An efficient approach for modeling feedback systems

There is help available. Design-Oriented Analysis (D-OA: don’t forget the hyphen!) is a paradigm based on the postulate that “design” is the reverse of “analysis.” With D-OA, the answer to an analysis forms the starting point for the design.

Let’s now turn the spotlight on a new approach to analysis and design of feedback systems, the “General Feedback Theorem” (GFT). The typical analysis procedure used by design, integration, and reliability engineers is to throw an entire circuit onto a breadboard, or into a simulator, to see how it survives, including attempts to measure the loop gain and/or real-world behavior. The design phase may consist of tweaking components and circuitry, guided by innumerable simulations, until final results are satisfactory.

A much more efficient approach is to begin with a simple circuit using device models, but postpone parasitic effects, which later will be added in sequentially. Even if you do little or no symbolic analysis, GFT circuit simulation tells you in what ways targeted elements affect the result.

When you finally substitute in device models, you’ll have a much better handle on how these components influence the design. Further, the procedure of D-OA by simulation with the incorporation of GFT models is significantly enhanced because the results for a feedback system are exact.

They are not impaired by approximations and assumptions inherent in a conventional single-loop model. Moreover, with the GFT, it may no longer be necessary to attempt hardware measurements of loop gain, which is a considerable saving of time and effort.

Shortcomings of the conventional approach

The well-established method of analyzing a feedback system begins with the familiar block diagram of Fig. 1, from which the feedback ratio K and the loop gain T=A K are calculated. The designer’s job is to set K and the forward gain A so that the final closed-loop gain H meets the specification, often with the help of circuit simulation software. Several design iterations, often aided by hardware measurements of the loop gain, may be needed before the closed-loop gain is satisfactory.

Unfortunately, this approach can give incorrect results. The block diagram of Fig. 1 demonstrates a conventional (incomplete) representation of the actual hardware system. Your immediate reaction may be, “any discrepancy between the predicted and actual results, is probably small enough to neglect.”

Possibly, but with some designs, such discrepancies can’t be ignored. In short, with the GFT, it’s easy to get the exact analysis results quickly and easily, to accurately predict system performance.

Let’s start by reviewing in more detail the conventional approach based on Fig. 1. Here, the closed-loop gain H (the “answer”) is given by

A “better” form is

where

It is convenient to define a discrepancy factor D as a unique function of T,

so that the closed-loop gain H can be expressed concisely as

Format (2) is better because H_∞ represents the specification. It’s the only known quantity at the outset, and K=1/H_∞ is designed to meet the specification. The only hard part is designing the loop gain T so that the actual closed-loop gain H meets the specification within the required tolerances. That is, the discrepancy factor D must be close enough to 1 over the specified bandwidth.

The form (2) or (6) is also better because it embodies one of the principles of D-OA, namely “get the quantities you want in the answer into the statement of the problem as early as possible.” In this case, H_∞ is the desired gain, and D is the discrepancy between H_∞ and the actual answer you’ll get.

Equally important, A is banished from the answer because it contributes only by way of T, and its own value is of no interest. Equations (5) and (6) expose the well-known and unique relationship between loop gain phase margin, and both the frequency and time domain responses of the closed-loop gain.

The model of Fig. 1 is incomplete because it does not account for bi-directional signal transmission in the boxes. If both boxes have reverse transmission, there is also a nonzero reverse loop gain, and it is convenient to lump together all the properties omitted from this block diagram under the label “nonidealities.” Consequently, all analysis based on this model ignores the nonidealities.

General Feedback Theorem unveiled

The GFT sweeps away all the a priori assumptions and approximations inherent in the conventional approach, and produces results directly in terms of the circuit elements. The formula is:

which is seen to be the same as (2) but with an extra term involving the null loop gain T_n. By analogy with T and D, a null discrepancy factor D_n can be defined:

The final result (7) can therefore also be written

The term D_n contains the first-order effects of the nonidealities, absent from the conventional result (6).

Although the first two factors in (9) appear to be the same as in (6), an essential feature is that they are calculated differently than by the conventional method of injecting a single test signal at an arbitrary point inside the loop.

In the GFT, the test signal is injected not only inside the major loop (although outside any minor loops), but also at the error signal summing point. In addition, since in general there is both an error voltage and an error current, both a test voltage and a test current need to be injected. This “dual test signal injection configuration” is depicted in Fig. 2.

All three factors H_∞, D, and D_n in the GFT of (9) are calculated subject to various sets of conditions imposed upon the injected test signals, and the input and output signals. Expressions are not displayed here because circuit examples are handled with Intusoft's ICAP/4 Spice simulator, which is equipped with GFT templates.

The GFT result of (7) or (9) is represented by the “augmented” block diagram of Fig. 3. Because superficially Fig. 3 differs from Fig. 1 only in the presence of an additional block containing the nonidealities, it is important to emphasize the fundamental difference between the conventional approach and that based on the GFT.

In the conventional approach, the block diagram of Fig. 1 is the starting point, in which reverse transmission in both boxes is ignored, and the result (2) is developed from Fig. 1.

In the GFT approach, Fig. 2 is the starting point, and the result (7) is developed directly from the complete circuit without any assumptions or approximations. Since (7) is represented by Fig. 3, the block diagram of Fig. 3 is part of the result. The boxes in Fig. 3 are unidirectional, and do not necessarily correspond to any separately identifiable parts of the circuit.

The values of these boxes, expressed in terms of H_∞, T, and T_n or H₀, automatically incorporate any nonidealities that may be present in the actual circuit. Although the nonidealities have first-order effects represented T_n or H₀, they may have second-order effects upon T, which in turn cause T and D to differ from the values obtained by the conventional approach.

Although the augmented block diagram of Fig. 3 exhibits a “loop,” it represents any linear system, even if there is not a physically discernible loop. An example is a Darlington Follower, for which the GFT affords a means of investigating the well-known potential instability.

Using circuit simulation with GFT templates

To use the ICAP/4 SPICE simulator rather than doing a symbolic analysis, simply choose an injection point inside the loop at the error summing point. Then, select the appropriate GFT icon. Its template invokes simulation runs to calculate H_∞, T, and T_n, and post-simulation calculations to produce the discrepancy factors D and D_n, and ultimately H.

A simple feedback amplifier example

A series-shunt feedback amplifier circuit model is shown in Fig. 4. The forward path is a simplified model of a typical IC. The voltage gain is achieved in the first two stages, which may be differential, and current gain is achieved in the final Darlington-Follower stage.

The frequency response is determined by the sole capacitance. Each active device is represented by a simple BJT T-model, which has the advantage of also representing an FET by setting the drain current equal to the source current.

Applying the GFT, the crucial first step is to choose the proper test-signal injection configuration. The error voltage is the voltage between the input and the fed-back voltage at the feedback divider tap point, and the error current is drawn from the feedback divider tap point, as indicated in Fig. 4.

To invoke the GFT Template, the appropriate dual-injection icon is selected to provide the test signals e_z and j_z. They’re connected so that v_y and i_y correspond to the error voltage and error current, and v_x and i_x correspond to the driving voltage and driving current to the forward path, as shown in Fig. 5.

The icon also provides the system input signal e_i. It also observes the output signal v₀ because it has to adjust the test signals relative to the input in order to establish the various required conditions for the simulations.

An objective of D-OA is to figure out as much as possible about the answer before plunging into the analysis. In our case, we expect H_∞ to be 1/K, the reciprocal of the feedback ratio that was initially chosen to meet the system specification. The injection configuration was specifically set up to achieve this. Here, 1/K = (R₁ + R₂)/R₁ = 10 or 20dB, flat at all frequencies.

We expect T to be large at low frequencies and to have a single pole determined by C_c . Consequently, D will be flat at essentially 0dB at low frequencies, with a pole at the crossover frequency of T, beyond which D will be the same as T. The GFT results are shown in Fig. 6, and the expectations are indeed borne out. Also shown is the final result for H, the closed-loop gain, whose bandwidth is determined by the T crossover frequency as in the conventional approach. Since the null discrepancy factor D_n is flat at 0dB, up to the much higher null loop gain crossover frequency, reverse transmission (“wrong way”) through the feedback path (the only nonideality present) does not have any significant effect upon H.

A more realistic feedback amplifier model

The more interesting and realistic model of Fig. 7 includes two added capacitances for each active device. Of course, even more complex simulation device models can be used. What are our expectations for the results, in comparison with those of Fig. 5?

Since all the extra elements are capacitances, we expect the low-frequency properties to remain the same, but the dominant pole and hence the loop-gain crossover frequency will be lowered. Therefore, to enable a more meaningful comparison between the two circuits, C_c in Fig. 7 has been reduced sufficiently to preserve the same loop-gain crossover frequency. Nevertheless, the extra capacitances create more poles and zeros, so the high-frequency loop gain is expected to be more complicated.

The major consequence of the presence of the extra capacitances is that there is now a second feedforward path (through a string of capacitances), in addition to one through the feedback path in the “wrong” direction, whereby the input signal can reach the output. Also, there is now nonzero reverse transmission through the forward path, which in turn creates a nonzero reverse loop gain. It is not necessary to separate these nonidealities, because luckily they are all automatically accounted for in the calculation of the loop gain T (usually little effect) and in the null loop gain T_n (the major effect).

The quantitative results of the GFT Template simulations for Fig. 7, shown in Fig. 8, bear out the expectations. The null-loop gain crossover frequency is drastically lowered. Even though the magnitude of the null discrepancy factor D_n is 0dB at both ends of the frequency range, its phase undergoes a huge lag that is transferred directly into a correspondingly huge phase lag in the final closed-loop gain H.

Although the magnitude of H at high frequencies may not be of much interest in the frequency domain, its corresponding phase has a controlling effect in the time domain. The step responses of Fig. 5 and Fig. 7 are shown in Fig. 9. The huge phase lag of H at high frequencies for the circuit of Fig. 7 causes the expected delay in the step response. However, the ensuing rise time is shorter than for Fig. 5, with the perhaps unexpected beneficial result that the final value is achieved sooner for Fig. 9 than for Fig. 7.

The bottom line is, the GFT of (9), whose principal difference from (6) of the conventional approach is the presence of the null discrepancy factor D_n, predicts a substantial modification of the closed-loop performance. The nonidealities represented by T_n or D_n are always present in a realistic model of an electronic feedback system, and at least in some respects can actually improve rather than degrade the performance. This rare exception to Murphy’s Law provides added incentive to utilize the GFT.

No need to measure loop gain directly

In the conventional approach, efforts are often made to measure loop gain on actual hardware in order to check that the phase margin is adequate, typically with little or no thought given to whether the measurement is consistent with the actual closed-loop gain. It is very awkward to inject even a single test signal into an IC, let alone dual or triple injection signals.

Fortunately, it is no longer necessary to measure loop gain directly. The GFT is exact with respect to the simulated model. All that’s required is to measure the final closed-loop gain H, which you would do anyway to check whether it meets the specification. If the simulated H differs from the measured H, you have to adjust the model until the two are the same. When this is achieved, you know the model is correct and the simulation tells you what T and T_n are individually. This is the culmination of the D-OA process.

Useful for all analog engineers

Analog, mixed-signal, and power supply design engineers are not the only ones to benefit from an ability to apply the powerful methods of D-OA. Those who review and verify designs of others, such as integration and reliability engineers, also need to know how design-oriented results should be presented. In fact, you can improve the effectiveness of the whole project by requiring that design engineers present their results according to the Principles of Design-Oriented Analysis.

For further information on the GFT, and Design-Oriented Analysis in general, see http://www.rdmiddlebrook.com. Intusoft maintains a GFT web page.

R. David Middlebrook is professor Emeritus of Electrical Engineering at the California Institute of Technology. Tim Ghazaleh is Marketing Director of Intusoft, Gardena, CA.