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Resonance

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

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  1. Steve Evans

    Steve Evans Guest

    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 Kelly

    Don Kelly Guest

    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 Phantom

    The Phantom Guest

    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 Evans

    Steve Evans Guest

    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. CBarn24050

    CBarn24050 Guest

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

    The Phantom Guest

    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 Kelly

    Don Kelly Guest

    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 Evans

    Steve Evans Guest

    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. 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. CBarn24050

    CBarn24050 Guest

    Subject: Re: Resonance
    They ARE both linear.
     
  11. Steve Evans

    Steve Evans Guest

    well I wonder why they don't mention that rather important little fact
    in the text books!!! :-(
     
  12. It all depends on the textbook.

    Tom
     
  13. Steve Evans

    Steve Evans Guest

    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. 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 Kelly

    Don Kelly Guest

    ----------
    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)
     
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