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Designing Stable Control Loops
The objective of this topic is to provide the designer with a practical review of loop compensation techniques applied to switching power supply feedback control. A top-down system approach is taken starting with basic feedback control concepts and leading to step-by-step design procedures, initially applied to a simple buck regulator and then expanded to other topologies and control algorithms. Sample designs are demonstrated with Math cad simulations to illustrate gain and phase margins and their impact on performance analysis. I. INTRODUCTION
Insuring stability of a proposed power supply solution is often one of the more challenging aspects of the design process. Nothing is more disconcerting than to have your lovingly crafted breadboard break into wild oscillations just as its being demonstrated to the boss or customer, but insuring against this unfortunate event takes some analysis which many designers view as formidable. Paths taken by design engineers often emphasize either cut-and-try empirical testing in the laboratory or computer simulations looking for numerical solutions based on complex mathematical models. While both of these approach a basic understanding of feedback theory will usually allow the definition of an acceptable compensation network with a minimum of computational effort. II. STABILITY DEFINED
Fig. 1. Definition of stability
Fig. 1 gives a quick illustration of at least one definition of stability. In its simplest terms, a system is stable if, when subjected to a perturbation from some source, its response to that perturbation eventually dies out. Note that in any practical system, instability cannot result in a completely unbounded response as the system will either reach a saturation level – or fail. Oscillation in a switching regulator can, at most, vary the duty cycle between zero and 100% and while that may not prevent failure, it wills ultimate limit the response of an unstable system. Another way of visualizing stability is shown in Fig. 2. While this graphically illustrates the concept of system stability, it also points out that we must make a further distinction between large-signal and small-signal stability. While small-signal stability is an important and necessary criterion, a system could satisfy thisrt quirement and yet still become unstable with a large-signal perturbation. It is important that designers remember that all the gain and phase calculations we might perform are only to insure small-signal stability. These calculations are based upon – and only applicable to – linear systems, and a switching regulator is – by definition – a non-linear system. We solve this conundrum by performing our analysis using small-signal perturbations around a large-signal operating point, a distinction which will be further clarified in our design procedure discussion。
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Fig. 2. Large-signal vs. small-signal stability
III. FEEDBACK CONTROL PRINCIPLES
Where an uncontrolled source of voltage (or current, or power) is applied to the input of our system with the expectation that the voltage (or current, or power) at the output will be very well controlled. The basis of our control is some form of reference, and any deviation between the output and the reference becomes an error. In a feedback-controlled system, negative feedback is used to reduce this error to an acceptable value –as close to zero as we want to spend the effort to achieve. Typically, however, we also want to reduce the error quickly, but inherent with feedback control is the tradeoff between system response and system stability. The more responsive the feedback network is, the greater becomes the risk of instability. At this point we should also mention that there is another method of control – feedforward.With feed forward control, a control signal is developed directly in response to an input variation or perturbation. Feed forward is less accurate than feedback since output sensing is not involved, however, there is no delay waiting for an output error signal to be developed, andfeedforward control cannot cause instability. It should be clear that feed forward control will typically not be adequate as the only control method for a voltage regulator, but it is often used together with feedback to improve a regulator’s response to dynamic input variations.
The basis for feedback control is illustrated with the flow diagram of Fig. 3 where the goal is for the output to follow the reference predictably and for the effects of external perturbations, such as input voltage variations, to be reduced to tolerable levels at the output Without feedback, the reference-to-output transfer function y/u is equal to G, and we can
With the addition of feedback (actually the subtraction of the feedback signal)
and the reference-to-output transfer function becomes y/u=G/1+GH
If we assume that GH __ 1, then the overall transfer function simplifies to y/u=1/H
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Fig. 3. Flow graph of feedback control
Not only is this result now independent of G,it is also independent of all the parameters of
the system which might impact G (supply voltage, temperature, component tolerances, etc.) and is determined instead solely by the feedback network H (and, of course, by the reference).Note that the accuracy of H (usually resistor tolerances) and in the summing circuit (error amplifier offset voltage) will still contribute to an output error. In practice, the feedback control system, as modeled in Fig. 4, is designed so thatG __ H and GH __ 1 over as wide a frequency range as possible without incurring instability. We can make a further refinement to our generalized power regulator with the block diagram shown in Fig. 5. Here we have separated the power system into two blocks – the power section and the control circuitry. The power section handles the load current and is typically large, heavy, and subject to wide temperature fluctuations. Its switching functions are by definition, large-signal phenomenon, normally simulated in most stability analyses as just a two states witch with a duty cycle. The output filter is also considered as a part of the power section but can be considered as a linear block.
Fig. 4. The general power regulator
IV. THE BUCK CONVERTER
The simplest form of the above general power regulator is the buck – or step down – topology whose power stage is shown in Fig. 6. In this configuration, a DC input voltage is switched at some repetitive rate as it is applied to an output filter. The filter averages the duty
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