Resonance

[Deniz]

-->------L--
I | |
C R
| |
= =

Circuit seen above (series RL and a parallel C) is supposed to
resonate somehow. There is a current source driving the circuit (since
I thought that a voltage source driving the circuit would be useless).

Overall impedance (Zo) is calculated as: 1/(R/(w^2*L^2 +
R^2)+j*w*(R^2*C - L + w^2*L^2*C)/(w*L^2 + R^2))

Now in order to maximize abs(Zo) we let imaginary part to be 0, by
letting wo = [1/LC - (R/L)^2]^0.5 (here i assume that abs(Zo) is
maximum when imaginary part is 0). Since Zo is maximized, I*Zo
(voltage across the capacitor (Vc)) will reach the maximum, and at wo
the circuit is said to be resonating?

My first question is: At resonance frequency wo, can we immediately
(without calculating time dependent expressions) say that when the
stored energy in the capacitor reaches maximum, the stored energy in
the inductor becomes 0 ?

2)If we want voltage across R to reach its maximum, we calculate
complex expression for Vr and if we let Vr's imaginary part to be 0,
we come up with a new resonance frequency w1 = (-R/L)^0.5, which is
meaningless. So can we immediately say or predict that voltage accross
R will reach its maximum when we set w = wo = [1/LC - (R/L)^2]^0.5
(which will reveal that finding resonance frequency has nothing to do
with letting the imaginary parts equal to 0)? If wo is making both the
voltage across Zo and R maximum, what is the reason for this?

(calculating the frequency which will make abs(Vr) maximum seemed to
be impossible, so i made a prediction)

3) Is there a series RLC equivalence of this above circuit (for Vr or
Vc)? If there is no such equivalence, how is parallel RLC equivalent
of this circuit calculated (for Vc and Vr) ?

 

[The Phantom]

On 20 Nov 2004 22:48:45 GMT, cbarn24050@aol.com (CBarn24050) wrote:

Subject: Resonance
From: daksoy@gmail.com (Deniz)
Date: 20/11/2004 22:04 GMT Standard Time
Message-id: <2502fe80.0411201404.3fc06c24@posting.google.com


My first question is: At resonance frequency wo, can we immediately
(without calculating time dependent expressions) say that when the
stored energy in the capacitor reaches maximum, the stored energy in
the inductor becomes 0 ?

NO, the enrgy in both components are equal but opposite.

What does it mean for the capacitor and inductor to have opposite
energies?

2)If we want voltage across R to reach its maximum, we calculate
complex expression for Vr and if we let Vr's imaginary part to be 0,
we come up with a new resonance frequency w1 = (-R/L)^0.5, which is
meaningless. So can we immediately say or predict that voltage accross
R will reach its maximum when we set w = wo = [1/LC - (R/L)^2]^0.5
(which will reveal that finding resonance frequency has nothing to do
with letting the imaginary parts equal to 0)? If wo is making both the
voltage across Zo and R maximum, what is the reason for this?

Maximum voltage accros R is at DC.



(calculating the frequency which will make abs(Vr) maximum seemed to
be impossible, so i made a prediction)

Maximun impedence for a parallel tuned circuit is at resonance.

Yes, but here the minimum impedence occurs at resonance.

 

[The Phantom]

On Mon, 22 Nov 2004 00:15:35 GMT, Steve Evans
<smevans@jif-lemon.co.mars> wrote:

On Sun, 21 Nov 2004 19:50:44 GMT, Robert Monsen
rcsurname@comcast.net> wrote:

At any given time, they don't have 'opposite' energy, because energy is
a scalar value.

However, the energy that they have is passed back and forth between the
two elements; thus, when the capacitor has maximal energy, the inductor
has minimal energy.

It is a fact that in a parallel resonant circuit, the impedance is at
a maximum. Is this due to the fact that - at resonance - the energy
flows between the cap and coil are so large as to be able to repell
any current from the external energy source, thereby rendering its
path effectively blocked?

Steve, I've been watching your postings as you strive to understand
a somewhat difficult subject--AC circuit theory. I can see that you
are struggling with it, but I admire your perseverance! Keep at it,
and you'll eventually get it. I'll add my input to the help others
have been giving you.

I think it's helpful to realize that what are called two-terminal
circuit elements (R, L, and C are the fundamental components)
*enforce* a relationship between voltage and current. The voltage
*across* a component and the current *through* it are not independent.
It is what two-terminal components do; they establish a relationship
between voltage and current, *for that component only*.

So, if some two-terminal circuit elements are in series, the
current in each of them *must* be identical. If they are in parallel,
the voltage seen (applied across) by each of them *must* be identical.

Thus, if they are in series, the currents must be the same and
*only* the voltages across each can be different. If they are in
parallel, only the *currents* in each can be different; the voltage
seen by each is the same.

For components in series, since the current in all of them is the
same, it makes sense to use current as a reference, and speak of the
phase of the voltages *across* (not to ground) each component with
respect to the current through all of them.

For components in parallel, it is appropriate to use the voltage
*across* them as the reference, and speak of the phase of the current
in each with respect to the voltage across all of them.

Now, since for a C and L in parallel the voltage across the two
components is the same, only the phase of the currents can differ.
The current in one is 180 degrees out of phase with the other, and
when those currents are added by the parallel connection, they tend to
cancel. If the magnitude of the currents is identical, which is what
happens at a frequency such that the reactance of each is the same
(this is resonance), then we get complete cancellation of the currents
(for ideal L and C). Thus the current into the parallel combination
of the L and C is zero, even though we have applied some non-zero
voltage to the two of them. When we have a circuit that has the
property that no current (or very little) is produced with a finite
applied voltage, we say that the impedance of that circuit is high.
It's not that the parallel combination of L and C at resonance repel
the applied voltage. In fact, a current does exist in both the L and
C, but the two currents are 180 out of phase, and completely add to
zero at the connection of the L and C.

The same thing happens in a series resonant L and C circuit, but
with the roles of current and voltage reversed. For a given current
through the L and C, if the applied current is at a frequency where
the reactance of the L and C is the same, then the magnitude of the
voltage *across* the inductor and *across* the capacitor is the same
and since these voltages are 180 degrees out of phase (for ideal
components), they completely cancel (add to zero). Remember that the
current in the L and in the C is the same, since they are in series.
Thus we have a circuit with (almost) no voltage across it even though
a current is passed through it. We say that such a circuit has a low
impedance. This circuit doesn't repel the applied current; it's just
that the voltage *across* one component cancels the voltage *across*
the other, giving a resultant of zero *across* the series combination.

I hope this helps.

 

[Rich Grise]

On Sun, 21 Nov 2004 19:50:44 +0000, Robert Monsen wrote:

The Phantom wrote:
On 20 Nov 2004 22:48:45 GMT, cbarn24050@aol.com (CBarn24050) wrote:


Subject: Resonance
From: daksoy@gmail.com (Deniz)
Date: 20/11/2004 22:04 GMT Standard Time
Message-id: <2502fe80.0411201404.3fc06c24@posting.google.com

My first question is: At resonance frequency wo, can we immediately
(without calculating time dependent expressions) say that when the
stored energy in the capacitor reaches maximum, the stored energy in
the inductor becomes 0 ?

NO, the enrgy in both components are equal but opposite.

What does it mean for the capacitor and inductor to have opposite
energies?

At any given time, they don't have 'opposite' energy, because energy is
a scalar value.

However, the energy that they have is passed back and forth between the
two elements; thus, when the capacitor has maximal energy, the inductor
has minimal energy.

You could say that the energy for each is a sinusoidal wave above the x
axis, 180' out of phase, and that the sum of the two sine waves is equal
to the total energy in the system. That energy can be increasing if
there is an impulse, decreasing if the oscillation is damped, or
'constant' if the damping and impulse balance out.

They're 90 degrees out of phase, because the capacitor stores energy as
charge, and the inductor stores energy as current. The current hits its
maximum as charge is going through zero, and vice-versa.

But yes, it's the same energy, just being sloshed back and forth. Think of
a spring and a weight. The spring's compression is the voltage, its
spring constant is the capacitance, the weight is inductance, and its
movement is current.

And somebody said something about a series resonant circuit and a parallel
resonant circuit - if you just look at the inner loop, a parallel resonant
circuit is just a series resonant circuit with its ends tied together. So
the difference really depends on your point of view.

Hope This Helps!
Rich

 

[The Phantom]

On Tue, 23 Nov 2004 04:32:52 GMT, Rich Grise <rich@example.net> wrote:

On Sun, 21 Nov 2004 19:50:44 +0000, Robert Monsen wrote:

The Phantom wrote:
On 20 Nov 2004 22:48:45 GMT, cbarn24050@aol.com (CBarn24050) wrote:


Subject: Resonance
From: daksoy@gmail.com (Deniz)
Date: 20/11/2004 22:04 GMT Standard Time
Message-id: <2502fe80.0411201404.3fc06c24@posting.google.com

My first question is: At resonance frequency wo, can we immediately
(without calculating time dependent expressions) say that when the
stored energy in the capacitor reaches maximum, the stored energy in
the inductor becomes 0 ?

NO, the enrgy in both components are equal but opposite.

What does it mean for the capacitor and inductor to have opposite
energies?

At any given time, they don't have 'opposite' energy, because energy is
a scalar value.

However, the energy that they have is passed back and forth between the
two elements; thus, when the capacitor has maximal energy, the inductor
has minimal energy.

You could say that the energy for each is a sinusoidal wave above the x
axis, 180' out of phase, and that the sum of the two sine waves is equal
to the total energy in the system. That energy can be increasing if
there is an impulse, decreasing if the oscillation is damped, or
'constant' if the damping and impulse balance out.

They're 90 degrees out of phase, because the capacitor stores energy as
charge, and the inductor stores energy as current. The current hits its
maximum as charge is going through zero, and vice-versa.

But yes, it's the same energy, just being sloshed back and forth. Think of
a spring and a weight. The spring's compression is the voltage, its
spring constant is the capacitance, the weight is inductance, and its
movement is current.

And somebody said something about a series resonant circuit and a parallel
resonant circuit - if you just look at the inner loop, a parallel resonant
circuit is just a series resonant circuit with its ends tied together. So
the difference really depends on your point of view.

If the L and C are all there is, and you don't make any connection
to the pair, then the series/parallel distinction doesn't exist. See
E. A. Guillemin's classic book on circuit theory. He talked about
what he called a "soldering iron entry" or a "pliers entry" into a
(sub) circuit. The L and C combination looks different to the
observer depending on whether you open the loop (pliers entry) and
make the connection, or leave the loop intact and tack on in parallel
(soldering iron entry).

Hope This Helps!
Rich

 

[Steve Evans]

On 23 Nov 2004 22:04:16 -0600, The Phantom <phantom@aol.com> wrote:

The detail Rich is missing is that if you plot the energy in L and
C separately, you will see that the energy vs. time plot is a *double*
frequency function, compared to the voltage or current. This is
because the energy involves the *square* of the voltage or current
(for C or L), which is always positive regardless of whether the
voltage (or current) is in the positive or negative direction.

Maybe you can assist me with one small thing. wherever I see a plot of
current v. frequency for capacitors and coils, one is always pretty
mcuh a straight line whereas the ohter looks like exponential. Give
that the formlulas are XC=1/wC and XL=wL and there's no square in
either, where does the exponential character of one of the curve come
from?
--

Fat, sugar, salt, beer: the four essentials for a healthy diet.

 

[John Popelish]

Steve Evans wrote:

Okay, thnx, guys.
well that explains the mathematics. Now on to the physics. I'd always
tought of caps and coils as like mirror-images of each other in the
way they act WRT signals applied. In every ohter respect AFAIA, this
is true. Why then, is the response of one linear and the other
hyperbolic?? I'm not asking for a re-iteration of the math here;
what's the pysical processes going on that account for it?
--

Fat, sugar, salt, beer: the four essentials for a healthy diet.

Plot their impedances on log linear paper (logarithmic frequency or
period) and their inverse relationship is obvious. It is the linear
frequency scale that is distorting the ratiometric relationship.

--
John Popelish

 

[Steve Evans]

On Fri, 26 Nov 2004 00:14:59 GMT, Tom MacIntyre
<tom__macintyre@hotmail.com> wrote:

On Thu, 25 Nov 2004 23:46:52 GMT, Steve Evans
smevans@jif-lemon.co.mars> wrote:

On Thu, 25 Nov 2004 09:17:29 -0500, John Popelish <jpopelish@rica.net
wrote:

Plot their impedances on log linear paper (logarithmic frequency or
period) and their inverse relationship is obvious. It is the linear
frequency scale that is distorting the ratiometric relationship.

well I wonder why they don't mention that rather important little fact
in the text books!!! :-(

It all depends on the textbook.

In all the ones I've seeen, they don't show any graduations on the
plot panes at all. Then they say they don't because "precise values
aren't important its the relationship between current and Xc and Xl
that we're looking at here."
--

Fat, sugar, salt, beer: the four essentials for a healthy diet.

 

[John Popelish]

Steve Evans wrote:

On Thu, 25 Nov 2004 09:17:29 -0500, John Popelish <jpopelish@rica.net
wrote:

Plot their impedances on log linear paper (logarithmic frequency or
period) and their inverse relationship is obvious. It is the linear
frequency scale that is distorting the ratiometric relationship.

well I wonder why they don't mention that rather important little fact
in the text books!!! :-(

Sorry for a mistake. That should have read "log log paper". In this
form, the slope of the line is proportional ot the power. 1st power
functions slope up to the right with a slope of one. Negative first
power functions (1/x) slope up to the left with a negative one slope.


--
John Popelish

 

[Don Kelly]

"Steve Evans" <smevans@jif-lemon.co.mars> wrote in message
news:hiebq0h1veivm6adr40kpndhbaqp5sep32@4ax.com...

Okay, thnx, guys.
well that explains the mathematics. Now on to the physics. I'd always
tought of caps and coils as like mirror-images of each other in the
way they act WRT signals applied. In every ohter respect AFAIA, this
is true. Why then, is the response of one linear and the other
hyperbolic?? I'm not asking for a re-iteration of the math here;
what's the pysical processes going on that account for it?
--

Fat, sugar, salt, beer: the four essentials for a healthy diet.

----------
At DC a capacitance looks like an open circuit but at high frequency its
impedance is very small so that the impedance drops from infinity to 0 as
the frequency goes from 0 to infinity. If you plotted 1/Xc against frequency
you would get a straight line.

At DC an inductor has an impedance of 0 and the impedance rises linearly
with frequency. Try plotting 1/Xl vs frequency.

As for the physics -calculus rears its ugl head:
Inductor:
v=Ldi/dt which for steady state sinusoidal AC leads to V/I =2*pi*f*L in
magnitude.
Capacitor:
v =charge /C = 1/C (integral of current) which leads, for steady state
sinusoidal AC to V/I =1/(2*pi*f*C)

--
Don Kelly
dhky@peeshaw.ca
remove the urine to answer