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.

**Figure 1**(

*a*) Voltage follower and (

*b*) its small-signal equivalent. Assume

*g*= 20 mA/V,

_{m}*r*

*= 2.5 kΩ,*

_{π}*r*= 50 kΩ, and

_{o}*R*

_{1}=

*R*

_{2}= 1.0 kΩ.

The voltage follower of **Figure 1 a** 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 1**, we use nodal analysis and readily find the familiar expression [2]:

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

*a*is the gain of the basic amplifier

_{mod}*modified*so as to take loading into account, and

*A*

_{ideal}*the closed-loop gain in the limit*

_{ }*a*→ ∞ The gain of this circuit takes on the familiar form [2]:

_{mod}where the ratio *a _{mod}*/

*A*

_{ideal}*is also called the*

_{ }*loop gain*. With reference to

**Figure 2**, we write, by inspection,

*b*

**Figure 2**(

*a*) Block-diagram for TP analysis, and (

*b*) circuit to find the modified gain

*a*=

_{mod}*v*/

_{o}*v*.

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

*A*

_{ideal}*is achieved by letting*

_{ }*g*→ ∞. This results in

_{m}*v*

*→ 0, so the current through*

_{π}*r*

_{π}(and, hence, through

*R*

_{1}) tends to zero, indicating that

*v*→

_{o}*v*. Consequently,

_{i}

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

*b*)

Substituting *a _{mod}* and

*A*

_{ideal}*into Equation (2), we get*

_{ }

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*above. Using the component values of

_{exact}**Figure 1**, we get

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

*A*= 0.93336 V/V (5)

_{TP}

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

**Figure 3**Including the capacitance

*C*

_{π}

*to investigate the frequency response*

_{}after which both *A _{exact}* and

*A*become functions of

_{TP}*j*

*ω*. Now, considering that for

*ω*→ ∞ we have

*g*

_{m}z*(*

_{π}*j*

*ω*) → 0, it follows that

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

*A*stems from the aforementioned denominator term of “0” instead of “1”. Physically, we justify by noting that at high frequencies, where

_{exact}*C*

_{π}approaches short-circuit behavior, both

*V*

_{π}and

*g*

_{m}V_{π}will approach zero, thus reducing the circuit of

**Figure 3**to a mere voltage divider, as confirmed by Equation (7

*a*). 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

*a*is the

_{ft}*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 *a _{ft}* for our voltage-follower example, refer to the ac equivalents of

**Figure 5**. In

**Figure 5**we apply a test current

*a**i*and find the return current

_{t}*i*as

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

*source is then [1, 2]*

_{π}**Figure 4**Block-diagram for RR analysis.

**Figure 5**Circuits to find (

*a*) the return-ratio

*T*of the source

*g*

_{m}v*, and (*

_{π}*b*) the feedthrough gain

*a*around the same source.

_{ft}In **Figure 5 b** we use the voltage divider formula to write

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

**Figure 1**and plugging into Equation (8) gives

*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

*A*

_{RR}*coincides*with the expression of

*A*. But, from a bookkeeping viewpoint,

_{exact}*A*is a bit more instructive because it shows separately the component due to forward gain and that due to feedthrough.

_{RR}[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.