You have completed your design for a computer switching-mode power supply and everything checks out fine and it is installed in the computer. Then a programmer changes a software program in the computer and the computer shuts down because of out-of-tolerance voltage levels. If you are lucky, then your overvoltage protection circuits have prevented the wipe-out of every logic circuit in the computer. If your not lucky, then the wrath of your management, system engineers, and logic designers are about to descend on your serene world.
You may be the victim of entrainment. You did check your breadboard, prototype, and first production power supply for entrainment -- didn't you?
If you did not test your power supply for entrainment, read on and become convinced that it is something you should do.
So what is entrainment?
First, it is a nonlinear phenomenon, so don't look for any simple analytical fixes. This is what Nicholas Minorsky had to say in his classic book, Nonlinear Oscillations:
"The phenomenon of synchronization or "entrainment of frequency" was the first to be studied among other nonlinear 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. These effects were rediscovered more than two centuries later in electrical circuits by a number of physicists among whom one should mention Lord Raleigh, Vincent, Moller, Appleton, van der Pol, and others." [Chapter 18, Synchronization]
Anytime you see a list of names like this you know the mathematics are far from simple -- so we will avoid them and concentrate on a qualitative description. For the math lovers, Minorsky will keep you going for quite a while -- reading his references can expand it to a life-time.
For a description of entrainment, we will start by quoting Minorsky again.
"The synchronization effect can readily be observed in electronic circuits. If one applies to the grid of an electron-tube oscillator oscillating, say with a frequency of wo, an extraneous electromotive force of frequency w, one observes the "beats" of the two frequencies. If the frequency w approaches the frequency wo, the frequency of the beats decreases but this happens only up to a certain value of the difference |w - wo| after which the beats disappear suddenly and there remains only the frequency w. Everything happens as if the "free" (autoperiodic) frequency wo were "entrained" by the extraneous (heteroperiodic frequency w." [Chapter 18, Synchronization]
Now entrainment is not going to occur if there is no noise frequencies in your system, either due to a switching load (the most common stimulus), noise on the input power lines, or electromagnetic interference. However, since most practical power supply applications have these, you need to test for entrainment.
Switching-mode power supplies often have step changes in load, both positive and negative. If the step loads are periodic and the fundamental, harmonics, sub-harmonics, or beat frequencies approach the switching frequency, then the power-supply switching frequency can entrain or synchronize to the switching load and be pulled away from its natural or clocked frequency.
When entrainment or synchronization happens, sometimes the effects are minimal, but often they are dramatic. The output ripple on a 5 V logic supply can increase from 50 mV peak-to-peak to several volts peak-to-peak and over-stress or destroy the logic circuits if not protected by effective over-voltage protection. Short of this, the computer becomes useless due to excessive errors. And it can be fairly insidious, the frequency may just momentarily entrain and upset or overstress the logic with no trace that this happened or was the cause of the upset. Although the load is primarily the culprit, line noise and EMI can also cause entrainment and the increase in output ripple. Besides upsetting or overstressing the load, the power supply output capacitors and other components may be overstress.
What can make the problem worse is that systems that have successfully operated in the field for years may have entrainment invoked by a software or firmware change.
Autonomous circuits (no external driving function) such as a ripple regulators, hysteretic converters, etc., are far more susceptible to this problem than driven circuits. In fact, changing an autonomous circuit topology to a clock-driven circuit topology is often the solution. However, driven circuits such as PWM converters, can also be susceptible and need to be tested. Not synchronizing the power supply frequency to the system clock frequency can also aggravate the problem due to increased beat frequencies.
This is a non-linear phenomena. To the extent that it can be solved analytically for any given circuit, the analysis is complex to the point of being impractical for most projects. The same could be said to apply to simulation. The only practical solutions is to test a circuit for the phenomenon, and if induced, fix the circuit and make sure that the inducing conditions can never occur in the system..
Analysis and simulation are not helpful for practical design schedules due both to the complexity of the mathematics for analysis and the time and resources to simulate accurately. The only practical method of discovering frequency entrainment is testing for it -- and you may not catch everything. What follows are a few practical suggestion to improve the probability that your power supply design will not entrain.
Here is the way I test for it. Load transients are used in the description, but testing for line transients and transients on the reference are similar. Don't forget to also test for these.
First, you want to find out what the ripple increase is for normal load transients. Starting at a 10% transient load at 50% loading, you want to measure the transient amplitude at low frequency (below the unity gain frequency) when the transient load is applied. You expect to see approximately the delta load change times the characteristic impedance of the equivalent output filter, (delta I)*(L/C). If you don't like what you see (excessive amplitude), you have to reduce the characteristic impedance, Zo, either by deceasing L or increasing C. You can then estimate what the peak-to-peak amplitude will be when you increase the frequency to that of the cross-over frequency at unity gain. You increase the switching load frequency until you get what approximates a sine wave on the voltage output and check the peak-to-peak voltage. If it is what you expected, no detrimental entrainment has occurred and the circuit is operating normally. You then sweep the load frequency up and down to makes sure no sub-harmonic or harmonic is causing entrainment.
Then you repeat, increasing the load transient in 10% increments until you are switching from zero to 100% (or your specification load limits), looking for entrainment. I have observed load transient of +/- 20% entrain and cause the voltage output to swing from zero to 10 V on a 5 V power supply. One case was on the prototype (100 produced and sent to various design agencies to test) of a proposed military standard power supply. I called the design engineers to discuss the problem. They previously had no idea it was a problem, but duplicated the problem in their lab. They changed the design so the production supplies did not entrain.
After running the above tests to get a feel for your design, you then have to construct a test plan that has a chance of catching most problems in a reasonable amount of test time. See the discussion at the end of Point Measurements in the estimating expert system for a discussion of this problem.
You want to (must) test your design to make sure it is not susceptible to entrainment.
A group of us started playing with switching-mode power supplies in the early 1960's. One of the engineers with whom I worked, Paul Beihl, who had spend a summer as an engineering student in Austria, found a buck converter described in a German language engineering magazine he subscribed to. Several us started to build breadboards and play with the circuits.
My first design was a buck converter operating at 1 kHz using a Germanium "door knob" 2N174 transistor. About a year later a germanium power transistor became available that would switch at higher frequency. We tried our first production design with hysteretic converters operating at a nominal 12 kHz (yes you could hear them and they annoyed virtually everyone.)
Everything went fine through the breadboard stage and the early prototypes with normal software programs in the computer. Then the test engineers became involved and set up a test to test the core memory. The program switched the memory circuits from a one to a zero and back at the 1 MHz clock frequency and counted down, cutting the loading frequency in half each time until the load switching frequency was 1 Hz and then it counted back up. This loaded the memory power supplies with a zero to maximum load with a frequency sweep that excited anything there was to excite.
That is when we first saw entrainment. Normally when you sweep a load frequency you see an increase in ripple at the point the feedback loop goes through zero gain, but you control this by keeping the characteristic impedance of the output filter, SQRT(L/C), sufficiently low. But this was something else. At certain frequencies, the load would pull the frequency of the converter so that it no longer ran at its natural frequency but entrained with a harmonic, fundamental, or sub-harmonic of the load switching-frequency. When this happened, the ripple amplitude would increase to the point that the protection circuits shut down the computer.
The immediate fix was to go back to running actual programs in the computer while the circuit designers scrambled to find out was going on. We started to test for entrainment and found that autonomous (free running) converters were much more sensitive than those clocked to a fixed frequency. Paul Beihl and his fellow designers changed the design to a fixed frequency PWM which solved the problem for the production design. The solution is in a fixed frequency PWM Voltage Regulator patent which describes the root of the problem. We never again used a free-running converter in a computer, and later, always synched the PWM frequency to the computer clock to minimize beat frequencies and electromagnetic interference. We also tested every new design for entrainment so we knew how susceptible it was.
Later, we also found that you could entrain to noise on the input power lines and to electromagnetic interference. We also found that we sometimes activated chaos in our testing.
Many years later when I was at the Naval Ocean Systems Center I gave a briefing to hundreds of Navy power supply design activities that always included a viewgraph on entrainment. Virtually no power supply designer I talked with was aware of it.
This is in contrast with the users of power supplies in digital systems. At the 2006 Asilomar Microcomputer Workshop I spoke shortly on entrainment. This workshop is by invitation only with the pioneers and leading developers of microprocessors usually in attendance. While speaking, I could see several in the audience nodding their heads in recognition of the problem and several of them shared their experience with me before the workshop was over.
When power supply users are more aware of a problem than the power supply designer, it is time to become aware of the problem and solve it. Make sure your power supply design is immune to entrainment.
On the Web
I found no key entrainment papers or websites on the Web. However, you can use search engines such as Google and Google Scholar to find related information on the Web as well as IEEE Xplore. You might want to explore with the following keywords combined with "power supplies".
I found tens of thousands of hits, with nothing useful in the first 20 results except this Web site and a reference to a graduate course in nonlinear circuits in the University of California, Berkeley catalog. A large number of the hits are related to medicine, fluid dynamics, and the impact of power supply noise on clock jitter.
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.
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
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- Richard Meyer (11/26/2007): Caught the first typo, entertainment used for entrainment. He also reported his observation of new kittens all nursing in synchronization. [Correction made. JF]
- Luca Benetti (11/26/2007): Recommended the Weischedel paper and its underlying patent. The paper was added as a strong addition to the references. Many thanks for pointing it out for inclusion. [Weischedel paper added to references.
- Rudy Severns (11/27/2007): "I have a nice clean copy of Minorsky which was my bible for sub-harmonic oscillations. The only problem with Minorsky is that there is so much detail, it takes lot of effort to extract practical info. There is also a very good book on non-linear oscillations by Chihiro Hayashi which has more electrical circuit examples. Still of course a tour de force of differential equations." [Hayashi added to references. JF]
- Paul Beihl(12/16/2007: "Good article on entrainment. Your memory is great and I would just add a couple of corrections re: my summer work in Austria (instead of a year in Germany)" [Correction made, JF]