Inductor & Capacitor in Parallel in AC Circuit

[Chris Barrett]

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.

 

[Don Kelly]

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.

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 [email protected]
remove the X to answer
----------------------------

 

[Don Kelly]

--

Don Kelly [email protected]
remove the X to answer
----------------------------

 

[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.

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).

 

[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.

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)