Typical circuits textbooks tell us that there are four basic feedback topologies. The impression is given (if not stated) that a given feedback circuit will conform to one of these four topologies. This is misleading. One of the tasks of feedback analysis, it would seem, is to identify to which of the four basic topologies a given feedback circuit conforms. This article attempts to clarify and correct this misconception.

**Feedback-Circuit Topologies**

The basic goal of feedback-circuit analysis is to take the circuit diagram, which is sometimes a spaghetti-like connection of components, and adduce from it the transmittances (or *gains or transfer functions* ) of the blocks *G* and *H* of the classical feedback topology from Feedback Circuits and Two-Port Blocks, Part 1: From Circuits to Blocks with Ports, repeated slightly differently below in block diagram (a) and signal flow-path (b) representations. (Voltages or currents are designated here as *x* quantities, where *x* is either *v* or *i* .)

As we saw in Feedback Circuits and Two-Port Blocks, Part 1: From Circuits to Blocks with Ports, actual circuits can also have additional associated transmittances. The more general feedback topology is shown below. *T _{i} * and

*T*are outside the feedback loop but are included because they commonly occur with feedback circuits. Sometimes it is not obvious from a circuit diagram that such blocks should be included.

_{o}Block diagrams do *not* represent circuit elements and interconnections (which is the *circuit structure* or *topology* ) but instead describe the flow of electrical cause and effect. Each block has an input (cause) and an output (effect) which is an electrical quantity. The arrows represent causal constraints, pointing from output to input. The input multiplied by the *transmittance* written in the block is the output. For example, *x _{f} * =

*G*x

*x*. The summing block, ÎŁ , adds its inputs according to the sign by the arrowhead. This block diagram is a graphic way of expressing the following algebraic equations:

_{E}The first two equations describe the feedback loop itself. The loop is closed and consists of *G* , *H* , and ÎŁ. Solving for the overall *closed-loop* gain of the feedback amplifier,

The middle factor in parentheses is the gain of the closed feedback loop itself. Given a circuit diagram, if the corresponding block transmittances can be found, the closed-loop feedback gain can be calculated from the above general expression. Circuits are usually not obviously decomposable into the block transmittances. What is needed is a procedure that derives the blocks from feedback circuits in equivalent circuit form so that circuit analysis can then be used to determine their transmittances.

**From Circuits to Block Diagrams**

What is usually hardest in going from circuit to block diagram is to identify the summing block, ÎŁ, (and *E* , the error quantity) from the circuit diagram, and the feedback circuit (and *x _{f} * ). Simplifying this task brings us to the â€śfour topologiesâ€ť of circuits textbooks. (A circuit

*topology*is its structure, through interconnection of its components. The word is usually used to refer to generalized circuits of a given structure.) The four topologies result from all combinations of current or voltage feedback and error summing. There are four possible error and feedback quantities of feedback circuits, given in the table.

What is misleading is the notion that a given circuit has one of these four topologies. In other words, it is conveyed that any given feedback circuit is constrained, for example, to have either a series or shunt circuit for its summing circuit, and either node or loop feedback for its feedback variable. The key insight here is that

*It is not the circuit but the analyzer of it that determines whether voltages or currents are summed and fed back! *

If you have been thinking that the circuit itself (and not you) determines the electrical quantity summed or fed back, then you would be looking for some characteristic of the circuit to determine which quantity this is. But you choose the quantities for summing and feedback and then analyze the circuit accordingly. The choices are arbitrary, though some choices lead to a simpler analysis than others. Only simple circuits with ideal sources constrain the choice to only one quantity because the other choice leads to a circuit degeneracy.

**Choosing Feedback Quantities**

Starting with the feedback circuit, what is essential is that an electrical quantity (current or voltage), *x _{f} * , be identified as the quantity,

*x*, to be fed back. The choice of

_{f}*x*is constrained to be the output quantity of

_{f}*G*and the input quantity of

*H*. The first step in the analysis is to choose

*x*. There is often more than one choice that will result in a correct analysis, but also, one choice many times leads to the simplest analysis.

_{f}Because the circuit is not the block diagram, it may not be obvious from the circuit which quantity to choose. To see what might happen, consider the block diagram below. *G* has been decomposed into cascaded blocks *G _{A} * and

*G*. Suppose you choose a quantity that is inside

_{B}*G*. The additional output block,

*T*, is included between the feedback and circuit output quantity,

_{o}*x*, when the chosen feedback quantity is not the output quantity.

_{o}If *x _{f} * is chosen too much toward the input within the forward path,

*G*, then common factors appear in the algebraic expressions for

*H*and

*T*. As shown,

_{o}*G*=

*G*x

_{A}*G*, where

_{B}*x*is chosen as the output of

_{f}*G*instead of

_{A}*G*. This results in the following feedback equations:

_{B}Note that *G _{B} * is a common factor of both the

*H*term of

*x*and

_{E}*T*in

_{o}*x*. By letting

_{o}*x*be the output of

_{f}*G*, as

_{B}*x*instead,

_{f opt}*G*appears as a factor in the first equation and disappears from the others.

_{B}If *x _{f} * is instead chosen too close to the output, so that

*T*=

_{o}*T*x

_{oA}*T*and

_{oB}*x*is the output of

_{f}*T*, then

_{oA}In this case, introducing factor *T _{oA} * into the third equation removes it from the first two.

The optimal feedback quantity is the one which minimizes common transmittances in these equations. However, it is not necessary to choose, or even identify, such a feedback quantity. Let your intuition pick a quantity which seems most appropriate. It need only be a quantity of the *G* x *H* feedback loop. It becomes the input quantity to *H* and *T _{o} * , if not

*x*(in which case

_{o}*T*= 1).

_{o}Suppose you choose *x _{f} * to be a voltage. Voltages occur at nodes. Consequently,

*x*is identified with a circuit node from which a connection to the feedback-path (

_{f}*H*) input is made. For a current, a feedback loop must exist in which this current flows. A loop of

*G*generates this current, and it flows through circuitry comprising the input of H.

**Choosing Summing Quantities**

Now apply the same kind of reasoning to the summing circuit. If *x _{E} * is chosen too close to the output, common factors occur in the two terms of

*x*. Let

_{E}*x*be the input to

_{E}*G*. Then

_{B}By letting *G* = *G _{A} * x

*G*,

_{B}*G*becomes a factor in the first equation and is eliminated from the second.

_{A}The other case is that of choosing xE too close to the input, as the input of *T _{iB} * . Then

*T*appears as a common factor with

_{iB}*G*and in the error term containing H:

By moving xE to the output of *T _{iB} * ,

*T*is eliminated from the first equation, and the

_{iB}*H*term of

*x*and becomes a factor in the first

_{E}*x*term so that

_{E}*T*=

_{i}*T*ď€ x

_{iA}*T*.

_{iB}*You* Choose Feedback and Error Quantities

The form of input circuit (series or shunt) to be used in feedback loop analysis is *not* generally determined by the given circuit. However, your choice of error quantity *x _{E} * affects the choice of input circuit. This can be seen from the following input circuit.

In this circuit, the output of the *H* block is represented by a generalized source (either Thevenin or Norton equivalent) consisting of transmittance x(*x _{f} * ) and source resistance

*R*. The feedback-circuit input is a voltage source in series with an input resistance across which is voltage

_{Ho}*v*.

_{1}If *v _{1} * is chosen as

*v*, the

_{E}*H*-path port is made a Thevenin circuit and the input forms a loop ď€ a series topology. If

*v*is chosen for

_{2}*v*instead, then converting the input and feedback ports to Norton equivalent circuits results in a common node with voltage

_{E}*v*— a shunt topology. The feedback circuit input topology used for analysis is determined by the choice of error quantity, and not by the circuit.

_{E}The same kind of argument applies to the feedback circuits. Once *x _{f} * is chosen, then either a loop (for

*i*) or node (for

_{f}*v*) as the feedback circuit results. Often, either choice can lead to a successful analysis.

_{f}The key steps in analyzing a feedback circuit are to choose feedback and error quantities. With an awareness that input and output blocks may be required for some choices, consistent analysis will subsequently produce correct transmittance equations for *G* and <> (and *T _{i} * and

*T*). But do not be confused into thinking that your choices of

_{o}*x*and

_{E}*x*are determined by the circuit itself. Which of the four topologies results in your analysis depends on your choice of error and feedback quantities, and not the given circuit.

_{f}
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