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    Requirements usually constrain the input and output stages of an opera­tional amplifier to be a differential amplifier and some type of buffer (normally an emitter-follower connection), respectively.

    It is in the intermediate stage or stages that design flexibility is evident, and the difference in performance between a good and a poor circuit often

    reflects the differences in intermediate-stage design. The primary perform­ance objective is that this portion of the circuit provide high voltage gain coupled with a transfer function that permits stable, wide-band behavior in a variety of feedback connections. Furthermore, the flexibility of easily and predictably modifying the amplifier open-loop transfer function in order to optimize it for a particular feedback connection is desirable for a general-purpose design.

    Design with Three Voltage-Gain Stages

    截屏2021-08-14 下午2.16.30.png
    Figure 8.1 One approach to operational-amplifier design.

    One much-too-frequently used design is shown in simplified form in Figure 8.1. The path labeled feedforward is one technique used to stabilize the amplifier, and is not essential to the initial description of operation. The

    basic circuit uses a differential input since this connection is mandatory for low drift and high common-mode rejection ratio. Two common-emitter stages (transistors \(Q_3\) and \(Q_4\)) are used to provide the high voltage gain characteristic of operational amplifiers. Some sort of buffer amplifier (shown diagrammatically as the unity-gain amplifier in the output portion) is used to provide the required output characteristics.

    Casual inspection indicates some merit for the design of Figure 8.1. Low drift is possible and d-c gains in excess of 10 can be achieved. The difficulty is evident only when the dynamics of the amplifier are examined. The trans­fer function \(V_o(s)/ [V_{i2} (s) - V_{i1} (s)]\) determines stability in feedback connec­tions. With typical element values, this transfer function has three or four poles located within a two-to-three decade range of frequency. It is not possible to achieve large loop-transmission magnitude and simultaneously to maintain stability with this type of transfer function. The designer of this type of amplifier should be discouraged when he compares his circuit with that of a phase-shift oscillator, where negative feedback is applied around three or more closely spaced poles.

    The problem can be illustrated by computing the transfer function for the amplifier shown in Figure 8.1 with component values listed in Table 8.1. The reasons for selecting these component values are as follows. Fifteen-volt supplies are used since this value has become the standard for many solid-state operational amplifiers. The quiescent operating current of the first stage is low to reduce input bias current.

    Relatively modest increases in quiescent currents from stage-to-stage are used to minimize loading effects. At these levels, circuit impedances are such that little change in the transfer function results if \(r_x\) is assumed equal to zero. However, \(r_x\) has been retained for completeness. Junction capaci­tances are dominated by space-charge layer effects at low operating cur­ rents, so equal values for all transistor capacitances have been assumed. Clearly any equal change in all capacitances simply frequency scales the transfer function. The resistors in the base circuits of \(Q_3\) and \(Q_4\) are assumed large to maximize d-c gain. In practice, current sources can be used to main­tain high incremental resistance and to establish bias currents. Resistor \(R_3\) is chosen to yield a quiescent output voltage equal to zero.

    Table 8.1. Parameter Values for Example Using Amplifier of Figure 8.1

    Supply voltages:
    \(\pm 15\ V\)
    Bias currents:
    \(I_{C1} = I_{C2} = 10\ \mu A\)
    \(I_{C3} = 50\ \mu A\)
    \(I_{C4} = 250\ \mu A\)
    Transconductance(1) implied by bias currents:
    \(g_{m1} = g_{m2} = 4 \times 10^{-4} \text{ mho}\)
    \(g_{m3} = 2 \times 10^{-3} \text{ mho}\)
    \(g_{m4} = 10^{-2} \text{ mho}\)

    (1) Recall that for any bipolar transistor operating at current levels where ohmic drops are unimportant, the transconductance is related to quiescent collector current by \(g_m = q|I_C|/kT \simeq 40 V^{-1} |I_C|\) at room temperature.
    Other transistor parameters:
    \(\beta = 100 \text{ (all transistors)}\)
    \(r_{\pi 1} = r_{\pi 2} = 250\ k\Omega\)
    \(r_{\pi 3} = 50\ k \Omega\)
    \(r_{\pi 4} = 10\ k \Omega\)
    \(r_{x} = 100 \Omega \text{ (all transistors)}\)
    \(C_{\mu} = C_{\pi} = 10\text{ pF (all transistors)}\)
    \(R_1\) and \(R_2\) large compared to \(r_{\pi 3}\) and \(r_{\pi 4}\), respectively.
    \(R_3 = 60\ k\Omega\)
    (Satisfying the inequalities normally requires that current sources be used rather than resistors in practical designs.)

    Buffer amplifier assumed to have infinite input impedance.

    截屏2021-08-14 下午2.30.34.png
    Figure 8.2 Transfer function for amplifier of Figure 8.1

    A computer-generated transfer function \(V_o (j\omega)/[V_{i2} (j\omega) - V_{i1} (j\omega)]\) for this amplifier is shown in Bode-plot form in Figure 8.2.(The gains of the amplifier for signals applied to its two inputs are not identical at high frequencies because a fraction of the signal applied to the base of \(Q_1\) is coupled directly to the base of \(Q_3\) via the collector-to-base capacitance of \(Q_1\). This effect, which is insignifi­cant until frequencies approaching the \(f_T\)'s of the transistors used in the circuit, has been ignored in calculating the amplifier transfer function so that a true differential gain expres­sion results.) Two important features of this transfer function are easily related to circuit parameters. The low-frequency gain can be determined by inspection. Invoking the usual assumptions, the incremental changes in first-stage collector current is related to an incremental change in differential input voltage as

    \[i_{c1} = - \left (\dfrac{v_{i2} - v_{i1}}{1/g_{m1} + 1/g_{m2}} \right ) \label{eq8.2.1} \]

    Since \(R_1\) is large compared to the input resistance of \(Q_3\), all of this incremental current flows into the base of \(Q_3\). This base current is amplified by a factor of \(\beta_3\), and resulting incremental current flows into the base of \(Q_4\). The incremental output voltage becomes

    \[v_o = i_{c1} \beta_3 \beta_4 R_3 \label{eq8.2.2} \]

    combining Equations \(\ref{eq8.2.1}\) and \(\ref{eq8.2.2}\) shows that the low-frequency voltage gain is

    \[\dfrac{v_o}{v_{12} - v_{i1}} = \dfrac{\beta_3 \beta_4 R_3}{(1/g_{m1} + 1/g_{m2})} \nonumber \]

    Substituting parameter values from Table 8.1 into this equation shows that the incremental d-c gain is \(1.2 \times 10^5\).

    截屏2021-08-14 下午2.44.24.png
    Figure 8.3 Model used to determine dynamics of final common-emitter stage of three-stage amplifier.

    The lowest frequency pole plotted in Figure 8.1 has a break frequency of \(1.36 \times 10^4\) radians per second. This pole results from feedback through the collector-to-base capacitance of \(Q_4\) (sometimes called Miller effect), as shown by the following development. An incremental model that can be used to evaluate the transimpedance of the final common-emitter stage is shown in Figure 8.3. This transimpedance is a multiplicative term in the com­plete amplifier transfer function.

    Node equations for this circuit are

    \[\begin{array} {rcl} {-I_{c3}} & = & {[g_{\pi 4} + (C_{\mu 4} + C_{\pi 4})s] V_a - C_{\mu 4} s V_o} \\ {0} & = & {(g_{m4} - C_{\mu 4})s V_a + (G_3 + C_{\mu 4} s) V_o} \end{array} \nonumber \]

    Solving for the transimpedance shows that

    \[\dfrac{V_o (s)}{I_{c3} (s)} = \dfrac{\beta R_3 [-(C_{\mu 4}/ g_{m4})s + 1]}{r_{\pi 4} R_3 C_{\mu 4} C_{\pi 4} s^2 + r_{\pi 4} \{[(g_{m4} + g_{\pi 4}) R_3 + 1]C_{\mu 4} + C_{\pi 4} \} s + 1}\label{eq8.2.5} \]

    The denominator of Equation \(\ref{eq8.2.5}\) is normally dominated by the term that in­cludes the factor \(g_{m4}R_3 C_{\mu 4}\), reflecting the importance of feedback through \(C_{\mu 4}\). Substituting values from Table 8.1 into Equation \(\ref{eq8.2.5}\) and factoring the de­ nominator polynominal results in

    \[\dfrac{V_o (s)}{I_{c3} (s)} = \dfrac{6 \times 10^6 (-10^{-9} s + 1)}{(10^{-9} s + 1)(6.08 \times 10^{-5} s + 1)}\label{eq8.2.6} \]

    This development shows that the output stage would have a dominant pole with a \(1.64 \times 10^4\) radians-per-second break frequency in its transfer func­tion if the other components in the circuit did not alter the location of this pole. This value agrees with the location of the dominant pole for the com­plete amplifier within approximately 20%.

    The algebra involved in getting this result can be circumvented by recog­nizing that a one-pole(P. E. Gray and C. L. Searle, Electronic Principles:Physics, Models, and Circuits, Wiley, New York, 1969, pp. 497-503. ) (or Miller-effect) approximation to the input ca­pacitance of transistor \(Q_4\) predicts a value

    \[C_T = C_{\pi 4} + C_{\mu 4} (1 + g_{m4} R_3)\label{eq8.2.7} \]

    The break frequency estimated at this node is

    \[\omega_h = \dfrac{1}{r_{\pi 4} C_T} = 1.66 \times 10^4 \text{ rad/sec} \nonumber \]

    While the d-c gain and the dominant pole location for this configuration are easily estimated, the location of other transfer-function singularities are related to amplifier parameters in a more complex way.

    The essential feature to be gained from the Bode plot of Figure 8.2 is that this transfer function is far from ideal for use in many feedback connec­tions. The amplifier is hopelessly unstable if it is operated with its non-inverting input connected to an incremental ground and a wire connecting its output to its inverting input, creating a loop with \(a\) as shown in the Bode plot and \(f = 1\). In fact, if frequency-independent feedback is applied around the amplifier, it is necessary to reduce the magnitude of the loop transmission by a factor of 50 below the gain of the amplifier itself to make it stable in an absolute sense, and by a factor of 2000 to obtain \(45^{\circ}\) of phase margin. The required attenuation could be obtained by means of resistively shunting the input of the amplifier or through the use of a lag network (see Section 5.2.4). Either of these approaches severely compromises desensitivity and noise performance in many applications because of the large attenuation necessary for stability. Better results can normally be obtained by modifying the dynamics of the amplifier itself.

    Compensating Three-Stage Amplifiers

    截屏2021-08-14 下午2.53.09.png
    Figure 8.4 Compensation by adding a shunt impedance.

    At least two methods are often used to improve the dynamics of an amplifier similar to that described in the previous section. One of these approaches recognizes that the poles in the amplifier can be modeled as occurring because of \(R-C\) circuits located at various amplifier nodes. This type of association was made in the previous section for the dominant ampli­fier pole. The transfer function for a gain stage includes a multiplicative term of the general form \(R_e/(R_eC_e s + 1)\), where \(R_e\) and \(C_e\) are the effective resistance and capacitance at a particular node (see Figure 8.4). If a com­pensating series \(R-C\) network to ground consisting of a resistor \(R_c \\ R_e\), and a capacitor \(C_c \gg C_e\) is added, the transfer function becomes

    \[\dfrac{V_o (s)}{I_i (s)} \simeq \dfrac{R_e (R_c C_c s + 1)}{(R_e C_c s + 1)(R_c C_e s + 1)} \nonumber \]

    截屏2021-08-14 下午2.59.11.png
    Figure 8.5 Effect of adding a shunt impedance on the transfer function of one stage.

    The single pole has been replaced by two poles and a zero. (Note that asymptotic behavior at high and low frequencies, which is controlled by \(R_e\) and \(C_c\), has not been changed.) Component values are chosen so that one pole occurs at a much lower frequency than the original pole and the other at a frequency above the unity-gain frequency of the complete ampli­fier, as illustrated in Figure 8.5. The positive phase shift of the zero often can improve the phase margin of the amplifier. This type of compensation can be viewed as one of combining the uncompensated transfer function with appropriately located lag and lead transfer functions. While the singulari­ties must be related so that the compensated and uncompensated transfer functions are identical at very low and very high frequencies, the second pole can always be moved to arbitrarily high frequencies by locating the first pole at a sufficiently low frequency.

    截屏2021-08-14 下午3.03.01.png
    Figure 8.6 \(s\)-plane plots illustrating effect of shunt impedance on three-stage amplifier transfer function.

    An alternative way to view this type of compensation is shown in the s-plane diagrams of Figure 8.6. It is assumed that the three-stage amplifier has three poles at frequencies of interest. The lowest-frequency pole of the triad is replaced by two poles and a zero by means of a shunt \(R-C\) network. One possible way to choose singularity locations is to use the zero to cancel the second pole in the original transfer function and to locate the high-frequency pole that results from compensation above the highest-frequency original pole. The net effect of this type of compensation is to increase the separation of the poles so that greater desensitivity can be achieved for a given relative stability.

    Several variations of the basic compensation scheme exist. It is possible to realize similar kinds of transfer functions by connecting a series \(R-C\) network from collector to base of a transistor rather than from its base to an incrementally-grounded point. The same kind of compensation can be used at more than one node, and this multiple compensation is frequently required in more complex amplifiers.

    While this general type of compensation is effective and has been suc­cessfully applied to a number of amplifier designs, it is less than ideal for several reasons. One of the more important considerations is that the deter­mination of element values that result in a given transfer function requires rather involved calculations. This difficulty tends to discourage the user from finding the optimum compensating-element values for use in other than standard applications. This type of compensation also requires large capacitors (typically \(1000\ pF\) to \(0.1 \mu F\)) when the network is shunted from base to an incremental ground. The energy storage of a large capacitor can delay recovery following an amplifier overload that charges the capacitor to the wrong voltage level.

    An alternative type of compensation that may be used alone or in con­junction with a shunt impedance is to "feed forward" around one or more amplifier stages as shown in Figure 8.1. Here a unity-voltage-gain buffer amplifier (not essential but included in some designs to prevent loading at the inverting input terminal) couples the input signal to the base of \(Q_4\) through capacitor \(C_f\). Since the first stages are bypassed at high frequency, the high-frequency dynamics of the operational amplifier should be essen­tially those of the output stage. The hope is that the output stage has only one pole at frequencies of interest, and therefore will be stable with any amount of frequency-independent feedback.

    Feedforward is not without its disadvantages. The frequency response of a feedforward amplifier is significantly lower for signals applied to the non- inverting input than for signals applied to its inverting input. Thus the

    amplifier has severely reduced bandwidth when used in noninverting con­nections. There are also problems that stem from the type of transfer func­tions that result from feedforward compensation. There is usually a second- or third-order rolloff at low frequencies, with the transfer function recov­ering to first order in the vicinity of the unity-gain frequency. Since this transfer function resembles those obtained with lag compensation, the settling time may be relatively long because of the small amplitude "tails" that can result with lag compensation (see Section 5.2.6). It is also possible to have these amplifiers become conditionally stable in certain connections (Section 6.3.4). This topic is investigated in Problem P8.3.

    截屏2021-08-14 下午3.08.31.png
    Figure 8.7 The 709 integrated-circuit operational amplifier.

    Before leaving the subject of three-stage amplifiers, the liberty that has been taken in the definition of a stage is worth noting. The stages are never as simple as those shown in Figure 8.1. The essential feature that characterizes a voltage-gain stage is that it generally introduces one pole at moderate frequencies. The 709 (Figure 8.7) is an example of an early integrated-circuit amplifier that is a three-stage design. While we do not intend to investigate the operation of this circuit in detail (several modern and more useful amplifiers are described in Chapter 10), the basic signal-flow path illustrates the three-stage nature of this design. Transistors \(Q_1\) and \(Q_2\) form a differ­ential amplifier. The main second-stage amplification occurs through the \(Q_4-Q_6\) Darlington-connected pair. Transistors \(Q_3\) and \(Q_5\) complete a dif­ferential second stage with the \(Q_4-Q_6\) pair and are included primarily to reduce amplifier drift. Transistors \(Q_8\) and \(Q_9\) are used for level shifting, with common-emitter stage \(Q_{12}\) the final stage of voltage gain. Emitter followers \(Q_{13}\) and \(Q_{14}\) function as a buffer amplifier. There is some minor-loop feedback applied around the output stage to linearize its performance and to modify its dynamics via \(R_{15}\).

    Compensation is implemented by connecting a series \(R-C\) network from the output to the input of the second stage. It is also necessary to use ca­pacitive feedback from the amplifier output to the base of \(Q_{12}\) (essentially around the output stage) to obtain acceptable stability in most applications.

    Two-Stage Design

    While a number of operational-amplifier designs with three (or even more) voltage amplifying stages exist, it is hard to escape the conclusion that one is fighting nature when he tries to stabilize an amplifier with three

    or more closely spaced poles. The key to successful operational-amplifier design is to realize that the only really effective way to eliminate poles in an amplifier transfer function is to reduce the number of voltage-gain pro­ducing stages. Stages that provide current gain only, such as emitter fol­lowers, generally have poles located at high enough frequencies to be ig­nored.

    截屏2021-08-14 下午3.11.21.png
    Figure 8.8 Basic two-stage ampifier.

    An amplifier with two voltage-gain stages results if one of the common-emitter stages of Figure 8.1 is eliminated, as shown in Figure 8.8.(The great value and versatility of this basic amplifier and its many variations were first pointed out to me by Dr. F. W. Sarles, Jr.) Again, tran­sistors \(Q_1\) and \(Q_2\) function as a differential amplifier. However, in contrast to the previous amplifier, note that the base of transistor \(Q_1\) is the inverting input of the complete amplifier, while the first-stage output is the collector of

    transistor \(Q_2\). This emitter-coupled connection assures low input ca­ pacitance (approximately \(C_{\mu 1} + C_{\pi 1}/2\)) at the base of \(Q_1\) since this device is operating as an emitter follower. Low input capacitance is an advantage in many applications since feedback is normally applied from the output of the amplifier to its inverting input terminal. The input capacitance at the inverting input can introduce an additional moderate-frequency pole in the loop transmission of the amplifier-feedback network combination with attendant stability problems. Thus low input capacitance increases the range of feedback impedances that can be used without deteriorating the loop transmission.

    The transfer function for this amplifier calculated using the parameter values in Table 8.2 is (As in the case of the three-stage amplifier, the slight input-stage unbalance that occurs at high frequencies because of signals fed directly to the base of \(Q_3\) via the collector-to-base capacitance of \(Q_2\) has been ignored in the analysis that leads to this transfer function. The error introduced by this simplification is insignificant at frequencies below the unity-gain frequency of the amplifier. Furthermore, the transfer function of interest in most feedback applications where the feedback signal is applied to the base of \(Q_1\) does not include the feed-forward term associated with \(C_{\mu 2}\) )

    \[\dfrac{V_o (s)}{V_{i2} (s) - V_{i1} (s)} = \dfrac{6 \times 10^3}{(3 \times 10^{-4} s + 1)(1.1 \times 10^{-8} s + 1)}\label{eq8.2.10} \]

    截屏2021-08-14 下午3.17.05.png
    Figure 8.9 Transfer function of two-stage amplifier.

    with all other singularities above \(5 \times 10^8 \text{ sec}^{-1}\). The corresponding Bode plot (Figure 8. 9) shows that a phase margin of \(75^{\circ}\) results even when the out­put of the amplifier is fed directly back to its inverting input. This type of transfer function, obtained without including any additional compensation components, contrasts sharply with the uncompensated three-stage-ampli­fier transfer function of the previous section.

    It is informative to see why the transfer function of this amplifier is dominated by a single pole and why the second pole is separated from the dominant pole by a factor of approximately 30,000. This separation, which permits excellent desensitivity in feedback applications while maintaining good relative stability, is a major advantage attributable to the two-stage design. The dominant pole is primarily a result of energy storage in the collector-to-base capacitance of transistor \(Q_3\). A \(C_T\) approximation to the input capacitance of this transistor is (see the discussion associated with Equation \(\ref{eq8.2.7}\))

    \[C_T = C_{\pi 3} + C_{\mu 3} (1 + g_{m3} R_2) = 6.02 \times 10^{-9}\ F \nonumber \]

    The corresponding time constant

    \[\tau_{B3} = C_T r_{\pi 3} = 3.01 \times 10^{-4} \text{ sec} \nonumber \]

    Table 8.2 Parameter Values for Example Using Amplifier of Figure 8.8

    Supply voltages:
    \(\pm 15\ V\)
    Bias currents:
    \(I_{C1} = I_{C2} = 10 \ \mu A\)
    \(I_{C3} = 50 \ \mu A\)
    Transconductances implied by bias currents:
    \(g_{m1} = g_{m2} = 4 \times 10^{-4} \text{ mho}\)
    \(g_{m3} = 2 \times 10^{-3} \text{ mho}\)
    Other transistor parameters:
    \(\beta = 100 \text{ (all transistors)}\)
    \(r_{\pi 1} = r_{\pi 2} = 250\ k\Omega\)
    \(r_{\pi 3} = 50\ k \Omega\)
    \(r_x = 100 \Omega \text{ (all transistors)}\)
    \(C_{\mu} = C_{\pi} = 10\text{ pF (all transistors)}\)
    \(R_1 \gg r_{\pi 3}\)
    \(R_2 = 300 \ k\Omega\)

    Buffer amplifier assumed to have infinite input impedance.

    agrees with the dominant time constant in Equation \(\ref{eq8.2.10}\). The essential point is that the feedback through \(C_{\mu 3}\), which is actually a form of minor loop compensation (see Section 5.3), controls the transfer function of the com­plete amplifier at frequencies between approximately \(3.3 \times 10^3\) and \(10^8\) radians per second. As we shall see, the minor-loop feedback mechanism that dominates amplifier performance in this case can be used to advantage for compensation of more complex amplifiers that share the topology of this circuit.

    Most modern high-performance operational amplifiers represent rela­tively straightforward extensions of the circuit shown in Figure 8.8, and this popularity is a direct consequence of the excellent dynamics associated with the topology. An important modification included in most designs is the use of a more complex second stage than the simple common-emitter amplifier shown in Figure 8.8 in order to achieve higher d-c open-loop gain. Other options exist in the way the output buffer circuit is realized and the drift-reducing modifications that may be incorporated into the first and second stages.

    This page titled 8.2: AMPLIFIER TOPOLOGIES 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.