# Resonance

Discussion in 'Electronic Basics' started by Deniz, Nov 20, 2004.

1. ### Steve EvansGuest

Well, they're *each* 90 degrees out of phase, so they're in complete
antiphase (180') WRT each other. The cap's eneryg is stored in an
electric field; the inductor's is stored in a magnetic field and when
one's at a maximum, the other's at a minimum and vice versa.

2. ### Don KellyGuest

cycle.
----------
There is real energy but it is being shuttled back and forth. The average
power over a cycle is 0 so that the total energy input during the cycle is
also 0. At any instant in time the power is not 0 nor is the sum of the
energies stored in the L and C.

Where does the energy come from?
The conditions when operating at steady state are not the same as when the
circuit is first energised. There is a transient period in which energy is
initially stored in the capacitor and inductor (not necessarily the same in
each). You can't handle this period with the concepts of AC steady state
analysis (phasors, reactive, etc) but need to consider the differential
equations involved.

3. ### The PhantomGuest

I think we are talking about energy here, and it is true that the
time functions of the two energies are sinusoids, 180 out of phase,
but the reason is slightly more complicated, Steve. The current in L
and C are each 90 degrees out of phase with the reference (voltage),
in opposite directions (so to speak), so the two currents are indeed
180 out. But one might expect that since the energy in C is a
function of voltage and the energy in L is a function of current, that
the energies might be 90 degrees out of phase, since the voltage
across the C is only 90 degrees out phase with the current in the L.

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.
Remember your trigonometry, specifically the formula: SIN^2(x) = (1 -
COS(2x))/2 When you square a sinusoid, you get a double frequency
sinusoid plus a constant (the constant is the *average* energy). When
you look at the squares of two sinusoids that are 90 deg out of phase,
you get a couple of double frequency sinusoids that are *180* out of
phase with each other. When you add these two double frequency
sinusoids (plus their constant terms), the sinusoidal portions cancel
and the contants add to give a constant equal to the total energy in L
and C.

4. ### Steve EvansGuest

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?

5. ### CBarn24050Guest

Subject: Re: Resonance
Draw the line y=1/x and all will be revealed.

6. ### The PhantomGuest

It's not exponential (though it does look somewhat like it), it's
hyperbolic; it's because the reactance of capacitors vary as 1/wC and
inductors as wL. The current in a cap is v/(1/wC) which is v*wC; the
current in an inductor is v/(wL) which will plot as a hyperbola. If
you plot y=1/x you will see a curve like you're describing.

7. ### Don KellyGuest

Plot 1/x vs x -it is an inverse function, not an exponential. Also
plotexp^-x
exp^-x will be 1 at x=0 while 1/x will be infinite at x=0. The curves are
quite different.[/QUOTE]

8. ### Steve EvansGuest

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?

9. ### John PopelishGuest

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.

10. ### CBarn24050Guest

Subject: Re: Resonance
They ARE both linear.

11. ### Steve EvansGuest

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

12. ### Tom MacIntyreGuest

It all depends on the textbook.

Tom

13. ### Steve EvansGuest

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."

14. ### John PopelishGuest

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.

15. ### Don KellyGuest

----------
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)