Feedback Circuits and Two-Port Blocks, Part 1: From Circuits to Blocks with Ports

What is hard about analyzing feedback circuits is the extraction from the circuit diagram of the simple block diagram found in active circuits and control textbooks. To start with a relatively easy example, the noninverting op-amp circuit is shown below.

The block diagram consists of the op-amp itself as the summer, Σ , (as its differential input) and forward path, G . The resistive voltage divider is the feedback path, H . The error summing block appears in the circuit from Kirchhoff’s Voltage Law (KVL) and consists of the sum of the voltages around the input loop: vi minus the divided output voltage, H x vo , is the error voltage. G amplifies the error voltage and outputs vo . The fed-back quantity is also vo .

In this example, the op-amp itself is a kind of block with input summer. In conventional feedback form, the summer is a subtractor, subtracting the fed-back quantity from the input quantity. A differential-input amplifier entails both G and summer functions and is almost a block of a diagram in itself. The notation used in the above block diagram is that of control-theory textbooks, where G is the forward path and H the feedback path. Many active-circuits textbooks continue the tradition of calling G A and H β. However, this is confusing once BJT circuit analysis is undertaken because of the β parameter of BJTs. For serious treatment of feedback, the control books are better and their notation does not cause this kind of symbolic ambiguity. I recommend using the control notation; eventually the circuits textbook writers will read enough control textbooks to also see the advantage.

Academic Aside

The distinctive contribution that active-circuits textbooks can make to an understanding of feedback is not the control analysis but how to abstract from a circuit diagram the above kind of block diagram that the control theory books begin with in analysis of dynamic response. This is the hard part of analyzing feedback circuits once you know control theory. Thus it seems to me that electronics could be better taught by offering courses in active circuits and control theory concurrently, recognizing that control theory is important enough to be a core course in the undergraduate analog electronics curriculum. In it, dynamic circuit response is covered implicitly while the active-circuits course merely needs to show how to formulate from a circuit the block diagram and derive the transfer functions of the blocks. This would leave more room for inclusion of circuits topics that are routinely omitted, such as translinear circuits and the effects of high-frequency impedance gyrations of active circuits above their bandwidths. Not only does control theory cover the dynamics of feedback, it also presents in its digital aspects various digital signal processing (DSP) concepts from a control viewpoint. These are often more useful for circuit design than from the filter-oriented viewpoint that is often found in DSP courses. Also useful for E.E. undergraduates to take is numerical analysis – the same concepts from the viewpoint of mathematics apart from specific applications.

The Two-Port Concept: Between Circuit and Block Diagrams

So how do we go from circuit to block diagram? In this article I do not intend to fully answer the question (as I try to in my book Designing Amplifier Circuits at SciTechPubsXS ) but to instead zoom in on a key aspect of the circuit-to-block conversion, two-port networks . To those who have been schooled in the mainstream E.E. view of electronics, “two-port networks” brings to mind pre-defined two-port blocks based on y , z , s , and especially h or “hybrid” parameters that appear in old behavioral models of BJTs. If you have that sinking feeling that because you cannot remember all those formulas for deriving two-port parameters that you will not gain an intuitive understanding of feedback-circuit abstraction, take heart; you do not need them. Anything you must know primarily by rote memorization is probably working against your ability to intuitively grasp higher-level concepts that obviate the need for rote memorization. The basic concept of the two-port block is simple to intuitively grasp and should be part of the circuit analyzer’s mental tool kit, feedback or otherwise. We will not use it to characterize devices but to modularize circuits, thereby performing a kind of problem reduction on them.

The most general two-port circuit is shown below. A port is a pair of terminals with defined polarities for voltage and current, as shown. Positive current flows into the + voltage terminal, making the resistance looking into the port positive. The same magnitude of current flows out of the negative terminal. A circuit or network with two ports, irrespective of what connects them inside, is a two-port block of circuitry.

To give this black box some internal definition, the most general circuit for a two-port network is shown below. The new symbol is a generalized source: either a Thevenin or Norton equivalent circuit. Because Thevenin and Norton circuits are interchangeable, we will use whichever is to our advantage in feedback analysis.

One constraint is applied to the relationship between the ports that modularizes the network as a self-contained block. The dependency of the sources for the two ports must be a quantity of either port. If the output source is a voltage, vout , (and the output equivalent circuit is Thevenized) let vout be dependent on one of the input port quantities, either vi or ii . In general, let v or i be represented by x . Then the output voltage source must be a function of an input quantity, or vout (x i ). If the output is a Norton circuit instead, with a dependent current source, then iout = iout (xi). More generally, whatever the output source, xout , make it dependent upon x i : xout (x i ).

The same is true for the input port. Input source x in is a function of (that is, is dependent on) x o , or is x in (x o ), where xo is a terminal quantity of the output port. This additional constraint makes the block modular because the source dependencies within it only depend on its own two-port quantities. This is an implicit assumption about the blocks in a block diagram. If it is not true, additional inputs to blocks are required to show external dependencies. Thus we now have a general, modularized two-port block, as shown below.

This block is the key to the transition between circuit and block diagram. It is not uncommonly the optimum tool for trying to unravel feedback circuits into forward and feedback paths for analysis.

In the context of feedback, the generalized feedback circuit can be diagrammed using two-port blocks.

The upper-center block is the forward-path G block and the lower is the H block representing the circuitry of the feedback path. The G block is a two-port network, although the input side lacks a source; rarely does significant feedback occur through the active path of the circuit and the input-side source can be omitted. The input and output quantities are connected to these two two-port blocks through input and output three-port networks, where analysis often becomes confusing.

What contributes to the confusion is the existence of additional circuit blocks not accounted for in the classic feedback loop block diagram at input or output. A fuller block diagram that should always be assumed (and then simplified by setting input or output blocks to 1 as appropriate) is shown below.

Neither Ti nor To are part of the feedback loop but are often needed to represent the given feedback circuit. To illustrate this, consider the other op-amp configuration, the inverting op-amp, as shown below.

The Ti block represents the input divider from vi to E = v , the op-amp inverting-input voltage, chosen to be the error quantity. Therefore, the more complete block diagram should be assumed when analyzing feedback circuits.

Two-Port Construction Example

To illustrate two-port block development, consider the following simple circuit, a voltage divider with two paths for the upper resistance.

By straightforward circuit analysis, the parallel upper resistors are combined and have a value of R 1 ||R 2 . Then applying the voltage-divider formula,

Now consider the parallel paths of the upper resistors as two separate paths of a block diagram, and that the circuit is to be decomposed into two two-port circuits. We can do this as follows. First, the input voltage source can be duplicated in parallel with itself (by the voltage-source shifting theorem : two identical voltage sources in parallel are equivalent to one voltage source), leading to the following equivalent circuit.

Each of the resistor branches can now be replaced by an equivalent two-port circuit, simplified by omitting the input-side source because it is not needed to find the voltage transfer function.

The output sources are Thevenin equivalent circuits. To find the Thevenin source and resistance values, the source is found when the output port is opened by disconnecting the resistor from the output. When the output port is opened at R 1 , the open-circuit voltage at the output port is simply that of the input-dependent output source, vi . The same applies for the R 2 path. The Thevenin resistance can be found by inspecting the output port resistance. With the input port source shorted, it is simply R 1 and R 2 , as shown in the above diagram.

What has been accomplished? Although the output ports of the two blocks are identical to the original circuit shown previously, the block decomposition (though trivial) has been achieved. (If the decomposition itself seems superfluous, see the following feedback circuit example to illustrate otherwise.) Now the contributions of each block can be found by superposition.

For the upper block, null the source of the lower block. To null a source is to make its output zero. If it is an independent source, it can be shorted (for a voltage source) or opened (for a current source). However, if it is a dependent source, it must be nulled by making its independent variable such that it is zero. For instance, current-dependent voltage source v (i ) is nulled by making v a value that makes v (i ) = 0 V. To null a source, the other port (upon which it is dependent) is either shorted if the dependency is a voltage or opened if a current.

For the example circuit, the lower block is nulled to find the contribution to the output by the upper block by setting vi of the lower block to zero to find vo . For the upper block, the divider formed is

and for the lower block, with the upper vi nulled,

The total transfer function is the sum of the contributions from each block, or

This can be reduced by applying the useful loaded-divider reversal formula,

The result is the same as for the simple divider-formula derivation. The two-port derivation is much harder and seems to have accomplished nothing useful. It is applied in the third part of this article to a feedback circuit where its use is not trivial and simplifies rather than complicates the analysis.

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