Switching-Mode Power Supply Entrainment (6 of 7)

References

The goal of the reference section is to present the one best reference for follow-up study. It is rarely achieved. Here we have four references. If you have some that you think are better replacements for these or an important addition, let me know. Here are the reasons I included them.

• The book by Minorsky gives a strong mathematical background to the subject. His writings on the subject are considered classics and were my guide when I first experienced the problem.
• The book by Hayashi was suggested by Rudy Severns and after buying a copy and reviewing it, I agree. I especially like the figures in the book.
• The patent by Beihl is included because it was the solution to the problem I discussed in my personal anecdote.
• The paper by Weischedel is an early paper on the subject recommended by a reader. It includes 14 references on the subject up to that time, including the Beihl patent, giving a feel for the development up to the publication date. It includes two papers from the Duke University power electronics program under Professor T. G. Wilson, whose Ph. D. candidates continued to write relevant papers on the non-linearities of switching-mode power supply circuits over the years.
• The Seidel paper is included as an example of the current interest in entrainment in a wide number of specialties.

Minorsky, Nicholas, Nonlinear Oscillations, D. Van Nostrand Company, Inc., Princeton, New Jersey, ©1962

Chapter 18, Synchronization: "The phenomenon of synchronization or "entrainment of frequency" was the first to be studied among other non-linear phenomena. Apparently, it was observed for the first time by Huygens (1629-1695) who reported that two clocks, which were slightly "out of step" with each other when hung on a wall, became synchronized when fixed on a thin wooden board." ... Summing up [last paragraph-JF], in all such schemes, whatever their nature may be, the origin of synchronization phenomenon lies in the existence of a point of stable equilibrium when the frequencies coalesce. This means that the phenomenon in such a case has a natural tendency to approach this point of coalescence of the two frequencies.

Chapter 23, Interaction of Nonlinear Oscillations: "The essential property of nonlinear oscillatory systems, on the contrary, is that they do not exhibit any superposition of component oscillations or, more specifically, that they exhibit always an interaction of some kind between these component oscillations."

Hayashi, Chihiro, Nonlinear Oscillations in Physical Systems, Princeton University Press, Princeton, New Jersey, 1964.

"The purpose of this book is to provide engineers and scientists with fundamental knowledge concerning the important subject of nonlinear oscillation i physical systems."

Beihl, Paul A., James R. Gandy, and Marvin W. Loosle, Voltage Regulator, Patent number: 3305767, Filing date: Sep 10, 1963, Issue date: Feb 1967

This patent is for a fixed frequency PWM voltage regulated power supply designed to overcome short comings of non-fixed frequency regulators entraining to load frequency and general noise. The following quotation from the patent discusses the entrainment problem, although the word entrainment is not used.

"In some prior art PWM voltage regulators, the switch is operated in response to the amplified ripple portion of the output voltage, However, it is difficult to maintain the switching frequency constant in such a regulator due to variations in gain of the amplifying feedback circuit. Moreover, for an A.C. load, the operation of the switch tends to be at the same frequency as the operation of the load. ... In other prior PWM voltage regulators, the timing period of each cycle is varied in response to variations in the output voltages through a voltage-sensitive circuit, such as a Schmitt circuit. However, such a feedback arrangement for voltage regulation has been found to be unsatisfactory for most applications because a voltage sensitive circuit of that type is unduly sensitive to transients or noise."

Weischedel, Herbert R. An Application of Frequency Entrainment in a DC-to-DC Converter, IEEE Transactions on Industry Applications, Vol. IA-8, No. 4, July/August 1972. pp .437-442.

Author Abstract: Inherent simplicity and high efficiency combined with an excellent dynamic behavior make the use of self-oscillating switching circuits an attractive alternative in the design of electronic power conditioning equipment. Unfortunately, these free-running circuits often suffer from large variations in operating frequency under varying line and load conditions, a deficiency which seriously limits their usefulness for many applications. A general approach is described by which this drawback can be remedied without impairing the aforementioned advantages.

In order to demonstrate the method in an exemplary fashion, a self-oscillating switching voltage regulator (a dc-to-dc converter) is presented whose frequency of operation is determined by an external periodic forcing function. Operation of the circuit is based on a principle of nonlinear oscillation theory commonly known as "synchronization phenomenon" or "frequency entrainment". Utilization of the synchronization phenomenon for the design of power conditioning equipment here is believed to be new.

Comment: Synchronization is often used to synchronize several converters to each other and a count-down frequency from the system clock. In this case, the synchronization signal pulls in a converter runner at a slightly lower frequency to the desired higher frequency. In Weischedel's paper, the free running converter is at a much higher frequency and is entrained to a lower frequency. Both higher and lower frequencies can entrain.

The reference below is not related to electronics but illustrates entrainment in another field, medicine.

Seidel, H., and H. Herzel, Analyzing entrainment of heartbeat and respiration with surrogates, Engineering in Medicine and Biology Magazine, IEEE Publication Date: Nov/Dec 1998, Volume: 17, Issue: 6. pages 54-57

Entrainment between two rhythms is a very well-known phenomenon in the theory of nonlinear dynamics. Although heart and respiration influence each other by several mechanisms, and though modulation of heart rate by respiration is a very well-known and investigated phenomenon, there are surprisingly few indications of true entrainment between the two rhythms. This absence might be due to an insufficient coupling strength or to disturbances by other physiological rhythms. Nonetheless, we sometimes observe intermittent phases where cardiac and respiratory rhythms run in parallel; i.e., where both rhythms seem to be entrained. However, it is not obvious how to decide whether this effect is true entrainment or whether it is just quasi-entrainment that occurs when two rhythms have an approximate frequency ratio of for example, 4:1. Therefore, we use a surrogate-data technique to determine the probability of quasi-entrainment as a function of its duration. This probability can be used to obtain a significance level for true entrainment