Feedback Circuit Misconceptions
Typical circuits textbooks tell us that there are four basic feedback topologies. The impression is given (if not stated) that any feedback circuit will conform to one of these four topologies. One of the tasks of feedback analysis, it would seem then, is to identify to which of the four basic topologies a given feedback circuit conforms. But this is misleading. This article attempts to clarify the resulting confusion.
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, shown below in block diagram (a) and signal flow-path (b) representations. (Generalized electrical quantities - voltages or currents - are designated here as "x" quantities, where x is either v or i.)
Actual circuits can also have additional associated transmittances. The more general feedback topology is shown below. Ti and To 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.
Block diagrams do not represent circuit interconnections (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, xf = G× xE. The summing block, S, adds its inputs according to the sign by the arrowhead. This block diagram is a graphic way of expressing the following algebraic equations:
The first two equations describe the feedback loop itself. The loop is closed and consists of G, H, and S. Solving for the overall closed-loop gain of the feedback amplifier, T = xo/xi, and is:
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, S , (and E, the error quantity) from the circuit diagram, and the pickoff circuit (and xf). Simplifying this task brings us to the "four topologies" of textbooks. (A circuit topology is essentially its schematic diagram, showing its structure through interconnection of its components.) The four topologies are all combinations of current or voltage pickoff and error summing. As such, they are four possible topological characteristics of the feedback circuit. The usual names given to the four are given in the following table.
What is misleading in too many circuits textbooks 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 comparison for its summing circuit, and either node or loop sampling for its pickoff circuit. The key insight here is that it is not the circuit but the analyzer of it that determines whether voltages or currents are compared and sampled!
If you have been thinking that the circuit itself (and not you) determines the electrical quantity summed or picked off, then you would be looking for some characteristic of the circuit topology to determine which quantity this is. But you choose the quantities for summing and pickoff and then analyze the circuit accordingly. The choices are arbitrary, though some choices lead to a simpler analysis than others, as we will see. Only simple circuits with ideal sources constrain the choice to only one quantity because the other choice leads to a circuit degeneracy.
Choosing Pickoff Quantities
Starting with the pickoff circuit, what is essential is that an electrical quantity (current or voltage), xf, be identified as the pickoff quantity, xf, to be fed back. The choice of xf is constrained to be the output quantity of G and the input quantity of H. The first step in the analysis is to choose xf. 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.
But the circuit is not the block diagram, and it may not be obvious from the circuit which quantity to choose. To see what might happen, consider the block diagram below. Suppose you choose a quantity that is inside G. G has been decomposed into cascaded blocks GA and GB. The additional output block, To, is included between the pickoff point and the circuit output quantity, xo, when the chosen pickoff quantity is not the output quantity.
If xf is chosen too much toward the input within the forward path, G, then common factors appear in the algebraic expressions for H and To. As shown, G = GA× GB, where xf is chosen as the output of GA instead of GB. This results in the following feedback equations:
Note that GB is a common factor of both the H term of xE and To in xo. By letting xf be the output of GB instead, GB appears as a factor in the first equation and disappears from the others.
If xf is instead chosen too close to the output, so that To = ToA×ToB and xf is the output of ToA, then:
In this case, introducing factor ToA into the third equation removes it from the first two.
The optimal pickoff quantity is the one which minimizes common transmittances in these equations. However, it is not necessary to choose, or even identify, such a pickoff quantity. Let your intuition pick a quantity which seems most appropriate. It becomes the input quantity to H and To, if not xo (in which case To = 1).
Suppose you choose xf to be a voltage. Voltages occur at nodes. Consequently, xf is identified with a circuit node from which a connection to the feedback-path (H) input is made. For a current, a pickoff 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 let's apply the same kind of reasoning to the summing circuit. If xE is chosen too close to the output, common factors occur in the two terms of xE. Let xE be the input to GB. Then:
By letting G = GA×GB, GA becomes a factor in the first equation and is eliminated from the second.
The other case is that of choosing xE too close to the input, as the input of TiB. Then TiB appears as a common factor with G and in the error term containing H.
By moving xE to the output of TiB, TiB is eliminated from the first equation and H term of xE and becomes a factor in the first xE term so that Ti = TiA× TiB.
YouChoose Pickoff and Error Quantities
The form of input-network topology (series or shunt) is not generally determined by the circuit. But the choice of error quantity xE affects the choice of input topology. This can be seen from the following input network.
In this circuit, the output of the H block is represented by a generalized source (either Thevenin or Norton equivalent) consisting of transmittance x(xf) and source resistance RHo. The feedback-circuit input is a voltage source in series with an input resistance across which is voltage v1.
If v1 is chosen as vE, the H-path port is made a Thevenin circuit and the input forms a loop - a series topology. If v2 is chosen for vE instead, then converting the input and feedback ports to Norton equivalent circuits results in a common node with voltage vE - a shunt topology. The feedback circuit input topology is determined by choice of error quantity, and not by the circuit.
The same kind of argument applies to pickoff circuits. Once xf is chosen, then either a loop (for if) or node (for vf) as the pickoff circuit results. Often, either choice can lead to a successful analysis.
The key steps in analyzing a feedback circuit are to choose pickoff 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 H (and Ti and To). But do not be confused into thinking that your choices of xE and xf are determined by the circuit topology. Which of the four topologies results in your analysis depends on your choice of error and pickoff quantities, and not the circuit itself.
ÓDennis L. Feucht, 2001