oscillator design issue

[frank]

Hello,

When designing a system with negative feedback, it is recommended
practice to design it with at least a phase margin of 45 degerees to
avoid unwanted oscillations.

Now, when designing oscillators, which are positive feedback systems,
is it recommended to design them with -45 degrees phase margin in
order to assure oscillations?


My electronics book mention the first recommendation but not the
secons one.
Why ?

-Frank

 

[John Larkin]

Hello,

When designing a system with negative feedback, it is recommended
practice to design it with at least a phase margin of 45 degerees to
avoid unwanted oscillations.

Now, when designing oscillators, which are positive feedback systems,
is it recommended to design them with -45 degrees phase margin in
order to assure oscillations?


My electronics book mention the first recommendation but not the
secons one.
Why ?

-Frank

Because things oscillate when loop gain is >=1 at zero degrees overall
phase shift.

John

 

[Kevin Aylward]

Because things oscillate when loop gain is >=1 at zero degrees overall
phase shift.

Not always. The relevant phrase is "conditionally stable systems". i.e.
stability is determined by the net encirclements around the minus 1
point. Indeed, *the* fundamental point, imo, of stability theory *is*
that systems with net positive feedback, and gain greater that one, can
be perfectly stable. It was for such strange behaviour that stability
theory was invented in the first place. If the greater than one gain at
0 degs was sufficient, why have a complicated theorem in the first
place.

In fact it reminds me of shrodingers cat. It was introduced to show that
the naive Copenhagen Interpretation of the quantum state vector was
actually wrong. i.e. no such thing as a cat in two states
simultaneously. Nowadays, it keeps getting presented as an example of
what QM says is true. Its as gibberish today as it was then.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[frank]

Because things oscillate when loop gain is >=1 at zero degrees overall
phase shift.

John

True. Totally agree with that. But when designing the oscillator, if
your simulations / calulations indicate that the OL gain is for
example 5(magnitude) and -5 degrees (phase), the real life scenario
migth not be the same. You could end up with an OL gain response of
4.2 (magnitude) and +5 degrees (phase). Therefor,e although the phase
shifts are similar in the simulation world and the real world, your
circuit will not be able to achieve steady-state oscillations.

-Frank

 

[John Larkin]

True. Totally agree with that. But when designing the oscillator, if
your simulations / calulations indicate that the OL gain is for
example 5(magnitude) and -5 degrees (phase), the real life scenario
migth not be the same. You could end up with an OL gain response of
4.2 (magnitude) and +5 degrees (phase). Therefor,e although the phase
shifts are similar in the simulation world and the real world, your
circuit will not be able to achieve steady-state oscillations.

-Frank

In most oscillators, if simulation shows 5 degrees phase shift around
the loop, it just means that you're simulating at the wrong frequency.
In other words, there is a nearby frequency at which the phase shift
is truly zero, and that's where it will really oscillate. If you build
a typical oscillator (with a reasonable amount of gain reserve), and
it runs OK, and you then add an extra 5 degrees to some element, the
frequency just shifts a bit, and it keeps going. It sort of adjusts
itself.

It's not as if it won't oscillate; it will likely just oscillate at a
different freq than the simulation predicts. How much different
depends on the slope of the phase vs freq curve around the whole loop.
If you have a high-Q resonator somewhere, say a quartz crystal, the
slope will be very high, and a 5 deg phase shift won't pull the
frequency much. And vice versa. That's why high phase/freq slopes make
for oscillators with low phase noise.

If there is *no* frequency that has zero degrees net phase around the
loop with positive gain, it won't oscillate. It might ring like hell,
but will eventually damp out.

John

 

[Marc H.Popek]

Hey John,

Been following this thread... and the tough part to think about is the
disconnect between numerical precision, i.e. +5 DEG OL versus a -5 Deg OL
phase margin... and how when the loopis closed, that the a excess loop gain
provides a large enough signal, that further changes the "bulk time
averaged" node capacitance, and that brings the oscillator to the "right"
frequency.

In very phase stable oscillators, the amplitude variations and overdrive
levels allowed on nodes(particularly the base or gate) are tightly
controlled. This reduces the amplitude to phase conversion, and makes a mo
betta oscillator in terms of phase sensitive applications

Marc P

 

[Kevin Aylward]

True. Totally agree with that.

Oh...You obviously did not read my post then

Of course in reality, its always the "Amplifiers do, Oscillators don't".

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Active8]

Not always. The relevant phrase is "conditionally stable systems". i.e.
stability is determined by the net encirclements around the minus 1
point. Indeed, *the* fundamental point, imo, of stability theory *is*
that systems with net positive feedback, and gain greater that one, can
be perfectly stable.

i'm sort of realizing a salient point, you may have just made.

think two port stability factors in rf amps for a second. you know,
Linvall/Rollet...

if a transistor is unconditionally stable, it isn't supposed to
oscillate when you design it into an amp. but can it still be made to
oscillate or rather, can it be used as *the* gain device in an
oscillator? i never thought about this before.

for an oscillator, you need a conditionally stable device preferably
operating at impedance levels (or just one level) that fall(s) in the
unstable areas, right?

i bet an active device operating in the stable region won't oscillate in
spice either.

mike

 

[ddwyer]

i'm sort of realizing a salient point, you may have just made.

think two port stability factors in rf amps for a second. you know,
Linvall/Rollet...

if a transistor is unconditionally stable, it isn't supposed to
oscillate when you design it into an amp. but can it still be made to
oscillate or rather, can it be used as *the* gain device in an
oscillator? i never thought about this before.

for an oscillator, you need a conditionally stable device preferably
operating at impedance levels (or just one level) that fall(s) in the
unstable areas, right?

i bet an active device operating in the stable region won't oscillate in
spice either.

mike

The oscillator "finds the frequency" where loop phase is zero.
if it starts at say 15 degrees then 0 degrees is sufficiently close
that oscillation will occur close by in frequency

 

[Kevin Aylward]

i'm sort of realizing a salient point, you may have just made.

think two port stability factors in rf amps for a second. you know,
Linvall/Rollet...

if a transistor is unconditionally stable, it isn't supposed to
oscillate when you design it into an amp. but can it still be made to
oscillate or rather, can it be used as *the* gain device in an
oscillator? i never thought about this before.

for an oscillator, you need a conditionally stable device preferably
operating at impedance levels (or just one level) that fall(s) in the
unstable areas, right?

This is not the same meaning of conditionally stability as I used above,
I believe. Conditional stability in feedback theory is the case where
the gain is > 1 and the phase goes to zero. Typically, an n order roll
off, with n > 3, where prior to the unity gain point, the roll off is
compensated out to a single order. The phase can drop way below 0
degrees and then be pulled back up again. Its called conditionally
stable because the value of the gain can make the system unstable. That
is, at low and high gains, the system can be stable, but at an
intermediate gains, the system is unstable.


Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Michele Ancis]

Not always. The relevant phrase is "conditionally stable systems". i.e.
stability is determined by the net encirclements around the minus 1
point. Indeed, *the* fundamental point, imo, of stability theory *is*
that systems with net positive feedback, and gain greater that one, can
be perfectly stable. It was for such strange behaviour that stability
theory was invented in the first place. If the greater than one gain at
0 degs was sufficient, why have a complicated theorem in the first
place.

This is interesting, I must confess I don't know much of stability
theory. When I hear this word I think in these terms: (amplitude)
bounded input implies (amplitude) bounded output. This brings, when we
consider lumped element circuits, to the requirement for the transfer
function to have no poles in the right half plane (with no concern
about their phase).
Coming to the feedback system, I've made a few simulations with an
amplifier and a delay line, connected such that they have a loop gain
greater than one. The delay line introduces a phase shift...and even
if strictly speaking we can't talk of poles and zeroes, I've noticed
that I get an infinitely growing output for any limited input, and for
any time delay.

This would bring me to the conclusion that, if I have a system giving
me an amplified (and time shifted, this is fundamental, it doesn't
work with static systems) replica of my input, which goes and sums
with it, then it doesn't matter about the phase with which it will sum
with the input: the result will always be an ever growing signal. That
is, the system is unstable.

This seems to be in contraddiction with what you say, but probably the
missing link between a *delay line* and a lumped element circuit phase
behavior is causing me problems...

Would you mind clarifying this? I would also be keen on reading some
good book on the subject. I hope you're not a state variable fanatic,
because this is exactly why I've never gone through with this theory.
I'm much more a Bode fan ;-)

Thanks,

Michele

 

[Michele Ancis]

Nope. Nope. Nope. That's the *whole* *f'ing* point of the Nyquist
stability theorem.
Dah...That why I stressed this in detail.

And that's why, indeed, I ask you

The loop gain argument is
false, erroneous, wrong, a dead parrot one. It is deceased. No more to
live again.

This morning I found him sitting on my desk chair..but thanks to your
help I've probably defeated him!

The original paper discusses this.

Will you give me the reference? Or maybe directly the paper?

The argument around the loop
consists of summing up each exp(jw) term. The issue here, is that for
gains greater than 1, the series is non convergent. This means that this
particular mathematical approach fails. One can not make deductions from
non convergent series. Again, it was precisely because, Black and
Nyquist had experimental proof that gains greater than one, at 0 net
phase were perfectly stable, that the had to develop a correct theory as
to when a system was stable. The answer is given by the knowledge, that
the Laplace transform of any rational transfer function, if there is a
pole in the rhp, there is an ever increasing oscillation. Since
factorising a large polynomial is impossible without numerical methods
for degrees greater than 4, Nyquist came up with a graphical technique
to determine whether or not there is a pole in the rhp. The result is
akin to the fact that the integral around a closed curve of a complex
variable is zero if the function is analytic, i.e, no poles, or 2.pi.i
otherwise. The summary is the number of net encirclements of the -1
point, that is you can circle around, and come back again, and still be
ok.

OK, I think I got it, now. Thank you for the thorough explanation.
However, I would take this occasion to try to put all the tiles in
order on the subject. Will you advice me some book where the thing is
explained?

See above

I think there's another point, here. It is that, with distributed
systems - i.e. the delay line - you can't speak of poles and zeroes,
can you? Therefore the whole theory is useless.

Hopefully I have.

Yes, you did! Thanks again!

M

 

[Active8]

[snip]

This is not the same meaning of conditionally stability as I used above,
I believe.

good. i was hoping i read too much into that thanks.

Conditional stability in feedback theory is the case where
the gain is > 1 and the phase goes to zero. Typically, an n order roll
off, with n > 3, where prior to the unity gain point, the roll off is
compensated out to a single order. The phase can drop way below 0
degrees and then be pulled back up again. Its called conditionally
stable because the value of the gain can make the system unstable. That
is, at low and high gains, the system can be stable, but at an
intermediate gains, the system is unstable.

great. the two concepts aren't related and i can forget about looking
into that any furthur.

mike

 

[analog]

Kevin Aylward, Michele Ancis wrote:
...

Yes, you can. A pure delay, has no zero's, therefore it is stable.
A delay fed back and summed, might be 1/(1+exp(tau.s)), which does
have a pole, in fact it has many.

...
If I remember right an ideal delay requires an infinite number of
poles evenly spaced in a vertical line in the left half of the
complex plane, but these are also mirrored by an equally dense
vertical array of zeros in the right half of the complex plane.
The amount of delay is a function of their distance from the
imaginary axis (closer equals more delay). Without the zeros to
cancel the effect of the poles the magnitude response of such a
network would fall at high frequency. A delay should pass all
frequencies.

The mathematically expression for this might be called the Pade
approximation if memory serves and I think it is more or less a
truncation of the Pi series representation of exp(-sT), the
expression for an ideal delay (right?).

analog

 

[Kevin Aylward]

Kevin Aylward, Michele Ancis wrote:
...
...
If I remember right an ideal delay requires an infinite number of
poles evenly spaced in a vertical line in the left half of the
complex plane, but these are also mirrored by an equally dense
vertical array of zeros in the right half of the complex plane.
The amount of delay is a function of their distance from the
imaginary axis (closer equals more delay). Without the zeros to
cancel the effect of the poles the magnitude response of such a
network would fall at high frequency. A delay should pass all
frequencies.

The mathematically expression for this might be called the Pade
approximation if memory serves and I think it is more or less a
truncation of the Pi series representation of exp(-sT), the
expression for an ideal delay (right?).

analog

Ahmmm....quite a description...never heard of the term Pade. However, it
would seem that the is some misinterpretation of "requires" here.

A pure delay is exp(-s.tau). This is because the laplace transform of
f(t-a) is exp(-s.tau)Laplace(f(t))

with s= a+jw

exp(-(a+jw)tau) = exp(-a.tau).(cos(w.tau) - j.sin(w.tau)), which is
never zero or infinite. It therefore never "requires" to have any poles
or zeros. It simply dos not have any.

However, having now just looked up "Pade approximation", I see what your
driving at. In general, it is possible to approximate functions by other
functions, and these other functions may have poles and zeros. In this
case, an approximation of a power series by a ratio of power series. It
doesn't mean that the original function has such poles and zeros. The
most obvious rational ratio of polynomials to approximate exp(-s.tau)
is, of course, the Bessel-Thomson approximation.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.