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Theory of phase-locked multivibrators?

Discussion in 'Electronic Design' started by Tim Shoppa, Apr 6, 2007.

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  1. Tim Shoppa

    Tim Shoppa Guest

    Is there a general theory of locking simple (two-transistor or two-
    tube or two-whatever) RC multivibrators to a sine wave or square wave
    reference? Especially when frequency multiplication or division is
    done in the process?

    What I'm thinking, is that someone must've worked out some theory that
    says if you have a multivibrator of a certain natural frequency, and
    that if you inject a sine wave of a given amplitude at a base or a
    cathode or whatever with a slightly different frequency, that you will
    be able to lock if the natural frequency is 10% high or 5% or low or
    whatever, and if the amplitude injected is bigger, then the lock range
    is bigger, etc.

    I'm using some very vague terms above. I've explored the topic using
    Spice and clearly there's some more general math rather than "run a
    metric buttload of Spice simulations" because there are obvious
    patterns in the regions of stability. The patterns are kind of pretty
    when you are locking to a harmonic or subharmonic.

    But rather than running a metric buttload of Spice simulations, I
    believe there must be a more general theory about amplitude of
    injected reference and range of lock.

    Maybe this is some class of PLL with very broad/nonexistent loop

    My old-fashioned references that talk about phase locking
    multivibrators, or phantastatrons, like the MIT Radiation Lab books,
    give some general guidance but not really the formulas I believe must

    Any clues for me?

  2. Hell, I avoid math if I can...
    I have played with that, as drop-out compensator in FM... for video
    tape recoding, if no signal the MVB takes over.

    I just look at the 2 stage mvb as a 2 stage amplifier with very high gain.
    If you want to 'take over' that is flip a stage, you will have to outdo the
    feedback from the other stage,
    As simple as that.
    Of course the network you use to feed into the base / whatever has some effect
    on the actual frequency and gain of the thing itself.

    I just look up am silent for a while, grab the right capa-citator or
    resi-sistor and it works.
    Spice is for rice.
  3. AFAIR any RC oscillator was synchronised by adding
    small spikes to the exponential timing waveform.
    The oscillator was always tuned to run too slowly
    and the last spike simply flipped it early.

    The spike amplitude was about 10-20% of the
    exponential amplitude, depending on the division
    ratio, which was usually /1 to /10. Division-only

    Single-RC oscillators (eg, blocking osc) would need
    only one timing spike, whereas the 2-RC cross-coupled
    multivibrator often had both RC's spiked.

    I don't recall any sums, all a bit cut and try, afair.
  4. Tim Shoppa

    Tim Shoppa Guest

    Yes, it is that simple. From this (and the fact that the "amplifiers"
    in the multivibrator are always running saturated or open) I can

    1. Higher your Vcc is (or B+ or whatever), higher your injected
    amplitude has to be to get a lock.

    1b. Changing Vcc or B+ also has some effect on the free-running
    frequency of the multivibrator.

    I can also conclude:

    2. The shape of the wave you inject matters too. Square pulses seem to
    be most effective for locking, because the time that they can "flip"
    the state is well defined, and the sudden edge is good too. Sines and
    Triangles seem to be about equivalent to each other for most purposes
    (their ratios of peak height to RMS height to peak slope at crossover
    are not identical but are pretty similar).

    3. Further your injection frequency is from the natural frequency, the
    more amplitude you need to inject to get a lock.

    4. Once you try locking to a harmonic or subharmonic, things get a
    little hairy.

    5. If you aren't injecting a pure simple waveform but have noise
    fuzzing things up, this makes life even hairier.
    This is certainly the traditional approach! The five rules-of-thumbs
    above with a scope seems to work out pretty well.
    Yeah, that's why I'm thinking there must be a more general theory for
    this. Modeling oscillators or locked oscillators in Spice is possible
    but obviously there has to be something more appropriate other than
    "run a buttload of simulations and look to see which ones locked".

  5. Jim Thompson

    Jim Thompson Guest

    What you are seeking is information on INJECTION-LOCKED oscillators.

    One of my bosses at Motorola, Jan Narud, wrote his PhD at Stanford
    (IIRC) on the topic... probably dated between 1950-1960.

    ...Jim Thompson
  6. Boris Mohar

    Boris Mohar Guest

    So what happens why you take two free running multivibrators and lock them to
    each other?
  7. Tim Wescott

    Tim Wescott Guest

    As Jim said, the name of the thing is an injection-locked oscillator.

    What you end up with isn't really phase-locked, because response of the
    oscillator to the injected signal is an instantaneous change in phase
    rather than an overall change in frequency. If you model the loop from
    a control-system point of view it's a type 1 loop in phase, so trying to
    pull the frequency requires a phase offset.

    The relationship between the amplitude of the injected signal and the
    amount of frequency pulling that you can do will vary quite a bit with
    the details of the circuit. In a system that has a mostly sinusoidal
    signal, in to which you are injecting a mostly sinusoidal signal, you'll
    find that the amplitude vs. pulling relationship will be fairly linear
    over some range. In an honest-to-god multivibrator you'll have to just
    get into the grit of how the oscillator works, and the shape of the
    synchronizing pulse, to quantify the effect. Were I doing the work, I'd
    probably use SPICE to get an offset phase vs. frequency plot for a given
    pulse amplitude and shape, then take that information off line to make a
    model of the oscillator as a summing junction followed by an integrator.


    Tim Wescott
    Wescott Design Services

    Posting from Google? See

    Do you need to implement control loops in software?
    "Applied Control Theory for Embedded Systems" gives you just what it says.
    See details at
  8. Jim Thompson

    Jim Thompson Guest

    My favorite "injection" locked oscillator...

    Really a "phase-jerked" PLL.

    ...Jim Thompson
  9. Tim Shoppa

    Tim Shoppa Guest

    Thank you, Jim. I knew that "injection" was part of the lingo but
    somehow kept on using the word "phase" in my verbiage :).

  10. Tim Shoppa

    Tim Shoppa Guest

    The injected pulse method is not the only possibilty. If you inject a
    sine or a triangle wave I think the math is a little harder. A fast-
    rise-(or fall)-time pulse is easy: it adds a quantity of charge to the
    capacitor and speeds it up. Slowly changing injected waves work a
    little more like what Jan and I were talking about, where it shifts
    the trigger point either earlier or later.

    With some of the circuits I've been playing around with, there are
    funky fractions that you can lock to (say 2/3 or 3/2 and higher-n/
    higher-m ones) but the funkier the fraction gets, the less easily
    things lock up.

    The MIT radiation lab books talk about these circuits rather broadly
    (and also divides them up into nomenclature we don't here today, like
    phantastatrons) but avoids the math I wanted to see about noise on the
    injected signal etc.

  11. Tim Shoppa

    Tim Shoppa Guest

    There are analogies in nature: lightning bugs, for example, will tend
    to synchronize their blinks with each other. (If you never lived
    anywhere with lightning bugs, my apologies) I believe I saw some
    discussions about this and chaos theory when applied to large groups
    of multivibrators (well, the articles called them "lightning bugs")
    with randomly spread free-running frequencies and couplings.

    When I saw Jeff Goldblum in Jurassic Park, any interest I had in chaos
    theory was instantly erased. Oh, man, did I want him to get eaten by a
    T. Rex. But he lived to the end! That sucked!

    I still think what Lorenz was doing in the 60's was interesting.

  12. John  Larkin

    John Larkin Guest

    An injection-locked multivibrator can be analyzed almost by
    inspection; the outside signal just pushes things to switch sooner.
    Injection locking a sinewave oscillator is a lot more complex. James
    knows a lot about this.

  13. Joerg

    Joerg Guest

    Also, check out the works by Barlow and Wadley (South Africa). Often
    referred to as the Wadley Loop. Ok, LC oscillator and not RC but the
    mechanism is similar. I've got a receiver with such a scheme here on the
    office desk. You can "ratchet" the range oscillator in 1MHz increments
    onto harmonics of a crystal controlled square wave. AFAIR the actual
    receivers first came out in the late 60's. For some reason receivers
    with that scheme were rare but IMHO they were about the only ones where
    they made an effort to design the enclosure less utilitarian. They could
    almost have enough WAF to reside in the living room.
  14. Joerg

    Joerg Guest

  15. Jim Thompson

    Jim Thompson Guest

    Who cares? It's only digital ;-)

    I have a better version that examines phase error and limits the
    amount of "jerk" per cycle... used in GSM telephones.

    ...Jim Thompson
  16. A E Neumann

    A E Neumann Guest

    There is a PhD dissertation at Stanford about how oscillators
    It should be available for download.
    The only thing to know about the oscillator is that they are like
    women - they either do or they don't.
  17. Joerg

    Joerg Guest

    That would also mean that they cannot be simulated.

    Genie to lucky winner: "What is your 2nd wish?" ... "Make me understand
    women" ... "Ahm, let's get back to the first wish that I had deemed too
    excessive. How many lanes do you want that bridge to Hawaii to have?"
  18. Jim Thompson

    Jim Thompson Guest

    But they can be stimulated ;-)

    ...Jim Thompson
  19. John F

    John F Guest

    Old/New meaning for _digital_ ... :)
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