# Theory of phase-locked multivibrators?

• posted

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 filter.

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 exist.

Any clues for me?

Tim.

• posted

On a sunny day (6 Apr 2007 06:46:00 -0700) it happened "Tim Shoppa" wrote in :

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.

• posted

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 afair.

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.

--
Tony Williams.
• posted

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

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:

1. 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).

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

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

1. 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".

Tim.

• posted

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

--
|  James E.Thompson, P.E.                           |    mens     |
• posted

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

--

Boris Mohar
• posted

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
• posted

My favorite "injection" locked oscillator...

Really a "phase-jerked" PLL.

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
• posted

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

Tim.

• posted

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.

Tim.

• posted

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.

Tim.

• posted

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.

John

• posted

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.

--
Regards, Joerg

http://www.analogconsultants.com
• posted

Couldn't there be a wee setup and hold violation at the 2nd LS74?

--
Regards, Joerg

http://www.analogconsultants.com
• posted

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

--
|  James E.Thompson, P.E.                           |    mens     |
• posted

There is a PhD dissertation at Stanford about how oscillators start. 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.

• posted

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?"

--
Regards, Joerg

http://www.analogconsultants.com
• posted

[snip]

But they can be stimulated ;-)

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |