Feedback Circuits and Two-Port Blocks, Part 1: From Circuits to Blocks with Ports introduced the concept of the port and applied it to a simple example. In this third part, a feedback circuit example is used to illustrate how ports simplify feedback analysis.

The following circuit is a shunt-feedback BJT amplifier, shown with the BJT replaced by a T-model equivalent.

The choice of output quantity is simple: *v _{o} * . The error quantity is chosen to be

*i*, the BJT base current.

_{b}This decomposition is shown by a flow graph, equivalent to a block diagram representation. The small circular nodes are summing points for arrows coming in to them. Consequently, flow graphs differ from block diagrams in that the output quantities at the nodes are block-diagram summer outputs for multiple inputs to the node. We will stay with the more familiar block diagrams, though flow graphs are found within some electronics subcultures.

The circuit is now decomposed by making the feedback path explicit with a two-port block. *H* is now identified by the two-port equivalent circuit, as shown below.

The *H* block is constructed as follows. By choosing *i _{b} * to be the error quantity, which is the input quantity to the

*G*input port, then as the output of the summing block, it must be the difference (or sum with negated output from

*H*) of two currents. One is

*i*or some fraction of it, which is

_{i}*T*x

_{i}*i*, and the other is the feedback current,

_{i}*i*, shown flowing in the negative direction for the output port of

_{B}*H*. To sum currents at a single node to produce the error current (by Kirchhoff’s Current Law, KCL) we choose the output source of

*H*to be a Norton source, as shown.

The Norton source is *i _{out} * =

*v*/

_{o}*R*. Its value is found by first noting that the input quantity at the input port of

_{f}*H*is

*v*, the feedback quantity. Then the output source dependency must be

_{o}*i*(

_{out}*v*). To find the value of

_{o}*i*(

_{out}*v*), the output port of

_{o}*H*is shorted so that the short-circuit current,

*i*, is

_{out}*i*. With

_{B}*v*shorted, it is

_{b}*v*/

_{o}*R*. Then the output-port resistance is found by nulling

_{f}*i*(

_{out}*v*) by nulling

_{o}*v*by shorting its node. Looking back into the

_{o}*H*output port circuit, the resistance is

*R*.

_{f}For the input port of *H* , either a Thevenin or Norton source would suffice; we will choose a Thevenin source. Then its voltage is the open-circuit voltage of the port. Looking into the port, we see *v _{b} * when

*R*is disconnected from the

_{f}*v*node.

_{o}*v*is the voltage across the output port with it open-circuited and thus qualifies as the Thevenin dependent source voltage. To find the

_{b}*H*input-port resistance, the input source must be nulled by setting

*v*to zero. This is accomplished by shorting the

_{b}*H*output node, the BJT base, to ground. Then looking into the

*H*input port from

*v*, we see a resistance of

_{o}*R*. The

_{f}*H*two-port block has been constructed.

Ti is an input current divider:

With *H* and *T _{i} * identified,

*G*is easier to determine.

*G*has two paths, an active

*G*path and a passive

_{A}*G*path. The active path is through the BJT; the passive path is through

_{P}*R*, and is the reverse path through

_{f}*H*. By including the passive path in the analysis, we use the source on the input port of

*H*so that the reverse path can be realized through it. Then

The first term is *i _{b} * , multiplied by

*β*to become

*i*. This current develops

_{c}*v*across the collector resistance

_{o}*R*||

_{f}*R*and is negative. By convention, the loading of the

_{L}*G*output by the

*H*input port resistance is included in calculating

*G*. This also applies to the

*H*output-port loading of the

*G*input and is included in

*G*. However, this convention need not necessarily be observed. All that is really required is consistency. If you calculate

*G*with

*H*loading its output, then do not include the output resistance of

*G*when calculating the transmittance of

*H*; the loading has already been taken into account in

*G*and the (loaded)

*G*output variable is the actual (loaded) voltage and is the input to

*H*.

The second term is the passive path from the base through *R _{f} * to the collector. The

*H*input source, on the right side of the block diagram for

*H*, is

*v*. Because

_{b}*E*is

*i*, then

_{b}*i*x

_{b}*r*=

_{π}*v*, is divided by

_{b}*R*and

_{f}*R*.

_{L}*v*is the

_{b}*R*and

_{f}*R*divider input voltage as already loaded by

_{L}*r*. The second term,

_{π}*G*, is thus (

_{P}*v*/

_{b}*i*) x (

_{b}*v*/

_{o}*v*).

_{b}For the feedback path,

By substituting *T _{i} * ,

*G*,

*H*, and

*T*= 1 into the feedback formula,

_{o}the closed-loop transfer function is obtained. If *R _{L} * <<

*R*, then

_{f}*G*≅ 0 and

_{P}*G*≈

*G*. Then for

_{A}*β*→ ∞,

This simple formula is the approximate closed-loop transresistance for the single-BJT shunt-feedback amplifier for implementations for which *R _{f} * is large. Error is often attributable to finite

*β*, causing

*r*to excessively shunt current fed back through

_{π}*R*.

_{f}Hopefully, this feedback circuit example convinced you of the utility of two-port analysis for feedback circuits. The port is the transitional construct from circuit to block diagram. Once a given block is defined by identifying port quantities, then two-port construction can proceed.

Additional examples using two-port blocks are worked out, along with a more detailed explanation of feedback analysis procedure, in *Designing Amplifier Circuits* by D. Feucht. Only quasistatic (low-frequency ac) analysis has been applied to keep from complicating the topological aspect of feedback analysis with dynamic concerns. In the succeeding book of the analog design series, *Designing Dynamic Circuit Response* , capacitors and inductors also appear. However, by then, you will have developed circuit-to-block transformation skills and find that they readily extend to reactive circuits. Two-port block construction also remains central in the analysis of frequency-dependent circuits.

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