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Two popular ways of measuring the loop gain *T* of a negative-feedback circuit are Middlebrook’s double-injection method [1] and Rosenstark’s open-circuit/short-circuit method [2]. Each method lends itself both to computer simulation and to testing at the bench. The aim of this blog is to dispel possible confusion between the two methods by pointing out similarities, differences, and peculiarities.

**Figure 1** A negative-feedback circuit.

**Figure 1** shows a negative-feedback circuit in bare-bone form: The gain element is (arbitrarily) assumed to be a voltage-controlled current source; moreover, all passive elements on the left side of the interconnection have been coalesced into an equivalent impedance *Z*_{1}, and those on the right side into *Z*_{2}. With no external input(s) present, the circuit is in its dormant state. We wish to find its loop gain *T*. I am a fan of return-ratio analysis [3, 4], so I’ll refer to the circuit rendition of **Figure 2a**. By inspection and Ohm’s law we have *I _{r}* =

**Figure 2** Finding the loop gain *T* of the circuit of Figure 1 via (*a*) return-ratio analysis, and (*b*) by breaking the loop and properly terminating it at the return side.

Return-ratio analysis requires that we have access to the dependent source modeling gain. This is certainly the case when we are dealing with an ac model on paper, as presently. However, when facing a circuit at the transistor level, whether in the course of computer simulation or while testing it at the bench, we do not have access to its dependent source because it is ‘buried’ inside the transistor(s) providing the gain. An alternate approach is to break the feedback path, inject a test signal in the forward direction, and measure the response at the return side. For this technique to succeed, we must terminate the return side on the *same impedance* that the signal would encounter if it continued around the loop. In the present case this impedance is *Z*_{1}, as depicted in **Figure 2b**. We now have *V _{r}* = –(

**Rosenstark’s loop gain measurement**

The terminating impedance *Z*_{1} is not always a self-evident or well-known quantity. Rosenstark’s brilliant idea was to perform a pair of loop measurements with the return side terminated first on an *open circuit* and then on a *short circuit*, and then to suitably combine the two measurements to obtain the desired loop gain *T*, regardless of *Z*_{1}. This method is depicted in **Figure 3**. By inspection, we have

**Figure 3** Finding the loop gain *T* of the circuit of Figure 1 via Rosenstark’s technique: Terminating the return side (*a*) on an open circuit to find *T _{oc}*, and (

Defining

we get

Substituting *G _{m}* =

Multiplying through, side by side, gives

which is readily solved for *T* to give the important relation

Interestingly, *T _{oc} *and

I am a fan of current-feedback amplifiers (CFAs), so I will use PSpice to apply Rosenstark’s method to the CFA discussed in a previous blog [5]. **Figure 4** shows the pair of ac models needed to measure *T _{oc}* and

**Figure 4** Applying Rosenstark method to the ac model of a current-feedback amplifier (CFA).

We observe that *T* is very close to *T _{sc}*, indicating that feedback at the input is in this case predominantly of the current type (hence the reason for the name of this amplifier type). To gain quantitative insight, we use Equation (3) to derive the following important relationship

In our circuit we have *Z*_{1} = *r _{n}* = 25 Ω, and

**Figure 5** Plots of T* _{oc}*, T

*T* with just the short-circuit measurement. By dual reasoning, breaking the loop at a point where |*Z*_{2}| << |*Z*_{1}| may provide an adequate estimate for *T* with just the open-circuit measurement. In our CFA example, the point where *r _{o}* joins

[Continue reading on EDN US: Do not disturb dc bias conditions]

*Sergio Franco is an author and (now emeritus) university professor.*