# 8.3: HIGH-GAIN STAGES

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As mentioned in the previous section, a high-gain second stage is usually used to provide the basic amplifier with the voltage gain normally required from an operational amplifier. As we shall see, high current gain or high power gain alone is insufficient. It is necessary to have stages with high voltage gain, high transresistance (ratio of incremental output voltage to incremental input current), or both included in an operational-amplifier circuit. Note that there is no restriction on the number of transistors used in the stage. The implication in our definition of stage is that its dynamics are similar to that of a single common-emitter amplifier, that is, it introduces only one pole at frequencies that are low compared to the \(f_T\) of the devices used.

Use of the usual hybrid-pi model for the analysis of the simple common-emitter amplifier of Figure 8.10 shows that the low-frequency incremental voltage is \(v_o/v_i = -g_mR_L\) and the incremental transistance is \(v_o/i_i = -\beta R_L\). The magnitude of either of these quantities can be increased (seemingly without limit) by increasing \(R_L\). In order to obtain high gains without high supply voltages [the voltage gain of the circuit of Figure 8.10 is \((q/kT) (V_C - V_O) \simeq 40(V_C - V_O)\)], a current source can be used as the collector load. We realize that this technique will not result in infinite voltage gain and transresistance in an actual circuit because the simplified hybrid-pi model does not accurately predict the behavior of circuits with voltage gains in excess of several hundred. In order to proceed it is necessary to develop a more complete hybrid-pi model.

## Detailed Low-Frequency Hybrid-Pi Model

(This material is covered in greater detail in P. E. Gray et al., *Physical Electronics and Circuit Models **for **Transistors,*Wiley, New York, 1964, Chapter 8, and C. L. Searle et al., *Elementary Circuit Propertiesof **Transistors,*Wiley, New York, 1964, Chapter 4.)

The simplified hybrid-pi model predicts that both the base current and the collector current of a transistor are independent of changes in collector-to-base voltage. Actually, both currents are voltage-level dependent because of an effect called base-width modulation, as illustrated by the following argument. Consider an NPN transistor operating at moderate current levels with fixed base-to-emitter voltage \(V_{BE}\) and collector-to-base voltage \(V_{CB}\). The approximate charge distribution in the base region for this transistor is shown by the solid line in Figure 8.11. In this figure, \(n_p\) is the minority-carrier concentration in the base region; \(N_{po}\) is the equilibrium concentration of electrons in the base region; and \(x\) is the distance into the base region with \(x = 0\) at the base edge of the emitter-base space- charge layer. The charge distribution drops linearly from its value \(n_p (0)\) at \(x = 0\) to essentially zero (if the collector-to-base junction is reverse biased by at least several hundred millivolts) at the edge of the collector space-charge layer. However, the width of the collector space-charge layer is monotonically increasing function of collector-to-base voltage. Thus, if the collector-to-base voltage is reduced, the collector space-charge layer becomes narrower. This narrowing increases the effective width of the base region from its original value of \(W\) to a new value \(W + \Delta W\). The resultant new charge distribution is shown by the dotted line in Figure 8.11.

Two changes in terminal variables result from this change in base width. First, the collector current (proportional to the slope of the distribution) becomes smaller. Second, the base current increases, since the total rate at which charge recombines in the base region is directly proportional to the total charge in this region. The magnitudes of these changes are calculated as follows.

The collector current of an NPN transistor is related to transistor and physical constants by

\[I_C = \dfrac{qN_{po} AD_e}{W} e^{qV_{BE}/kT}\label{eq8.3.1} \]

where

\(N_{po}\) is the equilibrium concentration of electrons in the base region.

\(A\) is the cross-sectional area of the base.

\(D_e\) is the diffusion constant for electrons in the base region.

The assumptions necessary to derive this relationship include operation under conditions of low-level injection but at current levels large compared to leakage currents, and that the ohmic drops in the base region are negligible. The assumption of negligible ohmic voltage drop in the base region results in no loss of generality, since a base resistance can be added to the model which evolves from Equation \(\ref{eq8.3.1}\).

Under conditions of constant base-to-emitter voltage and temperature, Equation \(\ref{eq8.3.1}\) reduces to

\[I_C = \dfrac{K}{W}\label{eq8.3.2} \]

where the constant \(K\) includes all other terms from Equation \(\ref{eq8.3.2}\). Differentiating yields

\[\dfrac{dI_C}{dW} = -\dfrac{K}{W^2} \nonumber \]

Differential changes in \(W\) are related to incremental changes in collector-to-base voltage as

\[\Delta W = \dfrac{dW}{dV_{CB}} v_{cb} \nonumber \]

Incremental changes in collector current can thus be expressed in terms of incremental changes in collector-to-base voltage as

\[i_c = -\dfrac{K}{W^2} \dfrac{dW}{dV_{CB}} v_{cb} \label{eq8.3.5} \]

Solving Equation \(\ref{eq8.3.2}\) for \(K\) and substituting into Equation \(\ref{eq8.3.5}\) yields

\[i_c = -\dfrac{I_C}{W} \dfrac{dW}{dV_{CB}} v_{cb} \label{eq8.3.6} \]

The transconductance of a transistor is related to quiescent collector current as

\[g_m = \dfrac{qI_C}{kT} \label{eq8.3.7} \]

Solving Equation \(\ref{eq8.3.7}\) for \(I_C\) and substituting this result into Equation \(\ref{eq8.3.6}\) shows that

\[i_c = \left [ -\dfrac{kT}{qW} \dfrac{dW}{dV_{CB}} \right ] g_m v_{cb} \label{eq8.3.8} \]

The bracketed quantity in Equation \(\ref{eq8.3.8}\) is called the *base-width modulation factor* and is denoted by the symbol \(\eta\). Introducing this notation and adding the familiar relationship between incremental components of collector cur rent and base-to-emitter voltage to Equation \(\ref{eq8.3.8}\) yields

\[i_c = g_m v_{be} + \eta g_m v_{cb}\label{eq8.3.9} \]

The quantity \(\eta \) is typically \(10^{-3}\) to \(10^{-4}\), indicating that the collector current is much more strongly dependent on base-to-emitter voltage than on collector-to-base voltage. This is, of course, the reason we are able to ignore the effect of collector-to-base voltage variations except in high-gain situations.

The change in base current as a function of collector-to-base voltage can be calculated with the aid of Figure 8.11. If reverse injection from the base into the emitter region is assumed small, the base current is directly proportional to the area of the triangle, since the total number of minority carriers that recombine per unit time and thus contribute to base current is proportional to the total number of these carriers in the base region. The geometry of Figure 8.11 shows that the magnitude of the fractional change in the area of the triangle is equal to the magnitude of the fractional change in slope of the distribution for small changes in \(W\). Furthermore, an increase in \(W\) decreases collector current and increases base current. Equating fractional changes yields

\[\dfrac{i_b}{I_B} = -\dfrac{i_c}{I_C} = -\dfrac{\eta g_m v_{cb}}{I_C}\label{eq8.3.10} \]

Rearranging Equation \(\ref{eq8.3.10}\) and recognizing that \(I_C/I_B = \beta\) yields for the incre mental dependence of base current on collector-to-base voltage at constant base-to-emitter voltage

\[i_b = -\dfrac{\eta g_m v_{cb}}{\beta} \label{eq8.3.11} \]

Adding the incremental relationship between base current and base-to emitter voltage to Equation \(\ref{eq8.3.11}\) results in

\[i_b = \dfrac{g_m}{\beta} v_{be} - \dfrac{\eta g_m}{\beta} v_{cb} \label{eq8.3.12} \]

It is necessary to augment the familiar hybrid-pi transistor model to in clude the effects of base-width modulation when the model is used for the analysis of high-gain circuits. While there are several model modifications that would accurately represent base-width-modulation phenomena, convention dictates that the model be augmented by the addition of a collector to-emitter resistor \(r_o\) and a collector-to-base resistor \(r_{\mu}\) as shown in Figure 8.12. The objective is to choose the four elements of the model so that the terminal relationships dictated by Equations \(\ref{eq8.3.9}\) and \(\ref{eq8.3.12}\) are obtained. Note that, since four degrees of freedom are required to match arbitrary two-port relationships, it may be necessary to have the dependent current-generator scale factor in Figure 8.12 differ from gm, and this possibility is indicated by calling this scale factor \(g_m'\).

The terminal relationships developed from the analysis of the effects of base-width modulation are repeated here for convenience:

\(i_c = g_m v_{be} + \eta g_m v_{cb}\)

\(i_b = \dfrac{g_m}{\beta} v_{be} - \dfrac{\eta g_m}{\beta} v_{cb} \)

The equations relating the same variables for the model of Figure 8.12 are (Recall that corresponding \(r\)'s and \(g\)'s are reciprocally related. Thus, for example, \(g_o = 1/r_o\).)

\[\begin{array} {rcl} {i_c} & = & {g_m' v_{be} + g_{\mu} v_{cb} + g_o (v_{bc} + v_{cb})} \\ {} & = & {(g_m' + g_o) v_{be} + (g_o + g_{\mu} ) v_{cb}} \end{array} \nonumber \]

\[i_b = g_{\pi} v_{be} - g_{\mu} v_{cb} \nonumber \]

Equationing coefficients in these two sets of equations yields

\[g_m' + g_o = g_m \nonumber \]

\[g_o + g_{\mu} = \eta g_m \nonumber \]

\[g_{\pi} = \dfrac{g_m}{\beta} \nonumber \]

\[g_{\mu} = \dfrac{\eta g_m}{\beta} \nonumber \]

These equations are readily solved to determine model element values:

\[g_m' = g_m \left [ 1 - \eta \left ( 1 - \dfrac{1}{\beta} \right ) \right ] \label{eq8.3.19} \]

\[r_{\pi} = \dfrac{1}{g_{\pi}} = \dfrac{\beta}{g_m} \nonumber \]

\[r_o = \dfrac{1}{g_o} = \dfrac{1}{\eta g_m [ 1 - (1/\beta)]} \label{eq8.3.21} \]

\[r_{\mu} = \dfrac{1}{g_{\mu}} = \dfrac{\beta}{\eta g_m}\label{eq8.3.22} \]

Since for any well-designed transistor \(|\eta| \ll 1\) (typical values are \(10^{-3}\) to \(10^{-4}\)) and \(\beta \gg 1\), the approximations

\[g_m' \simeq g_m = \dfrac{q|I_C|}{kT} \nonumber \]

and

\[r_o \simeq \dfrac{1}{\eta g_m}\label{eq8.3.24} \]

usually replace Equations \(\ref{eq8.3.19}\) and \(\ref{eq8.3.21}\), respectively.

It is instructive to examine the relative magnitudes of the model parameters for a transistor under typical conditions of operation. Assume that a transistor with \(\beta = 200\) and \(\eta = 4 \times 10^{-4}\) is operated at \(I_C = 1\ mA\) at room temperature. Then \(g_m = 40\text{ mmho}\), \(g_{\pi} = 200\ \mu \text{mho}\) or \(r_{\pi} = 5\ k\Omega\), \(g_o = 16\ \mu \text{mho}\) or \(r_o = 62.5\ k\Omega\) and \(g_{\mu} = 0.08\ \mu \text{mho}\) or \(r_{\mu} = 12.5\ M\Omega\). Note that all conductances in the intrinsic model are proportional to \(g_m\) and therefore to quiescent collector current.

## Common-Emitter Stage with Current-Source Load

In spite of the internal loading of \(r_o\) and \(r_{\mu}\), high voltage gain is possible with a current-source load for a common-emitter stage, and this connection is used in many operational-amplifier designs. Figure 8.13\(a\) shows a schematic for such a stage and Figure 8.13\(b\) is the corresponding low-frequency equivalent circuit. It is assumed that the incremental resistance of the cur rent source is infinite. (The problems associated with realizing a high-resistance current source will be described in Section 8.3.5.) It is also assumed that the base resistance of the transistor can be neglected. This assumption is best justified by considering a complete amplifier where the resistances at various nodes are known. In most anticipated applications \(r_x\) will either be small enough so that it can be neglected even for voltage-source drives at the base of the transistor in question, or the value of \(r_x\) will be masked by a large driving resistance connected in series with it.

The equivalent circuit of Figure 8.13\(b\) is easily analyzed by solving the output-node equation:

\[g_m v_i + g_o v_o + g_{\mu} (v_o - v_i) = 0 \nonumber \]

Since \(g_{\mu} \ll g_o\) See Equations \(\ref{eq8.3.22}\) and \(\ref{eq8.3.24}\) and \(g_{\mu} \ll g_m\),

\[\dfrac{v_o}{v_i} \simeq -g_m r_o \label{eq8.3.26} \]

With the equivalence of Equation \(\ref{eq8.3.24}\), \(r_o = 1/\eta g_m\), the volrage-gain of the circuit becomes simply \(-1/\eta\). As mentioned earlier typical values for \(\eta\) are \(10^{-3}\) to \(10^{-4}\), and therefore a voltage-gain magnitude of \(10^3\) to \(10^4\) is possible.

The incremental input current can be calculated as follows.

\[i_i = (g_{\pi} + g_{\mu})v_i - g_{\mu} v_o \nonumber \]

Substituting from Equation \(\ref{eq8.3.26}\) yields

\[i_i = (g_{\pi} + g_{\mu} + g_m r_o g_{\mu})v_i\label{eq8.3.28} \]

Recognizing that

\[g_m r_o g_{\mu} = g_{\pi} \nonumber \]

simplifies Equation \(\ref{eq8.3.28}\) to

\[i_i = (2g_{\pi} + g_{\mu}) v_i \simeq 2 g_{\pi} v_i\label{eq8.3.30} \]

This relationship indicates that the use of a current-source load halves the input resistance of a common-emitter amplifier compared to the value when loaded with a moderate-value resistor, since the currents flowing through \(r_{\pi}\) and \(r_{\mu}\) are equal in this high-gain connection.

Combining Equations \(\ref{eq8.3.30}\) and \(\ref{eq8.3.26}\) shows that the transresistance is

\[\dfrac{v_o}{i_i} = -\dfrac{r_{\pi} g_m r_o}{2} = -\dfrac{\beta r_o}{2} = -\dfrac{r_{\mu}}{2}\label{eq8.3.31} \]

The dominant pole for this amplifier, at least for realistic values of driving-source resistance, occurs at the input. Because of the high voltage gain, the input capacitance includes a component several thousand times larger than \(C_{\mu}\), and this effective input capacitance is the primary energy-storage element.

## Emitter-Follower Common-Emitter Cascade

The current-source-loaded common-emitter stage analyzed in the preceding section can be driven with an emitter follower to increase trans-resistance. Figure 8.14 illustrates this connection. Analysis is simplified by applying the results of the last section. Since the input resistance of the common-emitter amplifier is \(r_{\pi}/2\) (Equation \(\ref{eq8.3.30}\)), the transfer ratios \(v_a/v_i\) and \(v_a/i_i\) can be calculated by replacing the input circuit of \(Q_2\) with a resistor equal to \(r_{\pi 2}/2\). These results are combined with Equations \(\ref{eq8.3.26}\) and \(\ref{eq8.3.30}\) to determine gain and transresistance. Furthermore, it is not necessary to consider elements \(r_o\) and \(r_{\mu}\) in the model for transistor \(Q_1\) since the voltage gain of this device is low. An incremental equivalent circuit that relates \(v_a\) to \(v_i\) is shown in Figure 8.15.

The voltage-transfer ratio is

\[\dfrac{v_a}{v_i} = 1 - \dfrac{1}{1 + r_{\pi 2}/2r_{\pi 1} + g_{m1} r_{\pi 2}/2}\label{eq8.3.32} \]

For the circuit of Figure 8.14 the quiescent collector current of \(Q_2\) is \(I\), while that of \(Q_1\) is approximately \(I/\beta_2\). Therefore,

\[r_{\pi 2} = \dfrac{\beta_2}{g_{m2}} = \dfrac{q_2 k T}{qI} \nonumber \]

and

\[r_{\pi 1} = \dfrac{\beta_1}{g_{m1}} = \dfrac{\beta_1 \beta_2 kT}{qI} = \beta_1 r_{\pi 2}\label{eq8.3.34} \]

Equation \(\ref{eq8.3.34}\) shows that for reasonable values of 11, the term \(r_{\pi 2}/2r_{\pi 1}\) in Equation \(\ref{eq8.3.32}\) can be dropped.

Introducing this simplification and noting that \(g_{m2} = \beta_2 g_{m1}\), so that \(r_{\pi 2} = 1/g_{m1}\) reduces Equation \(\ref{eq8.3.32}\) to

\[\dfrac{v_a}{v_i} = \dfrac{1}{3} \nonumber \]

Therefore

\[\dfrac{v_o}{v_i} = -\dfrac{1}{3 \eta_2} \label{eq8.3.36} \]

Since \(v_a = \tfrac{1}{3} v_i\), the input resistance is

\[\dfrac{v_i}{i_i} = \dfrac{3}{2} r_{\pi 1} \label{eq8.3.37} \]

Combining Equations \(\ref{eq8.3.36}\) and \(\ref{eq8.3.37}\) shows that the transresistance is

\[\dfrac{v_o}{i_i} = -\dfrac{r_{\pi 1}}{2 \eta_2} \nonumber \]

This equation can be compared with Equation \(\ref{eq8.3.31}\) by noting that \(r_{\pi 1} = \beta_1 \beta_2 /g_{m2}\). Thus

\[\dfrac{v_o}{i_i} = -\dfrac{\beta_1 \beta_2}{2g_{m2} \eta_2} = -\dfrac{\beta_1 r_{\mu 2}}{2} \nonumber \]

Transistor \(Q_1\) simply improves the transresistance of the circuit by a factor of 11.

The dominant pole for this circuit is associated with the input of \(Q_2\), since the incremental resistance to ground at this point remains high even with the emitter follower included.

## Current-Source-Loaded Cascode

The gain limitations of the common-emitter amplifier stem from an internal negative-feedback mechanism related to transistor operation. As the collector-to-base voltage changes, the effective width of the base region also changes and resulting variations in collector- and base-terminal current oppose the original change. This effect is similar to that of the collector-to-base capacitance \(C_{\mu}\) that supplies charge to both the collector and base terminals in such a direction as to oppose rapid variations in collector voltage. The cascode connection, which is useful because it minimizes feedback through \(C_{\mu}\) at high frequencies, can also be used to minimize the effects of base-width modulation on circuit performance.

A connection that combines a cascode amplifier with a current-source load is shown in Figure 8.16. This circuit can be analyzed by brute-force techniques, or a little thought can be traded for a page of calculations. We have already shown that the voltage gain of a current-source-loaded common-emitter amplifier is \(-1/\eta\).

Therefore the transfer ratio \(v_o/v_a\) in Figure 8.16 is

\[\dfrac{v_o}{v_a} = \dfrac{1}{\eta_2} + 1 \simeq \dfrac{1}{\eta_2} \nonumber \]

We have also shown that the input resistance for the common-emitter amplifier is \(r_{\pi} /2\). Observe that since the incremental collector current of \(Q_2\) cannot change in the connection of Figure 8.16, the incremental ratio \(v_a/i_a\) must be the same as the input resistance of the common-emitter amplifier, or

\[\dfrac{v_a}{i_a} = \dfrac{r_{\pi 2}}{2} \nonumber \]

The voltage gain of \(Q_1\) can be calculated by simply assuming it is loaded with a resistor equal \(r_{\pi 2}/2\). Accordingly,

\[\dfrac{v_a}{v_i} = -g_{m1} \dfrac{r_{\pi 2}}{2}\label{eq8.3.42} \]

providing this gain is small enough so that \(r_{\mu 1}\) and \(r_{o1}\) are negligible. Equation \(\ref{eq8.3.42}\) can be simplified by noting that \(r_{\pi 2} = \beta_2/g_{m2}\), and that \(g_{m1} = g_{m2}\) since both devices are operating at virtually identical quiescent currents. With this relationship the voltage gain of the current-source-loaded cascode becomes

\[\dfrac{v_o}{v_i} = -\dfrac{\beta_2}{2\eta_2} \nonumber \]

Since the input resistance of \(Q_1\) is \(r_{\pi 1}\), the transresistance for the circuit is

\[\dfrac{v_o}{i_i} = -\dfrac{\beta_2 r_{\pi 1}}{2 \eta_2} = -\dfrac{\beta_2 \beta_1}{2\eta_2 g_{m1}} = -\dfrac{\beta_2 \beta_1}{2\eta_2 g_{m2}} = -\dfrac{\beta_1 r_{\mu 2}}{2} \nonumber \]

Comparing the cascode with the two previous circuits, we see that it provides the same transresistance as the circuit including the emitter follower and has significantly higher voltage gain than either of the other circuits. It is of practical interest to note that transistors are available that can provide voltage gains in excess of 10 in this connection.

The dominant pole occurs at the collector of \(Q_2\) because the incremental resistance at this node is extremely high. The use of the cascode reduces the capacitance seen at the base of \(Q_1\) so that even with a high source resistance, the time constant at this node is typically between 100 and 10,000 times shorter than the collector-circuit time constant.

## Related Considerations

The circuits described in the last three sections offer at least one further advantage that is useful for the design of operational amplifiers. The current source included in all of these circuits insures that the transistors operate at quiescent current levels that are essentially independent of output voltage. Large output-voltage swings are therefore possible without altering any current-dependent transistor parameters.

Care may be required in the design of a current source with sufficiently high output resistance to prevent significant loading of the high-gain stages. Figure 8.17\(a\) shows a transistor connected as a current source. The output resistance for this connection determined from the incremental circuit model is

\[\dfrac{v_o}{i_o} = r_{\mu} \left |\left | \left [ \dfrac{1 + (g_m + g_o)(r_{\pi} || R_E)}{g_o} \right ] \simeq r_{\mu} \right | \right | \left [ \dfrac{1 + g_m (r_{\pi} || R_E)}{g_o} \right \label{eq8.3.45}]

The output resistance varies from

\[\dfrac{v_o}{i_o} \simeq r_o \text{ for } R_E = 0 \nonumber \]

to

\[\dfrac{v_o}{i_o} \simeq r_{\mu} \left | \right | \dfrac{g_m r_{\pi}}{g_o} = \dfrac{r_{\mu}}{2} \text{ for } R_E \gg r_{\pi} \nonumber \]

This analysis indicates that it is not possible to build a current source of this type with an output resistance in excess of \(r_{\mu}/2\).

Since \(r_{\mu}\) is current dependent and since the current source operates at a current level equal to that of its driving transistor in the high-gain circuits, r, and r, for a current-source transistor will be comparable to those of the driving transistor. The analysis of Section 8.3.2 can be extended to show that the output resistance of the common-emitter stage is r, when driven from a voltage source and is \(r_o/2\) when driven from a high impedance source. Thus use of a common-emitter current source (\(R_E = 0\) in Figure 8.17) can reduce the gain of this stage by as much as a factor of two. Since the output resistance of the emitter-follower common-emitter cascode is \(2r_o/3\) when driven from a voltage source, the susceptibility of this stage to loading is comparable to that of the common-emitter stage.

The output resistance of the cascode is \(r_{\mu}/2\), so even the highest output resistance that can be achieved with a bipolar-transistor current source will halve the unloaded gain of this stage. A further practical difficulty is that approaching a current-source resistance of \(r_{\mu}/2\) requires \(R_E \gg r_{\pi}\) (Equation \(\ref{eq8.3.45}\)). If we assume the base-to-emitter voltage of the transistor is small compared to \(V\) in Figure 8.17\(a\),

\[R_E \simeq \dfrac{V}{I_E} = \dfrac{qV}{kT g_m} \simeq \dfrac{40 V r_{\pi}}{\beta} \nonumber \]

In order to satisfy the inequality \(R_E \gg r_{\pi}\), it is necessary to have \(V \gg \beta/40\).

The use of low 0 transistors is not the answer, since such transistors also have low \(r_{\mu}\). One way to avoid the requirement for high supply voltage is to use the connection of Figure 8.18. Cascoding serves the same function as it does in the amplifier, and provides an output resistance of approximately \(r_{\mu}/2\) with a total supply voltage of several volts.

The analysis presented above shows that the output resistance of a bipolar-transistor current source is bounded by \(r_{\mu} /2\), and that this maximum value occurs only when the base of the transistor is connected to a low resistance level relative to the emitter-circuit resistance. Field-effect transistors (FET's) can be used in the interesting connection shown in Figure 8.19\(a\) to increase the output resistance of a current source. A model that can be used for the linear-region analysis of the FET is shown in Figure 8.19\(b\). An incremental equivalent circuit of the cascoded source, assuming that the finite output resistance of the current source \(R_S = v_a/i_a\). completely describes this element, is shown in Figure 8.19\(c\). This equivalent circuit shows that the relationship between \(v_o\) and \(i_o\) is

\[v_o = i_o R_S + \dfrac{i_o}{y_{os}} + \dfrac{i_o R_S y_{fs}}{y_{os}} \nonumber \]

or that

\[\dfrac{v_o}{i_o} = \dfrac{1}{y_{os}} + R_S \left ( 1 + \dfrac{y_{fs}}{y_{os}} \right ) \nonumber \]

Since the quantity \(y_{fs}/y_{os}\) can be several hundred or more for certain FET'S, this connection greatly increases the incremental resistance of the current source itself. For example, by using a bipolar-transistor current source cascoded with a FET, incremental resistances in excess of \(10^{12} \Omega\) can be obtained at a quiescent current of \(10\ \mu A\). It is theoretically possible to further increase current-source output resistance by using multiple cascoding with FET's, although stray conductance limits the ultimate value in actual circuits.

Another problem that occurs in the design of high-gain stages is that the output of the stage must be isolated with a very high-input-resistance buffer to prevent loading that can cause a severe reduction in the voltage gain of the stage. One approach is to use a FET as a source follower, since the input resistance of this connection is essentially infinite. The use of a FET as a buffer or to cascode a current source is frequently the best technique in discrete-component designs. However, it is presently difficult to fabricate high-quality bipolar and field-effect transistors simultaneously in monolithic integrated-circuit designs; thus alternatives are necessary for these circuits.

If a bipolar-transistor emitter follower (Figure 8.20) is used, care must be taken to insure sufficiently high input resistance. The incremental input resistance for this circuit with no additional loading is

\[\dfrac{v_i}{i_i} \simeq r_{\mu} \left |\right | [r_{\pi} + \beta (r_o \left |\right | R_E)] \nonumber \]

In order to approach the maximum input resistance of \(r_{\mu} /2\) (particularly important if the buffer is to be used with the cascode amplifier), it is necessary to have \(R_E \gg r_o\). This inequality normally cannot be satisfied with reasonable supply voltages, so a current source is frequently used in place of \(R_E\). A further advantage of the current source is that the drive current that can be supplied to any following stage becomes independent of voltage level.

One design constraint for an emitter follower intended for use with the current-source-loaded cascode amplifier is that the quiescent operating current of this stage should not be large compared with that of the cascode or else the gain of the stage will be determined primarily by \(r_{\mu}\) of the emitter follower.