X

Two-port analysis is widely used in the study of negative-feedback circuits.

**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.

The voltage follower of **Figure 1 a** forms a

Next, we use **TP analysis** to put the circuit in the block-diagram form of **Figure 2 a**, where

where the ratio *a _{mod}*/

The condition *a _{mod} *→ ∞ needed to find

*A _{ideal}* = 1 V/V (3

Substituting *a _{mod}* and

where “0” has been deliberately shown in the expression for *A _{TP}* to contrast it with the corresponding “1” appearing in the expression for

*A _{exact}* = 0.93458 V/V

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 *C*_{π}, the parasitic element dominating the emitter follower’s dynamics. The expressions of Equations (1) and (4) still hold, provided we make the substitution

after which both *A _{exact}* and

Mathematically, the dramatic departure of *A _{TP}* from

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

To find *T* (also called the loop gain) and *a _{ft}* for our voltage-follower example, refer to the ac equivalents of

The return ratio *T* of the *g _{m}v*

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

Calculating *T* and *a _{ft}* with the component values of

*A _{RR}* = 0.91625 + 0.01833 = 0.93458 V/V (10)

which *coincides *with the value of *A _{exact}* of Equation (5). In fact, substituting Equation (9) into Equation (8), one can verify, with a bit of algebraic manipulations, that the expression of

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

*Sergio Franco is an author and emeritus university professor.*

**References**

[1] Two-port vs. return-ratio analysis

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

[3] Quest for the Ideal Transistor?

[4] In Defense of the Current-Feedback Amplifier

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