Inductor & Capacitor in Parallel in AC Circuit

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.
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On 4/17/07 6:15 AM, in article WN3Vh.5402$ snipped-for-privacy@newsfe24.lga, "Chris

At any frequency, calculate the impedance. For true (ideal) inductors and capacitors, that impedance is going to be imaginary. For real ones, the impedances will be complex. It is the complex impedances that combine for series and parallel combinations in the same way resistors do at dc. But you must use complex algebra.
There are operational forms for combining impedances. You start with pL and 1/pC as impedances for inductors and capacitors respectively and p is the derivative operator. The combination is going to be a quotient of two polynomials of p.
If you do not understand what I am saying, you are going to have to study more. See Bode's book on feedback amplifiers.
Bill -- Fermez le Bush--about two years to go.
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On 4/17/07 9:59 PM, in article qAhVh.89604$DE1.38152@pd7urf2no, "Don Kelly"

Maybe I am missing something. I always thought that the commonly used symbols p and s were both derivative operator symbols and pretty much equivalent. IIRC Heavyside used the symbol p, as did Bode without much explanation. More modern texts using Laplace transforms used s. Am I missing something?
I also sat in on a course by Erdelyi in which he had fields (like fields of numbers) of functions and derivatives intertwined in ways I have forgotten. It was a formal way of dealing mathematically with Heavyside calculus but without invoking Laplace transforms. There is some stuff about his methods in Wikiepedia.
Bill -- Fermez le Bush--about two years to go.
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wrote:

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).
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On 4/19/07 8:21 PM, in article OkWVh.100269$aG1.51668@pd7urf3no, "Don Kelly"

It should also be pointed out that Fourier transforms can also be used for an operational calculus. There, jw is the differential operator. Also, do not forget that most DE books use D as the differential operator. I learned operational calculus primarily using Laplace transforms.
Operational methods are mostly useful for linear Des with constant coefficients. I have worked some problems variable coefficients. In some cases, taking a transform for a different kind of DE leads to a DE for the transform of a lower order. Also, often taking a transform of a PDE will get you an ODE. I have also worked on some problems where you take a double transform. That seems to be more common for FTs especially when applied to optical images where two dimensions are involved.
I do not know how far Heavyside went in solving heat problems or diffusion problems using operational calculus. The classic problem that Heavyside might have tackled would be for a twisted telephone pair for which series resistance and distributed shunt capacitance predominate. This leads to a diffusion equation with terms containing sqrt(p). With modern transform theory, inverting such functions is relatively easy.
Bill -- Fermez le Bush--about two years to go.
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wrote:

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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