# Theory of phase-locked multivibrators?

T

#### Tim Shoppa

Jan 1, 1970
0
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.

J

#### Jan Panteltje

Jan 1, 1970
0
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?

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.

T

#### Tony Williams

Jan 1, 1970
0
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.

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.

T

#### Tim Shoppa

Jan 1, 1970
0
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.

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:

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.
I just look up am silent for a while, grab the right capa-citator or
resi-sistor and it works.

This is certainly the traditional approach! The five rules-of-thumbs
above with a scope seems to work out pretty well.
Spice is for rice.

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.

J

#### Jim Thompson

Jan 1, 1970
0
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.

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

B

#### Boris Mohar

Jan 1, 1970
0
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.

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

T

#### Tim Wescott

Jan 1, 1970
0
Tim said:
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.
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
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html

J

#### Jim Thompson

Jan 1, 1970
0
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.

My favorite "injection" locked oscillator...

http://analog-innovations.com/SED/ShiftRegisterPLL.pdf

Really a "phase-jerked" PLL.

...Jim Thompson

T

#### Tim Shoppa

Jan 1, 1970
0
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.

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

Tim.

T

#### Tim Shoppa

Jan 1, 1970
0
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.

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.

(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.

T

#### Tim Shoppa

Jan 1, 1970
0
So what happens why you take two free running multivibrators and lock them to
each other?

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
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.

J

#### John Larkin

Jan 1, 1970
0
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

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

John

J

#### Joerg

Jan 1, 1970
0
Tim said:
Thank you, Jim. I knew that "injection" was part of the lingo but
somehow kept on using the word "phase" in my verbiage .

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.

J

#### Jim Thompson

Jan 1, 1970
0
Couldn't there be a wee setup and hold violation at the 2nd LS74?

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

A

#### A E Neumann

Jan 1, 1970
0
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:

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 havenoise
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 generaltheoryfor
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.

There is a PhD dissertation at Stanford about how oscillators
start.
The only thing to know about the oscillator is that they are like
women - they either do or they don't.

J

#### Joerg

Jan 1, 1970
0
A said:
There is a PhD dissertation at Stanford about how oscillators
start.
The only thing to know about the oscillator is that they are like
women - they either do or they don't.

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?"

J

#### Jim Thompson

Jan 1, 1970
0
A E Neumann wrote:
[snip]
There is a PhD dissertation at Stanford about how oscillators
start.
The only thing to know about the oscillator is that they are like
women - they either do or they don't.

That would also mean that they cannot be simulated.

But they can be stimulated ;-)

...Jim Thompson

J

#### John F

Jan 1, 1970
0
Jim said:
A E Neumann wrote:
[snip]
There is a PhD dissertation at Stanford about how oscillators
start.
The only thing to know about the oscillator is that they are
like
women - they either do or they don't.

That would also mean that they cannot be simulated.

But they can be stimulated ;-)

Old/New meaning for _digital_ ...

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