (MIT OpenCourseWare) .

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