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Inductor & Capacitor in Parallel in AC Circuit

Discussion in 'Electrical Engineering' started by Chris Barrett, Apr 17, 2007.

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  1. If I have the following AC circuit:

    .. .. ..
    ||||--( V )---/\/\/-------((((()-------||------||||
    '' '' ''

    I can describe it with the following equations

    V_L + V_R + V_C = V
    L dq^2/dt^2 + R dq/dt + (1/C) q = V

    I now have to deal with the following AC circuit:

    .. | | .. ..
    ||||--( V )---/\/\/---| |---||------||||
    '' | .. | '' ''

    How do I treat the inductor and capacitor that are in parallel? My guess
    is that I have a term representing the inductor and capacitor together,
    but I'm not sure. How do I represent this with a differential, or
    coupled differential equation?

    Thanks for any help.
  2. Don Kelly

    Don Kelly Guest

    You have the basic KVL and KCL equations. Use them. It gets messy. In the
    case of steady state AC you have the phasor approach which deals with a
    frequency domain model rather than a time domain model in that the frequency
    domain model leads. through solution of simultaneous linear equations to an
    easy evaluation of what you are trying- solution of simultaneous
    differential equations.
    For transient conditions, it is messier-and the Heaviside operator which
    Bill mentions (p=d/dt) is still used for machine modelling although the
    closely related Laplace operator is more commonly used for control and
    general transient analysis.
    These both offer a reduction of simultaneous linear time domain differential
    equations to simultaneous frequency domain linear algebraic equations-
    allowing, as in the steady state AC analysis, computational advantages .

    For your parallel L.C then the Laplace model represents this as sL in
    parallel with 1/sC
    Compare this to the steady state AC situation where you have jwL in parallel
    with 1/jwC

    Don Kelly
    remove the X to answer
  3. Don Kelly

    Don Kelly Guest


    Don Kelly
    remove the X to answer
  4. Don Kelly

    Don Kelly Guest

    I don't think that you are missing anything. The Heaviside and Laplace
    operators are both derivative operator symbols. Heaviside was covered well
    in a series of articles in either AIEE or the British equivalent (IEE)- a
    long time back-40's??. Laplace, for reasons that I knew and now don't
    remember caught on while Heaviside didn't. Possibly something to do with
    either initial conditions or the inverse transformation. The Heaviside
    operator p was in vogue in the late 20's and early 30's where it was used
    mainly as a symbolic operator p=d/dt in dealing with machines (particularly
    transients in synchronous machines) and is still used in modern machine
    texts in that sense (As the equations are generally non-linear- that is
    about as far as it goes). Bode dates back to that time so that may be why he
    also used "p"
    Laplace, in engineering applications, appears to have become popular in the
    '50's and was well suited to dealing with transients in general. Both
    Heaviside and Laplace could be used for transfer functions or dealing with
    characteristic equations but, and I may be wrong here, Laplace could handle
    steady state situations better and common phasor analysis simply means
    walking along the s=jw line in the complex frequency plane (of course it may
    be that the mathematicians liked Laplace better).
  5. Don Kelly

    Don Kelly Guest

    I really don't know how far Heaviside went. I had a copy of Bode's book at
    one time but where it went, along with some others is lost in the past. I
    know I passed on the original IEEE publication of Fortescue's symmetrical
    component paper to a person who would value it and preserve it and deserved
    to have it.
    The "p" notation, is still used in many machine texts simply is inherited
    from early nomenclature and is not, in fact, a transformation, nor
    considered as such, except in cases which can be linearised. The early
    nomenclature was in the time that complex number theory had not really made
    its mark on circuit analysis - "j" simply treated as a shorthand for a 90
    degree phase shift akin to comsideration of vectors in a 2 dimensional
    world. ("i" taken granted, "j" at 90 degrees and "k" ignored.
    One hell of a lot was pulled into EE education in the '50's -e.g. in '55 I
    met Laplace in a graduate math course (and was frustrated in trying to apply
    it) and in 57-58 it was in a 3rd year EE text (admittedly without much of
    the contour integration material met in '55-later added)
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