Editor's note: I am posting this excellent tech article on EDN so that you will all be aware of Professor Sergio Franco's blog on EDN entitled Analog Bytes. This blog has some really great engineering analyses that provide an in-depth insight to engineers for their work.
Steve Taranovich

Two-port (TP) analysis is widely used in the study of negative-feedback circuits. This type of analysis requires that we first identify which of the four topologies the circuit at hand belongs to (series-shunt, shunt-series, series-series, or shunt-shunt), and then that we suitably modify the basic amplifier so as to account for loading by the feedback network [1]. Textbooks do specify that TP analysis postulates certain approximations so that its results are not necessarily exact. However, not many textbooks dwell further into this issue by showing actual examples [2] for which TP analysis is insufficient, if not utterly inadequate; so, after becoming proficient in TP analysis, one may erroneously be tempted to take its results as exact. Let us illustrate using the voltage follower of Figure 1 as a vehicle.

Figure 1 (a) Voltage follower and (b) its small-signal equivalent. Assume gm = 20 mA/V, rπ = 2.5 kΩ, ro = 50 kΩ, and R1 = R2 = 1.0 kΩ.

The voltage follower of Figure 1a forms a series-shunt configuration, and it is simple enough that we can find its exact closed-loop gain directly. With reference to its ac equivalent of Figure 1b, we use nodal analysis and readily find the familiar expression [2]:

Next, we use TP analysis to put the circuit in the block-diagram form of Figure 2a, where amod is the gain of the basic amplifier modified so as to take loading into account, and Aideal   the closed-loop gain in the limit amod → ∞ The gain of this circuit takes on the familiar form [2]:

where the ratio amod/Aideal  is also called the loop gain.  With reference to Figure 2b, we write, by inspection,

Figure 2  (a) Block-diagram for TP analysis, and (b) circuit to find the modified gain amod = vo/vi.

The condition amod → ∞ needed to find Aideal  is achieved by letting gm → ∞.  This results in vπ → 0, so the current through rπ (and, hence, through R1) tends to zero, indicating that vovi. Consequently,

Aideal = 1 V/V                                                                                                                (3b)

Substituting amod and Aideal  into Equation (2), we get

where “0” has been deliberately shown in the expression for ATP to contrast it with the corresponding “1” appearing in the expression for Aexact above. Using the component values of Figure 1, we get

Aexact = 0.93458 V/V     ATP = 0.93336 V/V                                                                   (5)

The difference is minimal in the present case, which pertains to low-frequency operation, but it becomes much more pronounced at high frequencies, as we are about to show.

To investigate the frequency behavior, we use the ac equivalent of Figure 3, which includes the base-emitter capacitance Cπ, the parasitic element dominating the emitter follower’s dynamics.  The expressions of Equations (1) and (4) still hold, provided we make the substitution

Figure 3  Including the capacitance Cπ to investigate the frequency response

after which both Aexact and ATP become functions of jω.  Now, considering that for ω → ∞ we have gmzπ(jω) → 0, it follows that

Mathematically, the dramatic departure of ATP from Aexact stems from the aforementioned denominator term of “0” instead of “1”.  Physically, we justify by noting that at high frequencies, where Cπ approaches short-circuit behavior, both Vπ and gmVπ will approach zero, thus reducing the circuit of Figure 3 to a mere voltage divider, as confirmed by Equation (7a).  Evidently, TP analysis fails to account for this physical reality, and therefore its results must be taken with a grain of salt.

An elegant alternative to TP analysis, and one that yields exact rather than approximate results, is return-ratio (RR) analysis.  This type of analysis is based on the bock-diagram of Figure 4, where T is the return ratio of the dependent source modeling the amplifier’s gain, and aft is the feedthrough gain, that is, the gain with the dependent source set to zero.  The gain of this circuit takes on the form [2]

To find T (also called the loop gain) and aft for our voltage-follower example, refer to the ac equivalents of Figure 5.  In Figure 5a we apply a test current it and find the return current ir as

The return ratio T of the gmvπ source is then [1, 2]

Figure 4  Block-diagram for RR analysis.

Figure 5 Circuits to find (a) the return-ratio T of the source gmvπ, and (b) the feedthrough gain aft around the same source.

In Figure 5b we use the voltage divider formula to write

Calculating T and aft with the component values of Figure 1 and plugging into Equation (8) gives

ARR = 0.91625 + 0.01833 = 0.93458 V/V                                                                        (10)

which coincides with the value of Aexact of Equation (5).  In fact, substituting Equation (9) into Equation (8), one can verify, with a bit of algebraic manipulations, that the expression of ARR coincides with the expression of Aexact.  But, from a bookkeeping viewpoint, ARR is a bit more instructive because it shows separately the component due to forward gain and that due to feedthrough.

[Continue reading on EDN US: The case of the current-feedback amplifier]

Sergio Franco is an author and emeritus university professor.

References

[2] Analog Circuit Design: Discrete and Integrated by Sergio Franco

[5]  R. D. Middlebrook, “Measurement of Loop Gain in Feedback Systems,” Int. J. Electronics, Vol. 38, no. 4, pp. 485-512, 1975.