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Resonant Circuit problem

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We should keep in mind that if a resonant circuit is put in a feedback loop with an amplifier, then the loop will oscillate (assuming the loop gain is 1 or greater) when the total loop phase shift is zero (360) degrees. If the amplifier has infinite bandwidth (zero delay), then the resonant circuit phase shift must be either zero degrees (for a noninverting amplifier) or 180 degrees (for an inverting amplifier). A resonant network which does not pass through zero (or 180) degrees will not oscillate.
Perhaps the definition of resonance depends on the application.
 
Hehe, yep I'm a little bit confused right now, but I think it will sort out ;).

I have been taught that you define the resonance circuit by:

Im{Y}=0 or Im{Z}=0

And I have always solved resonance circuits by doing this.

But this seems to be a tricky one, and therefore you can't use this definition?

MrAl:

Whats d|Z|/dC? Just a derivation in regard to C? Never seen that before :)
Is it used for looking max/min on the capacitor?

And the same with d|Z|/dF? As I said before, never seen them before. And my course book never says anything about it either, but it sounds logical.
I guess I have to solve d|Z|/dF=0, cause I don't think sweeping is the right way...

So if we start from the beginning:
  1. Start with Z(jw)=1/(1/R-j/(w*L2)+j*w*C)+j*w*L1
  2. Solve C
  3. Calculate the impedance with this C
  4. Then solve d|Z|/dF=0
  5. Solve C
  6. Calculate the impedance with this C

Have I understood it correctly?

Thanks! You guys have spent many hours on this problem. Great work!
 
Hi again,

This discussion is interesting.


Roff:
Ok, but i dont think we can be that critical on what will and will not oscillate
in the feedback path of an ideal 360 degree amplifier. This is especially true
with the claim that a resonant network which does not pass through zero will not
oscillate, because you can take the network we have been working with and change
L1 to 150nH and although its phase does not go through zero behold the circuit
still oscillates, albeit at a different frequency of course.

Still, i wont argue that Im(Z)=0 can't be called resonance anymore because we
can find other references which will take this definition to extreme. The
only way to explain this apparently socially driven phenomenon is that the energy
transfer between inductors and capacitors is perfect during this resonance.
I must stress however that this definition will not work in every circuit simply
because the electrical significance does not dictate such a success. I also
must stress that this definition may be totally acceptable in other forums where
other measures are being used but mistaken for Im(Z)=0 again probably because
of the carry over from an RLC circuit or two (someone turns a dial and thinks
they are tuning for Im(Z)=0 when really they are tuning for Max |Z|, but
no one ever bothers to measure this directly because it's not really needed
except in pure theory).


h3rroin:
I hope the above helped a little. We cant discount this definition because it
has been in such widespread use over the years, but we do have to realize its
limitations in that many times we will desire to know Max Z when we think we
need to know Im(Z)=0.

d|Z|/dC means the derivative of the magnitude of the impedance Z, with respect to
the capacitance C. This relates how |Z| changes with a change in capacitance
and is useful for finding a peak or dip in the response for a given frequency
(and other components of course) as C changes. When d|Z|=0 we find candidate
min and max responses of the circuit.

d|Z|/dF means the same as above except with respect to frequency F. This allows
us to find the candidate min and max of the response as frequency changes (ie
resonate peaks or dips).

To do this, you would take this:
Z(jw)=1/(1/R-j/(w*L2)+j*w*C)+j*w*L1
and plug in all the values except F.
Then, compute |Z(jw)| which we call |Z| for short.
Then, compute d|Z|/dF by taking the derivative of
|Z| with respect to frequency F.
Now set this d|Z|/dF equal to zero, then solve
for all the solutions of F. This will show where
the peaks and dips are.

Another interesting way to do this is to instead
use V/Z(jw), where V is a constant. This looks
for peaks and dips in the current.

I dont think you should give up on the frequency sweeping technique
however, because it does after all present a more simple way to
find peaks and dips.
 
Last edited:
Hi again,

This discussion is interesting.


Roff:
Ok, but i dont think we can be that critical on what will and will not oscillate
in the feedback path of an ideal 360 degree amplifier. This is especially true
with the claim that a resonant network which does not pass through zero will not
oscillate, because you can take the network we have been working with and change
L1 to 150nH and although its phase does not go through zero behold the circuit
still oscillates, albeit at a different frequency of course.
I don't see how this network (L1=150n, C=62p) will oscillate when put in a feedback loop with an ideal amplifier. Please explain.:confused:
 
I don't see how this network (L1=150n, C=62p) will oscillate when put in a feedback loop with an ideal amplifier. Please explain.:confused:

Hello again Roff,


I havent gone through the equations for this yet, but i assume that
it is because sufficient conditions for oscillation exist which dont have
to include an exact phase shift of 0 degrees in the network because
a circuit can oscillate at other points when the loop gain is sufficient.
Yes a circuit might oscillate fine with a perfect 360 degree phase
shift, but that's not the only time it can oscillate. I think the simplest
way of looking at this is that sine waves can add (reinforce) even
when there is a phase shift and they dont have to be perfectly in
phase to add up to a higher value, depending on the gain too.
We can push a swing with a small force F when it swings far back
toward us, right at the crest, or we can wait until it starts back
in its forward path and push a little harder. Either way, it keeps it
moving. I think we can even push when the chain goes through its
perfectly vertical position, although we would need a lot more force
(gain) then to keep it going.

I'll see if i can get the equations into the proper form and analyze for
oscillation, but in the mean time, stick this network with a zero degree
phase shift amplifier and gain of 1 into a circuit simulator and see what
frequency the output of the amp puts out.


BTW, i hate to add more confusion to the confusion, but i guess it's
never too late <little chuckle>, as we have yet to discuss a THIRD
type of resonance which hasnt been mentioned yet.
Hint: i can get the network to oscillate at a frequency other than
the Im(Z)=0 frequency or the Max|Z| frequency by driving it with
a short pulse. This oscillation is damped but nonetheless still exists,
and does not even require a feedback amplifier.
It's sometimes called the "free oscillation" resonant frequency.

This now leaves us with three candidate resonant frequencies for
this circuit:

Im(Z)=0
|Z|=max (or min)
FreeOsc Z

Just to note that with this particular circuit, i get these three actual
frequencies approximately:

Im(Z)=0: 124MHz
|Z|max: 128MHz
FreeOsc Z: 127MHz

Of these three, |Z|max seems to be the most important in electrical
resonant circuit calculations other than a pure LC circuit, but yes we
dont want to totally dismiss the other two.

BTW, you might be familiar with the w0 frequency used in many simple
circuit calculations. This w0 is actually the Im(Z)=0 frequency and
can be used in other ways too and is used in much literature,
but the importance of this variable drops off as the network becomes
more complex. In fact, i think that is what also happened here by
simply adding another inductor to the classic parallel RLC circuit!
 
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Hello again Roff,


I havent gone through the equations for this yet, but i assume that
it is because sufficient conditions for oscillation exist which dont have
to include an exact phase shift of 0 degrees in the network because
a circuit can oscillate at other points when the loop gain is sufficient.
Yes a circuit might oscillate fine with a perfect 360 degree phase
shift, but that's not the only time it can oscillate. I think the simplest
way of looking at this is that sine waves can add (reinforce) even
when there is a phase shift and they dont have to be perfectly in
phase to add up to a higher value, depending on the gain too.
We can push a swing with a small force F when it swings far back
toward us, right at the crest, or we can wait until it starts back
in its forward path and push a little harder. Either way, it keeps it
moving. I think we can even push when the chain goes through its
perfectly vertical position, although we would need a lot more force
(gain) then to keep it going.
No disrespect intended, but I think you're blowing smoke.:D
I'll see if i can get the equations into the proper form and analyze for
oscillation, but in the mean time, stick this network with a zero degree
phase shift amplifier and gain of 1 into a circuit simulator and see what
frequency the output of the amp puts out.
I tried this with the original circuit values by using a CCVS (H) as the amplifier (with and without a diode limiter in the loop), but the DC feedback (see edit at the end of this post) always made the output run off to teravolts. A cap in series added a third frequency where the phase passed through zero, so it wanted to oscillate at that frequency. As a test, I removed L2, making the network a simple damped series LC circuit. It still has a DC path, but I was able to get oscillations at the zero phase shift frequency:
Fres=90.394MHz (1/(2pi√LC))
Fpeak=90.356MHz (shift from Fres due to 139Ω damping resistor)
FΦ=88.4873MHz (zero phase shift)
Fosc=88.4877MHz (period measurement)
Fosc=88.4722MHz (FFT)
BTW, i hate to add more confusion to the confusion, but i guess it's
never too late <little chuckle>, as we have yet to discuss a THIRD
type of resonance which hasnt been mentioned yet.
Hint: i can get the network to oscillate at a frequency other than
the Im(Z)=0 frequency or the Max|Z| frequency by driving it with
a short pulse. This oscillation is damped but nonetheless still exists,
and does not even require a feedback amplifier.
It's sometimes called the "free oscillation" resonant frequency.

This now leaves us with three candidate resonant frequencies for
this circuit:

Im(Z)=0
|Z|=max (or min)
FreeOsc Z

Just to note that with this particular circuit, i get these three actual
frequencies approximately:

Im(Z)=0: 124MHz
|Z|max: 128MHz
FreeOsc Z: 127MHz
I had tried ringing the circuit's bell with a fast step also, but I could not get enough consistency in time domain period measurements or FFTs to be certain that the ring frequency was different from both the others.

Of these three, |Z|max seems to be the most important in electrical
resonant circuit calculations other than a pure LC circuit, but yes we
dont want to totally dismiss the other two.

BTW, you might be familiar with the w0 frequency used in many simple
circuit calculations. This w0 is actually the Im(Z)=0 frequency and
can be used in other ways too and is used in much literature,
but the importance of this variable drops off as the network becomes
more complex. In fact, i think that is what also happened here by
simply adding another inductor to the classic parallel RLC circuit!
EDIT: The "oscillator" went to teravolts on the original circuit because the phase shift and impedance of the network are both zero at DC.
 
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No disrespect intended, but I think you're blowing smoke.:D
Well, it cant be that because i quit smoking years ago :)
The swing may have been a bad example however.
I think what you are after is sustained oscillations, and i too am a little
puzzled about the results i am seeing. I would like to get the equation
into a form which allows me to look into this a little further but i
need to take an inverse Laplace transform of a very big equation and
i dont feel like doing it by hand.

I tried this with the original circuit values by using a CCVS (H) as the amplifier (with and without a diode limiter in the loop), but the DC feedback (see edit at the end of this post) always made the output run off to teravolts. A cap in series added a third frequency where the phase passed through zero, so it wanted to oscillate at that frequency. As a test, I removed L2, making the network a simple damped series LC circuit. It still has a DC path, but I was able to get oscillations at the zero phase shift frequency:
Fres=90.394MHz (1/(2pi√LC))
Fpeak=90.356MHz (shift from Fres due to 139Ω damping resistor)
FΦ=88.4873MHz (zero phase shift)
Fosc=88.4877MHz (period measurement)
Fosc=88.4722MHz (FFT)
Ok, when we say 'ideal' amp we mean one with a perfect gain constant
and infinite input impedance, which also brings up a problem with the
network because the coil L1 can not work into an infinite impedance,
so i tried various resistors of 1meg down to 10k and these provided
some impedance for L1 to work into. I got resonance even with L1
equal to 150nH, but i can not yet explain why this worked. BTW the
10k connects to a voltage source of 1v to kick start the oscillator.
Without a kick it wont start because everything is zero at first. In
the real world a little white noise would start up a normal osc like
this.

I think i can add a little more confusion by saying that we might
get unsustained oscillation with other values too, but i dont think
that was the point you were trying to make, and i think your
point is valid. I also think that w0 has significance that we shouldnt
overlook and that goes without saying even if a circuit oscillates or
not because there is a lot of literature which will resort to this
type of reasoning too. Maybe your circuit proves that there is
a very good physical significance to this w0, but what you would
need to do is to define your circuit a bit better...after all a perfect
feedback amp has infinite input impedance and so no network will work
into this input correctly. Using a CCVS
may not be appropriate because we never wanted to ground one
side of the network, which is the same as loading it with 0 ohms.
I thought we wanted to place this in parallel with a regular voltage
amplifier.

I had tried ringing the circuit's bell with a fast step also, but I could not get enough consistency in time domain period measurements or FFTs to be certain that the ring frequency was different from both the others.
It's not easy to measure, but perhaps you can try some other values where the |Z| max is
more different than the Im(Z)=0 frequency. What i do is run a simulation with a very short
pulse that goes positive, then at the end of the damped oscillations i inject a very short
pulse that goes negative, then right after that another pulse that goes positive again, then
repeat for several cycles. i then zoom in on one of the single cycles being very careful
to observe where the zero crossing is so as to measure the time between zero crossings.


EDIT: The "oscillator" went to teravolts on the original circuit because the phase shift and impedance of the network are both zero at DC.
Ok, mine broke into oscillation (150nH) but as i said i cant yet explain
exactly why this worked. If you want to try to explore this, try
a voltage gain of 1 and an input resistor of 10k.
Note that without any resistor a voltage on the right side of the
network produces exactly the same voltage on the left side of
the network with no phase shift or other alteration no matter what
the values of the components are because the cap is a short and
the two inductors dont have any difference in voltage across them.

I'd call on Barkie but this network seems to oscillate so :)


Conclusions so far:

1. The |Z| max frequencies seem very important because the amplitude is always important.
2. The Im(Z)=0 frequencies seem more historical than anything else, so far.
We do have this one circuit with the amp with gain=1 but we cant decide on
how exactly the circuit is constructed (yet). We need to nail this down before
we can proceed, but we also have to keep in mind that classical circuit theory
is going to have a lot of this in it. We also have to keep in mind that it is going
to cause confusion because there is a lot of theory that confuses this with |Z| max
and i believe this is because the resistance R is relatively small in many circuits so that
these two frequencies are very close to each other.
3. The ring frequency FreeOsc(Z) seems to be interesting and have physical significance
when the network is driven by a pulse such as in a converter circuit.


I guess there is always the possibility that we are getting too far off the track here,
and we should let the end user decide what way they want to analyze their circuits :)
But then again, do we have a definitive answer when someone asks: "What is the
resonant frequency of my network?" ?
 
Last edited:
Back again and hopefully with a solution.

w = 565000000 rad/s
L1 = L2 = L = 50*10^-9 H
R = 139 ohms

R^2 - w^2*L*C*R^2 + (R - w^2*L*C*R)^2 + w^2*L^2 = 0

Solving out C gives us:

C = (3*R +- sqrt(R^2 - 4*w^2*L^2))/(2*w^2*L*R)

@Series: C1 = (3*R + sqrt(R^2 - 4*w^2*L^2))/(2*w^2*L*R) = 122.6 pF
@Parallel: C2 = (3*R - sqrt(R^2 - 4*w^2*L^2))/(2*w^2*L*R) = 65.356 pF

Series impedance = (w^2*L^2*R)/((R-w^2*L*C1*R)^2 + (w*L)^2) = 6.0003 ohms
Parallel impedance = (w^2*L^2*R)/((R-w^2*L*C2*R)^2 + (w*L)^2) = 132.9995 ohms

Tot = Series + Parallel = 139 ohms

Well, what do you guys think? Is this the right solution?
 
Last edited:
Hi again,


That is the right solution yes, if you solve for w0 as Im(Z)=0.
As we talked about before, there are other options.

I ended up with the form:

(w^2*C*L-2)*(w^2*C*L-1)*R^2+w^2*L^2=0

and this results in the same formula for C as you got,

C0=(3*R-sqrt(R^2-4*w^2*L^2))/(2*w^2*L*R)
C0=(3*R+sqrt(R^2-4*w^2*L^2))/(2*w^2*L*R)

but of course this again is for the w0 frequency which is
obtained by Im(Z)=0 and will not be the max or min amplitude
frequency point. To get those, we can use a little magic trick:

CZ=(3*R-sqrt(R^2+4*w^2*L^2))/(2*w^2*L*R)
CZ=(3*R+sqrt(R^2+4*w^2*L^2))/(2*w^2*L*R)

and now we either place the peak Z at w or the relative min Z at w.

Just in case we want to set the ring frequency instead, we can use:
C=(2*R+sqrt(4*R^2-w^2*L^2))/(2*w^2*L*R)

and the circuit will ring at the frequency w now.
 
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Hi again,


Well ok, i am back again to cause some more confusion :)
I was thinking this wasnt possible, but it turns out that it is.

Here's why...

When fooling around briefly with a couple other circuits i found
that the interpretation of "Series Resonance" and "Parallel Resonance"
also sometimes varies!

The simplest example is where you have a resistor in series with
a cap and also in series with an inductor, and that is sometimes called
a series resonant circuit, yet if you connect those same parts in
parallel that is sometimes called parallel resonance.

The next example is a little more interesting however and deals with
a loaded and unloaded circuit.
Take a resistor R, inductor L, cap C1 and connect them in series,
with the two loose ends being the input/output of the circuit.
Now take another capacitor C2 three times bigger than C1
and connect that in parallel
with the whole series network, so you end up with a series RLC
circuit in parallel with another cap C2.
Now, connect one side of this network to an AC generator and
the other side to a very low value resistor like 0.001 ohms.
Now, look at the frequency response of the impedance Z between
the generator and the 0.001 resistor where you get a max
Z and note it.
Now, instead of looking at Z, look at the voltage across the
small 0.001 resistor (it will be very low of course), and note
that frequency response where the max occurs.
Now compare results.
It turns out that the lower frequency obtained when looking at
Z is called the Series Resonant point, and the higher frequency
when we look across the 0.001 resistor is called the Parallel
Resonant point.
And just in case you havent noticed, that network is a simple
model of a crystal, and that is when these terms are applicable.
I believe this is why a parallel resonant crystal has to be 'loaded'
when used in an oscillator.
 
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Hi MrAl,

I believe this is why a parallel resonant crystal has to be 'loaded'
when used in an oscillator.
I don't understand what "this" is in the above quote (do I sound like Bill Clinton?).
In a parallel mode crystal oscillator, one end of the crystal is driven. A frequency-dependent voltage must appear on the other end of the crystal. For this to happen, there has to be a load present. It is usually specified to be a capacitance which is large enough to swamp the intrinsic amplifier capacitance.
BTW, I am not a crystal oscillator expert.:p
 
Hi again,


By that i mean that the crystal will oscillate at the series resonate
frequency if it is not loaded properly. To get it to oscillate at the
parallel frequency it must be loaded properly, and many crystals
are cut to resonate at the parallel frequency and would be stamped
to reflect that frequency, not the series resonate frequency.
Attempting to use them in series resonance would mean
the in-circuit oscillator frequency would be a little off.
Many amplifiers used with oscillators have a high input impedance
so that would leave one end of the crystal unloaded, even though
the other end connects to the output of the amplifier. It would
be easy to swamp that input capacitance yet still not load
the crystal properly. I think some manufacturers even specify
the correct capacitance value for a given crystal.

This is exactly what i was seeing in my little crystal circuit.
One resonant frequency was 0.919MHz, and the other was 1.048MHz.
To get the higher one i had to load the crystal. The lower one
in this case would have been quite a bit off.

Microchip has a pretty good app note on crystal oscillators and
the effects of loading/unloading the crystal used for them.
I could try to find it if you are interested, or perhaps just do a
search on their site.
 
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hi Al,
Just posted this pdf on another thread.

May help others.
 

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Hi Eric,

Oh good, couldnt have been more timely :)
 
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