# An efficient approach for modeling feedback systems

Whether you’re an analog, mixed-signal or system engineer, you probably remember “falling off a cliff” when you discovered that the analysis methods learned in college simply didn’t work in the real world. As a more polished engineer, you still find that conventional design involving feedback becomes very difficult to arm against things like complex transfer functions, circuit/system sensitivity and even intricacies of 2-port networks.

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.